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4600	https://evidentscientific.com/en/insights/making-the-most-of-kohler-illumination	Making the Most of Köhler Illumination \| Blog Post \| Olympus IMS Evident has completed its acquisition of Pramana, Inc., a leading manufacturer of digital pathology solutions. 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Making the Most of Köhler Illumination Author(s): Robert Bellinger 15 July, 2016 Technique facilitates even illumination and excellent specimen contrast Critical illumination was the predominant technique in microscopy until August Köhler (1866–1948) developed a new method of illumination, now known as Köhler illumination. The problem with critical illumination was that the bright light lamp source created a filament image in the same plane as the specimen image. Visibility of the bulb filament in the final image caused uneven specimen illumination and introduced glare and shadowing artifacts. Köhler illumination resolved these artifacts by using a defocused light source image to evenly illuminate the specimen. Still relevant today, Köhler illumination should be applied whenever suboptimal specimen illumination is preventing proper observation. Köhler illumination can provide even illumination of specimens in brightfield, darkfield, and all variations of phase contrast microscopy using either transmitted or reflected light pathways. Köhler illumination facilitates even illumination, high resolution, and good specimen contrast. An image captured with reflected light microscopy. The above image was captured without Köhler illumination and the bottom with Köhler illumination. How do you set up and use Köhler illumination? Transmitted light microscopy: Köhler illumination requires several optical components of the microscope to function in between the light source and the specimen. These include the collector lens, field diaphragm, condenser diaphragm, and condenser lens. The collector lens acts to collect light from the light source and focus it on the plane of the condenser diaphragm. The condenser lens then projects this light through to the specimen. By adjusting the field diaphragm, the image of the field diaphragm aperture in the specimen plane is set to a size slightly larger than the imaged region of the specimen, which corresponds to the portion of the specimen image seen at the eyepiece field stop. As the field diaphragm, specimen, and eyepiece field stop all lie on the same conjugate image plane, this adjustment allows the illuminating rays to completely fill the eyepiece field of view, while minimizing the amount of extraneous light blocked by the eyepiece field stop. Reflected light microscopy: Reflected light microscopy or epi-illumination is the choice for illumination of opaque specimens like metals, ores, ceramics, polymers, semiconductors (e.g., unprocessed silicon, wafers, integrated circuits), slag, coal, plastics, and paint. The key optical components needed to set up Köhler illumination in reflected light microscopy are arranged in opposite orientation to those in transmitted light microscopy. The condenser aperture iris diaphragm is closer to the light source and the field iris diaphragm is closer to the specimen. In reflected light microscopy, the objective serves a dual function—on the way down, the objective serves as a properly aligned condenser, controlling the angle of the light striking the specimen. On the way up, the objective’s numerical aperture (NA) determines the angle of light which can be captured as it is reflected from the specimen. All things being equal, the higher the NA, the better the resolution of the objective, and the better the resolution of the specimen. Typical setup for a reflected light microscope. How are Köhler illumination techniques applied? Brightfield microscopy: The most commonly used optical microscopy technique. Illumination of the sample is transmitted from above via a tungsten-halogen lamp focused through the vertical illuminator positioned above the stage. Light reflected by a beam splitter through the objective illuminates the specimen. Light reflected from the specimen surface re-enters the objective and passes into the eyepiece or to a camera port. Absorption and diffraction of the incident light by the specimen often lead to discernible variations in the image. Specimens that show little difference in intensity or color require dark field microscopy or reflected differential interference contrast (DIC) (see below). An image of a microchip under brightfield observation. Darkfield microscopy: Well-suited for applications where contrast comes from light scattered by the specimen. Light not scattered by the specimen is not collected by the objective lens and not incorporated into the image. Consequently, the field around the specimen appears dark. The main limitation of darkfield microscopy is the low light levels seen in the final image. This is how Köhler illumination techniques contribute—the sample must be strongly illuminated. Raised features that are too smooth to cast shadows will not appear in brightfield images, but the light that reflects off the sides of the feature will be visible in the darkfield images. An image of a microchip under darkfield observation. Phase contrast microscopy: An optical microscopy technique that generates sample contrast from interference of different path lengths of light reflected from the specimen. Amplitude and phase shifts occur based on properties of the specimen. These changes show as variations in brightness from the scattering and absorption of light. Phase contrast microscopy is particularly important in industrial microscopy as it reveals many features or structures of a specimen not visible in brightfield microscopy. An image of cross-sectioned polished metal under brightfield observation with no phase contrast (top) and with phase contrast (below). Differential interference contrast (DIC microscopy): A new phase contrast imaging method. DIC microscopy enhances contrast by creating artificial shadows, as if the object is illuminated from the side. To achieve DIC microscopy, polarized light is separated into two orthogonally polarized parts. The mutually coherent parts, which are spatially displaced (sheared) at the specimen plane, are then recombined before observation. The interference of the two parts at recombination is sensitive to their optical path differences, the product of their refractive index and geometric path length. The observed contrast is proportional to the path length gradient along the shear direction, giving the appearance of a three-dimensional physical relief. An image of cross-sectioned polished metal under brightfield observation with no DIC (above) and with DIC (below). Article Tags Metal Manufacturing Academic Research Energy Electronics Robert Bellinger Product Applications Manager, Industrial Microscopes Rob Bellinger is a product applications manager for industrial microscopes at Evident. He has been part of Evident for more than 15 years. He currently provides application support for our industrial microscope systems in the US, Canada, and Latin America. Need Assistance? Contact EVIDENT About Evident Applications & Solutions Insights Learn Support Careers Clinical Microscopes Life Science Microscopes Materials Science Microscopes Objectives Objective Finder Image of the Year Microscope Resource Center Customer Service Microscope Software Downloads Discontinued Products Stay In Touch Subscribe to Evident Evident Scientific Looking for Industrial Measurement Solutions? Visit the IMS website. © Copyright EVIDENT, All rights reserved 2025. Terms of Use\|Privacy Notice\|Cookie Notice\|Report Compliance Concern\|German Imprint\|Cookies Settings
4601	https://www.quora.com/What-is-the-formula-for-the-sum-of-n-odd-numbers	Something went wrong. Wait a moment and try again. Adding Odd Numbers Sum of Series Arithmetic Number Theory Formulas in Maths Sums of Mathematics Mathematical Equations Positive Odd Integers Math Theory 5 What is the formula for the sum of n odd numbers? Sort Assistant Bot · Sep 5 The sum of the first n odd positive integers is S(n) = 1 + 3 + 5 + ... + (2n−1) = n^2. Key points and derivations: Proof by induction: base n=1 gives 1; assume S(k)=k^2; then S(k+1)=k^2+(2(k+1)−1) = k^2+2k+1=(k+1)^2. Algebraic sum: S = Σ_{i=1}^n (2i−1) = 2Σ_{i=1}^n i − Σ_{i=1}^n 1 = 2·(n(n+1)/2) − n = n^2. Visual/geometric: n^2 equals an n×n square built by layering successive odd-length “L” borders. If the sequence of odd numbers does not start at 1 (e.g., m, m+2, ..., m+2(n−1) with m odd), the sum is S = n·m + n(n−1). For m=1 this reduces to n^2. Promoted by Coverage.com Johnny M Master's Degree from Harvard University (Graduated 2011) · Updated Sep 9 Does switching car insurance really save you money, or is that just marketing hype? This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take. But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious. They claimed I could save over $1,000 per year. I ended up exceeding that number and I cut my monthly premium by over $100. That’s over $1200 a year. For the exact same coverage. No phone tag. No junk emails. Just a better deal in less time than it takes to make coffee. Here’s the link to two comparison sites - the one I used and an alternative that I also tested. If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying. Related questions What is the sum of odd numbers? What is the sum of first n odd numbers? What is the sum of first n even numbers? What is the formula for the sum of odd numbers? What is the formula for the sum of consecutive odd numbers? Related questions What is the sum of odd numbers? What is the sum of first n odd numbers? What is the sum of first n even numbers? What is the formula for the sum of odd numbers? What is the formula for the sum of consecutive odd numbers? What is the sum of odd numbers from 1 to n? What's the proof for why the sum of the first n odd natural numbers equals n^2? What is the sum of n odd natural number? What is the sum of odd numbers from 1 to 50? What is the sum of even numbers from 1 to 50? What is the sum of the first 50 odd natural numbers? What is the sum of the first 10 odd numbers? What is the formula to find the sum of the first and odd natural numbers? What is the sum of the odd numbers in the range from 10 to 20? What is the sum of the first n-odd natural numbers? What is the sum of the 1st 6 odd numbers? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
4602	https://www.youtube.com/watch?v=JTg8gyQ37pM	Decomposing shapes to find area (add) \| Math \| 3rd grade \| Khan Academy Khan Academy 850 likes 266389 views 13 Jun 2016 Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: Lindsay finds the area of an irregular shape by decomposing it into 3 rectangles and adding the area of the rectangles. Practice this yourself on Khan Academy right now: Watch the next lesson: Missed the previous lesson? Watch here: 3rd grade math on Khan Academy: We know you've been rocking through 2nd grade adding and subtracting all kinds of whole numbers (up to 2 digits, right?). That's awesome! In 3rd grade math we want you to start using bigger numbers and start multiplying and dividing, too. By the way, did you know that some numbers aren’t actually “whole?” They’re “partially whole.” We call them fractions! We want you to start playing around and having fun with those, too. There's also area, perimeter, and place value to be discovered. Whew. We have so much to do and can't wait to do it with you. Let's go! Want a virtual coach for 3rd grade? Unlock the 3rd grade mission on Khan Academy here: About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. For free. For everyone. Forever. #YouCanLearnAnything Subscribe to KhanAcademy: 0 comments
4603	https://www.purplemath.com/modules/sqrvertx.htm	Home Lessons HW Guidelines Study Skills Quiz Find Local Tutors Demo MathHelp.com Join MathHelp.com Login Select a Course Below Standardized Test Prep ACCUPLACER Math ACT Math ALEKS Math ASVAB Math CBEST Math CLEP Math FTCE Math GED Math GMAT Math GRE Math HESI Math Math Placement Test NES Math PERT Math PRAXIS Math SAT Math TEAS Math TSI Math VPT Math + more tests K12 Math 5th Grade Math 6th Grade Math Pre-Algebra Algebra 1 Geometry Algebra 2 College Math College Pre-Algebra Introductory Algebra Intermediate Algebra College Algebra Homeschool Math Pre-Algebra Algebra 1 Geometry Algebra 2 Completing the Square: Finding the Vertex Finding the Formula Purplemath The parabola is the curve that you get when you graph a quadratic equation of the form y = ax2 + bx + c. You've graphed these curves, and you've probably been introduced to the vertex, which is the uppermost (or lowermost) point on the parabola (depending on the direction in which the parabola opens). But now you're being asked to find the vertex specifically, and — of course! — they're making the vertex hard to find from the graph. To find the coordinates of the vertex, you're going to have to "complete the square". Content Continues Below MathHelp.com Graphing in y − k = a(x − h)2 Form Why do I have to complete the square? Completing the square, when dealing with a parabola's quadratic equation, reformats that quadratic equation into the vertex form; from this vertex form, you can read off the coordinates of the vertex of the parabola. When those coordinates are messy — say, (−¾, 3.7) — completing the square is the only way to be sure of the exact x- and y-values for the vertex. Can't I use my calculator to find the vertex? Advertisement Yes, you can probably use your graphing calculator to get at least close to the values, but simple fractional values like will often be stated as non-terminating decimals; in this case, the screen would show "0.1428571", which isn't very helpful, especially when your answers are required to be "exact". Also, your instructor may decide to have at least a portion of the test be no-calculators [and no phones], so you probably do need to know how to do this by hand. What is the vertex form of a parabola's quadratic? The vertex form of a parabola's quadratic equation looks like this: y = a(x − h)2 + k When the equation is reformatted as above, the point (h, k) is the vertex. Affiliate The a in the vertex form is the same a as in y = ax2 + bx + c; that is, both of the a's have exactly the same value. The sign on a (plus or minus) tells you whether the quadratic's parabola opens up or opens down. Think of it this way: A positive a draws a smiley face, and a negative a draws a frowny face. (Yes, it's a silly picture to have in your head, but it makes is very easy to remember how the leading coefficient works.) Content Continues Below When they're wanting you to find the exact coordinates of the vertex of a parabola, they don't usually give you the quadratic in vertex form; instead, they usually give you the quadratic in the regular y = ax2 + bx + c format. How do you convert from the regular format to the vertex format? By using the technique of completing the square. Here's an example: Find the vertex of y = 3x2 + 2x − 1 This is my original equation: y = 3x2 + 2x− 1 First, I'll move the loose number over to the other side of the equation, with the y: y + 1 = 3x2 + 2x Now I'll factor out whatever is multiplied on the squared term, keeping in mind that: "Factor" does not mean to "make disappear" or "divide off onto the other side"; "factor" means "divide out front". So I'll just pull that 3 out front, giving me: For the purposes of completing the square, I'm going to create space on the left-hand side with the y, and, if a is anything other than simply 1, I'll also put a copy of a in front of this space. I'll need this space and the copy of a to keep my equation "balanced". Now I'll take half of the coefficient of the linear x-term from inside the right-hand side parentheses (that is, I'll divide that coefficient by two), not forgetting its sign, square the result, and add it to both sides inside the parentheses. (This is why the parentheticals were created in the first place.) On the left-hand side of the equation, I'll multiply out the "a times the squared coefficient" part on the left-hand side, and convert the right-hand side to squared form. (This is where I use that sign that I kept track of earlier, putting that sign in the middle of the squared expression.) I'll simplify some more, as necessary: Now I'll move the loose number from the left-hand side back over to the right-hand side: To be thorough, I'll reformat into vertex form (which is a matter of making sure the signs match the vertex form's formula), and read off the vaues of h and k. In this case, since a = 3 (which is a positive number), then this is a right-side-up parabola (that is, it's an upward-opening curve), and the vertex, , is the lowest point on the graph. You may be wondering why I went to the trouble of reformatting the equation to "proper" vertex form: I did this because the formula for the vertex form is: y = a(x − h)2 + k I wanted to make very clear to myself that the value that was subtracted from x to result in the binomial was a negative value. Because, to get a "plus" in the simplified form, I had to have subtracted a "minus": The two minuses cancelled to give me the plus inside the parenthetical expression. Turning to the loose number tacked on after the squared parenthetical, I see that it is subtracted from the parenthetical. To make this fit the vertex form of the quadratic, I converted the subtraction to the addition of a negative, because: Affiliate Affiliate Warning: It's easy to confuse yourself at this final stage of the process, by trying to read off the vertex as "(h, k) = (whatever number is inside the squared part, whatever the other number is) ", without noticing the fact that the h-part is subtracted and the k-part is added. (This is part of why I wandered through that long "aside" above, about why the vertex being (h, k) made sense, if you thought about it. Because if you are aware of the logic behind (h, k) being the vertex, then it's generally easier for you to keep the formula straight.) If you take care to ensure that you have your quadratic completely converted to vertex form by being careful of the signs, then you'll be able to avoid one of the most commonly-made mistakes for these problems (namely, having the wrong signs on the coordinates of the vertex). Make sure you practice this until you can consistently interpret your results correctly. By the way, did you notice that the vertex coordinates weren't whole numbers? Instructors are starting to figure out that students are guessing the vertex from the pretty pictures in their graphing calculators, and they know that students often have the idea that all answers are always either whole numbers or "neat" fractions like one-half. For instance, if the calculator screen estimates a vertex as being at (0.48, 0.98), many students will assume that the answer must "really" be (0.5, 1), instead of, say, . So, in order to check that students really do know how to find the vertex (and not just guess a decimal approximation from a picture), teachers are giving more complicated exercises. If you have been told that you should know this technique for finding the vertex, rest assured that your teacher has ways of checking whether you have really learned this. Don't plan on using calculator cheats. What are the steps for completing the square? To take a general quadratic equation and reformat it so you can find the vertex of the associated parabola, follow these steps: Collect the x-containing terms on the right-hand side of the "equals" sign, with the y and any loose numbers on the left-hand side of the "equals". If any number a ≠ 1 is multiplied onto the x2 term, factor this out and move it in front of the parenthetical containing the x-terms. Rewrite each side of the equation to add room for completing the square. On the left-hand side with the y, add an empty parenthetical, multiplied by the factored-out value a, if applicable. On the right-hand side, add extra space inside the parenthetical. (This parenthetical already has the a multiplied onto it, if applicable.) Take the coefficient of the linear x-term — along with its sign! — and multiply by one-half. Square this new value, and add it inside the parentheticals on either side of the equation. Convert the right-hand side to a squared binomial (multiplied by a, if applicable). Move any loose numbers from the left-hand side over to the right-hand side. Make sure that your signs inside the vertex form are correct, and then read off the value of the vertex. URL: You can use the Mathway widget below to practice converting to vertex form. (Or skip the widget, and continue with the next page.) Try the entered exercise, or type in your own exercise. Then click the button and select "Find the Vertex Form" to compare your answer to Mathway's. Please accept "preferences" cookies in order to enable this widget. (Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.) Page 2 Select a Course Below Standardized Test Prep ACCUPLACER Math ACT Math ALEKS Math ASVAB Math CBEST Math CLEP Math FTCE Math GED Math GMAT Math GRE Math HESI Math Math Placement Test NES Math PERT Math PRAXIS Math SAT Math TEAS Math TSI Math VPT Math + more tests K12 Math 5th Grade Math 6th Grade Math Pre-Algebra Algebra 1 Geometry Algebra 2 College Math College Pre-Algebra Introductory Algebra Intermediate Algebra College Algebra Homeschool Math Pre-Algebra Algebra 1 Geometry Algebra 2 Share This Page Terms of Use Privacy / Cookies Contact About Purplemath About the Author Tutoring from PM Advertising Linking to PM Site licencing Visit Our Profiles © 2024 Purplemath, Inc. All right reserved. Web Design by
4604	https://openstax.org/books/university-physics-volume-1/pages/2-1-scalars-and-vectors	Skip to Content Go to accessibility page Keyboard shortcuts menu Log in University Physics Volume 1 2.1 Scalars and Vectors University Physics Volume 1 2.1 Scalars and Vectors Search for key terms or text. Learning Objectives By the end of this section, you will be able to: Describe the difference between vector and scalar quantities. Identify the magnitude and direction of a vector. Explain the effect of multiplying a vector quantity by a scalar. Describe how one-dimensional vector quantities are added or subtracted. Explain the geometric construction for the addition or subtraction of vectors in a plane. Distinguish between a vector equation and a scalar equation. Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, “a class period lasts 50 min” or “the gas tank in my car holds 65 L” or “the distance between two posts is 100 m.” A physical quantity that can be specified completely in this manner is called a scalar quantity. Scalar is a synonym of “number.” Time, mass, distance, length, volume, temperature, and energy are examples of scalar quantities. Scalar quantities that have the same physical units can be added or subtracted according to the usual rules of algebra for numbers. For example, a class ending 10 min earlier than 50 min lasts 50min−10min=40min50min−10min=40min. Similarly, a 60-cal serving of corn followed by a 200-cal serving of donuts gives 60cal+200cal=260cal60cal+200cal=260cal of energy. When we multiply a scalar quantity by a number, we obtain the same scalar quantity but with a larger (or smaller) value. For example, if yesterday’s breakfast had 200 cal of energy and today’s breakfast has four times as much energy as it had yesterday, then today’s breakfast has 4(200cal)=800cal4(200cal)=800cal of energy. Two scalar quantities can also be multiplied or divided by each other to form a derived scalar quantity. For example, if a train covers a distance of 100 km in 1.0 h, its speed is 100.0 km/1.0 h = 27.8 m/s, where the speed is a derived scalar quantity obtained by dividing distance by time. Many physical quantities, however, cannot be described completely by just a single number of physical units. For example, when the U.S. Coast Guard dispatches a ship or a helicopter for a rescue mission, the rescue team must know not only the distance to the distress signal, but also the direction from which the signal is coming so they can get to its origin as quickly as possible. Physical quantities specified completely by giving a number of units (magnitude) and a direction are called vector quantities. Examples of vector quantities include displacement, velocity, position, force, and torque. In the language of mathematics, physical vector quantities are represented by mathematical objects called vectors (Figure 2.2). We can add or subtract two vectors, and we can multiply a vector by a scalar or by another vector, but we cannot divide by a vector. The operation of division by a vector is not defined. Figure 2.2 We draw a vector from the initial point or origin (called the “tail” of a vector) to the end or terminal point (called the “head” of a vector), marked by an arrowhead. Magnitude is the length of a vector and is always a positive scalar quantity. (credit "photo": modification of work by Cate Sevilla) Let’s examine vector algebra using a graphical method to be aware of basic terms and to develop a qualitative understanding. In practice, however, when it comes to solving physics problems, we use analytical methods, which we’ll see in the next section. Analytical methods are more simple computationally and more accurate than graphical methods. From now on, to distinguish between a vector and a scalar quantity, we adopt the common convention that a letter in bold type with an arrow above it denotes a vector, and a letter without an arrow denotes a scalar. For example, a distance of 2.0 km, which is a scalar quantity, is denoted by d = 2.0 km, whereas a displacement of 2.0 km in some direction, which is a vector quantity, is denoted by →dd⃗ . Suppose you tell a friend on a camping trip that you have discovered a terrific fishing hole 6 km from your tent. It is unlikely your friend would be able to find the hole easily unless you also communicate the direction in which it can be found with respect to your campsite. You may say, for example, “Walk about 6 km northeast from my tent.” The key concept here is that you have to give not one but two pieces of information—namely, the distance or magnitude (6 km) and the direction (northeast). Displacement is a general term used to describe a change in position, such as during a trip from the tent to the fishing hole. Displacement is an example of a vector quantity. If you walk from the tent (location A) to the hole (location B), as shown in Figure 2.3, the vector →DD⃗ , representing your displacement, is drawn as the arrow that originates at point A and ends at point B. The arrowhead marks the end of the vector. The direction of the displacement vector →DD⃗ is the direction of the arrow. The length of the arrow represents the magnitude D of vector →DD⃗ . Here, D = 6 km. Since the magnitude of a vector is its length, which is a positive number, the magnitude is also indicated by placing the absolute value notation around the symbol that denotes the vector; so, we can write equivalently that D≡\|→D\|D≡∣∣D⃗ ∣∣. To solve a vector problem graphically, we need to draw the vector →DD⃗ to scale. For example, if we assume 1 unit of distance (1 km) is represented in the drawing by a line segment of length u = 2 cm, then the total displacement in this example is represented by a vector of length d=6u=6(2cm)=12cmd=6u=6(2cm)=12cm, as shown in Figure 2.4. Notice that here, to avoid confusion, we used D=6kmD=6km to denote the magnitude of the actual displacement and d = 12 cm to denote the length of its representation in the drawing. Figure 2.3 The displacement vector from point A (the initial position at the campsite) to point B (the final position at the fishing hole) is indicated by an arrow with origin at point A and end at point B. The displacement is the same for any of the actual paths (dashed curves) that may be taken between points A and B. Figure 2.4 A displacement →DD⃗ of magnitude 6 km is drawn to scale as a vector of length 12 cm when the length of 2 cm represents 1 unit of displacement (which in this case is 1 km). Suppose your friend walks from the campsite at A to the fishing pond at B and then walks back: from the fishing pond at B to the campsite at A. The magnitude of the displacement vector →DABD⃗ AB from A to B is the same as the magnitude of the displacement vector →DBAD⃗ BA from B to A (it equals 6 km in both cases), so we can write DAB=DBADAB=DBA. However, vector →DABD⃗ AB is not equal to vector →DBAD⃗ BA because these two vectors have different directions: →DAB≠→DBAD⃗ AB≠D⃗ BA. In Figure 2.3, vector →DBAD⃗ BA would be represented by a vector with an origin at point B and an end at point A, indicating vector →DBAD⃗ BA points to the southwest, which is exactly 180°180° opposite to the direction of vector →DABD⃗ AB. We say that vector →DBAD⃗ BA is antiparallel to vector →DABD⃗ AB and write →DAB=−→DBAD⃗ AB=−D⃗ BA, where the minus sign indicates the antiparallel direction. Two vectors that have identical directions are said to be parallel vectors—meaning, they are parallel to each other. Two parallel vectors →AA⃗ and →BB⃗ are equal, denoted by →A=→BA⃗ =B⃗ , if and only if they have equal magnitudes \|→A\|=\|→B\|∣∣A⃗ ∣∣=∣∣B⃗ ∣∣. Two vectors with directions perpendicular to each other are said to be orthogonal vectors. These relations between vectors are illustrated in Figure 2.5. Figure 2.5 Various relations between two vectors →AA⃗ and →BB⃗ . (a) →A≠→BA⃗ ≠B⃗ because A≠BA≠B. (b) →A≠→BA⃗ ≠B⃗ because they are not parallel and A≠BA≠B. (c) →A≠−→AA⃗ ≠−A⃗ because they have different directions (even though \|→A\|=\|−→A\|=A)∣∣A⃗ ∣∣=∣∣−A⃗ ∣∣=A). (d) →A=→BA⃗ =B⃗ because they are parallel and have identical magnitudes A = B. (e) →A≠→BA⃗ ≠B⃗ because they have different directions (are not parallel); here, their directions differ by 90°90°—meaning, they are orthogonal. Check Your Understanding 2.1 Two motorboats named Alice and Bob are moving on a lake. Given the information about their velocity vectors in each of the following situations, indicate whether their velocity vectors are equal or otherwise. (a) Alice moves north at 6 knots and Bob moves west at 6 knots. (b) Alice moves west at 6 knots and Bob moves west at 3 knots. (c) Alice moves northeast at 6 knots and Bob moves south at 3 knots. (d) Alice moves northeast at 6 knots and Bob moves southwest at 6 knots. (e) Alice moves northeast at 2 knots and Bob moves closer to the shore northeast at 2 knots. Algebra of Vectors in One Dimension Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. We can illustrate these vector concepts using an example of the fishing trip seen in Figure 2.6. Figure 2.6 Displacement vectors for a fishing trip. (a) Stopping to rest at point C while walking from camp (point A) to the pond (point B). (b) Going back for the dropped tackle box (point D). (c) Finishing up at the fishing pond. Suppose your friend departs from point A (the campsite) and walks in the direction to point B (the fishing pond), but, along the way, stops to rest at some point C located three-quarters of the distance between A and B, beginning from point A (Figure 2.6(a)). What is his displacement vector →DACD⃗ AC when he reaches point C? We know that if he walks all the way to B, his displacement vector relative to A is →DABD⃗ AB, which has magnitude DAB=6kmDAB=6km and a direction of northeast. If he walks only a 0.75 fraction of the total distance, maintaining the northeasterly direction, at point C he must be 0.75DAB=4.5km0.75DAB=4.5km away from the campsite at A. So, his displacement vector at the rest point C has magnitude DAC=4.5km=0.75DABDAC=4.5km=0.75DAB and is parallel to the displacement vector →DABD⃗ AB. All of this can be stated succinctly in the form of the following vector equation: →DAC=0.75→DAB. D⃗ AC=0.75D⃗ AB. In a vector equation, both sides of the equation are vectors. The previous equation is an example of a vector multiplied by a positive scalar (number) α=0.75α=0.75. The result, →DACD⃗ AC, of such a multiplication is a new vector with a direction parallel to the direction of the original vector →DABD⃗ AB. In general, when a vector →AA⃗ is multiplied by a positive scalar αα, the result is a new vector →BB⃗ that is parallel to →AA⃗ : →B=α→A. B⃗ =αA⃗ . 2.1 The magnitude \|→B\|∣∣B⃗ ∣∣ of this new vector is obtained by multiplying the magnitude \|→A\| of the original vector, as expressed by the scalar equation: B=\|α\|A. 2.2 In a scalar equation, both sides of the equation are numbers. Equation 2.2 is a scalar equation because the magnitudes of vectors are scalar quantities (and positive numbers). If the scalar α is negative in the vector equation Equation 2.1, then the magnitude \|→B\| of the new vector is still given by Equation 2.2, but the direction of the new vector →B is antiparallel to the direction of →A. These principles are illustrated in Figure 2.7(a) by two examples where the length of vector →A is 1.5 units. When α=2, the new vector →B=2→A has length B=2A=3.0units (twice as long as the original vector) and is parallel to the original vector. When α=−2, the new vector →C=−2→A has length C=\|−2\|A=3.0units (twice as long as the original vector) and is antiparallel to the original vector. Figure 2.7 Algebra of vectors in one dimension. (a) Multiplication by a scalar. (b) Addition of two vectors (→R is called the resultant of vectors →A and →B). (c) Subtraction of two vectors (→D is the difference of vectors →A and →B). Now suppose your fishing buddy departs from point A (the campsite), walking in the direction to point B (the fishing hole), but he realizes he lost his tackle box when he stopped to rest at point C (located three-quarters of the distance between A and B, beginning from point A). So, he turns back and retraces his steps in the direction toward the campsite and finds the box lying on the path at some point D only 1.2 km away from point C (see Figure 2.6(b)). What is his displacement vector →DAD when he finds the box at point D? What is his displacement vector →DDB from point D to the hole? We have already established that at rest point C his displacement vector is →DAC=0.75→DAB. Starting at point C, he walks southwest (toward the campsite), which means his new displacement vector →DCD from point C to point D is antiparallel to →DAB. Its magnitude \|→DCD\| is DCD=1.2km=0.2DAB, so his second displacement vector is →DCD=−0.2→DAB. His total displacement →DAD relative to the campsite is the vector sum of the two displacement vectors: vector →DAC (from the campsite to the rest point) and vector →DCD (from the rest point to the point where he finds his box): →DAD=→DAC+→DCD. 2.3 The vector sum of two (or more) vectors is called the resultant vector or, for short, the resultant. When the vectors on the right-hand-side of Equation 2.3 are known, we can find the resultant →DAD as follows: →DAD=→DAC+→DCD=0.75→DAB−0.2→DAB=(0.75−0.2)→DAB=0.55→DAB. 2.4 When your friend finally reaches the pond at B, his displacement vector →DAB from point A is the vector sum of his displacement vector →DAD from point A to point D and his displacement vector →DDB from point D to the fishing hole: →DAB=→DAD+→DDB (see Figure 2.6(c)). This means his displacement vector →DDB is the difference of two vectors: →DDB=→DAB−→DAD=→DAB+(−→DAD). 2.5 Notice that a difference of two vectors is nothing more than a vector sum of two vectors because the second term in Equation 2.5 is vector −→DAD (which is antiparallel to →DAD). When we substitute Equation 2.4 into Equation 2.5, we obtain the second displacement vector: →DDB=→DAB−→DAD=→DAB−0.55→DAB=(1.0−0.55)→DAB=0.45→DAB. 2.6 This result means your friend walked DDB=0.45DAB=0.45(6.0km)=2.7km from the point where he finds his tackle box to the fishing hole. When vectors →A and →B lie along a line (that is, in one dimension), such as in the camping example, their resultant →R=→A+→B and their difference →D=→A−→B both lie along the same direction. We can illustrate the addition or subtraction of vectors by drawing the corresponding vectors to scale in one dimension, as shown in Figure 2.7. To illustrate the resultant when →A and →B are two parallel vectors, we draw them along one line by placing the origin of one vector at the end of the other vector in head-to-tail fashion (see Figure 2.7(b)). The magnitude of this resultant is the sum of their magnitudes: R = A + B. The direction of the resultant is parallel to both vectors. When vector →A is antiparallel to vector →B, we draw them along one line in either head-to-head fashion (Figure 2.7(c)) or tail-to-tail fashion. The magnitude of the vector difference, then, is the absolute value D=\|A−B\| of the difference of their magnitudes. The direction of the difference vector →D is parallel to the direction of the longer vector. In general, in one dimension—as well as in higher dimensions, such as in a plane or in space—we can add any number of vectors and we can do so in any order because the addition of vectors is commutative, →A+→B=→B+→A, 2.7 and associative, (→A+→B)+→C=→A+(→B+→C). 2.8 Moreover, multiplication by a scalar is distributive: α1→A+α2→A=(α1+α2)→A. 2.9 We used the distributive property in Equation 2.4 and Equation 2.6. When adding many vectors in one dimension, it is convenient to use the concept of a unit vector. A unit vector, which is denoted by a letter symbol with a hat, such as ˆu, has a magnitude of one and does not have any physical unit so that \|ˆu\|≡u=1. The only role of a unit vector is to specify direction. For example, instead of saying vector →DAB has a magnitude of 6.0 km and a direction of northeast, we can introduce a unit vector ˆu that points to the northeast and say succinctly that →DAB=(6.0km)ˆu. Then the southwesterly direction is simply given by the unit vector −ˆu. In this way, the displacement of 6.0 km in the southwesterly direction is expressed by the vector →DBA=(−6.0km)ˆu. Example 2.1 A Ladybug Walker A long measuring stick rests against a wall in a physics laboratory with its 200-cm end at the floor. A ladybug lands on the 100-cm mark and crawls randomly along the stick. It first walks 15 cm toward the floor, then it walks 56 cm toward the wall, then it walks 3 cm toward the floor again. Then, after a brief stop, it continues for 25 cm toward the floor and then, again, it crawls up 19 cm toward the wall before coming to a complete rest (Figure 2.8). Find the vector of its total displacement and its final resting position on the stick. Strategy If we choose the direction along the stick toward the floor as the direction of unit vector ˆu, then the direction toward the floor is +ˆu and the direction toward the wall is −ˆu. The ladybug makes a total of five displacements: →D1=(15cm)(+ˆu),→D2=(56cm)(−ˆu),→D3=(3cm)(+ˆu),→D4=(25cm)(+ˆu),and→D5=(19cm)(−ˆu). The total displacement →D is the resultant of all its displacement vectors. Figure 2.8 Five displacements of the ladybug. Note that in this schematic drawing, magnitudes of displacements are not drawn to scale. (credit "ladybug": modification of work by “Persian Poet Gal”/Wikimedia Commons) Solution The resultant of all the displacement vectors is →D=→D1+→D2+→D3+→D4+→D5=(15cm)(+ˆu)+(56cm)(−ˆu)+(3cm)(+ˆu)+(25cm)(+ˆu)+(19cm)(−ˆu)=(15−56+3+25−19)cmˆu=−32cmˆu. In this calculation, we use the distributive law given by Equation 2.9. The result reads that the total displacement vector points away from the 100-cm mark (initial landing site) toward the end of the meter stick that touches the wall. The end that touches the wall is marked 0 cm, so the final position of the ladybug is at the (100 – 32)cm = 68-cm mark. Check Your Understanding 2.2 A cave diver enters a long underwater tunnel. When her displacement with respect to the entry point is 20 m, she accidentally drops her camera, but she doesn’t notice it missing until she is some 6 m farther into the tunnel. She swims back 10 m but cannot find the camera, so she decides to end the dive. How far from the entry point is she? Taking the positive direction out of the tunnel, what is her displacement vector relative to the entry point? Algebra of Vectors in Two Dimensions When vectors lie in a plane—that is, when they are in two dimensions—they can be multiplied by scalars, added to other vectors, or subtracted from other vectors in accordance with the general laws expressed by Equation 2.1, Equation 2.2, Equation 2.7, and Equation 2.8. However, the addition rule for two vectors in a plane becomes more complicated than the rule for vector addition in one dimension. We have to use the laws of geometry to construct resultant vectors, followed by trigonometry to find vector magnitudes and directions. This geometric approach is commonly used in navigation (Figure 2.9). In this section, we need to have at hand two rulers, a triangle, a protractor, a pencil, and an eraser for drawing vectors to scale by geometric constructions. Figure 2.9 In navigation, the laws of geometry are used to draw resultant displacements on nautical maps. For a geometric construction of the sum of two vectors in a plane, we follow the parallelogram rule. Suppose two vectors →A and →B are at the arbitrary positions shown in Figure 2.10. Translate either one of them in parallel to the beginning of the other vector, so that after the translation, both vectors have their origins at the same point. Now, at the end of vector →A we draw a line parallel to vector →B and at the end of vector →B we draw a line parallel to vector →A (the dashed lines in Figure 2.10). In this way, we obtain a parallelogram. From the origin of the two vectors we draw a diagonal that is the resultant →R of the two vectors: →R=→A+→B (Figure 2.10(a)). The other diagonal of this parallelogram is the vector difference of the two vectors →D=→A−→B, as shown in Figure 2.10(b). Notice that the end of the difference vector is placed at the end of vector →A. Figure 2.10 The parallelogram rule for the addition of two vectors. Make the parallel translation of each vector to a point where their origins (marked by the dot) coincide and construct a parallelogram with two sides on the vectors and the other two sides (indicated by dashed lines) parallel to the vectors. (a) Draw the resultant vector →R along the diagonal of the parallelogram from the common point to the opposite corner. Length R of the resultant vector is not equal to the sum of the magnitudes of the two vectors. (b) Draw the difference vector →D=→A−→B along the diagonal connecting the ends of the vectors. Place the origin of vector →D at the end of vector →B and the end (arrowhead) of vector →D at the end of vector →A. Length D of the difference vector is not equal to the difference of magnitudes of the two vectors. It follows from the parallelogram rule that neither the magnitude of the resultant vector nor the magnitude of the difference vector can be expressed as a simple sum or difference of magnitudes A and B, because the length of a diagonal cannot be expressed as a simple sum of side lengths. When using a geometric construction to find magnitudes \|→R\| and \|→D\|, we have to use trigonometry laws for triangles, which may lead to complicated algebra. There are two ways to circumvent this algebraic complexity. One way is to use the method of components, which we examine in the next section. The other way is to draw the vectors to scale, as is done in navigation, and read approximate vector lengths and angles (directions) from the graphs. In this section we examine the second approach. If we need to add three or more vectors, we repeat the parallelogram rule for the pairs of vectors until we find the resultant of all of the resultants. For three vectors, for example, we first find the resultant of vector 1 and vector 2, and then we find the resultant of this resultant and vector 3. The order in which we select the pairs of vectors does not matter because the operation of vector addition is commutative and associative (see Equation 2.7 and Equation 2.8). Before we state a general rule that follows from repetitive applications of the parallelogram rule, let’s look at the following example. Suppose you plan a vacation trip in Florida. Departing from Tallahassee, the state capital, you plan to visit your uncle Joe in Jacksonville, see your cousin Vinny in Daytona Beach, stop for a little fun in Orlando, see a circus performance in Tampa, and visit the University of Florida in Gainesville. Your route may be represented by five displacement vectors →A, →B, →C, →D, and →E, which are indicated by the red vectors in Figure 2.11. What is your total displacement when you reach Gainesville? The total displacement is the vector sum of all five displacement vectors, which may be found by using the parallelogram rule four times. Alternatively, recall that the displacement vector has its beginning at the initial position (Tallahassee) and its end at the final position (Gainesville), so the total displacement vector can be drawn directly as an arrow connecting Tallahassee with Gainesville (see the green vector in Figure 2.11). When we use the parallelogram rule four times, the resultant →R we obtain is exactly this green vector connecting Tallahassee with Gainesville: →R=→A+→B+→C+→D+→E. Figure 2.11 When we use the parallelogram rule four times, we obtain the resultant vector →R=→A+→B+→C+→D+→E, which is the green vector connecting Tallahassee with Gainesville. Drawing the resultant vector of many vectors can be generalized by using the following tail-to-head geometric construction. Suppose we want to draw the resultant vector →R of four vectors →A, →B, →C, and →D (Figure 2.12(a)). We select any one of the vectors as the first vector and make a parallel translation of a second vector to a position where the origin (“tail”) of the second vector coincides with the end (“head”) of the first vector. Then, we select a third vector and make a parallel translation of the third vector to a position where the origin of the third vector coincides with the end of the second vector. We repeat this procedure until all the vectors are in a head-to-tail arrangement like the one shown in Figure 2.12. We draw the resultant vector →R by connecting the origin (“tail”) of the first vector with the end (“head”) of the last vector. The end of the resultant vector is at the end of the last vector. Because the addition of vectors is associative and commutative, we obtain the same resultant vector regardless of which vector we choose to be first, second, third, or fourth in this construction. Figure 2.12 Tail-to-head method for drawing the resultant vector →R=→A+→B+→C+→D. (a) Four vectors of different magnitudes and directions. (b) Vectors in (a) are translated to new positions where the origin (“tail”) of one vector is at the end (“head”) of another vector. The resultant vector is drawn from the origin (“tail”) of the first vector to the end (“head”) of the last vector in this arrangement. Example 2.2 Geometric Construction of the Resultant The three displacement vectors →A, →B, and →C in Figure 2.13 are specified by their magnitudes A = 10.0, B = 7.0, and C = 8.0, respectively, and by their respective direction angles with the horizontal direction α=35°, β=−110°, and γ=30°. The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a) →R=→A+→B, (b) →D=→A−→B, and (c) →S=→A−3→B+→C. Figure 2.13 Vectors used in Example 2.2 and in the Check Your Understanding feature that follows. Strategy In geometric construction, to find a vector means to find its magnitude and its direction angle with the horizontal direction. The strategy is to draw to scale the vectors that appear on the right-hand side of the equation and construct the resultant vector. Then, use a ruler and a protractor to read the magnitude of the resultant and the direction angle. For parts (a) and (b) we use the parallelogram rule. For (c) we use the tail-to-head method. Solution For parts (a) and (b), we attach the origin of vector →B to the origin of vector →A, as shown in Figure 2.14, and construct a parallelogram. The shorter diagonal of this parallelogram is the sum →A+→B. The longer of the diagonals is the difference →A−→B. We use a ruler to measure the lengths of the diagonals, and a protractor to measure the angles with the horizontal. For the resultant →R, we obtain R = 5.8 cm and θR≈0°. For the difference →D, we obtain D = 16.2 cm and θD=49.3°, which are shown in Figure 2.14. Figure 2.14 Using the parallelogram rule to solve (a) (finding the resultant, red) and (b) (finding the difference, blue). For (c), we can start with vector −3→B and draw the remaining vectors tail-to-head as shown in Figure 2.15. In vector addition, the order in which we draw the vectors is unimportant, but drawing the vectors to scale is very important. Next, we draw vector →S from the origin of the first vector to the end of the last vector and place the arrowhead at the end of →S. We use a ruler to measure the length of →S, and find that its magnitude is S = 36.9 cm. We use a protractor and find that its direction angle is θS=52.9°. This solution is shown in Figure 2.15. Figure 2.15 Using the tail-to-head method to solve (c) (finding vector →S, green). Check Your Understanding 2.3 Using the three displacement vectors →A, →B, and →F in Figure 2.13, choose a convenient scale, and use a ruler and a protractor to find vector →G given by the vector equation →G=→A+2→B−→F. Interactive Observe the addition of vectors in a plane by visiting this vector calculator, and by engaging the Phet simulation below. Access multimedia content Previous Next Order a print copy Citation/Attribution This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax. Attribution information If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution: Access for free at If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution: Access for free at Citation information Use the information below to generate a citation. We recommend using a citation tool such as this one. Authors: William Moebs, Samuel J. Ling, Jeff Sanny Publisher/website: OpenStax Book title: University Physics Volume 1 Publication date: Sep 19, 2016 Location: Houston, Texas Book URL: Section URL: © Jul 8, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.
4605	https://www.chegg.com/homework-help/questions-and-answers/optimizing-economic-functions-622-given-profit-function-tt-160x-3x-2xy-2y2-120y-18-firm-pr-q45881940	Solved OPTIMIZING ECONOMIC FUNCTIONS 6.22. Given the profit \| Chegg.com Skip to main content Books Rent/Buy Read Return Sell Study Tasks Homework help Understand a topic Writing & citations Tools Expert Q&A Math Solver Citations Plagiarism checker Grammar checker Expert proofreading Career For educators Help Sign in Paste Copy Cut Options Upload Image Math Mode ÷ ≤ ≥ o π ∞ ∩ ∪           √  ∫              Math Math Geometry Physics Greek Alphabet Math Calculus Calculus questions and answers OPTIMIZING ECONOMIC FUNCTIONS 6.22. Given the profit function TT = 160x - 3x? - 2xy – 2y2 + 120y -- 18 for a firm producing two goods x and y, (a) maximize profits, (b) test the second-order condition, and (c) evaluate the function at the critical values i and .. Tx = 160 - 6x – 2y = 0 , = -2x – 4y + 120 = 0 When solved simultaneously, z = 20 and ✓ = 20. b) Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. See Answer See Answer See Answer done loading Question: OPTIMIZING ECONOMIC FUNCTIONS 6.22. Given the profit function TT = 160x - 3x? - 2xy – 2y2 + 120y -- 18 for a firm producing two goods x and y, (a) maximize profits, (b) test the second-order condition, and (c) evaluate the function at the critical values i and .. Tx = 160 - 6x – 2y = 0 , = -2x – 4y + 120 = 0 When solved simultaneously, z = 20 and ✓ = 20. b) Show transcribed image text There are 4 steps to solve this one.Solution Share Share Share done loading Copy link Step 1 Explanation: To maximize a function in two variables we find partial derivatives of given function with respect t... View the full answer Step 2 UnlockStep 3 UnlockStep 4 UnlockAnswer Unlock Next question Transcribed image text: OPTIMIZING ECONOMIC FUNCTIONS 6.22. Given the profit function TT = 160x - 3x? - 2xy – 2y2 + 120y -- 18 for a firm producing two goods x and y, (a) maximize profits, (b) test the second-order condition, and (c) evaluate the function at the critical values i and .. Tx = 160 - 6x – 2y = 0 , = -2x – 4y + 120 = 0 When solved simultaneously, z = 20 and ✓ = 20. b) Taking the second partials, Tixx = -6 Tyy = -4 Tixy = -2 With both direct second partials negative, and "xx Tyy> (TTX)", is maximized at x = y = 20 T = 2782 Not the question you’re looking for? Post any question and get expert help quickly. 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4606	https://www.engineeringtoolbox.com/specific-heat-capacity-water-d_660.html	Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications! Specific Heat Capacity of Water: Temperature-Dependent Data and Calculator Online calculator, figures and tables showing specific heat of liquid water at constant volume or constant pressure at temperatures from 0 to 360 °C (32-700 °F) - SI and Imperial units. Specific heat (C) is the amount of heat required to change the temperature of a mass unit of a substance by one degree. When calculating mass and volume flow in a water heating systems at higher temperature - the specific heat should be corrected according the figures and tables below. The specific heat is given at varying temperatures (°C and °F) and at water saturation pressure (which for practical use, gives the same result as atmospheric pressure at temperatures < 100 °C (212°F)). I sochoric specific heat (Cv) for water in a constant-volume , (= isovolumetric or isometric ) closed system. Isobaric specific heat (Cp) for water in a constant pressure (ΔP = 0) system. Online Water Specific Heat Calculator The calculator below can be used to calculate the liquid water specific heat at constant volume or constant pressure and given temperatures. The output specific heat is given as kJ/(kmolK), kJ/(kgK), kWh/(kgK), kcal/(kg K), Btu(IT)/(mol°R) and Btu(IT)/(lbm °R) Note! Temperature must be within the ranges 0-370 °C, 32-700 °F, 273-645 K and 492-1160 °R to get valid values. See Water and Heavy Water - thermodynamic properties. See also other properties of Water at varying temperature and pressure : Boiling points at high pressure, Boiling points at vacuum pressure, Density and specific weight, Dynamic and kinematic viscosity, Enthalpy and entropy, Heat of vaporization, Ionization Constant, pKw , of normal and heavy water, Melting points at high pressure, Prandtl number, Properties at Gas-Liquid Equilibrium Conditions, Saturation pressure, Specific gravity, Specific volume, Thermal conductivity, Thermal diffusivity and Vapour pressure at gas-liquid equilibrium, as well as Specific heat of Air - at Constant Pressure and Varying Temperature, Air - at Constant Temperature and Varying Pressure, Ammonia, Butane, Carbon dioxide, Carbon monoxide, Ethane, Ethanol, Ethylene, Hydrogen, Methane, Methanol, Nitrogen, Oxygen and Propane. Specific heat for liquid water at temperatures from 0 to 360 °C: For full table with Isobaric Specific Heat - rotate the screen! Water - Specific Heat vs. Temperature \| Temperature \| Isochoric Specific Heat (Cv) \| \| \| \| Isobaric Specific Heat (Cp) \| \| \| \| \| (°C) \| (J/(mol K)) \| (kJ/kgK) \| (kWh/(kg K)) \| (kcal/(kg K)) (Btu(IT)/lbm °F) \| (J/(mol K)) \| (kJ/kgK) \| (kWh/(kg K)) \| (kcal/(kg K)) (Btu(IT)/lbm °F) \| \| 0.01 \| 75.981 \| 4.2174 \| 0.001172 \| 1.0073 \| 76.026 \| 4.2199 \| 0.001172 \| 1.0079 \| \| 10 \| 75.505 \| 4.1910 \| 0.001164 \| 1.0010 \| 75.586 \| 4.1955 \| 0.001165 \| 1.0021 \| \| 20 \| 74.893 \| 4.1570 \| 0.001155 \| 0.9929 \| 75.386 \| 4.1844 \| 0.001162 \| 0.9994 \| \| 25 \| 74.548 \| 4.1379 \| 0.001149 \| 0.9883 \| 75.336 \| 4.1816 \| 0.001162 \| 0.9988 \| \| 30 \| 74.181 \| 4.1175 \| 0.001144 \| 0.9834 \| 75.309 \| 4.1801 \| 0.001161 \| 0.9984 \| \| 40 \| 73.392 \| 4.0737 \| 0.001132 \| 0.9730 \| 75.300 \| 4.1796 \| 0.001161 \| 0.9983 \| \| 50 \| 72.540 \| 4.0264 \| 0.001118 \| 0.9617 \| 75.334 \| 4.1815 \| 0.001162 \| 0.9987 \| \| 60 \| 71.644 \| 3.9767 \| 0.001105 \| 0.9498 \| 75.399 \| 4.1851 \| 0.001163 \| 0.9996 \| \| 70 \| 70.716 \| 3.9252 \| 0.001090 \| 0.9375 \| 75.491 \| 4.1902 \| 0.001164 \| 1.0008 \| \| 80 \| 69.774 \| 3.8729 \| 0.001076 \| 0.9250 \| 75.611 \| 4.1969 \| 0.001166 \| 1.0024 \| \| 90 \| 68.828 \| 3.8204 \| 0.001061 \| 0.9125 \| 75.763 \| 4.2053 \| 0.001168 \| 1.0044 \| \| 100 \| 67.888 \| 3.7682 \| 0.001047 \| 0.9000 \| 75.950 \| 4.2157 \| 0.001171 \| 1.0069 \| \| 110 \| 66.960 \| 3.7167 \| 0.001032 \| 0.8877 \| 76.177 \| 4.2283 \| 0.001175 \| 1.0099 \| \| 120 \| 66.050 \| 3.6662 \| 0.001018 \| 0.8757 \| 76.451 \| 4.2435 \| 0.001179 \| 1.0135 \| \| 140 \| 64.306 \| 3.5694 \| 0.000992 \| 0.8525 \| 77.155 \| 4.2826 \| 0.001190 \| 1.0229 \| \| 160 \| 62.674 \| 3.4788 \| 0.000966 \| 0.8309 \| 78.107 \| 4.3354 \| 0.001204 \| 1.0355 \| \| 180 \| 61.163 \| 3.3949 \| 0.000943 \| 0.8109 \| 79.360 \| 4.4050 \| 0.001224 \| 1.0521 \| \| 200 \| 59.775 \| 3.3179 \| 0.000922 \| 0.7925 \| 80.996 \| 4.4958 \| 0.001249 \| 1.0738 \| \| 220 \| 58.514 \| 3.2479 \| 0.000902 \| 0.7757 \| 83.137 \| 4.6146 \| 0.001282 \| 1.1022 \| \| 240 \| 57.381 \| 3.1850 \| 0.000885 \| 0.7607 \| 85.971 \| 4.7719 \| 0.001326 \| 1.1397 \| \| 260 \| 56.392 \| 3.1301 \| 0.000869 \| 0.7476 \| 89.821 \| 4.9856 \| 0.001385 \| 1.1908 \| \| 280 \| 55.578 \| 3.0849 \| 0.000857 \| 0.7368 \| 95.285 \| 5.2889 \| 0.001469 \| 1.2632 \| \| 300 \| 55.003 \| 3.0530 \| 0.000848 \| 0.7292 \| 103.60 \| 5.7504 \| 0.001597 \| 1.3735 \| \| 320 \| 54.819 \| 3.0428 \| 0.000845 \| 0.7268 \| 117.78 \| 6.5373 \| 0.001816 \| 1.5614 \| \| 340 \| 55.455 \| 3.0781 \| 0.000855 \| 0.7352 \| 147.88 \| 8.2080 \| 0.002280 \| 1.9604 \| \| 360 \| 59.402 \| 3.2972 \| 0.000916 \| 0.7875 \| 270.31 \| 15.004 \| 0.004168 \| 3.5836 \| Specific heat for liquid water at temperatures from 32 to 675 °F: For full table with Isobaric Specific Heat - rotate the screen! Water - Specific Heat vs. Temperature \| Temperature \| Isochoric Specific Heat (Cv) \| \| \| Isobaric Specific Heat (Cp) \| \| \| \| (°F) \| (Btu(IT)/(mol °R)) \| (Btu(IT)/(lbm °F)) (kcal/(kg K)) \| (kJ/kgK) \| (Btu(IT)/kmol °R) \| (Btu(IT)/lbm °F) (kcal/kg K) \| (kJ/kg K) \| \| 32.2 \| 40.0 \| 1.007 \| 4.217 \| 40.032 \| 1.008 \| 4.220 \| \| 40 \| 39.9 \| 1.005 \| 4.208 \| 39.916 \| 1.005 \| 4.208 \| \| 50 \| 39.8 \| 1.001 \| 4.191 \| 39.801 \| 1.002 \| 4.196 \| \| 60 \| 39.6 \| 0.996 \| 4.169 \| 39.739 \| 1.001 \| 4.189 \| \| 80 \| 39.2 \| 0.986 \| 4.128 \| 39.660 \| 0.999 \| 4.181 \| \| 100 \| 38.7 \| 0.975 \| 4.082 \| 39.643 \| 0.998 \| 4.179 \| \| 120 \| 38.3 \| 0.963 \| 4.033 \| 39.662 \| 0.999 \| 4.181 \| \| 140 \| 37.7 \| 0.950 \| 3.977 \| 39.702 \| 1.000 \| 4.185 \| \| 160 \| 37.2 \| 0.937 \| 3.923 \| 39.761 \| 1.001 \| 4.191 \| \| 180 \| 36.7 \| 0.923 \| 3.865 \| 39.835 \| 1.003 \| 4.199 \| \| 200 \| 36.1 \| 0.909 \| 3.805 \| 39.927 \| 1.005 \| 4.209 \| \| 212 \| 35.7 \| 0.900 \| 3.768 \| 39.993 \| 1.007 \| 4.216 \| \| 220 \| 35.5 \| 0.895 \| 3.745 \| 40.042 \| 1.008 \| 4.221 \| \| 240 \| 35.0 \| 0.880 \| 3.686 \| 40.186 \| 1.012 \| 4.236 \| \| 260 \| 34.4 \| 0.867 \| 3.629 \| 40.364 \| 1.016 \| 4.255 \| \| 280 \| 33.9 \| 0.854 \| 3.574 \| 40.580 \| 1.022 \| 4.278 \| \| 300 \| 33.4 \| 0.841 \| 3.522 \| 40.838 \| 1.028 \| 4.305 \| \| 350 \| 32.3 \| 0.813 \| 3.404 \| 41.685 \| 1.050 \| 4.394 \| \| 400 \| 31.3 \| 0.789 \| 3.302 \| 42.902 \| 1.080 \| 4.522 \| \| 450 \| 30.4 \| 0.767 \| 3.209 \| 44.009 \| 1.108 \| 4.639 \| \| 500 \| 29.7 \| 0.748 \| 3.130 \| 47.296 \| 1.191 \| 4.986 \| \| 550 \| 28.8 \| 0.725 \| 3.035 \| 51.318 \| 1.292 \| 5.410 \| \| 600 \| 28.3 \| 0.713 \| 2.987 \| 59.690 \| 1.503 \| 6.292 \| \| 625 \| 28.4 \| 0.716 \| 2.997 \| 66.611 \| 1.677 \| 7.022 \| \| 650 \| 28.9 \| 0.728 \| 3.047 \| 82.851 \| 2.086 \| 8.734 \| \| 675 \| 29.9 \| 0.754 \| 3.156 \| 126.670 \| 3.189 \| 13.353 \| Unit Converter . 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4607	https://www.youtube.com/watch?v=BqS9SOyYYJo	Finding the LCD for a group of Rational Expressions GreeneMath.com 181000 subscribers 2163 likes Description 213057 views Posted: 25 Feb 2016 0:00 - 2:05 Basic Definitions 2:06 - 9:46 Fraction Examples 10:09 - 10:54 Method 10:55 - 13:05 First Example 13:06 - 14:34 Second Example 14:35 - 16:03 Third Example 16:04 - 17:10 Fourth Example 17:11 - 18:11 Fifth Example In this video, we learn how to find the LCD or least common denominator for a group of rational expressions. 164 comments Transcript: Basic Definitions hello and welcome to Algebra 1 Lesson 45 in this video we're going to learn about finding the LCD that's the least common denominator for a group of rational expressions so now that we've mastered how to multiply and divide rational expressions kind of the next thing we're going to learn is how to add and subtract rational expressions now before we can do that we need to learn how to find the least common denominator for a group of rational expressions so I want you to think back to when we worked with fractions in algebra and we should all remember that adding or subtracting fractions requires a common denominator and you want to find the least common denominator because that's going to save you time and effort with regard to your simplification in the end so the LCD or least common denominator is the smallest number that is divisible by all denominators the LCD is the LCM of all denominators now let me kind of explain this a lot of students get confused when you start talking about LCD and LCM and GCF and gcd they always get those confused for the first part of kind of their math career so the LCM is the least common multiple so you have two three four numbers whatever it is and you want to find the least common multiple when we talk about the least common denominator that's when we're working with fractions and we just take the denominators and we find the least common multip multiple of the denominators and that is the LCD so you can see how the LCM and the LCD are related now the greatest common factor or the greatest common divisor those are the same thing okay if something is a factor it's also a divisor so when we talk about that we're looking for a number that is the largest factor or the largest divisor of a group of numbers now the way they're different is that the GCF or the gcd is going to generally be a smaller number and the LCM or the LCD is going to be a larger number so let's revisit real quick with this example with adding Fraction Examples fractions so I have something like 310 plus 71 15 now we know that we can't just add across like we do with multiplication 3 + 7 is 10 10 + 15 is 25 you know I see students do that that's wrong that is wrong we need to have a common denominator before we start and the most efficient common denominator is the least common denominator so the LCD or the least common denominator and this is equal to the least common multiple of the denominator so of 10 and 15 now hopefully you remember how to do this from pre-algebra but if you don't you basically just Factor each number so for 10 it's 5 2 5 is Prime and so is two for 15 it's what it's 5 3 five is Prime and so is three now each prime factor from every prime factorization is going to go in your list when you build it the only thing is if you have a duplicate prime factor it's only going to go in the largest number of times it occurs in either of the factorizations so in other words I have five and I have two I have five and I have three so five occurs in each prime factorization the largest number of repeats or the largest number of occurrences and any of the prime factorizations is one just have one here and I have one here so I'm just going to put one in one factor five in when I build that LCM or LCD or however you want to think about it now I'm going to multiply by I have a two here no two over here so I just throw it in there I have a three here no three over here so I just throw that in there so the LCD again is the LCM the least common multiple of 10 and 15 and that's 5 2 which is 10 3 which is 30 so once I have found that I go through and I rewrite each fraction as an equivalent fraction where 30 is the denominator so in other words with 3/10ths I would multiply the numerator by three and also the denominator remember I'm just multiplying by a complicated form of one so I'm not changing the value I'm just changing what it looks like then plus we have 7 15 and this is multiplied by 2 over 2 and so this is equal to 2 3 3 is 9 over 10 3 that's 30 then plus 7 2 is 14 over 15 2 that's 30 and then 9 + 14 is going to be 23 so we end up with 23 over 30 as my answer and that can't be simplified any further okay so we know how to find the LCD with regard to regular fractions it's going to get a little bit more complicated when we start thinking about rational expressions because we have some messy variables in right so it's a little bit more complex but the general idea is the same now before I move on I just want to show you so that I settle all confusion that this is not the same as the GCF now notice how you have the number 10 and you have the number 15 now the LCD or the LCM is a larger number 30 is the smallest positive number that is divisible by both 15 and 10 meaning I divide 30 by 15 I get two no remainder I divide 30 by 10 I get three no remainder okay if we talk about the greatest common factor or the greatest common divisor this is equal to what it's going to be a smaller number because we're looking for the largest number that each of these numbers is divisible by so it's kind of the same process you factor each number to start but all you're going to do is look for what's common to both okay so I can only put something in the greatest common factor listing if it's common to everything in this case I don't have a two that's common to everything I don't have a three that's common to everything the only thing that's common to both is five and so my greatest common factor is a five 10 divided 5 is two no remainder 15 divided 5 is three no remainder so you see that the greatest common factor or again the greatest common divisor is a smaller number and the LCD or LCM is a larger number so that's one of the tricks I used to remember it's taught to me by a teacher back when I was in eth grade but it's something that you should think about because a lot of students really struggle with this definition between again the GCF and the gcd same thing and then the LCD the least common denominator which is the LCM of the denominators so let's look at one more example with fractions real quick just to kind of get our feet going we have 1124 minus 92th so again if I want the LCD this is the least common multiple of these two denominators so of 24 and 20 so what do I do I factor 24 and I could do 4 6 4 is 2 2 6 is 2 3 for 20 I'll do 5 4 4 is 2 2 so now if I build my list every prime factor from each prime factorization is going to go in there I'm just paying attention for duplicate prime factors so with two I have one two three of those here and I have one two of those here so when I build my list how many am I going to put in here well I'm going to put three I go with the largest number of repeats from either prime factorization that occurs here I have three of them so I'm going to put 2 2 2 or eight again where students make a mistake is they throw all of them in there and that's going to give you a common multiple but it's not going to be the least common multiple and that's what you're looking for to reduce the amount of work when you go to simplify all right so now the next thing is I have a three not common to both so I'm going to throw that in there and then I have a five not common to both so I'm just going to throw that in there so now we just multiply 2 2 2 is 8 8 3 is 24 24 5 is 120 so that's your LCD or LCM now if we wanted to find the GCF the greatest common factor or the greatest common divisor what would that be well when we build that list it's only going to be numbers in the prime factorizations that's common to everything the only thing we have here that's common to everything is two now I have two factors of two here and again I have one two three factors of two here with the GCF it's got to be common everything so it's going to be two factors of two that's going to go into the GCF and so two factors of two is four so the greatest common factor is four again this is a smaller number than these two and then the LCD or LCM is a larger number than these two that's how you you kind of get an understanding of it 24 ID 4 is 6 no remainder 20 ID 4 is 5 no remainder with this one 120 ID 24 is 5 no remainder and 120 divid 20 is 6 no remainder so you can see the difference between the two let's go ahead and execute this problem just for the sake of completeness we know what the LCD is so let's transform each fraction so we'd have 11 over 24 we'd multiply this by 5 over 5 and then - 9/ 20 and we'd multiply this by 6 over 6 so I'd end up with 55 over 120 minus 54 over 120 and this would give me 1 over 120 as my answer and I can't simplify that any further so again similarly when we work with rational expressions we need to have a common denominator to add or subtract it's going to be the same process again it's going to be Messier because we have a lot of variable involved so to find the LCD for a group of rational expressions we're going to factor all denominators completely just like if I'm looking for the LCD for group of numbers I'm going Method to factor the denominators completely I'm get it down to the prime factorization of each number then I'm going to list each different denominator Factor when a factor occurs in more than one factorization list the largest number of repeats okay there it goes again among all factorizations so again this is the same process that we use when we're thinking about numbers if I have a prime factor of two in one prime factorization and a prime factor of two in another one I'm looking for the largest number of repeats between those two factorizations to put that in when I build my list all right then the last step is just to multiply the factors to obtain the LCD so again same process is when we work with fractions but it's just going to be a little bit more messy a little bit more timec consuming a little bit more tedious for the simple First Example fact that we have variables involved all right so let's start out with these two rational expressions we have 13 21x cubed and we have 72x to 5th power so I just want to find the LCD so I'm working with the denominators so the LCD is what it's the LCM of 21x cubed and of 12 x to the 5th power so pretty easy for the number parts we already know how to do that right we would just say okay well 21 is what it's 3 7 and 12 is 4 3 4 is 2 2 all right so to build the number part what would I do I list every prime factor I just watch out for duplicates if I have a duplicate prime factor meaning it occurs in more than one prime factorization looking for the largest number of repeats so I have a three here and here it occurs only once in each so I'm just going to put one in here not not two I have a seven here no seven over here I have two twos in here so I'm going to throw both of those in there and we know that's four so the number part is 3 7 which is 21 4 which is 84 so let's just go ahead and write 84 there and then what about the variable part well do I really need to go through and say well I have x x X and then I have x x x x X X again I want the largest number of repeats so if it's the same variable now I'm just going to go with the largest exponent remember when we talked about the greatest common factor we went with the smallest exponent okay now we're going with the largest exponent so the largest exponent is the five so it's going to be 84 x to the 5th power that constitutes the largest number of repeats for either of the prime factorizations so your LCD here is 84 x to the 5th power all right let's take a look at another one so we have 3x - 2 over 25 x^2 and then we also are looking Second Example at -7 over 10 x to the 4th power so again my LCD my LCD is the LCM of these two denominators so 25 x^2 and then 10 x to the 4th power so again we know how to do the number part can almost do that in our head we know 25 is what it's 5 Time 5 and 10 is 5 2 so I know that I have a duplicate prime factor of five it occurs twice here so one and then two two factors of five here only one here so I go with the largest number of repeats and that's in the prime factorization of 25 so I'm going to throw two of those in here 5 5 is 25 and then I'm going to throw a two in there so times two 25 2 is 50 so the number part is a 50 now for the variable part I have X2 and I have x to the 4th power again I want to go with the largest exponent on X and the reason for that is that's going to be the largest number of repeats between the two prime factorizations if I had X2 it's x X so two of those if I have x to the 4th power I have x x x X this is the largest number this is the largest number again as given by the largest exponent so I just use that so this is 50x to 4th power all right so let's take a look at a note we have 7 x Cub - 11 over 7x2 + 21x and then we have Third Example -4x 5th power -3 over 5x^2 + 15x now for this one you might start thinking about this is really complicated what do we do again you got to factor everything completely so I want the LCD of these two guys the first thing I'm going to do is I'm going to factor them so from this first guy here I can pull out a 7x so I can pull out a 7x and then what's left inside is x + three from this guy right here I can pull out a 5x so I pull out a 5x and what's left is x + 3 so now I'm looking at what's common so I have x + 3 and I have x + 3 so when I build my LCD my LCD I'm going to have a factor of X plus 3 in there it occurs only one in each so that's all I'm going to do now each one has an X I have an X here and I have an X in each case it's only one so I'm going to put One X in there and then I have a seven and I have a five each of those is a prime number 7 Time 5 is 35 so what this is going to be it's going to be 35x 35x time x + 3 that quantity and I would just leave it just like this in factored form you can go through and multiply it if you want but usually when you're finding an LCD it's just better to leave it in factored form in case you need to do some cancelling later on all right Fourth Example let's take a look at another one so we have 2X Cub - 5 over 5x - 10 then we have 11 x - 4 over 6 x - 12 so again I want to look at these two denominators and I want to factor them completely so for this one I could pull out a five and I would have xus 2 for this one I could pull out a six and I would have x - 2 so you can see you have x - 2 and x -2 so for the LCD you're going to have what you have a five and you have a six 5 6 is going to be what that's going to be 30 and then you have that xus 2 it only occurs once in each factorization so that's all I need to do when I put it in here I've got one factor of xus 2 then I've got one factor of five and one factor of six because 6 is 2 3 and five is prime so 5 2 3 would give me 30 that's how I got that and then I've got my one factor of x - 2 so you get 30 the quantity x - 2 and again I would leave this in factored form all right let's take a look at one more problem I think you get the general idea so we have n minus 5 I can't really factor Fifth Example that I'll just write n minus 5 I have 3 n Cub + 15 n^ 2 so I could pull out a 3 n 2 and I would get n + 5 and then over here I have n^2 - 25 that's the difference of two squares so this is n - 5 n + 5 so for my LCD what do I have I have n minus 5 here and also here and then I have n plus 5 here and also here so each only occurs once this occurs once this occurs once so I would just put one in there and then this occurs once and this occurs once so I would just put one in there and then we have this 3 n squ that's got to go in so let me kind of scooch this down so we like it to be in the front that's just more traditional and so our LCD is 3 n^ 2 the quantity n minus 5 the quantity n + 5
4608	https://physics.umd.edu/hep/drew/trig/	Trigonometry Without Pain Table of Contents Introduction to Trigonometry More Trig Functions Properties of Trig Functions Sine and Cosine of Sums of Angles## Introduction to TrigonometryBack to top Trigonometry is one of those subjects that will either bore you to tears, drive you crazy, or fascinate you. Either way, the bottom line is that trigonometry was discovered because it's useful, and has a purpose. And that means everyone can understand it if they start at the beginning and work their way through it. So, the beginning of trigonometry starts with the intersection of 2 lines, and in particular, when those 2 lines cross at right angles: The right angle is very special, because when lines cross at right angles, the lines are completely independent. That is, you can walk along either line and get nowhere along the other line. Try crossing 2 lines at an angle other than a right angle, and if you walk along one of the lines, you will slowly walk along the other. That's what it means to be independent. Now, if you connect the two open ends of the lines, that connection is called the "hypotenuse", which comes from the Greek word "stretch": Using more Greek letters, we define the angle $\theta$ like so: We could have defined $\theta$ using the angle between the hypotenuse and the vertical line, but it doesn't matter, as we will see later. Trigonometry, and all of the formulae that drive you crazy (or inspire you) involve understanding the relationship between all of the lengths, and the angle $\theta$: Figure 1. Right triangle. So we have 3 lengths, and 1 angle, so why don't we care about the other 2 angles inside the triangle? To understand this, first let's discuss how we measure angles, and to do that we first have to consider the circle. In the picture below, we have 2 lines that intersect at right angles (black), and a ray that points from the center outward (blue). The ray makes an angle $\theta$ with the horizontal, and when you go around one full turn, you get back to the same angle that you started with, as in the figure below: So we could define the angle $\theta$ as being between 2 limits, e.g. $0$ and $1$, or we can use some other measure. Thanks to the early Mesopotamian mathematicians, who used a number system in base 60, instead of $0\le\theta\le 1$ we say that $0\le\theta\le 360$ degrees, where $\theta=0$ is the same angle as $\theta=360\deg$, where the symbol $\deg$ means "degrees". Given this measure, we now can see that a right angle is $360/4=90\deg$, and halfway around is $180\deg$. Now back to the right triangle, and the question about why we are only considering one of the 3 internal angles (and the 3 sides): the reason is that we already know the right angle, which is $90\deg$, and in fact the sum of all the angles is equal to $180\deg$. For the proof, let's label the other angle between the hypotenuse and the vertical line as $\phi$, and then make a copy of the triangle but flip it over so that the vertical line is to the left, and the horizontal line is on the top. And then attach the 2 triangles at the hypotenuse, as in the following figure: Since we've flipped the triangle over, the 2 angles $\theta$ and $\phi$ are the same on both sides. Now, since the figure becomes a rectangle, then all of the interior angles of a rectangle are right angles ($90\deg$), and since there are 4 of them, they sum to $360\deg$. But since the rectangle is made of 2 identical triangles, then the sum of the interior angles of the triangle is $180\deg$, which means $\theta + \phi + 90\deg = 180\deg$ or $\theta + \phi = 90\deg$. So if we know $\theta$, we know $\phi$, they are not independent, and that's why we are only worrying about $\theta$ here. (Or as they say in geometry: QED). Now to make things even more complicated, let's consider any angle $\theta$ that defines an arc of a circle as in the figure below, where $r$ is the radius of the circle and $s$ is the length of the arc: What is the ratio of $r/s$? Well, one thing we know from the ancients is that if $\theta = 360\deg$, then the ratio of the circumference $s$ (the arc length of the complete circle) to the diameter $d=2r$ is $s/2r=\pi$, or $s\r=2\pi$ where $\pi=3.14159...$. This implies that we can measure angles in units (called "radians") such that $360\deg = 2\pi$ radians or $180\deg = \pi$, which is how we convert from degrees to radians. And this also says that the ratio of the arc length to the radius tells you the value of the angle, but only if you measure it in radians! So now we know how to measure angles, but we want to go back to trigonometry, which starts with right triangles. For our triangle in Figure 1, we want to investigate the relationship between the 3 sides and the interior angle $\theta$. From Pythagoras, we have the Pythagorean theorem, which states that $$c^2 = a^2 + b^2\nonumber$$ which means that in fact we only have 2 independent lengths. So now we are ready to learn about trigonometry and the trig functions. Since only 2 of the 3 lengths are independent, we can ask about the relationship between any two lengths. So let's keep $a$ constant, let $b$ vary, and require that the triangle stay a right triangle, which forces $a^2 + b^2 = c^2$ by the Pythagorean theorem. In the figure below you can see two examples. Notice that as $b$ gets larger, so does $\theta$. The question is: how does $\theta$ change when you change $b$? Figure 1. To understand this, let's take the ratios $a/c$ and $b/c$. If we use the Pythagorean theorem, we can easily show that $$\Big(\frac{a}{c}\Big)^2 + \Big(\frac{b}{c}\Big)^2 = 1\nonumber$$ And clearly, as the ratios change, so will the angle $\theta$ change. So we can define these ratios in the following way. (Note that the word "sine" is named after the Greek word "sine" which means "curve".) "Sine": $\sin\theta = $ opposite/hypotenuse "Cosine": $\cos\theta = $ adjacent/hypotenuse In the figure below, we plot the ratios (sine and cosine) as a function of angle, with sine in red and cosine in blue. Figure 2. The plot shows you $\sin(\theta)$ on the vertical axis in blue, and $\cos(\theta)$ on the vertical axis in red, vs $\theta$ on the horizontal, where $\theta$ is measured in radians. When $\theta=0$, that corresponds to a very small ratio of $b/c$, like the left triangle in Figure 1 above. In that case, when the opposite/hypotenuse is small, the adjacent/hypotenuse is large, as in the the right triangle in Figure 1. Since the adjacent/hypotenuse is the cosine, then for $\theta=0$ we expect the cosine go be at 1, as it is in the red curve above. You can make similar arguments for the case of $\theta=90\deg=\pi/2$, which has $b/c$ close to $1$, so opposite/hypotenuse (sine) goes to $1$ and adjacent/hypotenuse (cosing) goes to $0$. What you can also see from this diagram is that $\cos(\pi/2-\theta)=\sin(\theta)$ (and by the same argument, $\sin(\pi/2-\theta)=\cos(\theta)$). More on that below. So now we know how the sine and cosine are defined, and how they relate to the lengths of the adjacent and opposite sides of a triangle (adjacent and opposite to the angle $\theta$). But we've only defined these 2 functions for the situation where $\theta$ is between $0$ and $\pi/2$. What about the other 3/4 of a circle: $\pi/2\le\theta\le 2\pi$? To extend the definitions over the entire range of $\theta$, we introduce the concept of "the unit circle", as in the figure below: We define the angle $\theta$ in the same way as above: the angle that a ray with one end at the origin makes with the horizontal. In this figure, the circle has a radius $r$ and is centered at the origin, so that no matter what $\theta$ is, the ray touches the circle (circles are defined as the locus of all points that have a constant distance from some fixed origin). In this circle, we can choose the length $r$ so we choose $r=1$ to make things easier below. The point $p$ is the intersection of the ray with the circle. Notice that the ray forms a right triangle with adjacent side $x$ (instead of $a$) and opposite side $y$ (instead of $b$). The definition of the sine and cosine still hold here, however instead of saying that the sine is the ratio of opposite/hypotenuse, we say that the sine tells you the value of the position along the horizontal when you draw a perpendicular line connecting point $p$ to the horizontal line. So in this figure, $x$ is a coordinate along the horizontal axis, and can be positive ($x$ to the right of the vertical line) or negative ($x$ to the left). And similarly for y, which is the coordinate along the vertical of the perpendicular line from $p$ to the vertical axis. Note that $\sin\theta=y/r$ (opposite/hypotenuse) and $\cos\theta=x/r$ (adjacent/hypotenuse), but since $r=1$, we can write $\sin\theta=y$ and $\cos\theta=x$. Also, in green we have labeled the 4 quadrants of the circle, starting in the upper right quadrant and increasing as we go around counterclockwise. This will be used below. Now we can define sine and cosine by saying that no matter where $\theta$ is on the circle, if we drop perpendicular lines from the point $p$ to the vertical and horizontal axes, we will have the value for the $\sin\theta$ and $\cos\theta$, anywhere on the circle. Cute eh? In the figure below, you can click on the solid red circle and drag it around the blue (unit) circle and see how $\theta$ changes, and the projected $x$ and $y$ coordinates (given by $\sin\theta$ and $\cos\theta$) move. \| \| \| \| --- \| $\theta$ \| $=$ \| 45$\deg$ \| \| $\sin\theta$ \| $=$ \| 0.707 \| \| $\cos\theta$ \| $=$ \| 0.707 \| We now have a way of defining the sine and cosine functions over the full cycle $0$ to $2\pi$ ($0$ to $360\deg$): they represent the coordinates of points on the unit circle. Notice that all 4 quadrants of the unit circle are pretty much equivalent, with the only difference being that some are reflections of others. For instance, quadrant I (where $x$ and $y$ are both positive) is similar to quadrant II (where $x$ is negative but $y$ is still positive), but flipped over the vertical axis. So we would expect that the graph of $\sin\theta$ between $90\deg$ and $180\deg$ is similar to the graph of $\sin\theta$ between $0\deg$ and $90\deg$, except that in the latter as $\theta$ increases, $\sin\theta$ increases over the interval and in the former, it decreases. So it should be relatively easy to construct the curve of $\sin\theta$ vs $\theta$ over the entire interval using the curve in quadrant I as in Figure 2. above. In the figure below, we see the sine curve over the entire range of $\theta$ between $0$ and $2\pi$ radians. Notice the symmetry in the 4 quadrants: in quadrant I, $\sin\theta$, the $y$ coordinate, increases from 0 to 1 whereas in quadrant II, the $y$ coordinate decreases from 1 to 0. In quadrant III, the $y$ coordinate decreases from 0 to -1 and in quadrant IV, it increases from -1 to 0. So we could use the curve in Figure 2 and draw the entire sine curve over the full interval $0\le\theta\le 2\pi$, as in the figure below: Similarly, we can draw the full cosine curve in the same interval: One last note: above, we showed that since the interior angles of a right triangle add up to $180\deg$, that means $\theta +\phi=90\deg$, and that means that $\sin(\pi/2-\theta)=\cos(\theta)$. Graphically, this means that if you take the sine curve and move it to the left by $\pi/2$, you will get the cosine curve. And voila, that's what the 2 curves above show! More Trig FunctionsBack to top Now that we have the sine and cosine functions as the ratios of the opposite and adjacent lengths to the hypotenuse, we can form the remaining trig functions. The table below has all 6 trig functions: \| \| \| \| \| --- --- \| \| Sine \| $\sin\theta$ \| \| opposite/hypotenuse \| \| Cosine \| $\cos\theta$ \| \| adjacent/hypotenuse \| \| Tangent \| $\tan\theta$ \| $\sin\theta/\cos\theta$ \| opposite/adjacent \| \| Cotangent \| $\cot\theta$ \| $1/\tan\theta$ \| adjacent/opposite \| \| Secant \| $\sec\theta$ \| $1/\cos\theta$ \| hypotenuse/adjacent \| \| Cosecant \| $\csc\theta$ \| $1/\sin\theta$ \| hypotenuse/opposite \| Properties of Trig FunctionsBack to top Now that we have all of the 6 trig functions, we can explore properties. For instance, the Pythagorean theorem relates the 3 sides of the triangle $a$, $b$, and $c$ (where $c$ is the hypotenuse): $$a^2 + b^2 = c^2\nonumber$$ This also tells us that $$\sin^2\theta + \cos^2\theta = 1\label{fm}$$ and that this relationship is true no matter what value $\theta$ takes on. Note that $\sin^2\theta$ means the same as $(\sin\theta)^2$, and not $\sin\theta^2$. If we take equation $\ref{fm}$ and divide each side by $\cos^2\theta$, we get: $$\tan^2\theta + 1 = \sec^2\theta\nonumber$$ If we take equation $\ref{fm}$ and divide by $\sin^2\theta$, we get: $$1 + \cot^2\theta= \csc^2\theta\nonumber$$ Another useful property of the sine and cosine has to do with their "symmetry". That is, what is $\sin(-\theta)$ and $\cos(-\theta)$. There are many ways to think of how to do this, but one useful way to think of this is to realize that a negative angle is the same as $360\deg$ minus that angle (e.g. $-90\deg$ is the same as $270\deg$). And since we now know that the $\sin$ and $\cos$ are the same things as the $y$ and $x$ coordinates of the point on the unit circle respectively, if you go to negative angle then the $y$ coordinate changes sign and the $x$ coordinate stays the same, as in the figure below: So this says that we have the relations: $$\sin(-\theta) = -\sin(\theta)\label{esn}$$ $$\cos(-\theta) = +\cos(\theta)\label{ecn}$$ Sine and Cosine of Sums of AnglesBack to top We often want to take the sine or cosine of a sum of angles, e.g. $\sin(\theta + \phi)$ and $\cos(\theta + \phi)$. Below is a nice way to derive formulae on how to do calculate the sine and cosine of sums using the sine and cosines of the individual angles. We start with the figure below, which sets up the proof. We want to calculate $\sin(\theta + \phi)$, so we draw a rectangle (with 4 right angles), and inscribe a right triangle inside the square in blue. The hypotenuse of that triangle is set to 1 by fiat: The angle $\alpha$ is also shown, but since $\alpha$, $\theta$, and $\phi$ form a right triangle, we know that $\alpha = 90\deg - \theta - \phi$. And since $\alpha$ forms one of the angles of the right triangle that has the blue hypotenuse and the horizontal bottom of the rectangle as a base, we also know that the other angle of that triangle is $\theta + \phi$, as in the figure below: Using the fact that the hypotenuse is set to 1, and the definitions of the sine (opposite/hypotenuse) and cosine (adjacent/hypotenuse), we can then label the lengths of the 2 sides of the inscribed triangle as in the next figure: Now let's calculate $\sin(\theta+\phi)$, which is the length of the bottom horizontal leg of the rectangle, as labeled in the figure below. Also, we've labeled the interior angle $\beta$, which has a value $\beta = 90\deg - \phi$ and since $\beta$ and the right angle of the inscribed triangle and the interior angle (marked $\phi$ at the top) add up to $180\deg$, that means that the angle $\phi$ at the top is the same angle $\phi$ in the bottom left corner. And we've labeled the two legs of the upper side of the rectangle as $a$ and $b$. Now we are ready to calculate the two legs of the upper side of the rectangle. The length $a$ is related to the angle $\phi$ by the ratio of opposite to hypotenuse, so $a=\sin\phi \cdot \cos\theta$, and the length $b$ is related to the angle $\phi$ by the ratio of adjacent to hypotenuse, so $b=\cos\phi\cdot\sin\theta$. From the equivalence of the upper to the lower rectangle side, we have $$\sin(\theta+\phi) = a + b\nonumber$$ or $$\sin(\theta+\phi) = \sin\phi\cos\theta + \sin\phi\cos\theta\nonumber$$ Using similar logic on the two vertical sides of the triangle, we can equate the left side - $\cos\phi\cos\theta$ - to the sum of the two legs on the right side - $\sin\phi\sin\theta + \cos(\theta+\phi)$ - to get the relation:\ $$\cos(\theta+\phi) = \cos\phi\cos\theta - \sin\phi\sin\theta\nonumber$$ To calculate $\sin(\theta-\phi)$ and $\cos(\theta-\phi)$ we can use equations $\ref{esn}$ and $\ref{ecn}$ and subsitute $\phi\to-\phi$ to get: $$\sin(\theta-\phi) = \sin\phi\cos\theta - \sin\phi\cos\theta\nonumber$$ and $$\cos(\theta-\phi) = \cos\phi\cos\theta + \sin\phi\sin\theta\nonumber$$ So the final equations are: $$\sin(\theta\pm\phi) = \sin\phi\cos\theta \pm \sin\phi\cos\theta\label{es2}$$ and $$\cos(\theta\pm\phi) = \cos\phi\cos\theta \mp \sin\phi\sin\theta\label{ec2}$$ Equations $\ref{es2}$ and $\ref{ec2}$ are used quite a lot in trigonometry! For instance, we can set $\phi=\theta$ and use eqation $\ref{es2}$ (the positive one) to calculate: $$\sin 2\theta = 2\sin\theta\cos\theta\label{es2t}$$ and $$\cos 2\theta = \cos^2\theta - \sin^2\theta\label{ec2t}$$ If we combine equations $\ref{ec2t}$ and $\ref{fm}$, we get $$\cos^2\theta = \frac{1+\cos 2\theta}{2}\label{ec22t}$$ And since $\cos^2\theta+\sin^2\theta = 1$, we have $$\sin^2\theta = \frac{1-\cos 2\theta}{2}\label{es22t}$$ Back to top
4609	https://www.quora.com/Why-am-I-good-at-math-but-unable-to-convert-a-word-problem-into-an-equation	Something went wrong. Wait a moment and try again. Word Struggles Problem Solving Skills Learning Disabilities Cognitive Abilities Mathematics Word Problems Learning Issues 5 Why am I good at math but unable to convert a word problem into an equation? Allen Ries Math Major University of Alberta · Author has 25.1K answers and 9.7M answer views · Aug 29 Solving math problems requires a different set of skills than converting word problems into math equations. For one thing reading a word problem requires a sophisticated understanding of the language. If you cannot understand what you read you are going to have an uphill battle to understand which equation is needed to solve the problem. Related questions Why am I hopelessly dumb at math? I can't seem to understand how to solve an equation or inequality, and word problems are even more of a problem. I keep trying to push myself to do math and it makes me uncomfortable. I'm so behind. How do I stop this? How can I keep a leveled mind? Why am I unable to concentrate in problem solving, coding, reading, poor at math? What is a real-life word problem involving quadratic equations? What is the point of explaining yourself in math? Why not just solve the equation and be done with it? William Beeman Ph.D. in anthropology with extensive research on education · Author has 30.7K answers and 34.4M answer views · 5y This question is the bane of mathematics instruction. Many teachers teach formulas and students get it in their head that knowing mathematics is just plugging different numbers into the abstract formulas that they encounter in the classroom and solving the problems. They then have no ability to take real world situations and convert them into formulas that they can solve. You are not alone. It is This question is the bane of mathematics instruction. Many teachers teach formulas and students get it in their head that knowing mathematics is just plugging different numbers into the abstract formulas that they encounter in the classroom and solving the problems. They then have no ability to take real world situations and convert them into formulas that they can solve. You are not alone. It is partly the way you have been taught. You need to identify the variables in the word problem, abstract them and plug them into a formula—some formulas will be ones you have learned in ... Keith Ramsay Ph.D. in mathematics · Author has 5.4K answers and 8.2M answer views · 5y I don’t know; I find it puzzling when people have difficulty converting a word problem into an equation. It seems to be a common difficulty. I think you might be able to figure out what the obstacle is yourself, though. Suppose you have a word problem which you are trying to convert into an equation. Perhaps you have some examples where someone then shows you an equation for the same word problem. Does it seem to you like you know what the equation is saying? Does it seem to you like you know what the word problem is saying? If so, do they seem to be saying the same thing, just in two different I don’t know; I find it puzzling when people have difficulty converting a word problem into an equation. It seems to be a common difficulty. I think you might be able to figure out what the obstacle is yourself, though. Suppose you have a word problem which you are trying to convert into an equation. Perhaps you have some examples where someone then shows you an equation for the same word problem. Does it seem to you like you know what the equation is saying? Does it seem to you like you know what the word problem is saying? If so, do they seem to be saying the same thing, just in two different ways? Or are you thinking of one in one way, and the other in another way? If one or the other seems incoherent then that has something to do with why it’s confusing. Sometimes a student is dyslexic or functions differently from most students in some other way (maybe even just needing glasses) without realizing it. I don’t want to assume things about you but if there are other times when you run into difficulties maybe you could see whether there is a pattern. It does seem to happen sometimes that students learn how to do mathematical manipulations by applying procedures and not fully understanding what they mean. They may say, “I could do the calculation if you’d just give me the formula”. It seems to be a reasonably common difficulty, and I don’t really know what to suggest if it’s your own difficulty. If you are good at the part about applying rules, and the difficulty is with having something that isn’t a matter of applying rules, now is probably a good time to change your approach some. Take time to digest what the problem is saying and think of it as less a matter of your applying rules to it. A lot of mathematics is more like talking about a problem than like immediately creating a solution. I’ve heard that it sometimes helps to say that variables are names that you give to things. Part of turning a story problem into a formula is just to give names to things. Eventually it might turn out that not all of the names were necessary, but you don’t have to figure that out right away. One thing I used to do as a student was to try to write my final answer from the start, not using any scratch paper. This is a kind of perfectionism, and it isn’t helpful. In some of these problems, it’s useful to be able to tinker around some, and then take the pieces that turned out to be needed for the answer and copy them out. Draw diagrams if it helps. I will forget what variable I assigned to some quantity and have to re-read to see what I called it, or forget what a variable stands for and have to re-read to see what I meant by it. It can be surprising how little short-term memory we have. If your analysis of a problem is very roundabout and wanders around a lot before getting to the point, that’s okay. Afterward one can often then identify what the key bits were. Say train A is going from Denver to Chicago at 40 miles per hour, and train B, which started at the same time, is going from Chicago to Denver at 50 miles per hour, and on parallel tracks, which are 1000 miles long. How long after they leave do they meet? Well, they gave you the total distance, 1000 miles, but it’s okay to give that a name like D. You could call the distance that train A goes R and the distance train B goes S. So R+S=D, right? Then you could call the time from when they leave until they meet T. Sometimes I note, “want to solve for T”. We could call their speeds v and w, even though we know what they are. It makes sense to write each of these things down as it occurs to you. If we name all of these things, it means we wind up not just with one equation but with a bunch of them, but that’s okay. Does it now seem natural to say v=R/T? That is “speed is distance divided by time” for one train. One kind of difficulty is with knowing the operations only in the abstract and not as the mathematical counterpart to these real-world quantities. It also seems pretty common for students to get tripped up by feeling some anxiety. It doesn’t have to be very much. Don’t assume that you have any permanent obstacles here; people often have some temporary difficulty like this that goes away. And this particular one does seem to be pretty common. [Added in edit: note also the comment about badly formulated word problems. Sometimes the difficulty is not yours at all.] Assistant Bot · Sep 6 Strength in mathematics and difficulty translating word problems into equations are distinct but related skills. The first is often about procedural fluency, pattern recognition, and symbol manipulation; the second requires language comprehension, modeling, and problem-structuring. Common causes and concrete strategies to fix the gap: Why this happens Different cognitive demands: Symbolic manipulation taps procedural memory and rote practice (algebra rules, arithmetic). Word-to-equation translation requires reading comprehension, identifying quantities, relations, implicit assumptions, and mapping Strength in mathematics and difficulty translating word problems into equations are distinct but related skills. The first is often about procedural fluency, pattern recognition, and symbol manipulation; the second requires language comprehension, modeling, and problem-structuring. Common causes and concrete strategies to fix the gap: Why this happens Different cognitive demands: Symbolic manipulation taps procedural memory and rote practice (algebra rules, arithmetic). Word-to-equation translation requires reading comprehension, identifying quantities, relations, implicit assumptions, and mapping them to symbols. Over-reliance on templates: success with routine algebra can create a habit of applying memorized procedures rather than understanding the story structure. Language and conceptual bottlenecks: vague pronouns, passive phrasing, and implicit units hide the quantities that must become variables. Working-memory load: holding multiple quantities, units, and relationships while forming equations can exceed working-memory capacity, causing errors even when calculation skills are strong. Misreading relational words: “more than,” “times as much,” “difference,” and “of” have precise mathematical meanings that are often mistranslated. Lack of modeling practice: fewer opportunities to practice turning real descriptions into formal statements than to practice algebraic manipulation. Step-by-step method to improve (practical, repeatable) Read twice First: get the story and context. Second: underline all quantities, units, comparisons, and question asked. Define variables explicitly Give each unknown a clear label (x = number of shirts, y = price per shirt). Write units after variable names to avoid confusion. Inventory knowns and unknowns List given values, expressions, and what must be found. Translate relational phrases to math equivalents (cheat-sheet) “More than” → add: x is 5 more than y → x = y + 5 “Less than” → subtract: x is 3 less than y → x = y − 3 “Product/of” → multiply: product of x and y → x·y “Per/each/for every” → division or rate: miles per hour → distance = rate × time “Total/sum” → addition, “difference” → subtraction “As much as” and comparative structures often reverse order—write literal phrases into a skeleton equation first. Build equations from simple sentences Convert one sentence at a time into a short expression; then link them. Use placeholders “(this) = (that)” to force explicit relationships. Draw diagrams or tables For mixtures, rates, work, money or geometry, a sketch, table of values, or timeline clarifies who/what each quantity belongs to. Check units and plausibility Units that don’t match reveal wrong equations; estimated magnitudes can catch sign or order errors. Practice deliberately with variety Solve 10–15 word problems weekly that force variable definition and modeling: ages, mixtures, rates, area/volume, percentages, consecutive integers, and ratios. After solving, rewrite the English-to-equation mapping and compare different valid formulations. Reflexive debugging If stuck, rephrase the sentence in your own words, substitute simple numbers, or test with extreme values to see if an equation behaves plausibly. Use scaffolds initially Templates: “Let x = …” then “According to sentence 1: …” and “According to sentence 2: …” until internalized. Quick practice drills (5–10 minutes each) Translate 5 one-line statements into equations without solving (focus purely on mapping). Turn a solved algebra problem back into a short word problem; reverse practice strengthens modeling. Explain aloud (or write) why each term in your equation corresponds to a phrase in the problem. Expected progress and mindset Early improvement comes quickly with structured practice (weeks). Complex problem translation becomes fluent with months of varied exposure. Treat modeling as a separate skill from algebraic manipulation; deliberately practice modeling until the mapping feels automatic. If difficulty persists despite practice Check for language comprehension issues or working-memory limits; targeted reading comprehension exercises, chunking strategies, and external supports (notes, diagrams) help. Seek targeted tutoring or guided problem sets that force you to verbalize the mapping step-by-step. Summary Being “good at math” often means strong procedural skills; converting word problems is a modeling task that uses different abilities. Use explicit variable definition, sentence-by-sentence translation, diagrams, unit checks, and short daily drills to bridge the gap. Progress is rapid with focused, varied practice. Related questions What should you do if you cannot solve a problem that you are working on? What is the best way to solve math word problems? I am good at every subject in school except for Math. What should I do to get better at Math, since I feel stupid when I'm doing it? Why am I not understanding the mathematics subject, even after trying so much? I am not good at math, money, and word problems. What do I do? Richard McConnel M.S. in Aerospace and Aeronautical Engineering & Mathematics, US Naval Postgraduate School (Graduated 1966) · Author has 676 answers and 202.3K answer views · 5y OK - you have probably been good at “math” so far - but I am guessing all you have do far is arithmetic and basic algebra. Word problems are a specific problem area - it is your first introduction to true logic - not just memorization. But, understanding is essential your success at mathematics. So recognize this. You need to talk one-on-one with your teacher or tutor if you don’t get it. You will need to recognize that here are several specific type of word problems - time/distance, gallons/min, ratio, etc - You will need to work and rework these problems until you recognize the various types OK - you have probably been good at “math” so far - but I am guessing all you have do far is arithmetic and basic algebra. Word problems are a specific problem area - it is your first introduction to true logic - not just memorization. But, understanding is essential your success at mathematics. So recognize this. You need to talk one-on-one with your teacher or tutor if you don’t get it. You will need to recognize that here are several specific type of word problems - time/distance, gallons/min, ratio, etc - You will need to work and rework these problems until you recognize the various types and the approaches to the solutions. You should start a notebook and write the problems down (yes - every step as you understand) as you solve them - then for review for an exam you can review the various questions and the solutions. Sponsored by Morgan & Morgan, P.A. How do you know if you qualify for a claim? We’ve helped hundreds of thousands of people with their injury claims. Do you have a case? Manisha Sarkar Studied Bachelor of Technology in Electronics and Communications Engineering at Institute of Engineering and Management (IEM) (Graduated 2019) · 5y Hey, you have not mentioned what makes you think that you are good at math. But i am assuming that you are referring to your academics so far. In that case two things can happen You just think you are good at math ,but actually you are not, you may have scored high marks in academics but only by cramming. you are actually good at math but you are just lacking concentration. for the first case you really need to clear your basics once again, if you will be focused it won’t take much time. And for the second case you need to meditate to rebuild your focus, so that you can concentrate on the words o Hey, you have not mentioned what makes you think that you are good at math. But i am assuming that you are referring to your academics so far. In that case two things can happen You just think you are good at math ,but actually you are not, you may have scored high marks in academics but only by cramming. you are actually good at math but you are just lacking concentration. for the first case you really need to clear your basics once again, if you will be focused it won’t take much time. And for the second case you need to meditate to rebuild your focus, so that you can concentrate on the words of the problem more efficiently if your understanding of the words will be clear it would help you to convert it into an equation ,without much effort. Ron Brown Professor of Physics, Emeritus · Upvoted by Horst H. von Brand , PhD Computer Science & Mathematics, Louisiana State University (1987) · Author has 13.6K answers and 84.2M answer views · Updated Aug 30 Related Why do so many people struggle with translating word problems into mathematical equations? Great question. And I’m not sure I know the answer … other than that different people have different strengths and weaknesses. My late older brother was amazing at teaching himself different languages. I’m better at other things. I used to send out a diagnostic pretest (that didn’t count) to the students enrolled in my introductory level calculus-based university physics classes - mostly just math concepts I would expect them to be able use without needing any explanation. I always started with a couple of words-to-equation translation problems. It might go something like this: “Write the expre Great question. And I’m not sure I know the answer … other than that different people have different strengths and weaknesses. My late older brother was amazing at teaching himself different languages. I’m better at other things. I used to send out a diagnostic pretest (that didn’t count) to the students enrolled in my introductory level calculus-based university physics classes - mostly just math concepts I would expect them to be able use without needing any explanation. I always started with a couple of words-to-equation translation problems. It might go something like this: “Write the expression that states that w is proportional to x, the square of y, and inversely proportional to the cube of z.” Or “Write the expression that states that the time rate of change of some quantity N is proportional to the value of N itself.” Or “Write the expression that states that the change in position of some object is equal to the product of its initial speed times time plus half its acceleration times the square of time.” It was surprising to me how many of these 4.0+ GPA entering freshmen science and engineering students who had taken two courses of algebra and one of HS calculus or honors math or something would have trouble doing so. The ones that “got” that last one were often surprised when they realized it wrote an equation they had memorized from physics when doing kinematics problems. I would ask the class how many of them actually liked word problems. There might be at most a small handful that would raise their hands. I would tell them they would do fine in physics. Because nearly all real problems are word problems. That is, the mathematics is very useful in solving physics problems, but physics is about ideas that can be expressed in words. That is, it is more verbal than mathematical. But they needed to comfortable with doing the math. Sponsored by US Auto Insurance Now These companies are overcharging you for auto insurance! Say goodbye to high car insurance rates if you live in these ZIPs. Dennis Stevenson Emeritus Associate Professor of the School of Computing, Clemson University · Author has 2.3K answers and 992.3K answer views · 5y Solving word problems is a two-step process. The first is what I call the “modeling” step. In modeling you convert natural language statements to mathematical statements. Identify variables and their types, find statements that indicate relationships of variables and constants. Now step two: you should now have algebraic information to solve. Jack Fraser-Govil Doctor of Physics, Writer of Code, Player of Games · Author has 2.6K answers and 51.5M answer views · 8y Related I've gotten really bad at maths, what can I do to improve it? Easy: Practice I’ve seen people who think they can learn maths just by reading the textbook, and then writing some notes. WRONG! Maths is like playing the flute. You have to practice, and keep practising the bits you can’t do. It’s no good not practicing, or only doing the bits you can already do - you have to go and practice the stuff you find difficult. Back when I was about 15, I used to hate expanding fractions - i.e. writing something like: −x2+7x+152x2+3x−5→3x−1−x2x+5 I just didn’t get it. So what did I do? Every time I found myself with a spare moment, I would i Easy: Practice I’ve seen people who think they can learn maths just by reading the textbook, and then writing some notes. WRONG! Maths is like playing the flute. You have to practice, and keep practising the bits you can’t do. It’s no good not practicing, or only doing the bits you can already do - you have to go and practice the stuff you find difficult. Back when I was about 15, I used to hate expanding fractions - i.e. writing something like: −x2+7x+152x2+3x−5→3x−1−x2x+5 I just didn’t get it. So what did I do? Every time I found myself with a spare moment, I would invent myself a fraction, and attempt to solve it (I worked at the checkout of the local supermarket at the time, so I’d do it on the spare receipts in the lulls) Eventually, after a few weeks of doing 3 or 4 of these a day - I cracked it. I understood it! I went into school and aced the exam. But that’s not the end - you then have to keep practicing, maybe not as often as before, but you need to keep it somewhere in the back of your head. It took me a few extra seconds to solve the one I wrote above, since I haven’t done it in a long while! As I said, it’s the same as a musical instrument - when you’ve got really good at a piece, you can’t expect to not touch your flute for 2 years and then come back and play the piece perfectly! Trust me…..I used to play the flute quite well…..not so much any more! So you need to just start practicing as much maths as you can. And you need to make sure you’re practicing the right stuff. There’s no point spending hours a day practicing addition, when you need to be learning calculus! So: Just like learning a musical instrument, you can’t learn how to do maths just by reading about it! Maths is learned by doing, by exploring, by seeing what happens if I do this… Eventually it will come to you, and you will get really good - so you move on to the next topic (but still keep up a small amount of practice on the old topics!) and you keep doing this and keep doing this, until all of a sudden…. You actually enjoy learning new maths! The horror! Sponsored by Grammarly 92% of professionals who use Grammarly say it has saved them time Work faster with AI, while ensuring your writing always makes the right impression. Daniel H. Fong Self-taught myself Math. · 8y Related Why am I so bad at math? I used to suck at math. Like a LOT. I always try to get the passing mark and be done with it. That was me. But now I’m different; I love math, especially Calculus. Here’s what I can tell you after about 3/2 years of studying math extensively. Try not to remember formulas. Instead, what you should try is to find out how do those formulas came about? Every kind of operation you do in math is logical. For example, you can solve for x in 3x+5=0 by simply subtracting 5 from both sides the equation and subsequently dividing it by 3 which gets you to - 5/3, this is a completely legitimate thing to do I used to suck at math. Like a LOT. I always try to get the passing mark and be done with it. That was me. But now I’m different; I love math, especially Calculus. Here’s what I can tell you after about 3/2 years of studying math extensively. Try not to remember formulas. Instead, what you should try is to find out how do those formulas came about? Every kind of operation you do in math is logical. For example, you can solve for x in 3x+5=0 by simply subtracting 5 from both sides the equation and subsequently dividing it by 3 which gets you to - 5/3, this is a completely legitimate thing to do because no rules are broken; the equality still holds. Try to be a skeptic when someone shows you a formula they think is true. Ask them how did they get formula(though I suggest you to do this within the confines of the internet world or with someone who tolerates this because your school teacher and friends might get tired of you). Important thing is to not only realize your mistakes but to ask whether or not can you recover from these mistakes. I used to be so down about my mathematical abilities; some students are just better at it than me. You have the ones who’re born with it and the ones who worked hard for it. That’s how it’ll be. Can’t do anything about it, reality is unfair. What matters is that is what you’re learning enough to help you achieve your goals? If yes, great. If not, learn more. Learning is a journey, you’ll spend a lot of time in that journey so try to allocate your time carefully. Don’t dwell on the past too much. It’s a waste of time. And time is not a luxury. Visualize more. I can’t stress you enough how important this is to me. Yeah I get it, imagining mathematical expressions in your head won’t give you the answer you wanted. True. But what it does give you is the intuition on how to get that answer you wanted; an approximation. Solving inequalities like \|x+3\|> 4x^2 +9x + 4 with your rough sketch in your mind can really help you solve them. This is really helpful. Seriously. If you have doubts, verify them. In math, we’re very strict about our definitions. Don’t bottle up your doubts, it’ll accumulate more and more if you do. This could spell disaster as you’re learning math because you now have “holes” in your arguments . This could’ve been avoided if you had just ask the questions you wanted. Keep this up, and you’ll hit a “wall” during your mathematical studies which will impede you from gaining progress. Well that’s all the advice I can offer you. The journey never ends, man. Just like you, I still have a lot to learn. Mark Rogers Software Support Specialist (2008–present) · Author has 115 answers and 86.9K answer views · 6y Related Why am I so bad at math but very good with English? The answer to this question actually lies in what you tell yourself. Somewhere in your youth, you decided that math was hard for you. You repeated this train of thought many times over, and that became a solid belief. While this was going on, you also found that you liked language arts, and that it was less challenging for you. As you continued to tell yourself this, it became a solid belief. Math skills, and language skills develop in different parts of the brain. Your brain is essentially a muscle, though it does not flex in quite the same way as a bicep, or an abductor. When you worked out your The answer to this question actually lies in what you tell yourself. Somewhere in your youth, you decided that math was hard for you. You repeated this train of thought many times over, and that became a solid belief. While this was going on, you also found that you liked language arts, and that it was less challenging for you. As you continued to tell yourself this, it became a solid belief. Math skills, and language skills develop in different parts of the brain. Your brain is essentially a muscle, though it does not flex in quite the same way as a bicep, or an abductor. When you worked out your English “muscle”, it became stronger. As you avoided your Math “muscle”, it became weaker. There is nothing inherently wrong with this. It is just a subconscious choice that you made early on. If you choose to become good at Math, and do the work to make this happen, you will be good at both. The real key, is in what you tell yourself. Change your internal talk from “I’m bad at Math.” to “I like the challenge that Math brings to me.” Notice, I didn’t say change it to, “I’m good at Math.” That would be too much of a stretch for your mind to believe, right away. The phrase, “I like the challenge that Math brings to me.”, can be completely believable. Which makes the transition much less of a challenge. Change your self talk, and you will notice a change in your behavior. The change in your behavior will manifest in a change in skill level. The choice is yours to make, and there is no bad choice. Dennis Stevenson Computational Thinker · Author has 2.3K answers and 992.3K answer views · 10mo The first issue in solving word problems is to translate the problem from natural language into the language of mathematics. You must carefully identify the constants and variables. Get this wrong and it is impossible to get the correct answer. Anthony Hawken Author has 10.8K answers and 3.8M answer views · 10mo Originally Answered: I can do equations in math, but I can't do word problems or questions that are worded and why is this? · There could be a number of reasons. You find it difficult to read and comprehend what you have just read. You are unable to transform a worded problem into an equation. Related questions Why am I hopelessly dumb at math? I can't seem to understand how to solve an equation or inequality, and word problems are even more of a problem. I keep trying to push myself to do math and it makes me uncomfortable. I'm so behind. How do I stop this? How can I keep a leveled mind? Why am I unable to concentrate in problem solving, coding, reading, poor at math? What is a real-life word problem involving quadratic equations? What is the point of explaining yourself in math? Why not just solve the equation and be done with it? What should you do if you cannot solve a problem that you are working on? What is the best way to solve math word problems? I am good at every subject in school except for Math. What should I do to get better at Math, since I feel stupid when I'm doing it? Why am I not understanding the mathematics subject, even after trying so much? I am not good at math, money, and word problems. What do I do? I can't solve hard math problems. I always give up and look at the solution. What should I do to fix that? What are 5 examples of real-life word problems involving quadratic functions? Am I the only one who finds math problems easy but math word problems hard? How do some people calculate extremely fast? I can do equations in math, but I can't do word problems or questions that are worded and why is this? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
4610	https://magoosh.com/hs/ap/common-limits-ap-calc-exam/	Common Limits on the AP Calc Exam - Magoosh Blog \| High School High School Blog Magoosh has helped millions of students study since 2010. Our affordable SAT & ACT self-study plans includes exclusive practice questions, full-length practice tests, and a score improvement guarantee. Prep with Magoosh Top Resources:Free SAT Practice TestFree ACT Practice TestSAT Study SchedulesACT Study Schedules ACT®Blog Top Resources: Free SAT Practice Test Free ACT Practice Test SAT Study Schedules ACT Study Schedules Magoosh is the leader in ACT prep having helped millions of students study since 2010. Try ACT Prep with Magoosh Common Limits on the AP Calc Exam By Shaun Ault — Last updated on February 23, 2018 in AP As you prepare for the AP Calculus exam, it’s important to know what you’re getting into. Hopefully you’ve already taken a look at the list of topics, but this article is about one topic in particular: limits. Read on to familiarize yourself with the common limits that you’ll encounter on the test! The limit of a function at a particular point is the y-value that the function approaches near that point. What is a Limit? The concept of limits is fundamental to bridging the gap between elementary mathematics (arithmetic, geometry, algebra, trigonometry, etc.) and calculus. In fact limits provide the theoretic background on which the main tool of calculus, the derivative, is built. So it’s no surprise that limits show up on both the AB and BC versions of the AP Calculus exam. We won’t take time in this short article to review the formal definition of limit. Instead, you can take a look at a number of helpful resources to review the concept. What is the Limit of a Function? Limits on the AP Calculus Exam: A Review AP Calculus Exam Review: Limits and Continuity Common Limits on the Exam Now let’s get to some practice problems that highlight common limits on the AP Calculus exams. Answer Key Use the plug-in principle. Factor, cancel, then plug in. This one is easier than it appears. Because the variable is heading toward infinity, all we have to do is a highest-order term analysis. Now since the highest order term on top ( 3 x 2 ) has lower degree than the one on the bottom ( x 3 ), the limit must be 0. Again this is a limit as x → ∞, so use highest-order term analysis. But this time, the top has a higher degree than the bottom. Isolate the highest degree terms on top and bottom: ( x 7 ) / ( x 5 ) = x 2 Therefore, as x → ∞, our function behaves like x 2. So in the limit, the values simply grow without bound. The limit is ∞. Rounding out this trio of related examples, we have another limit in which x approaches one of the two infinities. Using highest-order term analysis this time, we see that the degree of the top is the same as the degree of the bottom. Let’s simplify. ( 7 x 4 ) / ( 5 x 4 ) = 7/5 This implies that as x → -∞, the function values get closer to the constant 7/5. That’s our limiting value! Notice that plugging in x = 2 results in nonzero/zero, so we know the answer will be infinite. When x is close to 2, but to the right of it, the top will be negative while the bottom is positive. Therefore, the limit is -∞. Absolute values must be treated with care. When x is less than -1, the quantity (x + 1) is negative. So you must switch the sign to get rid of the absolute value bars. You can either adjust the limit so that it has the form (sin θ)/θ, or just use l’Hospital’s Rule. The solution here uses the latter. Conclusion Now that you’ve seen a number of common limits, you can be that much more confident going into the test. But keep in mind, there are many variations possible on these limit problems, so be on your guard! By the way, if you want to review what else is on the exams, check out the following links. What Topics are on the AP Calculus AB Exam? What Topics are on the AP Calculus BC Exam? Author Shaun Ault Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music — almost as much as math! — and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed! 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4611	https://chem-textbook.ucalgary.ca/version2/chapter-7-main/collision-theory/activation-energy-and-the-arrhenius-equation/	Skip to the content Home → 7.5 Collision Theory → Activation Energy and the Arrhenius Equation Activation Energy and the Arrhenius Equation The minimum energy necessary to form a product during a collision between reactants is called the activation energy (Ea). How this energy compares to the kinetic energy provided by colliding reactant molecules is a primary factor affecting the rate of a chemical reaction. If the activation energy is much larger than the average kinetic energy of the molecules, the reaction will occur slowly since only a few fast-moving molecules will have enough energy to react. If the activation energy is much smaller than the average kinetic energy of the molecules, a large fraction of molecules will be adequately energetic and the reaction will proceed rapidly. The figure below shows how the energy of a chemical system changes as it undergoes a reaction converting reactants to products according to the equation A+B⟶C+D These reaction diagrams are widely used in chemical kinetics to illustrate various properties of the reaction of interest. Viewing the diagram from left to right, the system initially comprises reactants only, A + B. Reactant molecules with sufficient energy can collide to form a high-energy activated complex or transition state. The unstable transition state can then subsequently decay to yield stable products, C + D. The diagram depicts the reaction’s activation energy, Ea, as the energy difference between the reactants and the transition state. Using a specific energy, the enthalpy (see chapter on thermochemistry), the enthalpy change of the reaction, ΔH, is estimated as the energy difference between the reactants and products. In this case, the reaction is exothermic (ΔH < 0) since it yields a decrease in system enthalpy. The Arrhenius equation relates the activation energy and the rate constant, k, for many chemical reactions: k=Ae−Ea/RT(1) In this equation, R is the ideal gas constant, which has a value 8.314 J/mol/K, T is temperature on the Kelvin scale, Ea is the activation energy in joules per mole, e is the constant 2.7183, and A is a constant called the frequency factor, which is related to the frequency of collisions and the orientation of the reacting molecules. Postulates of collision theory are nicely accommodated by the Arrhenius equation. The frequency factor, A, reflects how well the reaction conditions favor properly oriented collisions between reactant molecules. An increased probability of effectively oriented collisions results in larger values for A and faster reaction rates. The exponential term, e−Ea/RT, describes the effect of activation energy on reaction rate. According to kinetic molecular theory (see chapter on gases), the temperature of matter is a measure of the average kinetic energy of its constituent atoms or molecules. The distribution of energies among the molecules composing a sample of matter at any given temperature is described by the plot shown in Figure 2(a). Two shaded areas under the curve represent the numbers of molecules possessing adequate energy (RT) to overcome the activation barriers (Ea). A lower activation energy results in a greater fraction of adequately energized molecules and a faster reaction. The exponential term also describes the effect of temperature on reaction rate. A higher temperature represents a correspondingly greater fraction of molecules possessing sufficient energy (RT) to overcome the activation barrier (Ea), as shown in Figure 2(b). This yields a greater value for the rate constant and a correspondingly faster reaction rate. A convenient approach for determining Ea for a reaction involves the measurement of k at two or more different temperatures and using an alternate version of the Arrhenius equation that takes the form of a linear equation lnk=(−EaR)(1T)+lnA(2) y=mx+b A plot of ln k versus 1T is linear with a slope equal to −EaR and a y-intercept equal to ln A. Determination of Ea The variation of the rate constant with temperature for the decomposition of HI(g) to H2(g) and I2(g) is given here. What is the activation energy for the reaction? 2HI(g)⟶H2(g)+I2(g) \| T (K) \| k (L/mol/s) \| --- \| \| 555 \| 3.52 × 10−7 \| \| 575 \| 1.22 × 10−6 \| \| 645 \| 8.59 × 10−5 \| \| 700 \| 1.16 × 10−3 \| \| 781 \| 3.95 × 10−2 \| Solution Use the provided data to derive values of 1T and ln k: \| 1T(K−1) \| ln k \| --- \| \| 1.80 × 10−3 \| −14.860 \| \| 1.74 × 10−3 \| −13.617 \| \| 1.55 × 10−3 \| −9.362 \| \| 1.43 × 10−3 \| −6.759 \| \| 1.28 × 10−3 \| −3.231 \| The figure below is a graph of ln k versus 1T. In practice, the equation of the line (slope and y-intercept) that best fits these plotted data points would be derived using a statistical process called regression. This is helpful for most experimental data because a perfect fit of each data point with the line is rarely encountered. For the data here, the fit is nearly perfect and the slope may be estimated using any two of the provided data pairs. Using the first and last data points permits estimation of the slope. Slope=Δ(lnk)Δ(1T) =(−14.860)−(−3.231)(1.80×10−3K−1)−(1.28×10−3K−1) =−11.6290.52×10−3K−1=–2.2×104K =−EaR Ea=−slope×R=−(−2.2×104K×8.314Jmol−1K−1) 1.8×105Jmol−1or180kJmol−1 Alternative approach: A more expedient approach involves deriving activation energy from measurements of the rate constant at just two temperatures. In this approach, the Arrhenius equation is rearranged to a convenient two-point form: lnk1k2=EaR(1T2−1T1)(3) Rearranging this equation to isolate activation energy yields: Ea=−R(lnk2−lnk1(1T2)−(1T1))(4) Any two data pairs may be substituted into this equation—for example, the first and last entries from the above data table: Ea=−8.314Jmol−1K−1(−3.231−(−14.860)1.28×10−3K−1−1.80×10−3K−1) and the result is Ea = 1.8 × 105 J mol−1 or 180 kJ mol−1 This approach yields the same result as the more rigorous graphical approach used above, as expected. In practice, the graphical approach typically provides more reliable results when working with actual experimental data. Check your learning The rate constant for the rate of decomposition of N2O5 to NO and O2 in the gas phase is 1.66 L/mol/s at 650 K and 7.39 L/mol/s at 700 K: 2N2O5(g)⟶4NO(g)+3O2(g) Assuming the kinetics of this reaction are consistent with the Arrhenius equation, calculate the activation energy for this decomposition. Answer: 1.1×105Jmol Check Your Learning Can you label a reaction coordinate diagram correctly? Test your understanding in this question below:
4612	https://study.com/skill/learn/how-to-identify-graph-a-hyperbola-not-centered-at-the-origin-explanation.html	How to Identify & Graph a Hyperbola Not Centered at the Origin \| Precalculus \| Study.com Log In Sign Up Menu Plans Courses By Subject College Courses High School Courses Middle School Courses Elementary School Courses By Subject Arts Business Computer Science Education & Teaching English (ELA) Foreign Language Health & Medicine History Humanities Math Psychology Science Social Science Subjects Art Business Computer Science Education & Teaching English Health & Medicine History Humanities Math Psychology Science Social Science Art Architecture Art History Design Performing Arts Visual Arts Business Accounting Business Administration Business Communication Business Ethics Business Intelligence Business Law Economics Finance Healthcare Administration Human Resources Information Technology International Business Operations Management Real Estate Sales & Marketing Computer Science Computer Engineering Computer Programming Cybersecurity Data Science Software Education & Teaching Education Law & Policy Pedagogy & Teaching Strategies Special & Specialized Education Student Support in Education Teaching English Language Learners English Grammar Literature Public Speaking Reading Vocabulary Writing & Composition Health & Medicine Counseling & Therapy Health Medicine Nursing Nutrition History US History World History Humanities Communication Ethics Foreign Languages Philosophy Religious Studies Math Algebra Basic Math Calculus Geometry Statistics Trigonometry Psychology Clinical & Abnormal Psychology Cognitive Science Developmental Psychology Educational Psychology Organizational Psychology Social Psychology Science Anatomy & Physiology Astronomy Biology Chemistry Earth Science Engineering Environmental Science Physics Scientific Research Social Science Anthropology Criminal Justice Geography Law Linguistics Political Science Sociology Teachers Teacher Certification Teaching Resources and Curriculum Skills Practice Lesson Plans Teacher Professional Development For schools & districts Certifications Teacher Certification Exams Nursing Exams Real Estate Exams Military Exams Finance Exams Human Resources Exams Counseling & Social Work Exams Allied Health & Medicine Exams All Test Prep Teacher Certification Exams Praxis Test Prep FTCE Test Prep TExES Test Prep CSET & CBEST Test Prep All Teacher Certification Test Prep Nursing Exams NCLEX Test Prep TEAS Test Prep HESI Test Prep All Nursing Test Prep Real Estate Exams Real Estate Sales Real Estate Brokers Real Estate Appraisals All Real Estate Test Prep Military Exams ASVAB Test Prep AFOQT Test Prep All Military Test Prep Finance Exams SIE Test Prep Series 6 Test Prep Series 65 Test Prep Series 66 Test Prep Series 7 Test Prep CPP Test Prep CMA Test Prep All Finance Test Prep Human Resources Exams SHRM Test Prep PHR Test Prep aPHR Test Prep PHRi Test Prep SPHR Test Prep All HR Test Prep Counseling & Social Work Exams NCE Test Prep NCMHCE Test Prep CPCE Test Prep ASWB Test Prep CRC Test Prep All Counseling & Social Work Test Prep Allied Health & Medicine Exams ASCP Test Prep CNA Test Prep CNS Test Prep All Medical Test Prep College Degrees College Credit Courses Partner Schools Success Stories Earn credit Sign Up How to Identify & Graph a Hyperbola Not Centered at the Origin Precalculus Skills Practice You must c C reate an account to continue watching Register to access this and thousands of other videos Are you a student or a teacher? I am a student I am a teacher Try Study.com, risk-free As a member, you'll also get unlimited access to over 88,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed. Get unlimited access to over 88,000 lessons. Try it risk-free It only takes a few minutes to setup and you can cancel any time. It only takes a few minutes. Cancel any time. Already registered? Log in here for access Back What teachers are saying about Study.com Try it risk-free for 30 days Already registered? Log in here for access 00:04 How to identify &… 03:06 How to identify &… Jump to a specific example Kathryn Boddie, Andrew Noble Instructors Kathryn Boddie Kathryn has taught high school or university mathematics for over 10 years. She has a Ph.D. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M.S. in Mathematics from Florida State University, and a B.S. in Mathematics from the University of Wisconsin-Madison. View bio Andrew Noble Andrew has taught early algebra through advanced calculus to students for over 10 years. He has a bachelor's and master's degree in electrical engineering from Colorado State University. He also has 6 years of experience as a software developer. View bio Example SolutionsPractice Questions Steps for Identifying and Graphing a Hyperbola Not Centered at the Origin Step 1: Identify the center of the hyperbola. The center of the hyperbola is (h,k) when the hyperbola is written in standard form: (x−h)2 a 2−(y−k)2 b 2=1 is a horizontal hyperbola. (y−k)2 b 2−(x−h)2 a 2=1 is a vertical hyperbola. Step 2: Plot the center, a point a units to the left of the center, a point a units to the right of the center, a point b units above the center and a point b units below the center. Connect the four points surrounding the center with a dashed rectangle. Connect the diagonals of the rectangle using dashed lines to form the asymptotes for the hyperbola. Step 3: Sketch the hyperbola. If the hyperbola is horizontal, draw the branches following the asymptotes and passing through the points plotted to the left and right of the center. If the hyperbola is vertical, draw the branches following the asymptotes and passing through the points plotted above and below the center. Vocabulary and Equations for Identifying and Graphing a Hyperbola Not Centered at the Origin Hyperbola: A hyperbola is a graph formed in which the difference between the distances from any point to two special points called the foci is constant in absolute value. The graph of a hyperbola is made of two disjoint branches that are symmetrical. Equation for Hyperbola: The standard form for an equation of a hyperbola centered at (h,k) is (x−h)2 a 2−(y−k)2 b 2=1 or (y−k)2 b 2−(x−h)2 a 2=1. If the positive term contains x, then the hyperbola is horizontal, and if the positive term contains y, then the hyperbola is vertical. We will use these steps, definitions, and equations to identify and graph a hyperbola not centered at the origin in the following two examples. Example Problem 1 - Identifying and Graphing a Hyperbola Not Centered at the Origin - Horizontal Graph the hyperbola given by the equation (x−3)2 9−(y−1)2 4=1. Step 1: Identify the center of the hyperbola. The center of the hyperbola is (h,k) when the hyperbola is written in standard form: (x−h)2 a 2−(y−k)2 b 2=1 is a horizontal hyperbola. (y−k)2 b 2−(x−h)2 a 2=1 is a vertical hyperbola. Our equation is of a horizontal hyperbola since the positive term contains x. The center of the hyperbola is (3,1). Step 2: Plot the center, a point a units to the left of the center, a point a units to the right of the center, a point b units above the center and a point b units below the center. Connect the four points surrounding the center with a dashed rectangle. Connect the diagonals of the rectangle using dashed lines to form the asymptotes for the hyperbola. In our equation, we have: a 2=9→a=3 b 2=4→b=2 We need to plot: the center (3,1) two points that are a=3 units to the left and the right of the center, so (0,1) and (6,1) two points that are b=2 units above and below the center, so (3,3) and (3,−1) And then connect the points with a dashed-line rectangle, and form the asymptotes of the hyperbola by drawing the diagonals of the rectangle in dashed lines. Setting up Asymptotes Step 3: Sketch the hyperbola. If the hyperbola is horizontal, draw the branches following the asymptotes and passing through the points plotted to the left and right of the center. If the hyperbola is vertical, draw the branches following the asymptotes and passing through the points plotted above and below the center. The hyperbola is horizontal, so we need to sketch it passing through (0,1) and (6,1), following the shape of the asymptotes. Sketching the Hyperbola Then, the final graph of the hyperbola can be found by erasing the dashed lines, center, and any points not on the hyperbola. Final Graph of the Hyperbola Example Problem 2 - Identifying and Graphing a Hyperbola Not Centered at the Origin - Vertical Graph the hyperbola given by the equation (y+2)2 16−x 2 25=1. Step 1: Identify the center of the hyperbola. The center of the hyperbola is (h,k) when the hyperbola is written in standard form: (x−h)2 a 2−(y−k)2 b 2=1 is a horizontal hyperbola. (y−k)2 b 2−(x−h)2 a 2=1 is a vertical hyperbola. The center of the hyperbola is (0,−2) and the hyperbola is vertical. Step 2: Plot the center, a point a units to the left of the center, a point a units to the right of the center, a point b units above the center and a point b units below the center. Connect the four points surrounding the center with a dashed rectangle. Connect the diagonals of the rectangle using dashed lines to form the asymptotes for the hyperbola. In our equation, we have: a 2=25→a=5 b 2=16→b=4 Setting up the graph by plotting points and connecting a rectangle to set up the asymptotes, we have: Setting up Asymptotes Step 3: Sketch the hyperbola. If the hyperbola is horizontal, draw the branches following the asymptotes and passing through the points plotted to the left and right of the center. If the hyperbola is vertical, draw the branches following the asymptotes and passing through the points plotted above and below the center. The hyperbola is vertical, so we need to sketch it passing through (0,6) and (0,−2), following the shape of the asymptotes. Sketching the Hyperbola The final graph of the hyperbola is below: Final Graph of the Hyperbola Get access to thousands of practice questions and explanations! Create an account Table of Contents Steps for Identifying and Graphing a Hyperbola Not Centered at the Origin Vocabulary and Equations for Identifying and Graphing a Hyperbola Not Centered at the Origin Example Problem 1 - Identifying and Graphing a Hyperbola Not Centered at the Origin - Horizontal Example Problem 2 - Identifying and Graphing a Hyperbola Not Centered at the Origin - Vertical Test your current knowledge Practice Graphing a Hyperbola Not Centered at the Origin Recently updated on Study.com Videos Courses Lessons Articles Quizzes Concepts Teacher Resources The Invention of Writing Ethnic Groups in Indonesia \| Demographics & People Gods of the Winter Solstice Holes by Louis Sachar \| Themes, Quotes & Analysis Aurangzeb \| Empire, Achievements & Failures Libya Ethnic Groups \| Demographics, Population & Cultures Cold War Lesson for Kids: Facts & Timeline The Chronicles of Narnia Series by C.S. Lewis \| Overview... 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4613	https://libguides.nwpolytech.ca/math/factoring	Skip to Main Content Math Hub Math Language Toggle Dropdown Glossary Common Math Verbs Word Problems Math Anxiety Study Strategies Toggle Dropdown Learning Math Preparing for Tests The Guide Sections The Fundamentals Toggle Dropdown Order of Operations Calculating Percentages Fractions Dividing Fractions Transposition Inequalities Functions Laws of Exponents Absolute Value Significant Figures Rational Expressions and Equations Toggle Dropdown Rational Expressions and Non-Permissible Values Simplifying Rational Expressions Multiplying Rational Expressions Dividing Rational Expressions Adding and Subtracting Rational Expressions Rational Equations and Problem Solving Rational Expressions Quiz Logarithmic and Exponential Functions Toggle Dropdown Defining Logarithms Solving Exponential Equations Solving Problems Involving Exponential Functions Solving Problems Involving Logarithmic Functions Solving Quadratic Equations Factoring Quadratic Equations Solving Quadratic Equations by Completing the Square The Quadratic Formula Sequences and Series Toggle Dropdown Arithmetic Sequences Arithmetic Series Geometric Sequences Geometric Series Set Theory Toggle Dropdown Understanding Sets and Set Notations Solving problems Involving Sets Probability Toggle Dropdown Odds and Probability Mutually Exclusive and Non-Mutually Exclusive Events Dependent and Independent Events Permutations and Combinations Toggle Dropdown The fundamental Counting Principle Permutations Combinations Trigonometry Toggle Dropdown Angles in Standard Position Trigonometric Ratios Transformations of Functions Composition of Functions Operations with Functions Inverse Functions Limits Toggle Dropdown Tangents Limit of a Function Properties of Limits Additional Evaluation Techniques Continuity Derivatives Toggle Dropdown Function of a Derivative Derivative Rules Trigonometric Functions Exponential and Logarithmic Functions Equation of Tangent Line Implicit Derivatives The Antiderivative Statistics Toggle Dropdown Normal Distribution Central Limit Theorem Confidence Intervals Sample Size Hypothesis Testing Hypothesis Testing Process Additional Resources Learning Portal Attribution Some of the content of this guide was modeled after a guideoriginally created by the Openstaxand has been adapted for the GPRC Learning Commons in October 2020. This work is licensed under a Creative Commons BY 4.0 International License. Factoring Quadratic Equations One way to solve a quadratic equation is by factoring the equation. A general quadratic equation is given by: In order to factor a quadratic equation, one has to perform the following steps: Step 1) Find two numbers whose product is equal to ac, and whose sum is equal to b. Step 2)Write the middle term, bx, as the sum of two terms. The coefficients of the two terms are the two numbers found in Step 1). Step 3)Factor the first two terms and the second two terms separately. Step 4)Factor the common term and set different factors equal to zero and solve for the variable. Example:Solve by factoring. Step 1)We look for two numbers whose product is 63 and whose sum is -16. Those two numbers are -9 and -7. Step 2)We rewrite the quadratic equation as: Step 3)We factor the first two terms and the second two terms separately: Step 4)We factor the equation and solve: Example:Solve Step 1)We have to find two numbers whose product is 9 and whose sum is 6. Those two numbers are 3 and 3. Step 2)We rewrite the equation as: Step 3)We factor the first two terms and write the second two terms separately: Step 4)We factor the equation and solve: Example:Solve Step 1)Two numbers that multiply to give -30 and add to give 1 are 6 and -5. Step 2)Using the numbers from step 1), we rewrite the equation as: Step 3)We group every two terms and factor them: Step 4)We factor the equation and solve: << Previous: Solving Quadratic Equations Next: Solving Quadratic Equations by Completing the Square >> Last Updated: Aug 26, 2025 3:02 PM URL: Print Page Login to LibApps Report a problem Subjects: Multidisciplinary Tags: math Grande Prairie Campus 10726 106 Avenue Grande Prairie, AB T8V 4C4 780-539-2911 1-888-539-4772 (Toll-Free) StudentInfo@NWPolytech.ca Fairview Campus 11235-98 Avenue PO Bag 3000 Fairview, AB T0H 1L0 780-835-6600 1-888-999-7882 (Toll-Free) StudentInfo@NWPolytech.ca National Bee Diagnostic Centre 1 Research Road PO Box 1118 Beaverlodge, AB T0H 0C0 780-357-7737 NBDC@NWPolytech.ca We acknowledge the Indigenous people and their ancestors, whose land we are on. © Northwestern Polytechnic - All rights reserved
4614	https://www.youtube.com/watch?v=WuV1jQPKnJQ	How to Find Equivalent Fractions for 2/8 MagnetsAndMotors (Dr. B's Other Channel) 86900 subscribers 59 likes Description 28933 views Posted: 3 Mar 2023 To find three equivalent fraction for 2/8, we can multiple 2/8 by the equivalent of 1/1. For example: 2/8 x 1/1 = 2/8 since 1/1 = 1 and multiplying by 1 does not change the value. If we multiply: 2/8 x 2/2 = 4/16 Here again, since 2/2 = 1 we aren't changing the value, just the way it is written. Note we could also simplify 2/8 down to 1/4 which is also an equivalent fraction for 2/8. We could also multiply by 3/3, 4/4, 5/5 and so on to find more fractions that are equal to 2/8. As long as we are multiplying by the equivalent of 1, we'll get a fraction that is the same as 2/8. 5 comments Transcript: let's find three equivalent fractions for two eighths here's how we'll do that so we want to find another fraction that's equivalent it has the same value as 2 divided by eight just that the fraction has a different numerator and denominator so first off I can see that we can simplify to eight two goes into two one time and two goes into eight four times so 1 4 and 2 8 they are equivalent fractions if you divide two by eight or one by four you get the same thing 0.25 because they're equivalent we could also take 2 8. and instead of dividing to simplify we could multiply we could multiply it by one but that would just give us 2 8. let's multiply it by two over two two over two that's just one two divided by two but now we have two times two that gives us four eight times two that gives us 16. so 4 16 that's another equivalent fraction to 2 8 and 1 4 and if you divide four by sixteen you get 0.25 same thing as we did before one last one we can multiply it by something like seven over seven that's one we get 14.56 that's another equivalent fraction for 2 8. this is Dr B thanks for watching
4615	https://oertx.highered.texas.gov/courseware/lesson/89/overview	Preview Please log in to save materials. Log in Report Details Resource Library Author: : Amy Petros Subject: : Chemistry Material Type: : Module Level: : Academic Lower Division, Academic Upper Division Tags: : - AP Chemistry - Chemistry - Molar Mass Log in to add tags to this item. License: : Creative Commons Attribution Language: : English Media Formats: : Text/HTML, Video Show More Show Less How big is a mole? Ted Ed View How Small is an Atom? Ted Ed View MP4 Using Moles and Avogadro's number as a conversion factor Download General Chemistry for Science Majors Unit 1 Unit 1 Unit 2 Unit 3 Moles as a conversion factor: How do we get from grams to number of atoms? Significant Digits and Accuracy Introduction to Matter: Mixtures, Elements, and compounds Physical properties: thermal conductivity Physical Properties: Specific Heat Capacity Kinetic and Potential Energy Naming Binary Compounds Moles as a conversion factor: How do we get from grams to number of atoms? Empirical Formula, molar mass, and mass percent Moles as a conversion factor: How do we get from grams to number of atoms? Overview This module has two parts: 1- Introduction to the concept of a "mole" and Avogadro's number. 2- Application of the Mole (Avogadro's number) as a conversion factor relating mass in grams to number of atoms or molecules (formula unit for ionic compounds) Introduction to the MOLE (chemist's dozen) How do we count atoms? Since atoms are so small, we can't! We'd literally die first before we could count them all. Check out the Ted Ed video, How Small is an Atom? for a taste of the relative size of these tiny particles. So, how about using mass in grams instead? We can easily weigh things, and since every pure element or compound has its own mass based on the number of protons and neutrons, we can convert from grams to atoms or molecules. We use the MOLE, which is 6.022 x 1023 of anything. The Ted Ed video on What's a Mole? Is a great place to start. How big is a mole? Ted Ed View How Small is an Atom? Ted Ed View Using Avogadro's number as a conversion factor Now that you understand what moles represent and why it's useful to scale to that unit, let's explore the application of this new conversion factor. In the video, we demonstrate converting atoms to moles to grams and grams to moles to atoms. If you need a refresher on Dimensional Analysis and Unit Conversions, skip over to the module on Unit Conversions. All our same rules apply. Video also includes molar mass of compounds. MP4 Using Moles and Avogadro's number as a conversion factor Download
4616	https://pmc.ncbi.nlm.nih.gov/articles/PMC9683083/	Importance of ECG in the Diagnosis of Acute Pericarditis and Myocardial Infarction: A Review Article - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. 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Learn more: PMC Disclaimer \| PMC Copyright Notice Cureus . 2022 Oct 24;14(10):e30633. doi: 10.7759/cureus.30633 Search in PMC Search in PubMed View in NLM Catalog Add to search Importance of ECG in the Diagnosis of Acute Pericarditis and Myocardial Infarction: A Review Article Aditya K Sarda Aditya K Sarda 1 Medicine and Surgery, Jawaharlal Nehru Medical College, Datta Meghe Institute of Medical Sciences, Wardha, IND Find articles by Aditya K Sarda 1,✉, Preeti Thute Preeti Thute 2 Anatomy, Jawaharlal Nehru Medical College, Datta Meghe Institute of Medical Sciences, Wardha, IND Find articles by Preeti Thute 2 Editors: Alexander Muacevic, John R Adler Author information Article notes Copyright and License information 1 Medicine and Surgery, Jawaharlal Nehru Medical College, Datta Meghe Institute of Medical Sciences, Wardha, IND 2 Anatomy, Jawaharlal Nehru Medical College, Datta Meghe Institute of Medical Sciences, Wardha, IND ✉ Aditya K. Sarda adiksarda15@gmail.com ✉ Corresponding author. Received 2022 Sep 15; Accepted 2022 Oct 24; Collection date 2022 Oct. Copyright © 2022, Sarda et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. PMC Copyright notice PMCID: PMC9683083 PMID: 36426313 Abstract A promising scientific field is health monitoring and associated technology. A standard testing method to evaluate and identify heart issues is the electrocardiogram (E.C.G.) and diagnose cardiovascular diseases (CVDs). E.C.G. monitoring technologies are becoming more and more prevalent in publications at an exponential rate. E.C.G.is the most crucial tool for screening cardiology and other medical specialities. Twelve leads can be recorded by traditional E.C.G. equipment, while current E.C.G. systems allow for extra leads also with fewer electrodes. Furthermore, “smart” gadgets allow patients to take an E.C.G. at residence. Presenting different ischemia-related symptoms on the E.C.G. by the most recent recommendations. Presentation of contemporary E.C.G. systems and their possible benefit in identifying ischemia-related E.C.G. symptoms based on recent study findings. The identification of ischemia E.C.G. abnormalities can be facilitated and optimised by current E.C.G. systems using vector-based electrocardiography. Although they can be effective for documenting transient E.C.G. abnormalities, especially inside the S.T. segment, smart non-vector-based devices for patients are primarily beneficial for the diagnosis of arrhythmias and cannot substitute the 12-lead E.C.G. for the diagnosis of ischemia. The electrocardiogram (E.C.G.) is inexpensive and easily accessible, but because of its alleged limited specificity, its utility as a screening tool for early detection of athletes with a cardiac condition in danger of immediate cardiac death is contentious. The interpreting parameters have been continuously evolving over the past 10 years as various efforts have been made to better the separation between healthy and pathological E.C.G. abnormalities in athletes. Electrocardiographic abnormalities that are unrelated to cardiac electrical activity are known as electrocardiographic artefacts. E.C.G.elements, including the baseline and waves, can become altered as a result of artefacts. Keywords: pr segment and t wave, q wave, st segment, electrocardiograph, acute pericarditis, acute myocardial infarction Introduction and background Acute myocardial infarction (MI) is a word coined for a match of coronary assault, which is caused by the deposition of plaques or cholesterol in the inner surface of the coronary arteries resulting in decreased blood flow to the heart and scarring the cardiac muscle mass due to insufficient oxygen supply. Though MI primarily affects patients over the age of 45, it can also affect young men or women. Fortunately, patients under 45 years old do not often experience its occurrence. Cardiovascular disease has a fatality rate that is greater than twice as high as cancer in the U.S., making it the most common cause of death, and over half of these vascular fatalities are attributable to acute myocardial attack/infarction. The discussion of acute attack/MI patient management during and after hospitalisation places a focus on primary and secondary prevention, patient autonomy, and decision-making. An assessment of the future directions for treating acute MI is also included. Viral infections are frequently linked to acute pericarditis. The typical symptom of infection is chest discomfort, which manifests in conjunction with or shortly after the indications of infection. About 5% of patients who are brought to the emergency medicine department with thoracic discomfort not related to an acute myocardial attack/infarction are found to have acute pericarditis, which is referred to as pericardial inflammation. Most typically, it affects males between the ages of 20 and 50. The most prevalent symptom in people with an acute MI is chest pain, which is commonly described as tightness in the chest, pressure, or a squeezed feeling. The majority of MI with non-obstructive coronary artery (M.I.N.O.C.A.) patients who experience an acute myocardial attack/infarction with S.T. segment rise from the baseline typically have a pre-existing coronary or myocardial aetiology, most frequently plaque disruption or myocarditis. Acute pericarditis, an acute inflammatory sickness that affects the pericardium, can result from several different illnesses (both non-infectious and infectious). The diagnosis is often made based on symptoms (chest discomfort, difficulty breathing), electrocardiogram (E.C.G.)abnormalities (S.T. amplification), physical assessment (pericardial friction rubbing), and elevation of cardiac biomarkers. It could take either alone or in tandem with an involved inflammatory disorder. Acute pericarditis and acute myocarditis can cohabit in routine clinical practice due to their convergent aetiologies. Inflammation of the pericardium (heart covering), also known as acute pericarditis, can result in cardiac involvement and pericardial effusion (peri myocarditis). Systemic inflammatory rheumatic illnesses can cause pericarditis. However, it can also represent a distinct disease entity. Non-S.T.-segment elevation MI (NSTEMI) and S.T.-segment elevation MI (STEMI), in which S.T.-segment elevation is observed on the E.C.G. graph, are the two kinds of acute MI . A frequent condition with numerous causes that can manifest in both primary and secondary care settings is pericarditis. The sampling and analysis of pericardial fluid have been improved by new diagnostic techniques, which enable thorough cause characterisation. Despite these developments, pericarditis still most frequently arises from idiopathic causes; they include radiation therapy, heart surgery, and percutaneous procedures. Non-steroidal anti-inflammatory drugs continue to be the primary choice of treatment for simple instances of pericarditis because it typically has a self-limiting course. Integrating new imaging techniques makes it easier to accurately identify and treat complications like acute pericardial effusion or constriction. An echo-guided percutaneous technique can be used to safely treat the majority of pericardial effusions. In most situations, pericardiectomy relieves symptoms and is still the only effective treatment for constrictive pericarditis. A thorough physical examination, an E.C.G., an echocardiogram, blood work, and still a chest x-ray is frequently necessary to make the clinical finding of acute pericarditis. Acute pericarditis and S.T.-elevated myocardial infarct share comparable E.C.G. features, making the distinction between the two frequently challenging (STEMI) (S.T. segment change). Review ST-segment elevation Cases of S.T.-segment upliftment from baseline in MI with S.T.-segment upliftment from the baseline in the anterior and anteroposterior territories, and less frequently, concurrent S.T.-elevations in the anterior and inferior electrocardiography leads. To identify this cardiac emergency, which frequently requires urgent coronary revascularisation, accurate interpretation of the E.C.G. is essential. Additionally, E.C.G. improves prognostic value and helps localise the artery connected to the infarct. Because of epidemiologic information and arteriographic correlative investigations, the S.T.-segment reaction to exercise is frequently utilised to identify people with ischemic heart disease. Abrupt S.T.-segment rise (RT checkbox indication), horizontal or upwards S.T.-segment convexity, lead III (three/3) S.T.-segment rise greater than lead II (two/2) S.T.-segment elevation and lead III > II To assess the strength of the link, we utilised the incidence rate with such a 95% confidence interval (CI). Every year, approximately seven million people worldwide are identified with ACS. In STEMI patients, mortality is reduced by cardiac catheter and PCI within one or 20 minutes of presentation; fibrinolytic therapy is only administered to those who are unable to undergo prompt PCI. Low death rates are associated with rapid invasive coronary angiography (RICA), followed by interventional or surgical revascularisation, in high-risk NSTE-ACS patients. The 12-lead E.C.G.continues to be the mainstay of early STEMI diagnosis, and it also serves as the key indicator for S.T.-elevation MI patients requiring emergency reperfusion treatment. These are characterised as S.T. rise MI mimics and includes, for example, left ventricular dilatation, acute pericarditis, and benign early repolarization. In rare clinical circumstances, a patient's E.C.G. may mimic S.T. elevation MI yet exhibit S.T.-segment elevated from a non-coronary-based illness. The ability of pericarditis and early repolarization syndrome (ERS) and takotsubo cardiomyopathy to resemble S.T. elevation MI is well documented (STEMI). Acute chest discomfort and an S.T. rise of a fraction of 1 mV were present in all individuals. For the existence of S.T.-segment convexity, each E.C.G. was evaluated. The physical sign of acute pericarditis is the pleural frictional rub, which is often audible around the lower lateral sternal border. An easy tool for diagnosing acute pericarditis is the E.C.G.Diffuse concave-upwards S.T.-segment rise and, in rare instances, P.R.-segment depression are common E.C.G. abnormalities. Early repolarisation and acute MI both have E.C.G. abnormalities that resemble acute pericarditis. Q-wave Q-waves may indicate acute MI when seen on an E.C.G.There is no distinction between transmural and non-transmural infarction in the E.C.G. alterations. On the other hand, a Q-wave's existence or absence corresponds with several features of a patient's post-MI clinical history and hence has predictive significance. Congestive heart failure is more likely to worsen Q-wave infarctions while they are being treated in the hospital. Congestive heart failure was more likely to occur later in participants who had their first Q-wave infarction, although coronary insufficiency was more common in participants who had their first non-Q-wave infarction. A chest pain attack caused aberrant Q waves to arise in Leads V2 and V3, but they quickly vanished after the discomfort subsided. Although MI could not be ruled out, the serial electrocardiographic alterations and the clinical history were consistent with angina or coronary insufficiency. We think that the myocardium's transient ischemia was what caused these Q waves. Q waves and amplified CKMB activity are the presentations of pericarditis, which must be separated from MI brought on by coronary artery disease. Infarction-associated pericarditis's pericardial effusion and elevated extravascular lung water were not brought on by left ventricular failure but rather by other causes that showed a greater infarct. Acute MI typically begins with aberrant Q waves, which are frequently seen. However, there is not any evidence that abnormal Q waves following thrombolytic treatment have a huge negative impact on the ability to reduce infarct size. Although reversible Q.R.S. complex alterations linked to an S.T.-segment shift are a hallmark of acute myocardial ischemia (AMI), these changes have yet to be comprehensively examined in a significant patient cohort. Over the past four years, a deliberate hunt for E.C.Gs with confirmed reversible Q.R.S. alterations connected to any acute injury type has been made. PR segment depression Significant P.R.-segment depressed > or = 1.2 millimetres in the inferior leads was linked to an uncomfortable hospital stay and a subpar short-term prognosis in acute inferior MI. These individuals had a high chance of developing heart free-wall rupture, atrioventricular block, and supraventricular arrhythmias. When there were aberrant P waveforms or P.R. dislocation in either lead, the duration of stay was longer. Patients with an aberrant P-wave had a higher likelihood of having left main cardiovascular disease. Higher one-month and 12-month death rates were linked to aberrant P-wave shapes in any lead (OR 3.09 [1.35-7.05] and 5.33 [2.74-10.36], respectively). Additionally, increased one month(OR 2.33; 1.03-5.28) and 12-month mortality were linked to P.R. displacement in just about any lead. P.R. deficiency in II, III, and AFV, as well as elevation in AVR or AVL and P.R. depression in the precordial leads, were all linked to an increase in one-year fatality (OR 12.49). (5.2-30.0). When age, ejection fraction, peak biomarker troponin I, and left the primary illness were taken into account, it was shown that P.R. displacement in either lead was linked to higher one-year mortality rates. When present in 31% of STEMI patients, P.R. segment relocation in any lead independently predicted 12-month mortality. The distinctive E.C.G. signs of acute pericarditis are P.R.-segment distress (found rarely), multi-lead S.T. segment rise, and S.T.-segment distress in the lead named aVR. Reciprocal T-wave changes The behaviour of main T-wave abnormalities following exercise does not modify the interpretation of the ischemic alterations, according to exercise-induced T-wave normalisation studies. T-wave abnormalities are typically non-specific. However, isoproterenol treatment prevents post-MI T-wave modifications while treating a variety of functional and neurogenic T-wave abnormalities. A thallium-201 scan was positive in 49 of 52 myocardium segments with controlled T impulses and 43 of 45 individuals (95%) overall. When the regions of engagement were linked to thallium scanning and T-wave changes, 76% of lead aV's T-wave normalisation was related to positive inferior-posterior scans, and 77% of V1 and V5's T-wave normalisation was connected with positive anterior-septal scans. Only after comparing the precordial leads to the baseline, E.C.G. was the little T-wave alterations visible. The goal of the current investigation was to see if reciprocal alterations could occur in place of or in addition to any other symptoms of AMI . The normalisation method for normalisation could include producing an isoelectric-electric S.T. segment and upright T-wave by adding algebraically the immediate ischemia-induced S.T. segment and T-wave magnitude in addition to the pre-existing S.T. segment lowering, and T-wave inversion. Recordings of reciprocal changes, such as MI, are possible. T-wave inversion Inverted T waves provide diverse anatomical lead groups with varied prognostic data. A higher risk of CHD is associated with a T-wave reversal in the lateral and anterior lead groups, and an increased risk of death is associated with lateral T-wave reversal. It was discovered that the inferior lead group's inverted T-wave was a harmless occurrence. It is challenging to interpret the T waveform on an E.C.G. It relies on changes to cell membranes because it represents ventricular repolarisation. It is influenced by several physiological, central nervous system, and physical health aspects. It is the most unstable part of the QRST complex and the most sensitive sickness indicator. The huge intensification of the T waveform, which is monstrously blown out, typically inverted, and covers an area many times that of the Q.R.S. complex, is one of the most striking and least understood oddities. Heart illness has frequently been brought to light, particularly whenever the T waveforms are inverted. Conventionally thin and symmetric inverted T waves are brought on by myocardial ischemia. The isoelectric S.T. section is frequently concave when a T-waveform inversion (TWI) is associated with acute coronary syndrome(ACS), and this is followed by a powerful symmetric downstroke. Patients having anterior STEMI who simultaneously exhibited inverted T waves inside the leads and S.T. rise might be identified by an accurate indicator of spontaneous reperfusion of an infarct-related artery on the E.C.G.. Value of the recently found T-wave inversion as a predictor in patients who have undergone first percutaneous coronary interventions and have elevated S.T.-segment MI (PCI). New T-wave inversion is the beginning of T-wave inversion during the percutaneous coronary intervention (PCI)without the presence of negative T-waves only on the displayed E.C.G. Following the first PCI, a recently created T-wave inversion was linked to a positive long-term result. Table 1reflects changes in E.C.G. for differentiating acute pericarditis and acute MI based on the five features that include depression in the P.R. segment, changes in Q-wave, elevation of S.T. segment, inverted changes of T-wave and Inversion of T-wave. Table 1. Differentiating between acute pericarditis and acute myocardial infarction. Sr. No.ECG Features Acute Pericarditis Acute Myocardial Infarction 1.Depression in the PR Segment Can be seen in various patients Very rare to be noticed 2.Changes in Q-Wave Not seen in patients with the above condition Seen in patient with above condition 3.Elevation of ST-Segment Presented with concave elevation Presented with convex elevation 4.Inverted changes of T- Wave Not presented in patients with the above condition Seen in patients with the above condition 5.Inversion of T-Wave Seen in ECG once when the changes of ST-Segment are normalized Seen in ECG once when the changes of ST-Segment are normalized Open in a new tab Conclusions Normal E.C.G.has 12 leads which are used to measure the cardiac abnormalities in a person. There is a baseline in E.C.G., P-wave, Q.R.S. complex, and T-wave. The basic analysis of acute pericarditis and acute MI can be diagnosed on the E.C.G. It can also be differentiated based on some factors in E.C.G.The S.T.-segment upliftment denotes the rise of the S.T.-segment portion from the baseline. If the elevation of the S.T. segment is convex at the top of the elevated line, it is diagnosed as an acute MI, which is further studied in the limb for precisely conforming to the artery that is blocked, while in the case of the S.T. segment elevation if the elevated segment forms the concave curve at the top of the elevation, it is diagnosed as acute pericarditis. Q-wave is the starting negative bend of the rise to the Q.R.S. complex. Q-wave is the representation of the depolarisation of the ventricular in E.C.G. Q-wave is absent in the E.C.G.of patients suffering from acute pericarditis, while the Q-wave is present with no or minimal changes in the patient suffering from sudden MI. The P.R. segment is the amount of time that passes between the atrium and ventricle depolarising. One of the most crucial elements in the diagnosis of acute pericarditis, along with depression, is the P.R. segment in the P.R.segment is the most widely presented feature in E.C.G.of the patient who is suffering from acute pericarditis, but on the other hand, the P.R.-segment depression is one of the rarest phenomena presented by the patients of acute MI. T-wave represents ventricular myocardium repolarisation. Reciprocal T-wave is a very abnormal change of T-wave in the aVL, which is very typically transmural. These reciprocal T-wave changes are usually not presented in the E.C.G.of the patient with acute pericarditis. At the same time, the reciprocal T-wave changes are considered one of the most important changes in E.C.G.for the diagnosis of acute MI in patients. Inversion of T-wave is well-defined as the negative movement of T-wave in more than two contagious leads, excluding the leads like III, aVR, and V1. This T-wave cannot be considered an important factor for the differentiation of acute pericarditis and acute MI, as it is seen in both patients suffering from the above disorders. But the important thing is that it is seen only after the S.T.-segment upliftment is normalised in the E.C.G.of the patient suffering from the above disorder. The content published in Cureus is the result of clinical experience and/or research by independent individuals or organizations. Cureus is not responsible for the scientific accuracy or reliability of data or conclusions published herein. All content published within Cureus is intended only for educational, research and reference purposes. Additionally, articles published within Cureus should not be deemed a suitable substitute for the advice of a qualified health care professional. Do not disregard or avoid professional medical advice due to content published within Cureus. Footnotes The authors have declared that no competing interests exist. 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4617	https://www.britannica.com/summary/rock-geology	rock summary Our editors will review what you’ve submitted and determine whether to revise the article. rock, In geology, a naturally occurring and coherent aggregate of minerals. The three major classes of rock—igneous, sedimentary, and metamorphic—are based on the processes that formed them. These three classes are further subdivided on the basis of various factors, especially chemical, mineralogical, and textural attributes (see e.g., acid and basic rocks; crystalline rock; extrusive rock). See also felsic rock; intrusive rock; mafic rock.
4618	https://dein-sprachcoach.de/der-superlativ-im-deutschen/	Über Mich Kurse A2-Kurs B1-Kurs B2-Kurs C1-Kurs Crashkurs Blog Sprache allgemein Prüfungsvorbereitung Phonetik & Aussprache Grammatik & Interpunktion Rechtschreibung Merch Kurslogin Der Superlativ im Deutschen Dein Sprachcoach-Blog September 8 Share0 Wie kannst du am besten Deutsch lernen? Welche Grammatikthemen übst du am liebsten und welche Tipps helfen dir am meisten, wenn du einmal nicht weiter weißt? In diesen Sätzen findest du drei Superlative. Hast du sie erkannt? Wenn du dir jetzt die Frage stellst: „Was ist ein Superlativ?“, dann lies unbedingt weiter! Hier erfährst du nämlich alles zum deutschen Superlativ und wie du ihn richtig bildest. Was sind Superlative? Mit einem Superlativ beschreibst du Dinge oder Personen, die nicht zu übertreffen1 sind. Es geht also um die höchste Eigenschaft von etwas, das du miteinander vergleichst: Beispiel: Superlativ Wie lernst du neue deutsche Wörter am schnellsten? Am einfachsten ist es, wenn du neue Wörter in Gesprächen verwendest. Der Superlativ ist in der deutschen Grammatik die zweite Steigerungsform. Was meint „Steigerung“? Du steigerst deutsche Adjektive oder Adverbien entweder mit dem Komparativ oder mit dem Superlativ. Der Komparativ ist die erste Form der Steigerung und der Superlativ die zweite. Damit du mit dem Komparativ und Superlativ nicht durcheinander kommst, schauen wir uns zuerst den Komparativ genauer an. Was ist „Komparativ“? Mit dem Komparativ vergleichst du zwei Dinge oder Personen miteinander. Und wenn es bei diesem Vergleich einen Unterschied gibt, dann setzt du das deutsche Adjektiv oder Adverb in den Komparativ: Beispiel: Komparativ Am Morgen lerne ich leichter als am Abend. Tim ist schlauer als Kathrin. Den Komparativ bildest du mit dem Suffix –er. Achte bei einigen bestimmten Adjektiven auf den Umlaut, wenn du den Komparativ bildest. Den Umlaut bildest du bei einsilbigen Adjektiven mit dem Stammvokal a: alt – älter, arm – ärmer, hart – härter, kalt – kälter, krank – kränker, lang – länger, nah – näher, scharf – schärfer, schwach – schwächer, stark – stärker, warm – wärmer o: grob – gröber, groß – größer, hoch – höher u: dumm – dümmer, jung – jünger, klug – kluger, kurz – kürzer Alle anderen deutschen Adjektive – auch die mehrsilbigen – haben keinen Umlaut. Merke dir diese Regel schonmal für später, wenn wir uns den Superlativ genauer anschauen. Du wirst diese Regel nämlich noch brauchen! Wichtig ist, dass du nicht mit allen deutschen Adjektiven den Komparativ bilden kannst. Das ist der Fall, wenn Adjektive einen eindeutigen Inhalt haben und es keine Möglichkeit gibt, einen Unterschied auszudrücken, wie ewig, voll, tot, leer, rechteckig, doppelt oder allein. Ich kann zum Beispiel nicht sagen: Deine Aussprache ist falscher ist als die von deiner Freundin. Wenn ein Wort falsch ausgesprochen wird, dann ist es einfach falsch und kann nicht noch gesteigert werden. Übrigens: Fragst du dich auch manchmal, ob du in einem Vergleichssatz die Wörter wie oder als verwenden sollst? Dann schau dir den Artikel Vergleichssätze: mit wie oder als? auf jeden Fall an! Unregelmäßige Steigerungsformen und die Komparation der Adverbien Einige Adjektive im Komparativ bilden unregelmäßige Steigerungsformen. So veränderst du bei manchen Steigerungsformen das gesamte Wort, wie bei gut – besser oder viel – mehr. Oder es ändert sich der Konsonant bei der Steigerung der Adjektive: hoch – höher oder nahe – näher. Auch unregelmäßig steigerst du einige deutsche Adverbien: bald – eher (oder früher, schneller), gerne – lieber, sehr – mehr. Du musst wissen, dass du im Deutschen insgesamt nur wenige Adverbien steigern kannst. Regelmäßig steigerst du zum Beispiel die Adverbien oft – öfter und wohl – wohler. Superlativ und Komparativ im Vergleich Lass uns nun beide Formen, den Superlativ und den Komparativ, direkt miteinander vergleichen. Hier kommt das Wichtigste auf einen Blick: Adjektive gibt es im Deutschen in der Grundform und in den zwei Steigerungsformen. Die Grundform nennt sich auch Positiv: Lisa ist groß. Mit dem Komparativ vergleichst du zwei ungleiche Dinge miteinander: Lisa ist groß, aber Lea ist größer. Und durch den Superlativ drückst du bei einem Vergleich die höchste Stufe aus: Lisa ist groß, aber Lea ist am größten. Formen im Komparativ und Superlativ bildest du mit starken oder schwachen Flexionsendungen: das dickere Buch ein dickeres Buch das dickste Buch sein dickstes Buch ihr Buch ist am dicksten Wenn du die Adjektivdeklination zur richtigen Steigerung der Adjektive noch einmal wiederholen möchtest, dann lies dir gerne den Beitrag Adjektivdeklination durch! Hier erfährst du alles zu den richtigen Endungen des Adjektivs! Den Superlativ richtig bilden Die Höchststufe kannst du laut Duden mit am oder dem bestimmten Artikel (der, die, das) bilden. Wenn du den Superlativ mit am formulierst, hängst du an die Grundform des Adjektivs die Endung -sten an: Diese Schuhe finde ich am schönsten. Bildest du die Steigerung mit dem bestimmten Artikel, hängst du an das Adjektiv die Endung -ste an: Das Schönste am Urlaub ist es, lange zu frühstücken. Ist dir etwas aufgefallen? Richtig, die Großschreibung der Höchststufe! Superlativ: groß oder klein? Weißt du auch, warum du die Höchststufe im Satz „Das Schönste am Urlaub ist es, lange zu frühstücken“ großschreiben musst? Das liegt an der Nominalisierung des Adjektivs. Durch die Nominalisierung wird das Adjektiv schön in ein Nomen umgewandelt. Ein Adjektiv wird nominalisiert, indem du einen bestimmten Artikel voranstellst: das Schönste an diesem Urlaub ist, dass ich lange schlafen kann. Merke dir: beide Formen (Grundform + Endung -sten und bestimmter Artikel + Endung -ste) drücken die Höchststufe aus, werden aber unterschiedlich geschrieben. Die Regeln zum Superlativ auf einen Blick Wenn Adjektive auf -d, -t, -s, -ss, -ß, -z, -tz, -x, -sk oder -sch enden und auf der letzten Silbe einen Vollvokal2 haben, dann bildest du den Superlativ mit dem Suffix -est: fett – fetteste, nass – nasseste, süß – süßeste, spitz – spitzeste, frisch – frischeste. Steht in der letzten Silbe kein betonter Vokal oder endet das Adjektiv auf -isch, dann wählst du das Suffix -st: passend – passendste, verbreitet – verbreitetste, fantastisch – fantastischste. Bei diesen Adjektiven gelten sogar beide Formen mit -est oder -st als richtig: das neueste / neuste Produkt die genaueste / genauste Lösung das freieste / freiste Gefühl das roheste / rohste Gemüse die schlankste / schlankeste Frau Besonders sind Adjektive, deren Grundform auf -el oder -bel enden und bei denen diese Endungen unbetont sind. Hier entfällt das -e bei der Bildung des Komparativs, aber im Superlativ bleibt es erhalten: Die Hose ist dunkel, die Jacke ist dunkler, die Bluse ist am dunkelsten. Erinnerst du dich noch an den Umlaut beim Komparativ? Super, denn diese Regel brauchen wir auch für den Superlativ. Das sind die Formen Positiv, Komparativ und Superlativ mit Umlaut: a: alt – älter – älteste, arm – ärmer – ärmste, hart – härter – härteste, kalt – kälter – kälteste, krank – kränker – kränkste, lang – länger – längste, nah – näher – nächste, scharf – schärfer – schärfste, schwach – schwächer – schwächste, stark – stärker – stärkste, warm – wärmer – wärmste o: grob – gröber – gröbste, groß – größer – größte, hoch – höher – höchste u: dumm – dümmer – dümmste, jung – jünger – jüngste, klug – kluger – klügste, kurz – kürzer – kürzeste Test jetzt starten Gratis Sofortiges Ergebnis Zum gratis E-Book 10 GEHEIME Tipps, um die deutsche Sprache zu meistern! Besondere Adverbien und Adjektive Wie du bereits weißt, bilden nur wenige Adverbien im Deutschen den Komparativ. Das gilt dann natürlich auch für den Superlativ. Hier sind noch einmal die drei Stufen Grundform, Komparativ und Superlativ von Adverbien im Überblick: sehr – mehr – am meisten gern(e) – lieber – am liebsten bald – eher – am ehesten oft – öfter – am öftesten Diese Formen nutzt du, wenn du unregelmäßige Adjektive steigern willst: gut – besser – am besten viel – mehr – am meisten hoch – höher- am höchsten nahe – näher – am nächsten Fragen & Antworten Was heißt Superlativ und was ist ein absoluter Superlativ? Der Superlativ drückt die höchste Eigenschaft von etwas aus. Der absolute Superlativ wird auch Elativ genannt und ist die absolut höchste Steigerungsform der Adjektive: Wir arbeiten mit modernster Technik. Wie bilde ich den Superlativ? Mit dem Superlativ kannst du ein Verb näher beschreiben: Lisa läuft schnell, Andrea läuft schneller, Marina läuft am schnellsten. Adjektive nach sein/werden/bleiben können den Superlativ mit am und dem bestimmten Artikel bilden: Der Urlaub ist am schönsten. Die Erinnerung ist das Schönste. Wird der Superlativ dekliniert? Ja, du musst bei der Bildung des Superlativs den Kasus und Numerus des Nomens beachten: Ich erinnere mich gern an den schönsten Urlaub. Wortschatz übertreffen: besser sein als jemand Vollvokal (der): betonter oder betonbarer Vokal Lass dich nicht entmutigen, wenn es länger dauert, die deutsche Sprache zu meistern. Ein tolles Deutsch zu erreichen, ist ein fortlaufender Prozess, aber du kannst es schaffen! Wenn du mehr Hilfe brauchst, habe ich sehr viel kostenloses Lernmaterial, mit dem du effektiv lernen kannst. Die Inhalte haben auch mir geholfen, perfektes Deutsch zu sprechen. Über 100 PDF's mit Übungen, Beispielen, Wortschatz und Grammatik warten auf dich. 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4619	https://courses.lumenlearning.com/calculus1/chapter/extrema-and-critical-points/	Module 4: Applications of Derivatives Extrema and Critical Points Learning Outcomes Define absolute extrema Define local extrema Explain how to find the critical points of a function over a closed interval Describe how to use critical points to locate absolute extrema over a closed interval Absolute Extrema Consider the function [latex]f(x)=x^2+1[/latex] over the interval latex[/latex]. As [latex]x\to \pm \infty[/latex], [latex]f(x)\to \infty[/latex]. Therefore, the function does not have a largest value. However, since [latex]x^2+1\ge 1[/latex] for all real numbers [latex]x[/latex] and [latex]x^2+1=1[/latex] when [latex]x=0[/latex], the function has a smallest value, 1, when [latex]x=0[/latex]. We say that 1 is the absolute minimum of [latex]f(x)=x^2+1[/latex] and it occurs at [latex]x=0[/latex]. We say that [latex]f(x)=x^2+1[/latex] does not have an absolute maximum (see Figure 1). Figure 1. The given function has an absolute minimum of 1 at [latex]x=0[/latex]. The function does not have an absolute maximum. Definition Let [latex]f[/latex] be a function defined over an interval [latex]I[/latex] and let [latex]c\in I[/latex]. We say [latex]f[/latex] has an absolute maximum on [latex]I[/latex] at [latex]c[/latex] if [latex]f(c)\ge f(x)[/latex] for all [latex]x\in I[/latex]. We say [latex]f[/latex] has an absolute minimum on [latex]I[/latex] at [latex]c[/latex] if [latex]f(c)\le f(x)[/latex] for all [latex]x\in I[/latex]. If [latex]f[/latex] has an absolute maximum on [latex]I[/latex] at [latex]c[/latex] or an absolute minimum on [latex]I[/latex] at [latex]c[/latex], we say [latex]f[/latex] has an absolute extremum on [latex]I[/latex] at [latex]c[/latex]. Before proceeding, let’s note two important issues regarding this definition. First, the term absolute here does not refer to absolute value. An absolute extremum may be positive, negative, or zero. Second, if a function [latex]f[/latex] has an absolute extremum over an interval [latex]I[/latex] at [latex]c[/latex], the absolute extremum is [latex]f(c)[/latex]. The real number [latex]c[/latex] is a point in the domain at which the absolute extremum occurs. For example, consider the function [latex]f(x)=\frac{1}{(x^2+1)}[/latex] over the interval latex[/latex]. Since [latex]f(0)=1\ge \dfrac{1}{x^2+1}=f(x)[/latex] for all real numbers [latex]x[/latex], we say [latex]f[/latex] has an absolute maximum over latex[/latex] at [latex]x=0[/latex]. The absolute maximum is [latex]f(0)=1[/latex]. It occurs at [latex]x=0[/latex], as shown in Figure 2b. So remember: the maximum/minimum = [latex]y[/latex]; the location of the maximum/minimum = [latex]x[/latex]. A function may have both an absolute maximum and an absolute minimum, just one extremum, or neither. Figure 2 shows several functions and some of the different possibilities regarding absolute extrema. However, the following theorem, called the Extreme Value Theorem, guarantees that a continuous function [latex]f[/latex] over a closed, bounded interval [latex][a,b][/latex] has both an absolute maximum and an absolute minimum. Figure 2. Graphs (a), (b), and (c) show several possibilities for absolute extrema for functions with a domain of latex[/latex]. Graphs (d), (e), and (f) show several possibilities for absolute extrema for functions with a domain that is a bounded interval. Extreme Value Theorem If [latex]f[/latex] is a continuous function over the closed, bounded interval [latex][a,b][/latex], then there is a point in [latex][a,b][/latex] at which [latex]f[/latex] has an absolute maximum over [latex][a,b][/latex] and there is a point in [latex][a,b][/latex] at which [latex]f[/latex] has an absolute minimum over [latex][a,b][/latex]. The proof of the extreme value theorem is beyond the scope of this text. Typically, it is proved in a course on real analysis. There are a couple of key points to note about the statement of this theorem. For the extreme value theorem to apply, the function must be continuous over a closed, bounded interval. If the interval [latex]I[/latex] is open or the function has even one point of discontinuity, the function may not have an absolute maximum or absolute minimum over [latex]I[/latex]. For example, consider the functions shown in Figure 2(d), (e), and (f). All three of these functions are defined over bounded intervals. However, the function in graph (e) is the only one that has both an absolute maximum and an absolute minimum over its domain. The extreme value theorem cannot be applied to the functions in graphs (d) and (f) because neither of these functions is continuous over a closed, bounded interval. Although the function in graph (d) is defined over the closed interval [latex][0,4][/latex], the function is discontinuous at [latex]x=2[/latex]. The function has an absolute maximum over [latex][0,4][/latex] but does not have an absolute minimum. The function in graph (f) is continuous over the half-open interval [latex][0,2)[/latex], but is not defined at [latex]x=2[/latex], and therefore is not continuous over a closed, bounded interval. The function has an absolute minimum over [latex][0,2)[/latex], but does not have an absolute maximum over [latex][0,2)[/latex]. These two graphs illustrate why a function over a bounded interval may fail to have an absolute maximum and/or absolute minimum. Before looking at how to find absolute extrema, let’s examine the related concept of local extrema. This idea is useful in determining where absolute extrema occur. Local Extrema and Critical Points Consider the function [latex]f[/latex] shown in Figure 3. The graph can be described as two mountains with a valley in the middle. The absolute maximum value of the function occurs at the higher peak, at [latex]x=2[/latex]. However, [latex]x=0[/latex] is also a point of interest. Although [latex]f(0)[/latex] is not the largest value of [latex]f[/latex], the value [latex]f(0)[/latex] is larger than [latex]f(x)[/latex] for all [latex]x[/latex] near 0. We say [latex]f[/latex] has a local maximum at [latex]x=0[/latex]. Similarly, the function [latex]f[/latex] does not have an absolute minimum, but it does have a local minimum at [latex]x=1[/latex] because [latex]f(1)[/latex] is less than [latex]f(x)[/latex] for [latex]x[/latex] near 1. Figure 3. This function [latex]f[/latex] has two local maxima and one local minimum. The local maximum at [latex]x=2[/latex] is also the absolute maximum. Definition A function [latex]f[/latex] has a local maximum at [latex]c[/latex] if there exists an open interval [latex]I[/latex] containing [latex]c[/latex] such that [latex]I[/latex] is contained in the domain of [latex]f[/latex] and [latex]f(c)\ge f(x)[/latex] for all [latex]x\in I[/latex]. A function [latex]f[/latex] has a local minimum at [latex]c[/latex] if there exists an open interval [latex]I[/latex] containing [latex]c[/latex] such that [latex]I[/latex] is contained in the domain of [latex]f[/latex] and [latex]f(c)\le f(x)[/latex] for all [latex]x\in I[/latex]. A function [latex]f[/latex] has a local extremum at [latex]c[/latex] if [latex]f[/latex] has a local maximum at [latex]c[/latex] or [latex]f[/latex] has a local minimum at [latex]c[/latex]. Note that if [latex]f[/latex] has an absolute extremum at [latex]c[/latex] and [latex]f[/latex] is defined over an interval containing [latex]c[/latex], then [latex]f(c)[/latex] is also considered a local extremum. If an absolute extremum for a function [latex]f[/latex] occurs at an endpoint, we do not consider that to be a local extremum, but instead refer to that as an endpoint extremum. Given the graph of a function [latex]f[/latex], it is sometimes easy to see where a local maximum or local minimum occurs. However, it is not always easy to see, since the interesting features on the graph of a function may not be visible because they occur at a very small scale. Also, we may not have a graph of the function. In these cases, how can we use a formula for a function to determine where these extrema occur? To answer this question, let’s look at Figure 3 again. The local extrema occur at [latex]x=0[/latex], [latex]x=1[/latex], and [latex]x=2[/latex]. Notice that at [latex]x=0[/latex] and [latex]x=1[/latex], the derivative [latex]f^{\prime}(x)=0[/latex]. At [latex]x=2[/latex], the derivative [latex]f^{\prime}(x)[/latex] does not exist, since the function [latex]f[/latex] has a corner there. In fact, if [latex]f[/latex] has a local extremum at a point [latex]x=c[/latex], the derivative [latex]f^{\prime}(c)[/latex] must satisfy one of the following conditions: either [latex]f^{\prime}(c)=0[/latex] or [latex]f^{\prime}(c)[/latex] is undefined. Such a value [latex]c[/latex] is known as a critical point and it is important in finding extreme values for functions. Definition Let [latex]c[/latex] be an interior point in the domain of [latex]f[/latex]. We say that [latex]c[/latex] is a critical point of [latex]f[/latex] if [latex]f^{\prime}(c)=0[/latex] or [latex]f^{\prime}(c)[/latex] is undefined. As mentioned earlier, if [latex]f[/latex] has a local extremum at a point [latex]x=c[/latex], then [latex]c[/latex] must be a critical point of [latex]f[/latex]. This fact is known as Fermat’s theorem. Fermat’s Theorem If [latex]f[/latex] has a local extremum at [latex]c[/latex] and [latex]f[/latex] is differentiable at [latex]c[/latex], then [latex]f^{\prime}(c)=0[/latex]. Proof Suppose [latex]f[/latex] has a local extremum at [latex]c[/latex] and [latex]f[/latex] is differentiable at [latex]c[/latex]. We need to show that [latex]f^{\prime}(c)=0[/latex]. To do this, we will show that [latex]f^{\prime}(c)\ge 0[/latex] and [latex]f^{\prime}(c)\le 0[/latex], and therefore [latex]f^{\prime}(c)=0[/latex]. Since [latex]f[/latex] has a local extremum at [latex]c[/latex], [latex]f[/latex] has a local maximum or local minimum at [latex]c[/latex]. Suppose [latex]f[/latex] has a local maximum at [latex]c[/latex]. The case in which [latex]f[/latex] has a local minimum at [latex]c[/latex] can be handled similarly. There then exists an open interval [latex]I[/latex] such that [latex]f(c)\ge f(x)[/latex] for all [latex]x\in I[/latex]. Since [latex]f[/latex] is differentiable at [latex]c[/latex], from the definition of the derivative, we know that [latex]f^{\prime}(c)=\underset{x\to c}{\lim}\dfrac{f(x)-f(c)}{x-c}[/latex] Since this limit exists, both one-sided limits also exist and equal [latex]f^{\prime}(c)[/latex]. Therefore, [latex]f^{\prime}(c)=\underset{x\to c^+}{\lim}\dfrac{f(x)-f(c)}{x-c}[/latex], and [latex]f^{\prime}(c)=\underset{x\to c^-}{\lim}\dfrac{f(x)-f(c)}{x-c}[/latex] Since [latex]f(c)[/latex] is a local maximum, we see that [latex]f(x)-f(c)\le 0[/latex] for [latex]x[/latex] near [latex]c[/latex]. Therefore, for [latex]x[/latex] near [latex]c[/latex], but [latex]x>c[/latex], we have [latex]\frac{f(x)-f(c)}{x-c}\le 0[/latex]. From the equations above we conclude that [latex]f^{\prime}(c)\le 0[/latex]. Similarly, it can be shown that [latex]f^{\prime}(c)\ge 0[/latex]. Therefore, [latex]f^{\prime}(c)=0[/latex]. [latex]_\blacksquare[/latex] From Fermat’s theorem, we conclude that if [latex]f[/latex] has a local extremum at [latex]c[/latex], then either [latex]f^{\prime}(c)=0[/latex] or [latex]f^{\prime}(c)[/latex] is undefined. In other words, local extrema can only occur at critical points. Note this theorem does not claim that a function [latex]f[/latex] must have a local extremum at a critical point. Rather, it states that critical points are candidates for local extrema. For example, consider the function [latex]f(x)=x^3[/latex]. We have [latex]f^{\prime}(x)=3x^2=0[/latex] when [latex]x=0[/latex]. Therefore, [latex]x=0[/latex] is a critical point. However, [latex]f(x)=x^3[/latex] is increasing over latex[/latex], and thus [latex]f[/latex] does not have a local extremum at [latex]x=0[/latex]. In Figure 4, we see several different possibilities for critical points. In some of these cases, the functions have local extrema at critical points, whereas in other cases the functions do not. Note that these graphs do not show all possibilities for the behavior of a function at a critical point. Figure 4. (a–e) A function [latex]f[/latex] has a critical point at [latex]c[/latex] if [latex]f^{\prime}(c)=0[/latex] or [latex]f^{\prime}(c)[/latex] is undefined. A function may or may not have a local extremum at a critical point. Later in this module we look at analytical methods for determining whether a function actually has a local extremum at a critical point. For now, let’s turn our attention to finding critical points. We will use graphical observations to determine whether a critical point is associated with a local extremum. Example: Locating Critical Points For each of the following functions, find all critical points. Use a graphing utility to determine whether the function has a local extremum at each of the critical points. [latex]f(x)=\frac{1}{3}x^3-\frac{5}{2}x^2+4x[/latex] [latex]f(x)=(x^2-1)^3[/latex] [latex]f(x)=\dfrac{4x}{1+x^2}[/latex] Show Solution The derivative [latex]f^{\prime}(x)=x^2-5x+4[/latex] is defined for all real numbers [latex]x[/latex]. Therefore, we only need to find the values for [latex]x[/latex] where [latex]f^{\prime}(x)=0[/latex]. Since [latex]f^{\prime}(x)=x^2-5x+4=(x-4)(x-1)[/latex], the critical points are [latex]x=1[/latex] and [latex]x=4[/latex]. From the graph of [latex]f[/latex] in Figure 5, we see that [latex]f[/latex] has a local maximum at [latex]x=1[/latex] and a local minimum at [latex]x=4[/latex]. Figure 5. This function has a local maximum and a local minimum. 2. Using the chain rule, we see the derivative is [latex]f^{\prime}(x)=3(x^2-1)^2(2x)=6x(x^2-1)^2[/latex] Therefore, [latex]f[/latex] has critical points when [latex]x=0[/latex] and when [latex]x^2-1=0[/latex]. We conclude that the critical points are [latex]x=0,\pm 1[/latex]. From the graph of [latex]f[/latex] in Figure 6, we see that [latex]f[/latex] has a local (and absolute) minimum at [latex]x=0[/latex], but does not have a local extremum at [latex]x=1[/latex] or [latex]x=-1[/latex]. Figure 6. This function has three critical points: [latex]x=0[/latex], [latex]x=1[/latex], and [latex]x=-1[/latex]. The function has a local (and absolute) minimum at [latex]x=0[/latex], but does not have extrema at the other two critical points. 3. By the chain rule, we see that the derivative is [latex]f^{\prime}(x)=\dfrac{(1+x^2 \cdot 4)-4x(2x)}{(1+x^2)^2}=\dfrac{4-4x^2}{(1+x^2)^2}[/latex] The derivative is defined everywhere. Therefore, we only need to find values for [latex]x[/latex] where [latex]f^{\prime}(x)=0[/latex]. Solving [latex]f^{\prime}(x)=0[/latex], we see that [latex]4-4x^2=0[/latex], which implies [latex]x=\pm 1[/latex]. Therefore, the critical points are [latex]x=\pm 1[/latex]. From the graph of [latex]f[/latex] in Figure 7, we see that [latex]f[/latex] has an absolute maximum at [latex]x=1[/latex] and an absolute minimum at [latex]x=-1[/latex]. Hence, [latex]f[/latex] has a local maximum at [latex]x=1[/latex] and a local minimum at [latex]x=-1[/latex]. (Note that if [latex]f[/latex] has an absolute extremum over an interval [latex]I[/latex] at a point [latex]c[/latex] that is not an endpoint of [latex]I[/latex], then [latex]f[/latex] has a local extremum at [latex]c[/latex].) Figure 7. This function has an absolute maximum and an absolute minimum. Watch the following video to see the worked solution to Example: Locating Critical Points. Closed Captioning and Transcript Information for Video For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end. You can view the transcript for this segmented clip of “4.3 Maxima and Minima” here (opens in new window). Try It Find all critical points for [latex]f(x)=x^3-\frac{1}{2}x^2-2x+1[/latex]. Hint Calculate [latex]f^{\prime}(x)[/latex]. Show Solution [latex]x=-\frac{2}{3}[/latex], [latex]x=1[/latex] Try It Locating Absolute Extrema The extreme value theorem states that a continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. As shown in Figure 2, one or both of these absolute extrema could occur at an endpoint. If an absolute extremum does not occur at an endpoint, however, it must occur at an interior point, in which case the absolute extremum is a local extremum. Therefore, by Fermat’s Theorem, the point [latex]c[/latex] at which the local extremum occurs must be a critical point. We summarize this result in the following theorem. Location of Absolute Extrema Let [latex]f[/latex] be a continuous function over a closed, bounded interval [latex]I[/latex]. The absolute maximum of [latex]f[/latex] over [latex]I[/latex] and the absolute minimum of [latex]f[/latex] over [latex]I[/latex] must occur at endpoints of [latex]I[/latex] or at critical points of [latex]f[/latex] in [latex]I[/latex]. With this idea in mind, let’s examine a procedure for locating absolute extrema. Problem-Solving Strategy: Locating Absolute Extrema over a Closed Interval Consider a continuous function [latex]f[/latex] defined over the closed interval [latex][a,b][/latex]. Evaluate [latex]f[/latex] at the endpoints [latex]x=a[/latex] and [latex]x=b[/latex]. Find all critical points of [latex]f[/latex] that lie over the interval latex[/latex] and evaluate [latex]f[/latex] at those critical points. Compare all values found in (1) and (2). From the location of absolute extrema, the absolute extrema must occur at endpoints or critical points. Therefore, the largest of these values is the absolute maximum of [latex]f[/latex]. The smallest of these values is the absolute minimum of [latex]f[/latex]. Now let’s look at how to use this strategy to find the absolute maximum and absolute minimum values for continuous functions. Example: Locating Absolute Extrema For each of the following functions, find the absolute maximum and absolute minimum over the specified interval and state where those values occur. [latex]f(x)=−x^2+3x-2[/latex] over [latex][1,3][/latex]. [latex]f(x)=x^2-3x^{\frac{2}{3}}[/latex] over [latex][0,2][/latex]. Show Solution Step 1. Evaluate [latex]f[/latex] at the endpoints [latex]x=1[/latex] and [latex]x=3[/latex]. [latex]f(1)=0[/latex] and [latex]f(3)=-2[/latex] Step 2. Since [latex]f^{\prime}(x)=-2x+3[/latex], [latex]f^{\prime}[/latex] is defined for all real numbers [latex]x[/latex]. Therefore, there are no critical points where the derivative is undefined. It remains to check where [latex]f^{\prime}(x)=0[/latex]. Since [latex]f^{\prime}(x)=-2x+3=0[/latex] at [latex]x=\frac{3}{2}[/latex] and [latex]\frac{3}{2}[/latex] is in the interval [latex][1,3][/latex], [latex]f(\frac{3}{2})[/latex] is a candidate for an absolute extremum of [latex]f[/latex] over [latex][1,3][/latex]. We evaluate [latex]f(\frac{3}{2})[/latex] and find [latex]f\left(\frac{3}{2}\right)=\frac{1}{4}[/latex] Step 3. We set up the following table to compare the values found in steps 1 and 2. \| [latex]x[/latex] \| [latex]f(x)[/latex] \| Conclusion \| --- \| 0 \| 0 \| \| [latex]\frac{3}{2}[/latex] \| [latex]\frac{1}{4}[/latex] \| Absolute maximum \| \| 3 \| -2 \| Absolute minimum \| From the table, we find that the absolute maximum of [latex]f[/latex] over the interval [latex][1,3][/latex] is [latex]\frac{1}{4}[/latex], and it occurs at [latex]x=\frac{3}{2}[/latex]. The absolute minimum of [latex]f[/latex] over the interval [latex][1,3][/latex] is -2, and it occurs at [latex]x=3[/latex] as shown in the following graph. Figure 8. This function has both an absolute maximum and an absolute minimum. 2. Step 1. Evaluate [latex]f[/latex] at the endpoints [latex]x=0[/latex] and [latex]x=2[/latex]. [latex]f(0)=0[/latex] and [latex]f(2)=4-3\sqrt{4}\approx -0.762[/latex] Step 2. The derivative of [latex]f[/latex] is given by [latex]f^{\prime}(x)=2x-\dfrac{2}{x^{\frac{1}{3}}}=\dfrac{2x^{\frac{4}{3}}-2}{x^{\frac{1}{3}}}[/latex] for [latex]x\ne 0[/latex]. The derivative is zero when [latex]2x^{\frac{4}{3}}-2=0[/latex], which implies [latex]x=\pm 1[/latex]. The derivative is undefined at [latex]x=0[/latex]. Therefore, the critical points of [latex]f[/latex] are [latex]x=0,1,-1[/latex]. The point [latex]x=0[/latex] is an endpoint, so we already evaluated [latex]f(0)[/latex] in step 1. The point [latex]x=-1[/latex] is not in the interval of interest, so we need only evaluate [latex]f(1)[/latex]. We find that [latex]f(1)=-2[/latex] Step 3. We compare the values found in steps 1 and 2, in the following table. \| [latex]x[/latex] \| [latex]f(x)[/latex] \| Conclusion \| --- \| 0 \| 0 \| Absolute maximum \| \| 1 \| -2 \| Absolute minimum \| \| 2 \| -0.762 \| We conclude that the absolute maximum of [latex]f[/latex] over the interval [latex][0,2][/latex] is zero, and it occurs at [latex]x=0[/latex]. The absolute minimum is −2, and it occurs at [latex]x=1[/latex] as shown in the following graph. Figure 9. This function has an absolute maximum at an endpoint of the interval. Watch the following video to see the worked solution to Example: Locating Absolute Extrema part (a). Closed Captioning and Transcript Information for Video For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end. You can view the transcript for this segmented clip of “4.3 Maxima and Minima” here (opens in new window). Try It Find the absolute maximum and absolute minimum of [latex]f(x)=x^2-4x+3[/latex] over the interval [latex][1,4][/latex]. Hint Look for critical points. Evaluate [latex]f[/latex] at all critical points and at the endpoints. Show Solution The absolute maximum is 3 and it occurs at [latex]x=4[/latex]. The absolute minimum is -1 and it occurs at [latex]x=2[/latex]. At this point, we know how to locate absolute extrema for continuous functions over closed intervals. We have also defined local extrema and determined that if a function [latex]f[/latex] has a local extremum at a point [latex]c[/latex], then [latex]c[/latex] must be a critical point of [latex]f[/latex]. However, [latex]c[/latex] being a critical point is not a sufficient condition for [latex]f[/latex] to have a local extremum at [latex]c[/latex]. Later in this module, we show how to determine whether a function actually has a local extremum at a critical point. First, however, we need to introduce the Mean Value Theorem, which will help as we analyze the behavior of the graph of a function. Try It Candela Citations CC licensed content, Original 4.3 Maxima and Minima. Authored by: Ryan Melton. License: CC BY: Attribution CC licensed content, Shared previously Calculus Volume 1. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at Licenses and Attributions CC licensed content, Original 4.3 Maxima and Minima. Authored by: Ryan Melton. License: CC BY: Attribution CC licensed content, Shared previously Calculus Volume 1. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. 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4621	https://math.stackexchange.com/questions/2426892/between-which-two-integers-does-sqrt2017-fall	elementary number theory - Between which two integers does $\sqrt{2017}$ fall? - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Between which two integers does 2017−−−−√2017 fall? Ask Question Asked 8 years ago Modified5 years, 9 months ago Viewed 924 times This question shows research effort; it is useful and clear 4 Save this question. Show activity on this post. Between which two integers does 2017−−−−√2017 fall? Since 2017 2017 is a prime, there's not much I can do with it. However, 2016 2016 (the number before it) and 2018 2018 (the one after) are not, so I tried to factorise them. But that didn't work so well either, because they too are not perfect squares, so if I multiply them by a number to make them perfect squares, they're no longer close to 2017.2017. How can I solve this problem? Update: Okay, since 40 2=1600 40 2=1600 and 50 2=2500 50 2=2500, I just tried 45 45 and 44 44 and they happened to be the answer - but I want to be more mathematical than that... elementary-number-theory contest-math radicals prime-factorization square-numbers Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications edited Jan 1, 2020 at 1:50 Simon Fraser 2,563 13 13 silver badges 27 27 bronze badges asked Sep 12, 2017 at 17:52 spacespace 4,879 3 3 gold badges 31 31 silver badges 62 62 bronze badges 5 Hint: 2025=45 2 2025=45 2 Blex –Blex 2017-09-12 17:55:10 +00:00 Commented Sep 12, 2017 at 17:55 It's enough to show that 44≤2017−−−−√≤45 44≤2017≤45 and that there are no other integer numbers between 44 and 45 (use the fact that 45 is successor of 44).Blex –Blex 2017-09-12 17:57:51 +00:00 Commented Sep 12, 2017 at 17:57 Why don't you ask a calculator, or your computer?Gribouillis –Gribouillis 2017-09-12 18:00:59 +00:00 Commented Sep 12, 2017 at 18:00 2017=1936+81=44 2+9 2 2017=1936+81=44 2+9 2 confirming Fermat theorem about 4 k+1 4 k+1 primes Raffaele –Raffaele 2017-09-12 18:02:12 +00:00 Commented Sep 12, 2017 at 18:02 1 Mathematically, you could say that 2017−−−−√2017 definitely falls between the integers 0 0 and 1000000 1000000 :). But seriously, there is absolutely nothing non-mathematical about what you did in your update. Pat yourself on the back and stop worrying that guessing and checking is somehow not a part of mathematics.Erick Wong –Erick Wong 2017-09-18 22:32:57 +00:00 Commented Sep 18, 2017 at 22:32 Add a comment\| 13 Answers 13 Sorted by: Reset to default This answer is useful 13 Save this answer. Show activity on this post. 2017−−−−√≈2000−−−−√=20 5–√≈20⋅2.236≈45 2017≈2000=20 5≈20⋅2.236≈45 and 44 2=1936,45 2=2025 44 2=1936,45 2=2025 hence 2017−−−−√∈(44,45)2017∈(44,45). Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Sep 12, 2017 at 17:56 Jack D'AurizioJack D'Aurizio 372k 42 42 gold badges 419 419 silver badges 886 886 bronze badges Add a comment\| This answer is useful 8 Save this answer. Show activity on this post. 44 2=1936<2017<2025=45 2 44 2=1936<2017<2025=45 2. Really, I don't think there's much to this one except for "try squaring small integers until you find the right ones". Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Sep 12, 2017 at 17:55 Duncan RamageDuncan Ramage 7,123 1 1 gold badge 21 21 silver badges 39 39 bronze badges Add a comment\| This answer is useful 6 Save this answer. Show activity on this post. You could use the root extraction algorithm to find it directly. It's sort of like long division. Starting from the decimal place, divide the number into pairs of digits. So 20 17 20 17. Find the largest integer whose square is less than the first pair. 4 2<20<5 2 4 2<20<5 2. This is the first digit of the square root. Subtract off the square and bring down the next two digits. 20−4 2=4⟹417 20−4 2=4⟹417. This is our remainder. Find the largest d d such that the product (20⋅r+d)⋅d(20⋅r+d)⋅d is less than the current remainder, where r r is the part of the root already found. (20⋅4+4)⋅4<417<(20⋅4+5)⋅5(20⋅4+4)⋅4<417<(20⋅4+5)⋅5, so d=4 d=4. Add d d to the end of the digits already found, subtract the product from the current remainder, and bring down the next two digits. So we have 44 for our root and 417−84⋅4=81⟹8100 417−84⋅4=81⟹8100 for our new remainder. Repeat 4 and 5 until you have enough digits or the remainder and all remaining digits of the number are zero. Since we now have enough digits for the integer part, we can stop here. So 44<2017−−−−√<45 44<2017<45. I do want to comment on one suggestion you had, though. However, 2016 (the number before it) and 2018 (the one after) are not, so I tried to factorise them. But that didn't work so well either, because they too are not perfect squares 2016=2 5⋅3 2⋅7 2016=2 5⋅3 2⋅7, which is divisible by a lot of perfect squares. Trial division by some of those perfect squares may result in a quotient that is itself close to a perfect square, which would give a guess as to what 2016−−−−√2016 is. 2016=2 2⋅504=3 3⋅224=4 2⋅126=6 2⋅56=12 2⋅14 2016=2 2⋅504=3 3⋅224=4 2⋅126=6 2⋅56=12 2⋅14. We should then recognize 224≈225=15 2 224≈225=15 2 and 126≈121=11 2 126≈121=11 2. This gives 44 2<2016<45 2 44 2<2016<45 2, so clearly 44 2<2017<45 2 44 2<2017<45 2 as well. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Sep 12, 2017 at 18:41 answered Sep 12, 2017 at 18:12 eyeballfrogeyeballfrog 23.3k 20 20 silver badges 51 51 bronze badges Add a comment\| This answer is useful 3 Save this answer. Show activity on this post. Try graphing x−−√x for x≥0 x≥0. You should see a fairly smooth curve that goes upwards, which means that if a<b a<b and they're both positive, then a−−√<b√a<b. From this, it's clear that the integers you want are ⌊2017−−−−√⌋⌊2017⌋ and ⌈2017−−−−√⌉⌈2017⌉. A calculator readily tells us that 2017−−−−√2017 is approximately 44.911, so the answer is 44 and 45. If you really want to do it by prime factorization, look at the divisors of 2016. Notice that 2016=42×48 2016=42×48. Then 43×47=2021 43×47=2021 and 44×46=2024 44×46=2024, which should strongly suggest 45 is the greater of the integers you're looking for. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Sep 12, 2017 at 18:14 Robert SoupeRobert Soupe 15k 2 2 gold badges 36 36 silver badges 65 65 bronze badges Add a comment\| This answer is useful 3 Save this answer. Show activity on this post. You can also use the least significant digits to get your bearings. Since 2016≡16(mod 100)2016≡16(mod 100), if 2016 2016 is a perfect square, then n n in n 2=2016 n 2=2016 is an integer satisfying n≡4,6(mod 10)n≡4,6(mod 10). Clearly n=4 n=4 or 6 6 is too small. Then, working our way up, we get (196,256),(576,676),(1156,1296),(1936,2116)(196,256),(576,676),(1156,1296),(1936,2116), the last two corresponding to 44 44 and 46 46. Of course 45 2≠2017 45 2≠2017, but maybe it's 2025 2025. Notice then that 2116−2025=91 2116−2025=91 and 91 91 is the 45 45 th odd number. Likewise, 2025−1936=89=2×44+1 2025−1936=89=2×44+1, so it checks out. So the answer is 44<2017−−−−√<45 44<2017<45. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Sep 12, 2017 at 21:21 Mr. BrooksMr. Brooks 1,156 2 2 gold badges 17 17 silver badges 44 44 bronze badges Add a comment\| This answer is useful 3 Save this answer. Show activity on this post. You can use the Newton square root method wikipedia for integers: x n+1=1 2(x n+S x n)x n+1=1 2(x n+S x n) Let us start with a crappy guess x 1=500 x 1=500: x 2=1 2(500+2017/500)=252 x 2=1 2(500+2017/500)=252 x 3=1 2(252+2017/252)=130 x 3=1 2(252+2017/252)=130 x 4=1 2(130+2017/130)=72.7 x 4=1 2(130+2017/130)=72.7 x 5=1 2(73+2017/73)=50 x 5=1 2(73+2017/73)=50 x 6=1 2(50+2017/50)=45 x 6=1 2(50+2017/50)=45 Now as the iterations seem to converge we can try 45 2=2025 45 2=2025 and we have our answer. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Sep 18, 2017 at 22:24 orangeskid 57.3k 3 3 gold badges 51 51 silver badges 121 121 bronze badges answered Sep 12, 2017 at 19:05 mathreadlermathreadler 26.6k 9 9 gold badges 40 40 silver badges 107 107 bronze badges Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. As to the next question, Pell -1 and Pell +1..Hmm, seems to be showing the exponents incorrectly. 106515299132603184503844444 2−2017⋅2371696115380807559791481 2=−1 106515299132603184503844444 2−2017⋅2371696115380807559791481 2=−1 22691017898615873418283839489716246568157231499338273 2−2017⋅505243842362839347335084683756885179819279000763128 2=1 22691017898615873418283839489716246568157231499338273 2−2017⋅505243842362839347335084683756885179819279000763128 2=1 So, 2017−−−−√2017 is above 106515299132603184503844444 2371696115380807559791481 106515299132603184503844444 2371696115380807559791481 and below 22691017898615873418283839489716246568157231499338273 505243842362839347335084683756885179819279000763128 22691017898615873418283839489716246568157231499338273 505243842362839347335084683756885179819279000763128 parisize = 4000000, primelimit = 500509 ? a = 106515299132603184503844444 %1 = 106515299132603184503844444 ? b = 2371696115380807559791481 %2 = 2371696115380807559791481 ? a^2 - 2017 b^2 %3 = -1 ? ? c = 22691017898615873418283839489716246568157231499338273 %4 = 22691017898615873418283839489716246568157231499338273 ? d = 505243842362839347335084683756885179819279000763128 %5 = 505243842362839347335084683756885179819279000763128 ? c^2 - 2017 d^2 %6 = 1 ? ? ? a^2 + 2017 b^2 - c %7 = 0 ? 2 a b - d %8 = 0 ? ? Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Sep 12, 2017 at 18:19 answered Sep 12, 2017 at 18:01 Will JagyWill Jagy 147k 8 8 gold badges 156 156 silver badges 285 285 bronze badges Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. I can only imagine this was intended to be about (10 a+5)2=100 a(a+1)+25,(10 a+5)2=100 a(a+1)+25, 15 2=225,15 2=225, 25 2=625,25 2=625, 35 2=1225,35 2=1225, 45 2=2025.45 2=2025. Then 44 2=2025−2⋅45+1=2025−90+1<2017.44 2=2025−2⋅45+1=2025−90+1<2017. EXAMPLE: factor 10001=10 4+1 10001=10 4+1 105 2=11025 105 2=11025 105 2−10001=1024 105 2−10001=1024 1024=32 2 1024=32 2 10001=105 2−32 2 10001=105 2−32 2 10001=(105−32)(105+32)=73⋅137 10001=(105−32)(105+32)=73⋅137 Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Sep 12, 2017 at 18:59 answered Sep 12, 2017 at 18:34 Will JagyWill Jagy 147k 8 8 gold badges 156 156 silver badges 285 285 bronze badges Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. There are well enough good answers here, but no one's suggested this method yet, so I'll add it. Method 1 1 - (Secant Approximation) We can take two simple perfect squares that straddle 2017 2017. We're just getting a rough estimate, so we'd prefer numbers to be easy to work with than near 2017 2017. For example, 1600≤2017≤3600 1600≤2017≤3600. The secant line passing (1600,40)(1600,40) and (3600,60)(3600,60) should approximate x−−√x in the interval [1600,3600][1600,3600]. This gives us: x−−√=24+x 100+ε 1(x)x=24+x 100+ε 1(x). So, 2017−−−−√≈44 2017≈44. Finally, we can check the squares of 43,44,45,…43,44,45,… and find our answer. Method 2 2 - (Mean of 2 2 nd Degree Taylor Polynomials) We can make method 1 1 slightly more accurate, though it's not really vital to do so. The 2 2 nd degree Taylor expansion of x−−√x at x=a x=a is T a(x)=a−−√+x−a 80+(x−a)2 8 a a√T a(x)=a+x−a 80+(x−a)2 8 a a. Then the mean of T 1600 T 1600 and T 3600(x)T 3600(x) should be a good estimate for x−−√x near the centre of the interval [1600,3600][1600,3600]. Then x−−√=60+x−2600 80−1 2⋅8((x−1600)2 40 3+(x−3600)2 60 3)+ε 2(x)x=60+x−2600 80−1 2⋅8((x−1600)2 40 3+(x−3600)2 60 3)+ε 2(x). Hence, 2017−−−−√≈45 2017≈45 and we can check the squares of nearby integers, as in method 1 1. These methods would also work for any 2000<x<3000 2000<x<3000 and could easily be adapted for other values of x x. They also give a convenient way of finding an initial guess, for methods such as Newton-Raphson (detailed in @mathreadler's answer). Accuracy of Methods The error term reaches its maximum at ε 1(2500)=1 ε 1(2500)=1. So, in the worst case scenario, we'd need to check the 3 3 numbers (y−1),y,(y+1)(y−1),y,(y+1), where y y is the estimate of x−−√x from method 1 1 and where x∈[1600,3600]x∈[1600,3600]. For x∈(1769,3110)x∈(1769,3110), we have ε 2(x)<0.67<ε 1(x)ε 2(x)<0.67<ε 1(x) but for x<1769 x<1769 or x>3110 x>3110, we have ε 2(x)>ε 1(x)ε 2(x)>ε 1(x). In other words, method 1 1 is more accurate than method 2 2 in the centre of [1600,3600][1600,3600], but the opposite is true near the bounds of the interval. However, since we're interested in the centre of the interval, this is good. The error term, ε 2(x)ε 2(x), reaches a minimum of 0 0 around x=2351 x=2351. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Jun 12, 2020 at 10:38 CommunityBot 1 answered Sep 14, 2017 at 1:33 JamJam 10.7k 4 4 gold badges 30 30 silver badges 45 45 bronze badges Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. it is 44<2017−−−−√<45 44<2017<45 since 44 2=1936 44 2=1936 and 45 2=2025 45 2=2025 Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Sep 12, 2017 at 17:55 Dr. Sonnhard GraubnerDr. Sonnhard Graubner 97.5k 4 4 gold badges 42 42 silver badges 80 80 bronze badges Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. Knowing the answer lies between 40 40 and 50 50 the rational thing to do would be to try the number in the middle of 40 40 and 50 50 i.e 45 45. If you don't feel comfortable with multiplying 45⋅45 45⋅45 you can use the formula for (40+5)2=40 2+2⋅40⋅5+5 2=1600+400+25=2025(40+5)2=40 2+2⋅40⋅5+5 2=1600+400+25=2025 Now you can use that 45 2−44 2=(45−44)(45+44)=89 45 2−44 2=(45−44)(45+44)=89 so one can see that 45 2>45 2−7=2017>45 2−89 45 2>45 2−7=2017>45 2−89. It helps speed up a little of calculation for contests and such. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Sep 12, 2017 at 18:06 kingW3kingW3 13.7k 11 11 gold badges 38 38 silver badges 61 61 bronze badges Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. For a rough estimate, I'd first divide by 100 100 and think about 20.17 20.17, for 2017−−−−√=10 20.17−−−−√2017=10 20.17. In fact, I would just consider 20−−√20: You know 4 2=16 4 2=16 and 5 2=25 5 2=25, so 20−−√20 is between 4 4 and 5 5, and is in fact close to the middle of them, i.e close to 4.5 4.5. Hence, 2017−−−−√2017 will be close to 45 45. So, see what 45 2 45 2 is ... and proceed from there. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Sep 12, 2017 at 18:48 Bram28Bram28 104k 6 6 gold badges 76 76 silver badges 123 123 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. 1600−−−−√=40 1600=40 and 2500−−−−√=50 2500=50. (40+4)2=40 2+8⋅40+16=1936(40+4)2=40 2+8⋅40+16=1936 and (40+5)2=40 2+10⋅40+25=2025(40+5)2=40 2+10⋅40+25=2025. Hence the desired answer is 44 44 and 45 45. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Jan 1, 2020 at 1:57 Simon FraserSimon Fraser 2,563 13 13 silver badges 27 27 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. 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4622	https://oeis.org/A002450	The OEIS is supported by the many generous donors to the OEIS Foundation. a(n) = (4^n - 1)/3. (Formerly M3914 N1608) 0, 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525, 1398101, 5592405, 22369621, 89478485, 357913941, 1431655765, 5726623061, 22906492245, 91625968981, 366503875925, 1466015503701, 5864062014805, 23456248059221, 93824992236885, 375299968947541 (list; graph; refs; listen; history; text; internal format) For n > 0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3) = 21, since the numbral square of 5 is 5()5 = 101()101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman, Dec 18 2001 a(n) is composite for all n > 2 and has factors x, (3x + 2(-1)^n) where x belongs to A001045. In binary the terms greater than 0 are 1, 101, 10101, 1010101, etc. - John McNamara, Jan 16 2002 Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - R. H. Hardin, Mar 16 2002 The Collatz-function iteration started at a(n), for n >= 1, will end at 1 after 2n+1 steps. - Labos Elemer, Sep 30 2002 [corrected by Wolfdieter Lang, Aug 16 2021] Second binomial transform of A001045. - Paul Barry, Mar 28 2003 All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast, Apr 10 2003 Also sum of squares of divisors of 2^(n-1): a(n) = A001157(A000079(n-1)), for n > 0. - Paul Barry, Apr 11 2003 Binomial transform of A000244 (with leading zero). - Paul Barry, Apr 11 2003 Number of walks of length 2n between two vertices at distance 2 in the cycle graph C_6. For n = 2 we have for example 5 walks of length 4 from vertex A to C: ABABC, ABCBC, ABCDC, AFABC and AFEDC. - Herbert Kociemba, May 31 2004 Also number of walks of length 2n + 1 between two vertices at distance 3 in the cycle graph C_12. - Herbert Kociemba, Jul 05 2004 a(n+1) is the number of steps that are made when generating all n-step random walks that begin in a given point P on a two-dimensional square lattice. To make one step means to mark one vertex on the lattice (compare A080674). - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Mar 13 2005 a(n+1) is the sum of square divisors of 4^n. - Paul Barry, Oct 13 2005 a(n+1) is the decimal number generated by the binary bits in the n-th generation of the Rule 250 elementary cellular automaton. - Eric W. Weisstein, Apr 08 2006 a(k) = [M^k]_2,1, where M is the 3 X 3 matrix defined as follows: M = [1, 1, 1; 1, 3, 1; 1, 1, 1]. - Simone Severini, Jun 11 2006 a(n-1) / a(n) = percentage of wasted storage if a single image is stored as a pyramid with a each subsequent higher resolution layer containing four times as many pixels as the previous layer. n is the number of layers. - Victor Brodsky (victorbrodsky(AT)gmail.com), Jun 15 2006 k is in the sequence if and only if C(4k + 1, k) (A052203) is odd. - Paul Barry, Mar 26 2007 This sequence also gives the number of distinct 3-colorings of the odd cycle C(2n - 1). - Keith Briggs, Jun 19 2007 All numbers of the form m4^m + (4^m-1)/3 have the property that they are sums of two squares and also their indices are the sum of two squares. This follows from the identity m4^m + (4^m-1)/3 = 4(4(..4(4m + 1) + 1) + 1) + 1 ..) + 1. - Artur Jasinski, Nov 12 2007 For n > 0, terms are the numbers that, in base 4, are repunits: 1_4, 11_4, 111_4, 1111_4, etc. - Artur Jasinski, Sep 30 2008 Let A be the Hessenberg matrix of order n, defined by: A[1, j] = 1, A[i, i] := 5, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 1, a(n) = charpoly(A,1). - Milan Janjic, Jan 27 2010 This is the sequence A(0, 1; 3, 4; 2) = A(0, 1; 4, 0; 1) of the family of sequences [a, b : c, d : k] considered by G. Detlefs, and treated as A(a, b; c, d; k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010 6a(n) + 1 is every second Mersenne number greater than or equal to M3, hence all Mersenne primes greater than M2 must be a 6a(n) + 1 of this sequence. - Roderick MacPhee, Nov 01 2010 Smallest number having alternating bit sum n. Cf. A065359. For n = 1, 2, ..., the last digit of a(n) is 1, 5, 1, 5, ... . - Washington Bomfim, Jan 21 2011 Rule 50 elementary cellular automaton generates this sequence. This sequence also appears in the second column of array in A173588. - Paul Muljadi, Jan 27 2011 Sequence found by reading the line from 0, in the direction 0, 5, ... and the line from 1, in the direction 1, 21, ..., in the square spiral whose edges are the Jacobsthal numbers A001045 and whose vertices are the numbers A000975. These parallel lines are two semi-diagonals in the spiral. - Omar E. Pol, Sep 10 2011 a(n), n >= 1, is also the inverse of 3, denoted by 3^(-1), Modd(2^(2n - 1)). For Modd n see a comment on A203571. E.g., a(2) = 5, 3 5 = 15 == 1 (Modd 8), because floor(15/8) = 1 is odd and -15 == 1 (mod 8). For n = 1 note that 3 1 = 3 == 1 (Modd 2) because floor(3/2) = 1 and -3 == 1 (mod 2). The inverse of 3 taken Modd 2^(2n) coincides with 3^(-1) (mod 2^(2n)) given in A007583(n), n >= 1. - Wolfdieter Lang, Mar 12 2012 If an AVL tree has a leaf at depth n, then the tree can contain no more than a(n+1) nodes total. - Mike Rosulek, Nov 20 2012 Also, this is the Lucas sequence V(5, 4). - Bruno Berselli, Jan 10 2013 Also, for n > 0, a(n) is an odd number whose Collatz trajectory contains no odd number other than n and 1. - Jayanta Basu, Mar 24 2013 Sum_{n >= 1} 1/a(n) converges to (3(log(4/3) - QPolyGamma[0, 1, 1/4]))/log(4) = 1.263293058100271... = A321873. - K. G. Stier, Jun 23 2014 Consider n spheres in R^n: the i-th one (i=1, ..., n) has radius r(i) = 2^(1-i) and the coordinates of its center are (0, 0, ..., 0, r(i), 0, ..., 0) where r(i) is in position i. The coordinates of the intersection point in the positive orthant of these spheres are (2/a(n), 4/a(n), 8/a(n), 16/a(n), ...). For example in R^2, circles centered at (1, 0) and (0, 1/2), and with radii 1 and 1/2, meet at (2/5, 4/5). - Jean M. Morales, May 19 2015 From Peter Bala, Oct 11 2015: (Start) a(n) gives the values of m such that binomial(4m + 1,m) is odd. Cf. A003714, A048716, A263132. 2a(n) = A020988(n) gives the values of m such that binomial(4m + 2, m) is odd. 4a(n) = A080674(n) gives the values of m such that binomial(4m + 4, m) is odd. (End) Collatz Conjecture Corollary: Except for powers of 2, the Collatz iteration of any positive integer must eventually reach a(n) and hence terminate at 1. - Gregory L. Simay, May 09 2016 Number of active (ON, black) cells at stage 2^n - 1 of the two-dimensional cellular automaton defined by "Rule 598", based on the 5-celled von Neumann neighborhood. - Robert Price, May 16 2016 From Luca Mariot and Enrico Formenti, Sep 26 2016: (Start) a(n) is also the number of coprime pairs of polynomials (f, g) over GF(2) where both f and g have degree n + 1 and nonzero constant term. a(n) is also the number of pairs of one-dimensional binary cellular automata with linear and bipermutive local rule of neighborhood size n+1 giving rise to orthogonal Latin squares of order 2^m, where m is a multiple of n. (End) Except for 0, 1 and 5, all terms are Brazilian repunits numbers in base 4, and so belong to A125134. For n >= 3, all these terms are composite because a(n) = {(2^n-1) (2^n + 1)}/3 and either (2^n - 1) or (2^n + 1) is a multiple of 3. - Bernard Schott, Apr 29 2017 Given the 3 X 3 matrix A = [2, 1, 1; 1, 2, 1; 1, 1, 2] and the 3 X 3 unit matrix I_3, A^n = a(n)(A - I_3) + I_3. - Nicolas Patrois, Jul 05 2017 The binary expansion of a(n) (n >= 1) consists of n 1's alternating with n - 1 0's. Example: a(4) = 85 = 1010101_2. - Emeric Deutsch, Aug 30 2017 a(n) (n >= 1) is the viabin number of the integer partition [n, n - 1, n - 2, ..., 2, 1] (for the definition of viabin number see comment in A290253). Example: a(4) = 85 = 1010101_2; consequently, the southeast border of the Ferrers board of the corresponding integer partition is ENENENEN, where E = (1, 0), N = (0, 1); this leads to the integer partition [4, 3, 2, 1]. - Emeric Deutsch, Aug 30 2017 Numbers whose binary and Gray-code representations are both palindromes (i.e., intersection of A006995 and A281379). - Amiram Eldar, May 17 2021 Starting with n = 1 the sequence satisfies {a(n) mod 6} = repeat{1, 5, 3}. - Wolfdieter Lang, Jan 14 2022 Terms >= 5 are those q for which the multiplicative order of 2 mod q is floor(log_2(q)) + 2 (and which is 1 more than the smallest possible order for any q). - Tim Seuré, Mar 09 2024 The order of 2 modulo a(n) is 2n for n >= 2. - Joerg Arndt, Mar 09 2024 REFERENCES A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112. J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..200 Peter Bala, A characterization of A002450, A020988 and A080674. Henry Bottomley, Illustration of initial terms Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75. - From N. J. A. Sloane, Dec 24 2012 Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 8. Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498. D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy) David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018. C. Ernst and D. W. Sumners, The Growth of the Number of Prime Knots, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987 Ernesto Estrada and José A. de la Peña, Integer sequences from walks in graphs, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84. Rigoberto Flórez, Robinson A. Higuita, and Antara Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014). Enrico Formenti and Luca Mariot, Exhaustive Generation of Linear Orthogonal CA, Automata 2023, Univ. Twente (Netherlands), see p. 22 of 38. Mattia Fregola, Elementary Cellular Automata Rule 1 generating OEIS sequence A277799, A058896, A141725, A002450 A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1. Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 373 Petro Kosobutskyy and Nataliia Nestor, On the mathematical model of the transformation of natural numbers by a function of a split type, Comp. Des. Sys. Theor. Practice (2024) Vol. 6, No. 2. See p. 7. Petro Kosobutskyy, Anastasiia Yedyharova, and Taras Slobodzyan, From Newton's binomial and Pascal's triangle to Collatz's problem, Comp. Des. Sys., Theor. Practice (2023) Vol. 5, No. 1, 121-127. Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. J. V. Leyendekkers and A.G. Shannon, Modular Rings and the Integer 3, Notes on Number Theory & Discrete Mathematics, 17 (2011), 47-51. Luca Manzoni, Luca Mariot, and Giuliamaria Menara, Combinatorial Designs and Cellular Automata: A Survey, arXiv:2503.10320 [math.CO], 2025. See pp. 16, 27. Luca Mariot, Cryptography by Cellular Automata, 2017. Luca Mariot, Orthogonal labelings in de Bruijn graphs, IWOCA 2020 - Open Problems Session, Delft University of Technology (Netherlands). Luca Mariot, Connections between Latin squares, Cellular Automata and Coprime Polynomials, Univ. Twente (Netherlands, 2023). See p. 16/37. Luca Mariot and Enrico Formenti, The number of coprime/non-coprime pairs of polynomials over F_2 with degree n and nonzero constant term. Luca Mariot, Enrico Formenti and Alberto Leporati, Constructing Orthogonal Latin Squares from Linear Cellular Automata. In: Exploratory papers of AUTOMATA 2016. Luca Mariot, Maximilien Gadouleau, Enrico Formenti, and Alberto Leporati, Mutually Orthogonal Latin Squares based on Cellular Automata, arXiv:1906.08249 [cs.DM], 2019. Luca Mariot, Counting Coprime Polynomials over Finite Fields with Formal Languages and Compositions of Natural Numbers, Univ. Twente (Netherlands 2023). See p. 11. Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3. Mező István, Several Generating Functions for Second-Order Recurrence Sequences , JIS 12 (2009) 09.3.7. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992. Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, The sequence of higher order Mersenne numbers and associated binomial transforms, arXiv:2307.08073 [math.NT], 2023. D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4. A. G. Shannon, Some recurrence relations for binary sequence matrices, NNTDM 17 (2011), 4, 913. - From N. J. A. Sloane, Jun 13 2012 T. N. Thiele, Interpolationsrechnung, Teubner, Leipzig, 1909, p. 35. Eric Weisstein's World of Mathematics, Repunit, Rule 250, Prime Knot. Wikipedia, Elementary cellular automaton Wikipedia, Lucas sequence: Specific names. Michael Williams, Collatz conjecture: an order isomorphic recursive machine, ResearchGate (2024). See pp. 8, 13. Index entries for linear recurrences with constant coefficients, signature (5,-4). FORMULA From Wolfdieter Lang, Apr 24 2001: (Start) a(n+1) = Sum_{m = 0..n} A060921(n, m). G.f.: x/((1-x)(1-4x)). (End) a(n) = Sum_{k = 0..n-1} 4^k; a(n) = A001045(2n). - Paul Barry, Mar 17 2003 E.g.f.: (exp(4x) - exp(x))/3. - Paul Barry, Mar 28 2003 a(n) = (A007583(n) - 1)/2. - N. J. A. Sloane, May 16 2003 a(n) = A000975(2n)/2. - N. J. A. Sloane, Sep 13 2003 a(n) = A084160(n)/2. - N. J. A. Sloane, Sep 13 2003 a(n+1) = 4a(n) + 1, with a(0) = 0. - Philippe Deléham, Feb 25 2004 a(n) = Sum_{i = 0..n-1} C(2n - 1 - i, i)2^i. - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004 a(n+1) = Sum_{k = 0..n} binomial(n+1, k+1)3^k. - Paul Barry, Aug 20 2004 a(n) = center term in M^n [1 0 0], where M is the 3 X 3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n [1 0 0] = [A007583(n-1) a(n) A007583(n-1)]. E.g., a(4) = 85 since M^4 [1 0 0] = [43 85 43] = [A007583(3) a(4) A007583(3)]. - Gary W. Adamson, Dec 18 2004 a(n) = Sum_{k = 0..n, j = 0..n} C(n, j)C(j, k)A001045(j - k). - Paul Barry, Feb 15 2005 a(n) = Sum_{k = 0..n} C(n, k)A001045(n-k)2^k = Sum_{k = 0..n} C(n, k)A001045(k)2^(n-k). - Paul Barry, Apr 22 2005 a(n) = A125118(n, 3) for n > 2. - Reinhard Zumkeller, Nov 21 2006 a(n) = Sum_{k = 0..n} 2^(n - k)A128908(n, k), n >= 1. - Philippe Deléham, Oct 19 2008 a(n) = Sum_{k = 0..n} A106566(n, k)A100335(k). - Philippe Deléham, Oct 30 2008 If we define f(m, j, x) = Sum_{k = j..m} binomial(m, k)stirling2(k, j)x^(m - k) then a(n-1) = f(2n, 4, -2), n >= 2. - Milan Janjic, Apr 26 2009 a(n) = A014551(n) A001045(n). - R. J. Mathar, Jul 08 2009 a(n) = 4a(n-1) + a(n-2) - 4a(n-3) = 5a(n-1) - 4a(n-2), a(0) = 0, a(1) = 1, a(2) = 5. - Wolfdieter Lang, Oct 18 2010 a(0) = 0, a(n+1) = a(n) + 2^(2n). - Washington Bomfim, Jan 21 2011 A036555(a(n)) = 2n. - Reinhard Zumkeller, Jan 28 2011 a(n) = Sum_{k = 1..floor((n+2)/3)} C(2n + 1, n + 2 - 3k). - Mircea Merca, Jun 25 2011 a(n) = Sum_{i = 1..n} binomial(2n + 1, 2i)/3. - Wesley Ivan Hurt, Mar 14 2015 a(n+1) = 2^(2n) + a(n), a(0) = 0. - Ben Paul Thurston, Dec 27 2015 a(kn)/a(n) = 1 + 4^n + ... + 4^((k-1)n). - Gregory L. Simay, Jun 09 2016 Dirichlet g.f.: (PolyLog(s, 4) - zeta(s))/3. - Ilya Gutkovskiy, Jun 26 2016 A000120(a(n)) = n. - André Dalwigk, Mar 26 2018 a(m) divides a(mn), in particular: a(2n) == 0 (mod 5), a(3n) == 0 (mod 37), a(5n) == 0 (mod 1131), etc. - M. F. Hasler, Oct 19 2018 a(n) = 4^(n-1) + a(n-1). - Bob Selcoe, Jan 01 2020 a(n) = A178415(1, n) = A347834(1, n-1), arrays, for n >= 1. - Wolfdieter Lang, Nov 29 2021 a(n) = A000225(2n)/3. - John Keith, Jan 22 2022 a(n) = A080674(n) + 1 = A047849(n) - 1 = A163834(n) - 2 = A155701(n) - 3 = A163868(n) - 4 = A156605(n) - 7. - Ray Chandler, Jun 16 2023 EXAMPLE Apply Collatz iteration to 9: 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5 and hence 16, 8, 4, 2, 1. Apply Collatz iteration to 27: 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5 and hence 16, 8, 4, 2, 1. [Corrected by Sean A. Irvine at the suggestion of Stephen Cornelius, Mar 04 2024] a(5) = (4^5 - 1)/3 = 341 = 11111_4 = {(2^5 - 1) (2^5 + 1)}/3 = 31 33/3 = 31 11. - Bernard Schott, Apr 29 2017 MAPLE [seq((4^n-1)/3, n=0..40)]; A002450:=1/(4z-1)/(z-1); # Simon Plouffe in his 1992 dissertation, dropping the initial zero MATHEMATICA Table[(4^n - 1)/3, {n, 0, 127}] ( Vladimir Joseph Stephan Orlovsky, Sep 29 2008 ) LinearRecurrence[{5, -4}, {0, 1}, 30] ( Harvey P. Dale, Jun 23 2013 ) PROG (Magma) [ (4^n-1)/3: n in [0..25] ]; // Klaus Brockhaus, Oct 28 2008 (Magma) [n le 2 select n-1 else 5Self(n-1)-4Self(n-2): n in [1..70]]; // Vincenzo Librandi, Jun 13 2015 (PARI) a(n) = (4^n-1)/3; (PARI) my(z='z+O('z^40)); Vec(z/((1-z)(1-4z))) \ Altug Alkan, Oct 11 2015 (Haskell) a002450 = (div 3) . a024036 a002450_list = iterate ((+ 1) . ( 4)) 0 -- Reinhard Zumkeller, Oct 03 2012 (Maxima) makelist((4^n-1)/3, n, 0, 30); / Martin Ettl, Nov 05 2012 / (GAP) List([0..25], n -> (4^n-1)/3); # Muniru A Asiru, Feb 18 2018 (Scala) ((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)(_ _)).scanLeft(0: BigInt)(_ + _) // Alonso del Arte, Sep 17 2019 def A002450(n): return ((1<<(n<<1))-1)//3 # Chai Wah Wu, Jan 29 2023 Partial sums of powers of 4, A000302. When converted to binary, this gives A094028. Subsequence of A003714. Primitive factors: A129735. Cf. A000225, A002446, A006995, A007583, A018215, A020988, A024036, A047849, A048716, A080355, A080674, A112627, A113860, A139391, A155701, A156605, A160967, A163834, A163868, A178415, A263132, A281379, A347834. Sequence in context: A255451 A028948 A084241 A187063 A026855 A272832 Adjacent sequences: A002447 A002448 A002449 A002451 A002452 A002453 nonn,easy,nice N. J. A. Sloane
4623	https://worksheetzone.org/pet-survey-bar-graph-activity-printable-interactive-626a12e601eea17118d96ae7	Pet Survey Bar Graph Activity Worksheet Enter Code Login 0 Like 0 Comment 0.0 Rating Pet Survey Bar Graph Activity Worksheets 9 views 1 downloads Download Amy M Hardage Description Teach kids how to interpret bar graphs with this pet survey worksheet. Analyze the data and answer engaging questions about pet preferences! Grade Grade 2Grade 3 Tags MathData and GraphingGraphing DataBar GraphsFill in The BlanksData Analysis Other similar worksheets Home Explore Create a Quiz Library Reports
4624	https://www.youtube.com/watch?v=PRDe9a1HJoU	How to - avoid negative number mistakes in algebra maths520 80700 subscribers 33 likes Description 2903 views Posted: 14 May 2013 A discussion of four places students find negative numbers impact upon their success in algebra! 3 comments Transcript: quadratic equations hi this video is all about common places to make sign errors when you're doing algebra particularly things like quadratic equations um so i'm going to just launch straight into it the first place that i always find people have a bit of difficulty is when they're solving a quadratic equation using the formula and negative numbers in here really do cause the majority of the issues that people have with this formula once they've learned how to substitute numbers in and they know what using a formula is about the main thing that gives people difficulty is the negative numbers so i'm just going to take you through a couple of the things that cause people a bit of problems first of all you have to know that this formula is at the beginning of your exam if you're doing gcse and therefore you can always just look it up you don't have to remember this formula and what the formula does is it works out the values of x in a quadratic equation like this a stands for the coefficient of x squared b is the coefficient of x and c is the constant number on its own at the end there so what we need to do is we need to fill in in this position all of the numbers now the first thing the first sign mistake that people will make is here they see a minus b okay and they put down minus five at this point okay that's a mistake that's a mistake because b is already negative b is a negative five okay and therefore when you do negative negative five or take away subtract negative five it ends up being a positive five so you really need to be careful with that the first mistake that you're going to make if you're not so good at your negative numbers is you're going to put a negative 5 there when you need a positive now this plus or minus symbol um you know we went well on here now but it just means there's going to be two up two answers a positive version and an answer where we use the negative at that point this is the position where the second mistake is going to happen and people quite often use their calculators for this sort of um calculation and even with calculators they get this wrong because what they do is they type in minus 5 squared okay and actually they need to be making sure that they put brackets around that so if you're doing this on a calculator make sure you use the brackets function because it's going to tell the calculator that you're trying to square negative 5 not that you're trying to do negative 5 squared and so you really need to make sure that at that point you're doing negative 5 times by itself negative 5 multiplied by negative 5 makes positive 25 of course so that is that can be a real mistake and another thing that people write here they write the four correctly they write a which is three and they write the value of c which is negative seven okay all so far excellent and if you do that on your calculator it will sort it out no problem but what people if they're not doing it on their calculator they're just not thinking about it um they will put negative and they'll just do 4 times 3 which is 12 times 7 which is 84. so they'll be writing minus 84 here that's a big no-no okay the reason being um that of course you're taking away negative 84. this number here if you ignore the minus 4 times 3 times negative 7 is negative 84. and so just trying to use brackets even if you're writing it down without using your calculator if you're just writing it down on the piece of paper the brackets are going to remind you of what you're doing so we've got three possible places where you can make a mistake using this formula you could make a mistake by putting the wrong sign at the front you could make a mistake by not including brackets on that number there and you could make a mistake if you just did that as a -84 and not obviously a plus 84. so this all simplifies down to minus 5 squared is 25 that's plus 84 over 6. okay and then in here 25 and 84 makes 109 i hope okay and then we've got over six now at that point you're going to put it on your calculator or whatever but you might put it on your calculator earlier just bear in mind you're going to make mistakes if you don't think about those negative numbers and if you're not conscious that negative numbers often lead to mistakes okay right the second place where people simplifying expressions make mistakes with negative numbers in algebra is in questions like this one okay they're asked to simplify an expression or they need to simplify it in the course of a an equation or something and what we're looking for is to multiply out the brackets so i normally use these little arrows here and i say right what's that going to be 3 times x plus 4 well i know that 3 lots of x plus four i need to multiply both parts by three and in here i need to multiply all parts by two so i get six x minus fourteen can you spot it the error there the error there is a sign error again all right this is really crucial this one um there are two ways of thinking about it i think i'm taking away two lots of negative seven if i'm taking away negative numbers it should be a plus okay taking away negative numbers has the result of adding those numbers the other way to think about it of course is to do -2 times minus seven which makes a positive fourteen okay so just be really careful when you're doing that if you see a couple of negatives in a bracket situation like this you've got to really think to yourself am i doing this correctly how i checked all of the signs are correct okay this then of course you can put the 3x and the minus 6x now this is another problem isn't it obviously 3x and minus 6x some people would just put 3x there but they have to think really carefully get minus 3x and plus 26. that's how you simplify that one key point being this positive 14 you need to get a positive at that point because we're doing a minus times or minus quadratic equation okay the third one is really one that i just talked about on the last one but i put it in here as a quadratic equation because this is quite often something that people make a mistake on they look at this they do all the right things they say yeah my 6x squared i've got 7 there i've got to put these two together somehow and i'm going to get 7x so that's what i do okay but they're wrong they're not really thinking it through okay if you go back to the very basics here we've got a positive 8 and we've got a negative 15 x and when i'm teaching people how to do negative numbers i use a number line if i'm starting at eight and it doesn't matter if it's in the 8x let's use an ax okay and now i'm taking away 15x okay that's going to be one two three four five six seven eight nine 10 11 12 13 14 15 i make that minus 7x so the number here sorry the symbol here needs to be a minus okay 8x take away 15x don't fall into the trap okay last one then um and this one is sort of similar to something again that we've already looked at um this is an equation we're not going to go on and solve it we're just going to point out the problems with these negative numbers um i've got two i think here i could multiply by negative 7x both sides of the equation okay so what i've got is i've got minus 7x multiplied by the left-hand side equals and what the minus 7x here is doing is it's cancelling out that divide by minus 7x so we just end up with the 5. and at this point people will start to make mistakes um if they're not being really careful okay i've done that one correctly but lots of people might be tempted to put a plus there so just be really really super careful around your negatives the negative numbers the mistakes that people make with negative numbers in algebra and do cost quite a lot of marks in exams and you want to be wary of that you
4625	https://files.eric.ed.gov/fulltext/ED173139.pdf	To -DoOmin TITLE' N$TtTATION. -, ONSAGENCY DATE ESUNE Runior High School Ma. hematics Units- Volume # GROmetry. Commentary. for Teachers Stanford Univ., Calif School Mathematics Study -Group., National Science. Foundation," ilashingoil, D.C. / 59 107p.:-1 F9r related'document see SE 027, 971-973; N.. DRS Flagg -DESCRIPTORS b1011110-1107--'ABSTRACt This is oliime two of a tOre voli using SMSG junior high school teit,materia . Each cotmentarron the text, answers'to all the exercise questionnaire used for evaluating the material, and comments by the,.teachers using the t. topics, non -metro geom4try: (2) informatl. geometry;. and (3) -a2Trqxj-mati0A:AMPY__ MF01/PC05 Plus'Postags. -Curriculum; Curticulpm Guide Instruction; Juniox High Schools, Education; MeasUrement; Secondary. Secondary School math matips EttiMation (Math4ma 'SchoOl. Group metry; thematicv ducation:. Kithemat chers t taint-a a.-"copy. of the. mary.of de:,(1) meat and =L .AItt__t vxi 4, R pr6ductions -supplied bir. EHRS,are the best that can he 'made ,., from the original document _a.- I - U S EIEPARTMENTOF ALTH = ED'JCATION ',Wit RE NATIONAL rN$TIT-UTE OF TliS DOCUMENT T-eA5 MEEN-4EIEPROr OuCtCr EXACTLY A5 aECRIVED cRom fl-iTeRSON OA ORGAPYI4ATION-CIFelt,IN fTiNCyri .POINTS C/PI OPCNION5 5TATEL3 CPO NOT_AltL5sAP it Y PtPPE-5QP4:0OFF ICIAt NA T01441. !ton-HITT OF g 411,04 PO SI Nfl r! nrI Pra r 5 . 1 PERMISSION TO-REPRODUCE THIS MATERIAL HAS BEEN GANTED BY TO 'ME EDUCATIONAL RESOURC INFORMATION CENTER (ERIC) lie Vr=,, Pr red by the SCH \OL MATHEMATICS STLiDY. GROUP, Under a grant frvtn t ATIONAL SCIENCE FOUNDATION PHOTOLIIHOPRINTED Di GUSHING' t, I AN71 ARBOR: MICHIGAN. UNITEa STATC-5 0 AMVUICA I/S9 ere prepaed in the ,sunimer o f 1958 . . y-by a School Mathematics 'Study Group 'clasaes,:during the .academic editi4g, :Of them has ben attempte , z and--Other errors have been. ±corrected.: = he .unitS were u edi the teachers were in _ . submit comments- c oricerni g them xperiences . A.,questionnaire was;--11.11ed wit- by teachers_ -for 'each A copy or -this queStiOnnaire and a summary of .commentS- by" the t eechei-.5 is .includ -each In addition,. °sec. gases 'where unit tess wee epared,tduring the-year,--a coll ction Of sug-, ted test, items _appears at the end of the cciimuen a the unit ---.= This' volute: iVicludds the units ccincenned -geometry. These area units VI II, I__ numberipg ,system, originally used.. e CONTENTS Informal eome = --ry InfOrmal 'deome_ II = e! Mau meat _an -Approxl _,. NoN4stIC4GEOMETR7 7 he Intersection of Sets Tolima .ze of :angle in ideas 'of t has been made to lead the pupil to consider _trio prepertiee fere he studiea_metri prOpe t of polygons, 'circles, and solids'. This is lone iri the spirit of tea which there is an attempt to lead the pups r cower y_ of-un g concepts ap_a basis-for learn computation or in the 4-- -and ai3plications of these to a AraiTiett.o.f p specific: et: the-dis some otthe_more . _ of geometric properties--\ dem situations -.. y of this ma al shoal ad to better undelgtanding of ., _ . propertfes of-geometry when they are introduced, Irth-in_the: or high school and in the -30th grade. It has been traditional --.. -ach these- ideas. hey-are' needed . ; ear witheu .-bOnsideF, 1 , . eacber has assumed these prdpetrt Obvious oning --them. Also ?there should he- s his gro p o closely r tioniaMo The s mathe.tialedscipll e aspec Ofi hoe-c geometry has become' ri fl.S -projective geometry. Thii Acind of . csteristic of ?much of mathematiCs in try -ileust-;7. a pj . . 4. , hat)it 40. its Origin iii a very practical problem bu ben developed 7 1 , --------that------f , ___ .. _ al-theerx--way . -pe eg aw _a a-615'o .some Te -ab-stract--ma n-ofi-projec --ge erne try- is= --------im.ater lel -the traced'to so the Renaissance -pAriod about perspective drawing. limy elemen int ereStin re , it i proi letine geometry. For example I . in.pojecti geometry po ints of: , es. which obtained as sec-tidn .. .g of a c ne rabclas, ellipses, constructed use - f-Straigh e st-,M-6 f e rdina es ean r dpced without the_use or a is or any mensuillMent7'-w tecever. The teacher may indo it helpful to read one of the following: rdy -and HerbertZRObbina -.WHAT I .MATHEMAT TOS? !Oxford Oniver esn, 1941 Chapter IV. Umm14 Johns We'sLI PROJECTIVE GEOMETRY. Th of Ame .1930. II, and rIA 1 Association, '7 T -VI-3 CONCEPTS Sets and -the Intersection of Sets la, ,We Use'li-tb mean straight line. and we think of :line as main:flied in extent. sit% prbpert3r. A line this at are pain empty set0 The coismen.element is said -:to be a set of points. The elem4nta of, IL set may v no-be called = in two or more sots make up the elements of the in ersection set..ofthe two det'may be the empty set. -.Line Segments and Half-L%nes. , . A.line sent is determined-by tiro points on the is ced. -the or more sets. .-The intersectiiOn endpoints of the segment. We use the idea of , lirie segment to represent such - things as a mark on the blackboard drawn along -a straight edge-or the edge of a book or of a dela:. Thete is alway A third .point between two points on a. line-A point- on a line separates the. line into three sets: the point, f-lines determined by the point. If the point is joined:,--set of points On a half-line we haVe a ray. The point .is. d the endpoint of the ray. The_ intersection set of these two raYeon-the line is the point-whichris their co on endpoint. Two half -lids determined by one point on a limaintersect inthe empty set. on a Oust as we "a point lies on a line." we also ea "a line lies _ ..... on a. point,. In a plane a set of lines lies on a point. A plane is .our- ideal way of chinking about a flat surface like a table top. . A plane itiunlimited in extent. 9 T-V1.4 There is -kerie o one. correspondente tween h -lines on a p and the Pointeton a-line. osedt o n lined such an hpae-wbich we, see cs P_ elite are '1180 Curves, 10. Circlebi trianglea-CrO. figures 1_ impleclosediuve-Al des,the <set of points simple 0Osect:eurvte, the inside the hurve and the c in a plane 11%0_3. the points outside. the curve. elf. The curve, is called the boundary of- -the inside and.the outsideOL If a poin .A, is outside a clobpd e and a point, Br is inside the curve, then the intersection bet of AB and 'tie. curve contain s at least one element. _ 11. Closed curves which are not simple closed curves- appear to cross over each other, .lake a figure bight. on a Line a, line is a act. of planes. the intersecti6 set of two planes is not-the empty set, it a line. .a plane, or that a plane lies oh a line. Two lines which' do not have the same direction intersection set,is the empty bet are called eke in different planes'. SpeCialligures 140, A complete quadrilateral consists of our lines and six points on which the lines lie so that on each lire are three of the six points and on each point are two of the four lines.. A figure con-sisting of 10 lineaAand.10points such that ort.each line lie 3 of the 10:pointaand on each point lie 3 of thc10 lines. is called 1 = 10 :Use the word. line correct 2 .Beabie.to.thiinkistout thei,popt the ueeOithe term, 'empty set". ---Be,able to find the intersection set of two or more sets of points i on. Recognize line:segments and t .Be able to identify half- linen, determine the intersectionsete d to f two or mote of these sets. Know Whatva-triangle-is.' Know what-en-exigl Use the word plane- correctly. Be able on a point. Be able to identi Points and lies s one talk abou set of aims correspondence, -9.- Be to identify si6ple closed curves-and to talk -abot he inside and the outside of simple closed curves in a plane. 10. Be able to identify closed curves whiph,are not simple closed cure -11. Recognize that -on a line is a--set Of planes the intersection set of two or, o e planes. 14-Recognize examples of skew lines. Be able-to -iden 1 3. Re able-to follow directions n drawing :figures tot involving metric. -ni-or IlneS, and pCInts,on a ,line. line on and mark pO'inta on it. With each example- talk about z ' k about- a 'line la unlimited in extent. Emphasize fret . t we upe word, line to mean straight line. Talk about net of a variety of .kinds such as the points op a line,-the pupils In Roosevelt Nigh ScbaoLF the members of Congress; kinds _ttea in a. 'television -programs: and numhers Ask the Clue to find the intersection set of two. .or more' Sets. y might- use examples like pe foll9 The elements which-are Boon to the set of natural _n between 19 and 7. re.. and the of .natural numbers which are-(b) The girls in the rocm and °the pupils The three se The cities'Of pC state. with brain ey be cities in yourSt of their names before The cities in= your st on Inver 20,009 in)your, h first...letter alphabet; whilchare south obi , the city In which_ you live. Choose' slime examples in which the intersection. set is the empty, set. a les, given of thIngs thich represent line 'segments and for -idthwy- the endpoints. of t e. segment. In 'a - number of the lee suggest that the pupils imagine a-llne segment:being extend a orm-a- 11116;-__ -7=-00M h xamples_o =1 ne__ segments ask_ the on the4 segment between the endpoints. An exampler_might be ;point _class to identify , T-VI-7 felled telephone-ea cable is-faste to two poles, a bird sitting on :the cable. 0 with the points. -being where the or between an _two poles thete.might Ask a number of ObiTiti xis about the number whichiare between,wo rational numberd. Try.ic . it the eeOeept of points'on a line id,quite similar to the con-3f 'infinite. set Ot,nuMbersv... n the. blackboard a number of nes point out bAlfIines, rays, and' endpoints.. Ask ,7 a /' line can be uaed We elpre of course safe in using this_te k points on,th the term half--. no one can ever measure or compare two hiailf-lines to see if their lengths are equal; Have children home to:theNblacitboareto ill4s-trate these termi., Give eamples and ask the 'Children to give others of the use of the term "ray" in ll the endpoint o en sbOut the inters In each cas all of the'± wo ha a ray andithe.line. Include in.. e, of the x ises Air questions din a ray and a half is of two rays ,The intersection set on which it lies; is of course the ray itself. questioning examples of this kind. Discuss in class. One of the purposes of this unit is to develop use of voca and to encourage the disposition and ability to make simple, direct statements Of mathematical properties. Pupils should be strong encouraged to use expressions like "a point lies on a line" and "a line lies on a point." The language of sets should also be used whenever possible. Considerations up to this point have been in one,dimension. One can use the example of a bug crawling on a wire or something of this kind for the line, and then change to life examples hich movement is restricted to a flat surface, 7 two diMensiods. If possible. make_ user of the soap film and a wire since phis More-ne arly approaChes the mathematic_ If pupils have pot Vith numbers it wi e ence'with one to one correspondence I be well to-talk about examples with numbers before taking reference to the sets on point and on a line. Some bring up the notion that th line of the set on A which to the line on which a correspondence with points is to Westablished, vial not intersect the line and hence there will be one exception in the one to .one correspondence. if this4c _es up the teacher might mention that in pro eotive geometry where these correspondences play an important part, one assumes that parallel lines meet in an "Ideal" point and only one such point, in order to maintain the one to one correspoidence. Parallel iincia "are not mentioned in the text material since this involves a metric property. If "parallel" is brought into the discussion by the the special -,uses might well be explored to same extent and it Should he po &nted out that these cases will be treated in detail in a later unit. There would be no harm_it the cites wished, to assume that parallel lines meet in an ideal point. Emphasise our broad use of the word °curve" and that a straight . line is one kind of 1. As geometry has been traditionally taught concepts of closed curves and the iaside and outside of a curve have not had a very important role. However these are very useful notions when the set language ,fla used in discussing triangles, area, and angles for exEemple. Die-tinctione between simple closed curves and closed curves should -made only by example with no attempt made to point out some of ,the 8 difficulties inherent in Curves like the figur 4 A, ise a circle is actually a met ince a circle eight. The teacher concept. However, a simple closed curve so 'useful in examples it consideration here. Any other simple closed brough curves like; , would of course be just as useful for our purpOseistut we don't have familiar names for as that "if A is outside a simple closed curve and B intersection set of A B and the curve contains at The notion is inside, the q.) least one element is an important idea and many questions should be asked about this relationship. The concept that a line which does not lie on two , vertices has at most two points of intersection with a triangle, as we define a triangle, will come up in later considerations of geometry and hence should be made entirely clear. Work-with planes and lines in space is often more difficult the other considerations up to this point. MOdelswill be very helpful.to many children. Pupils should be encouraged to make their own models. The teacher can use his own judgment about encouraging the drawing of figures representing intersecting planes. This may be helpful to some. Emphasise _he analogy between lines on a point and planes on a line, also that the intersection set of two lines is a point and that the intersection set of two plane is a line. Seek opportunities to make analogous or "parallel" statements likethi 13. The question of parallel lines will probably come up again when skew lines are dis- seed. Parallel lines are not skew lines. One:_ ion which:might be pointed out is that two parallel 1 li,wo ntersectinA lines. lie in a common plane, or determine a lane. Two 'skew lines do not lie in a common plane and do noto e a plane YOU shouldpoint out that some of the edges of a chalk box for example, are skew 1 s in pairs, and others are parallel in pairs (or groups of 4). A complete discussion of this situation ould be helPful. You probably will not want to teach the section on :Special VigIes to al pilso Pupils may not be able to see the sIg0ificance of the complete' quadrilateral and the Desargues configuration but they should enjoy working with these figures, and if they do, they will gain valuable experience with-the in which '&) po to thedbrighte the beduty of figures It may be interesting lete quadrilateral is the section i complete 4-plane in space 1.snguage and nt or line haa I come to see cial role. pupil° to observe that the by a fifth and that the Desargue_ configuration ia the ection of a complete 5-plane in space b i a sixth planP, A ccimpleQ.L4=plane in 'space consists of 4 planes (no 3 on the same line and not all on the same point)- and the 6 lines of intersection of t 6. The intersection set of a fifth plane-and each of/ the 4 Is a line (hence the 4 linen of quadrilateral) and the intersection set of the fifth plane and/each of 6 lines is s-point (hence the 6 points of the quadrils e While the complete quadrilateral and seem like very different figures one the other and both are among the mos mathematics. 10 )' Desargues configuration in a sense an extension of neral figures of all of Exert .1). pe6 18-0:1% Ak)/, 21, 22 thb empty set square, rectan 31 5, 7, (or any Answers' brapezoid, parallelogram, rhaktibus7 other odd -numbers-b) 0, 5, 19, 15, orany e) Q, R, U (The R, 3, etc.. =his 9, 11, 12 For exaMple, Exercises -1 ) The en0 with the pp. 6-7 otAi,multiOlp of 5 tinclikling student cou1diabel seme:p0ints of rm telephone number is 5-0724, an wer. oT the street, pbssibly the intersection-of Main Street city boundary, if Main Street goes through- the town. c) yes d) no f) E B Line seElt,ei _s AB AE AL) AF ep FE FE DE DB -11 / Point ein segments A, F, D,, E, B A, F, D, E A, F, D, A, F F, D F, D, E F, D, E, B D E D, E, B e tw h l a tines deter fled by point A and the two half-1.1nes deterdine4:by the point ,B. The setkeig. points determined by 1) the point A ed 2) the tioint B' 3)--line segment AB the half-line from A throujh B 5). the 'half line from B through A 6):----the-half-line from A opposite_ _4 aove 7) the ,halfr-line from. B opposite 5 a,boves 8) and 9) the two rays determined by Ad 1'0) and -111 the two rays determined by B 12) the intersebtion of 4) and 5), (set of s oatwe and B ekclusy-e. of Ac___and B) 13) the line: on A, and B 14) the .points -A and .B . d the half-line froi on B and the half-lLne from B'on A 5) We'dmig t think of a ray of light as a straight line at 4 °urge (the °Sun) and ettending Ifithout ( eginning 6). %he Y could be described as the boundary line starting where It intersects with the Mississippi,(inclusive) and extending east-ward without limit, orat One of the end points of the'line seg-ment. 1 the 'half-line would be the same as the deAcription aboVes exce t it would not include the ihtersection of the boundary linewi the Mississippi River. One could also describe a ray and a half-line starting at the shore of Lake Michigan. a) M b)' L, M .'die Set of endpoint. M d) The set 'of. endpoint L a) b) points of ,line ent-or L4 and the ray with points of line segment ICM.or Le and the ray with, The empty set. The intersection set of the and the ray determined by R line. frOM_K:.tPiihe right. 9 The intersection set of the the ray Tletei7inined by R (to 12 lied half line from- R to th right . to the right of R the half half line fircti-R-td the- right-and----------the left of .4 - is' the empty set. 11 clock handy airplane o Lars, -ePe le. top 'walls ceillhgd, floc's, blackbgard'; / -) a) unlimiteenumber, unlimited number set;-foUrreleme the line on A and. B f J intersection set intersection set o intersection set the sets of point s determined' by 11 the ray with endpoint A 2) the ray with entwipt 'B 3) the half line ono A 4) the. half lin from B 9) line segmen 6) intersection of -5) and 2) (point B) 7) point A on ray from )3 A.. quadrilateral is a figure composed of four points, no three on a sir le straight line, and four line segments 'joining 'them In pairs, s 'Uch that no three line segments Are 'on one point. 10) yes; if the :three points are no collinear, 12) See dia ram above (8.d) ." . op.respon yes, if all the deskt ark filled, and desk and one desk for 'every pupil. BC , Exe-c4es -,pp. 15-17 e-set,consisting of a point on A- 'and between' A and = e set consisting of a line intersebting line Segment AB and ndt,containing A or B. . You could also have, A AB , B -1- BC,. C A 4+ AC B B. :quadrilateral . The set .-consisting of The set consist 'he Point W s, e, line determined endpoint Y), The Point X is the line segment WY pentagon points R, F, an U of points D acid E ntersection set of the line segmen-I and the po nts' X and Y (or the ray thro X with intersection set or you could use or 'or of' the 1 segment . KL and lip ray throu W with endpoint Y rr it y 16T line 1 determined by points; W and.Y The POInt:Y.is intersection set of the ray through L . and Ole Pay through X' , with endpoint V. ,,. fn all of ab6ye descriptions you could use half471ines with endpoint M. in place of rays, onsider three si uation %here on side the cireae. inekseigment ions simila point -lies it angent 17) 1 . . , fkle se of podnts qoutside the' -ii.ole an i ontaining k and 2. ,= the,set of points acid in the half-plane-not con Xining A. I, Tie must also include tETpOin of the cir These points wetre'not shaded in a or b .,. one popsibili-y is ,one pos the half lane. inside the circle and the, line either:. illy is: :8 po .'he set of points on the line - bo ary l ne The set of aints in the half4la on one side bf boundary line The set o points in the half-plane on the other side f boundary ine. p woto ---The inside is the set of points inthe ntersecotia set of the two half planes determined by the lines on BA, dontaining B and, On PB containing A. The outside is theset'of all other points except ?those on the angle. Pt).:19-0, , b . intersection , set-Would be straight line (horizontal) e the, ceiling meets the front wall; 4 The intersection set would be the straight line (vertical) where -the front wall meets the side wall.,_ / Yes, t e corner where the ceiling, runt and side w7 ill inters This-is the one common int.-The intersection set would be a point Ding -segment AB and the plane. b). The null set or empty set There are many pairs.. One such pai the line which is, the intersectioi ceiling and the line which is the i :11!-s. back will -. would be , et of the front. wall and tersection set of a side Revolving doors, pages in an open book 5 The Intersection sets (lines) mentioned in e3. aboVe furnish, many examples. Buildings have numerous skew lines. 0 The .emp r null set, , 7) We-bight 'call-the points-lying-outside the=pilane, two-half-spaces, and the points on the plane the points Of We boundary plane. Th'e faces are sets of points which (if extended) determine p anes. There are six of these planes. The edges are. sets of points which (if extended) determine lines TheO .81.6.12 such lines These. lines or edges) re intersection sets,of pairs of planes. Ihe'cornert _or which ere are 8) are the intersection sets of three planes'not all intersecting on one line. These intersection sets are points. 10) The faces are sets of points, each-set (if extended) detertniping'a plane. The planes determined by these sets of points intersect in pairs fvming 6 intersection sets which are lines (if extended) The planes Intersect in threes, determining'4.intersection sets which are points. Exercises - pp. 22-23 1) d) yes, D, E, and F are collinear 2) 3) 26 tiaea-:.pp..2R2 \co 4 fee, klte 17-5 From the figure above, we see that the vertices of II AAIE:and L BB1F' lie on the three lines passing .through D. AA' meets BB' at 0, AE intersects BF -ait' C, and A'E intersects B'F at CI. 12 can be seen, 0, C and..C1 lie on the same line (are tollinea .1 4 IT VIT Sample. Tist Questions of pain s sketched e.1ow representra The set. of points s etc4pdbelow represent line et of paints RS represents a pine se The se curve) R S 4. points sketched below represents a simple closed Set A consists of the elements 4, 6, 8; set p consists of the elements 2, 10, 14. What elements are in the inte-;section set of Set A wid 6 t 13? 2, 6) A set must have at least two elements. 7. k line segment has twO e d p0ints. :8. In the sketch t elQw, L and the line Llic InLe.csection set of the line on In ,.tie sketch above, name h ee 1 ne.segments having L as end point. (LM) LD) 10. The line on C separates the set of points on the paper into how many sets? 3) 11. Look at the figure below. Find in column B the name of T et which illustrates 'ech rite be t-he EU) Column A A. line B. intersection of two 2. RS rays.-with endpoint,T. 3. intdrsection of RS and 4. C or o- 4 RT. D endpoint of a ray which contains T. ,line segment. F. intersebtion,01 a ray with endpoint R and a ray with endpOint..T. G. endpoints of a line segment, 1 The set of ):)oins on two ray with A do _on-_en dpOint called A, a triangle. B. an angle. C. a vertex. side of a triangle. 13. :Which, one of the following intersection sets is impossible a line and a square. A. the empty. set. B. a set of 1 element, C. a set of two elements, D. .a set of 3 elements 14. The intersection set of two planes it 19 onsist of 3 planes. half pl 2 half planes and a line, _3 lines. es. Separates the planeinto 3 ae 16. How Many -if lines are there in the figure below. A 17. many rays are there in the figure aboite?' How. many line segments? 19, It is impossible for the common Intersect on of three planes have an intersection set that is A. the empty sets a set with one elemen A set with two elements. 20. How many triangles does the figure contain ?. A. /4 B. 6 -C. 8 D. 10. E. 12 21. Make sketches in which he intersection set of a straight lint and a.- circle is the set described. If it is Im--possible to-draw such a sketch, write -impossible'. " 20 a one element . of two elements. A set of 3 elethen ate 'Of Sdhool; -City: Number .f" days to the teaching. ciud n sting- -Of-this -unit. ate datei: State. USE THE. BACK 'OF. THIS SHEET IP -YOU NEED EXTRA SPACE TO AN S ANY OF THE QUESTIONS BELOW Ms e a _ tatement about the ability level of the pupils in the class and state' whether yoUr school uses some plan of homo geneoup growing. What parts of h e unit proved' to be the Most teachable? What pats or e unit proved to be the mo t difficu -0 teach? id you emit any part?' Did. yob use y supplementary developmental mater If so what. ere they, and at what points were they used?. Did:yoU find it necessary Proiridethe pupils with addition-al material' s 0_ textbookb-o --did-yaw--Do-yau.think that a unit on this topic_ should be -included it regular textbooks for 7th -aid 8th) grades? _LPleaseLmake_ANY.= additional= c omment s= about teaching-perience with this unit which you think would be helpful to e Panel responsible for preparing and experimenting with tbook materials_for grades 7 and ypypils,seemed to be in the more able group, although, ,,seililere_reported as medium of low in ability. Five --iiitee& part-of' the 'unit. A majority felt'that-the Od.--Iiiifficient-develdpmentad. and practice. material. The'-i6hOeyeiv'almDst in complete agreement that the unit should be naiilaid7in the seventh grade curriculum. Opinions concerning the individual sections follow: SeCtion Sets and points 'on A line-. '-'-lntersectibn Line segments., half-lines Lines on a pdint One to tone-correspondence Cloted-curveb adrildteral Planes and skew SPecial figures' No parts of unit gle :"-EaSiest--tO Teach 10 Most-DiffituIt to teach 0 2 0 6 4 1 2 0 18 1 ' 0 a -6 0 ents,;of 18-teachers classes NonLMetric-GeometrY)-_fall into two patterns as Indicated by the following abbreviations: "Understandable and 'clear"; "a good unit";'" "refreshingly different"; "interesting-to teach-; "students enjoyed the unit"; "both ekcitingand frustrating"; -"beSt adapted-to better students"; Nore-developmental material' needed" "some .part.. too difficult"; "lacked motivation"; "students had considerable difficulty", Other comments in the nature of advice to other Students and -teaeller -learn together. teachers include: Students with I. Q. .of 751-learned some of the definition and participated-in the class exercises. -An-additional -helpful- reference-is: -Hardy...a d-Wrlght. theory of Numbers.. Students who are unfamiliar with the idea of set.neecLa little more work with "sets" prior .to the ihtroduction-of. "intersection sets." Pupils may show a preference for traditional material un-. less the teacher is conscious of his responsibility for=ma n7 taining interest'initially obtained by the "differentnese the material. 23 e c].a&e p ticulaxly In the fire 'need'time to ,develop concepts. e:,DesargueErConfiguration haThe difficult 24 geomet we mean the stuay of theproiertieS,of ough experimenting) measuring, and formUlating-tanclUsions tiim,,,reasoning This, is in contrast to the of_a logical, organisation -of theorems about tie. properties, by deductive reasoning from undefined terms and-stated assumptions. ln-ihii Amidition toexperimenting, measuring, and inductive reasoning, the pupils are introduced to the use of deductive thinking ai a-method of reaching conclusions, arguing from previously stated Primeiples-and definitions. We reserve for later the systematic organization of ,geometry as a deductive science, Starting-with postulates and undefided 'terms, and developing theorems and definitions- on this basis. The purposes gfor Aibiehl,he'unit was plAnned were these: -- To awareness of theoCeurreide iii the environment or -illustrationa of points, 1 regions-bounded by Pants lined To introduce certain geometric concepts and relations. d:planes and alko-of three-dimen-To give the pupils experience in discovering empirical laws about al relations on thy-basis of systematic observations of geometric To give encejn verification ofexPerimental esnl"Ps and in informal deductive argument ogthe basis of-previous principles. he-major topics are (1 e angle relationships formed- by-parallel lines and a transversali.and (2)- angle-relation-Jape in triangles., It is assumed that the pupils are already fami With the'doncept of angle, conventional angle notation, angle Measure---25 n addition, they shou this ries TheyOnee' d-mts frequently in this one are thOfe, nt, plane, and half-plane. Some notat,ion -.-also used in this one. ,The,introductory section serves as epoint of,depar e -for the -Pliyeical representations of geometrit concepts-. 'cussions of this natdre should take place frequently. Whenever a new arrangement of points and lines is studied it is helpful to ask the pupils to find 'illustrations in the environm the first discussion, a chalk or in cp tires. In and other solids may be used to otions of plane surfaced which intersect or do not intersec es whieh do-iir do not intersect, d the like. some .hints about applications for the purpose of .supplement- these remarks by pointing out the WC also give motivation. °metrical knowledge involved in designing an airplane. The main out how air will flow abouten'airplane of a given hape moving 3n a given direction at. a given speed. From this we can also use lculate the lifting force and the air resistance,. I t he p lerEigram of forces as an 'illustration. order ,to find_ -single force equivalent to two given force point we can-draw a diagram like this acting simultaneously at a which the given forces are represented in magnitude and direction .H,..We_complete_the_parallelograia,_ancL_the_d -26 Ude and direction of the.,-riabuitent force". Geoletry out the forces in an. electromagnetic field,. and, _ , psychologist, and the military strategist use geometry in ways elistAcf,and how an oil company should schedule AS pro-ddt Conn'Lthe theorl of f-relativity and in. the design of agricultural . _ - - -E,-- ---c o m p l i t efferion c -one-4U- of-- space-aire- use-diad ciet, the chemist, tie biologist, the engineer the ecoinorniat, Owed fro Keying, some -6Me of which were not -everf (TrOA-Med -Of -15- Thel,bisterical_fe As_serv_e both_as_mo.timtation_And_also_to_give the children a feeling for.mathemat a living, growing science, -as- an:integral-part of our- culture. The -pupils should lookup the =4 names in 'books and encyclopedias and their reports should be _posted where -all can to them The annual supplements to some of the 'encyclopedias often list,. under the heading. of "Mathema cant. important recent developments. These Are u can, be used to show who is doing outstanding Work today. The Scientific .American often publishes excellent expository y too technical fOr the childrenN ._art-i_cles, and recently several popular magaranes such_as_ Reader's Digest and Esquire have carried articles on mathematics and mathematicians. provideibackground for the itiicry of a AnsvorsariFifer-sectTh-e paralleInines, arid of triangles. It dead with t&.ee tires which intersect in one, two, three-, or no pOints, and the associated gle relationships. The concepts of vertical angles'and of, adjacent e introduced, and are defined by using -the concip is of ra d half-plane which were developed in the unit on non-metric geometry. 27 dliere to tha: un :angles or segmeDte' ve; the same measure, and-Will then be , -, -..a. .. d to tial measure. Later we shall call such f_gurie cbn-. . - r hip point it is more important to instil-1 ln the pupils gs -habits' than to Spend much time on a formal dieouss ien- 0 of sie es of vortical angles is develppe in tw drawing two intersecting ng the measures First the pupils' + given exercis lines and in measuring the angles formed. 6btained and than comparing their results with those obtained by' other pupils they should arrive at a tentative conclusion. hen re-examined; certain properties of adjacent anglea-are noted, the same conclusion is reached by deductive reasoning. The use of cm-measurements as a basii.for a tentative conelusion fdllowed aforal dediactiVe Argument, occurs at several other The emphasis should be on t discovering -ships by their own observations. lung them the answers. -Be patiei idt id, if necessary, ask leading 0 0. questions, but try to draw the answers out of the Students. It will be stimulating to, the whole- class if one of them makes the discovery ad of- your telling. them. You should guide the clam discussion -so make the children realize the necessity of saying exactly'what ,they mean. }lave the .youngsters criticize each other:' s formulations of the generalizations until they arrive at a statement of each principle - -,a_Then may show_ them.o fomulation for ateiaents as models in their :fr work on thiS unit. When they have finished the unit, you should di- -hute.,our formulation of._the principles so that the student insert them in -their notebooks ,for future referencew The practice in 0tMtinga pkOpOsition clearly and precidely is important or:all future A procedure which is, helpful in guiding pupils iii formulating tentative conclusions on the badis of experimentation and measurement. he illustrated in connection vith'Excrcise 5 in Part A. As the ptIn,are measuring-their-emgles, a-ta Ie-may-blackboard, with the names of the pupils in the first ,Coliamn\and the names 611-tbe-anglea Measured- a a op. L. -, pupil-Axigle B Angle CAB. _Angle DAC Angle BAB As each pupil-completeihis measuring, he-can report his results, record,them himself inn the table. After a few pupils hive renorteds the class can obserfre the relationship tetwean the-measurements recorded in the table, and Uotb that while the figures reported by difrerent _ . pupils are not the' same, the measurements obtained by each pupil for r. oh -pair of-vertical - angles Ithe- same---, or early- to The-obsery tIon pf the measureMents obtained by several pupils supportstha conClusion- about the equality:of the measures of vertical angles; and the procedure suggests to the pupils a method of organizing dat4.0,.. this kind. B, -the set .of three line's in which' wo lines' (not neces-sarily parallel). intersect a transversal-is studied. The 'angle identified-earlier (vertical adjacent) are noted, and the new angle pair corresponding, alternate interior) are defined. Lle studying-the tud ng he measures of unlimited .in length) _s Saida to divide the'plane .(also:nlimited in e hafPiiitea;-the-lint-beinuthebounda two halflplane In Part .- VIII -6 114., u. become acquainterlyiththis they, can readily recognize each kind angle pair. In # Exercise 3 -.the symbol used to eXplained;-.Ve:have used the symbol HAng are equal.. 4 can readi,'-"The measure .0 angle, . . . . denote equality of mizapures when the easves.pf two he -o angle ci." The 'discussion in the paragraph between exercise .5 and 6 maY require revieji of the concept of half-plane. -Recall that a line relation between W.hich tVio. lines. fOrm with a transversal and the intersection or pon-intersection of two .lilies is explored through measurement.of angles. The tentative conclusion is formulated, and acceptedi-that when a pair-et-co responding-angles have the same measure, the lines do.not intersect and are parallel. : This, yields the construction for parallel lines whiChiaused in the unit, with:a second proposed in exercise-8. in. Part b -informal dedue.. establish the principle that -when a--pair bf-alternate--interior angles have.the same measure be lines are parallel.' -In Exercisesn following Part D, pupils may hive difficulty with exercises One, way to.help them to recognize 4 familiar figure within a more complex one is ie cover up one of the transversals and extend the remain sversal and segment RS. In exercise 6, if a T-square or plastic triangle is available either may be used to demonstrate the application of Principle 2. w on so. to discuss many illustrations to show that the converse of-__ rent may be true or mav be-false4 ,._an.Rart_F,_the coiverses of, frinciple 2 and_Princliple, estigated. The converse of Principle 2 is stated as a principle, on= the basis of measurement and the converse of Principle 4 is then eddeductively.__ Some of the_exerciseS,dive experience in identi-fying angle pairgvin more coop Xi-figure In Part G the angle relationshipa in a-triangle ire studied. Exercises are designed to lead to the discovery ofthe angle-side _ relationahips in an isosceles triahglefrowwhich the corresponding relationships in an equilateral triangle may-be deduced. In exercise. _3-it would-be well to have the pupils-use compasses to:draw equilateral triangles,. _t-the gompaesesare available. Exercises 10-11, and-also, exer s,12-13, provide an opportunity to note that-one-conclusioh-le the converse of,the other. In Part H the angle-sum for -a triangle is developed, first experimentally and then deductiiely. by the use of parallel lines. few exercises are provided,to shold-the significance of-this principle, additibnal ones may weft be used. % Throughout the_stuy of this un ort should .be-4a4e tO-Jencourage_. the_punile to discover relationships for themselves, 'both by measureMent and observation of results, and ..by informal deduction, ,,and to state cleaily, in their own words, what their conclusions are and their reasons.: Some pupils will find exercises Mud e aster than. otherp. find them,. Some will need help in reading and dllowing directions, e keetier space perception thin others I 6 Prbbible 31 do Ships among three lines) and, with relativdly de, ::disdoer,mdst-of the relationships included in unite us ? cin of lines ii,thiCh-:are intersections" of planes =i; _ rally d oiling, walls end bard _Over_ motebtor___ stapding on the de skvaurfaces -f filing ,cagnet, desk surfaces. _ IllUstrations at pointig' are intersections of lines corners of froom edge! at windows lines on. note paperi Illustrations of three planes intersecting in 4 line--olc- three cards- attached with-adhesive-tepei parallel planbs/,-.-- opposite walls of roof gee_ of -a iai4strations o site sides or desk --covers of .tiobk, two eurfaces of window pane. 3 adjacent Illustrations of parallel lines and skew iines 7- 9pposite edges oft room, opposite sidel of sheet of paper, edges of haphazard pilei interaection.of hat o a third .14411 with floor Illustrations of gurved._surfaces_pipe _reflet surfaces, curled paper, clock face glass, globe lensed' of IllUstrations of planes on the street Tsiduale P ill ehhlo_ se's. -se. Example of curved sUifacea 'window light 'poles. From the Greek words mean .t Part A eTree Lines. it _ eirorated Intersecting straight-llnewv. _ -Angle BAE'ang angle!DAC are 4186 vertical angles., Angles ._BAD and_DEjare.adjagent anglea. Vertical angles have the.aeme-Measaree, 5., Each ,pair of vertOalfanglei have the same -me the,size o the angles in the figures vac cement even 100° h 1809 oot. 500_ (d) 'x +y =-1 +y= 180 x =.1 = 140 = Y._ C." 7C-= Z. Yea. of vertic--gleS 1-, 10. .Three angle- l' 'On one side a Each verteXlaall fciur-"alIgie'd. -Angle pairs .vertical or adjacent. 4 pairs of vertical angles 8 pairs of adjadent angles. a cOmon-vertex are c 19. The figures are alike in tha 4 plane. e .all forme by ant in the relationships between pairs of tree points or napoInts. akeW, that is they may" not lie in one plane. The lines may be skew to each other and 44- have k Two m intersect and-both be skew to the third, no intertectiops. yield _g only one intersett both tftiOne Iwo may be skew:and'the third may ti wing two intersections. These.may be iIlustrsted with sticks., -or curtain rods. Part B. Two Linea and a Transversal_ J7igures ftr problems 14 and 17. Problem 14 " intersects m2 and m3 intersects ml and m3 intersects mi and m2 Problem 17 --intersects ti mod. t2 8 angles., .4 pairs of vertical angles, e= angle g, angle f 11 angle h. Angle a and gle d and angle a, angle g !and angle h, 6. angle b and angle angle e and angle angle h and angle e. c, angle c and angle d, f, angle f and angle g, All-theAdjacdnt angle pairs of number 4. a and angle e, angle. c and angle g and h. Angle angles b' and f, angles 35 m atleb, our pairs of verticel angles in xple e, by Principle 1. Angle kr Angle a Angle w Angles r and p 10. rand p, angles x s tr. Angle r and angle z, angle p and'angle y. c. Angle x and angle w, angle p and angle; Angle and angle x, angle x and angle p, angle v and angle pp angle r. and angle v, angle z and angle e, angle s and angle yy angle y and angle V, angle w and angle z. le s d angle v and angle w Part C. Parallel Lines (Correspondin gles 0 1. Yea. On the sage side of t as the two angles. 2. The roads will intersect on the left side of t. 4. Corresponding angle. 7. a. Belo e. Parallel i. Above The distances between the intersectio_ is thdlength of C D, then ,e (CD This formula is for the information Parallel c. Above t Above g. Below Parallel k. Below d. Belo Parall Above are e-ualo If (CD n 40° ay the teak TVIII -13 ensure, of Angle y in degrees , 1 AB 10 3.7 20 1.9 30 1.3 40 1.0 50 .84 60 .74 70 .68 80 .65 -90 .64 100 .65 110 .68 120 .74 130 .84 140 1.0 150 1.3 160 1.9 170 3.7 The distance from C to D depends only on the angle y. The minimum distance occurs when angle y 10. If, (AB) is doubled, theni(CD) is also doubled. 11. Let, be the given line.m Draw a perpendicular line m 1. A Nark off a point B on m whose distance section A of -, L with m is one inch. Draw the line n through B perpendicular to Part D. Parallel Line (A ernate- Interior gles) 2. Angle C 2 150°, angle d T 1500 angle 37 rn Principle 1. Yes, to the tight- of T-Vill.14 If angle b is larger than angle c, then mi anv2 intersect on the side as angle e, but if angle e is larger, then they intersect on the s side as angle b, by principles land 3. If angles b ,e have the same principles 1 and 2. a then ml and m2 are parallel, by Parallel, 4 b. Parana, 2 c. Above to 3 d. Below 16arailel, 2 or 4 f. Above t, 2 or 4 g. Above t,. h. Above t, 1 and 2 f.. Above , 4 j. Parallel, 2 Exercises 1. Vertical, adjacent, corresponding, alternate interior 2. a. a and.d or b and e or c and f b. c. a and b, b and c, b, and c, or b, c, and a or a and g, b and h, c and e, f and d. dr, fda and e, etc. b. a and h, h and g, g and b, b and a, c and f, f and a, e and d d and c same as b. a nd c pr b and d d h b and f or c and g Angle x T 75°, by principles 4 and 5 Angle z = 4592 by principle 4. measure, then __st intersect c. 180° 60° . a. 180° eb. 65° BC is parallel to DE by principle 2. In.theae cases, use is made of the principle supplementary:., angle z has some other 38 adjacent angles are or 4 T -VXU -15 art E. Turning a Statement Armed (Converse converse m]ay be false. be false; converse -T_ converse may be false. d. Ma be true or false (Mary may no also undetermined. True; converse also true,, Statemen Converse -5. 6. 7. c. T d. e. T 1 T go to school); converse Part F. Converses of Principle 2 and Principle 4. Corresponding angles or alternate interior angles. -Corresponding angles angles should have the samemoasuros. and u, x and s, z z and s. nund1 Corresponding ogle pairs y Angle pairs m = Angle should have the same, meaeuies. angle w d w,_ w and r. corresponding angles Angle w g angle a vertical angles -.Atig let z angle s both have- the 9. Angle pairs b and c, g and h. same measure as-angle we_ 10i Angle pairs a and,h, b and e, f and c, g and d corresponding ng ea. 39 45-T- VUX -16 m. d and f = 68° -Angles b, 2 13 Alternate g = 1 erier. 14. Corresponding angles, alternate interior angles, vertical angles. 15.' 2 Corresponding angles at x and s R and W. 16. Angle a + angle b g loo since 2- is a straigh angle b 12 angle c - alternate interior angle angle a angle c g 180° = angle b angle d. 17. Same number 16. 18. . is perpendicular to t 19. W., By the definition of perpendicular Corresponding angles have the same measure. c. Angle b has the same measure as angle a. d. By the'aiefinition of perpendicular lines. 20. a. 4 transversals. intersects c and d. a and b are parallel lines and c and d d d line intersects a and b and a (and b) are,parallerlines. ending angle pairs are: n,p), (t,r), (s,q (e (k,$), j r) '0. Alternate interior angle pairs are: (1,n (j,p), 0). Vertical angle pairs are: (e, -(o-q)i-(pyr e. Angle pairs whose measures have a sum of 180° area h), (m,p), (t,4), ( (e,') f,h), (14) kfiX, g ) (hop ), (1, k,g (g,i), ,h (f,g) (10. (X,11 Angles 'I' 1 4181PP_Xig klj) (g, h, j (h4q), (i,p). t, p, r each g 280.. ,6.0711 each g 4O 5,0 G. TViahgles glee. f contains a trian 2 linesegments. c and d are encased by 4 line segments. prieere equilateral 4 sears isosceles by cy and e appear' Sc enee 1, g 7, and 1, 6, 9; The Sum of t : length or two sides of a triangle is greater .than the lettigth of the, third side. The sum of the measures f any two sides musc always be greater, than the measure, of the third side. The length of AC mus.t be less then 7 inches and more than 3 inche 5 A 12. 'kngle a gangl sew measure. b, angle bra C A 1- ---a, thus all three IS gles have the 13. If angle a g angle b, then sides opposite these angles have the 14. 15, The angle opposite the larger side has the larger measure. same measure. If angle b 11 angle c, then sides opposite these angles have the same 4easure. Thus all three sides have the sane measure, No. length of AC alone is doubled, then the angle ABC does not in a Simple way. If all three sides are doubled, the angles 1 the triangle do not change in measure. la Fart H. Angle of a Triangle he measures of thia angles of a triangle is\180 Angles EAB and ABC Are alternate interior angles, and are.therefore equal in measure. Angles DAL and ACB are corresponding angles and are also -equal 'in measure. The sum of the measures of angles DAE, EAR, And-BAC is 180° since DC is a straigheline. Therefore the --sum-crtbmimeasures -of-the-angles of the triangle-is-40°. es and ACB have equal measures by principle 6. her 500 and-80°, m C 42 UNIT VIII Sample Test Questions,-PART I. TRUE - FALSE Li tle.has been added to geometry since it was invented.. igle of 'an isosceles triangle .is equal to the measure of 660 one. of the other .two:.angles. must be equal to the.measure of 66° A statement may be true 'while its converses false,-Equal pairs of corresponding angles are formed when a transversal intersects two parallel lines in the-same plane. T A statement and its converse may both be true. F-Th-einterseCtion- set of three lines-inA plane 'must be three points. F 7 Many modern geometry textbooks are basedon one written. by Euclid. If a trianglb h two equal sides, it has-two equal angles, F 9. All isosceles triangles have the same Marie regal'dless of size. J 10.., The sum of the measures of the three interior angles of a, triangle is equal to the measure of 180°. F11. An equilateral triangle is also a scalene triangle. F 1. converse of a false statement is always false.. F 13. If a triangle has only two equal sides, It can have three three equaI'angles. T 14: An equilateral triangle is also an isosceles. Jangle. F 15. If a--figUrtis acloSed curve, it is a circle. 43 he figure at the right,, As Bs WIC are, symbols for the verticies or the triangle. ansversal intersects any pair-df,lines in plane the alternate interior angles are equal. a _F. 16 In the figure-shown at the right, . = if all measures are held equal ex-cept; angle MON and side MN, as angle -he same N increases in size the length of DIN will decrease. 19. r One method of prom'.' is to reduce a statement to a pre4 vioudly proved statement. F 20. It is possible to draw a triangle whoae sides measure 4 inches, 2 inches, and 1 inch.' The figure shown at the-right consists of two-pairs of parallel lines. Use the figure in marking 21: 26 true or false. T1-21. Angle a 5 angle 8. 22. Angle 3,Pit angle h. F 23., The measure of angle a + measure of angle d measure of angle 3 + mea-sure of angle'2. T 24. The Measure of angle c + measure angle h = measure of angle 2 + measure' of angle 5. T 25. The figure 'contains/ft-aft than 16 pairs of. equal angles. T26. If angle 7 angle 8 then all the measures of the angles 44 MULTIPLE.CHOIcg. ansversal intersects two lies. the same plane the measures ofthe co responding angle-e equals then the two lines are.; parallel C. pe rpendicular lines. intersecting lines. 4 E. Iltine of the above answers is correct. If one angle of a scalene triangle I 50° which of' lowing statements is-always true? One of the other angles 900 . One of the other angles -4 50° The sum of the measures of the' other two Two of the sides are equal. One of the other angles M.1 0° D. E. In the figure shown at the righ how many transversals intersect lines m and n? A. 1 B. 2 C. 3 If-'the--measure of one angle of a,triangle is equal tq the measure of another angle in the triangle... es of the tiliatigie are equal; one :!,Of the measures of. the sides. s.re s4ual. of the sides vppoai the orgies hose . urea are,equal4are not equ the measures of two sides are equal. E. none of the, above statements, is correct.. _ If two sides of a triangle measure three inche' and four _inches, the_ third. side could measure... A. one inch. B.. Aeren inches. less thari one in D. more than seven inches. E. none-,9f the above answers is correct. If three lines are drawn ol the 61114 plane and lines are parallel the figure formed, could in A. exactly three angles. B. exactly two points of intersection. C. a triangle. D. two closed curves. E. a_rectangl A distinct point A. a point you can see. B. .a sharp object. C. 'an important stateident. D-,a. circular mark made, with a, pencil.. E. the intersection of two'4ines. 46 the transversal t intersects Whqh angle forms with. e pair of alternate le0? Al4eotetric plane an airplane, carpenter's tool. , flat. surface like that a.window pane, a curved sUrface like ,;that of a round- lamp shade. an object ilte,a. :sheet; bf gywood. surface is 4 0 A. a . geometric a 'set Of a set of lines. D. a set of lines and pOin s .measure of area. , -the figure on the right, which f a pair of corresponding -047e1PS? A. a B. C he figure above 11nts x and are parallel . the sae measure angle d. angle a c.ha he- same measure as angle e. le b has the same measure as anglp-.dj. the re abdve, e dMeasure as angle d. same measure as anglv_ the measure n. In 'the b e ..abov angle adjapent- (3 angle.. and d are all .angles . adjacent. to igle ;a. none of the above answers is correct. PAR1 COMPUTiON, at .the right, are parallel and intrersect trans--ve a7 What ares.:the mea-each of the follow-the measures of one -he 'angles _is In th line 6. One meaeu 'ice _ 'IS at the right a pali. 3 1 s: intetspot.:oh,a' formed are,-equali- what. Is, Two Anes intersect on point formed :adjacent angle M In the tr le at the ri asU es 30° Angle BOA W 70° is the measure angle CAB?:, Corresponding angles are On (s 13 - 15 In the fi re: at the right, 'lines x, y and z, .are parallel; What is the measure of ttie-. . lowing angler? side sv r841 11. angle a 0°) lg. angle b (60° 13. angle _(700 ankle d (1300) .15. angle e (500) 49 --intOr- ct1en witl 0 u , the spac:e'whIch cv ectly r-f.. f011owingmktateliten -17::. angle "a 5 and . le- e-1 75 the lines the lines the libes . 4k. of 'd:1-1-1 °, and angle _e the lines PART Foreach of the following select the statement , a the"-imum or 1 aunt, of essential necehary) , of wh ch contains e at the right, d'.2 intersected by gl 6. -the s:um of the .measures of angles -2 and 4 equals the sum of the measures of angles 8. 2. In the figure at the Sight line o intersects lines In and n. Angle m angle 7 if A. angle 5 - angle angle 7 = angle 4. angle angle 2. ii = angle. '4 angle 5. angle tr. 1-11, angle 2 angle 8. In he figure at the right, angle ,10 g angle 11, and lines a and b are .parallel. Angle angle ,o because. Date e of eadhein o iriatud -City: = t a e; days given to the teaching testing) of this !unit: USE THE BACK OF THIS SHEET. IV You NEED EXTBA:8PACE10:. ANSWER ANY OF TREHQUBSTIONS BELOW e a statement .about the .ability- level of the pupils in the 01a ss.tiand -State%whether your sdhOol uses some 'Plan of hcitho gedeous'grouping. _pats of the ullit.Paved most_tea What parts of /the unit proved tb be the MD :teach vs' Did You omit any part icult to Did you use any suppiemt developmental materials? If so, -what were they, and at what points were they used? Did-you find lit necessary to provide the pupils with-addition-al practice material? If so, was it from textbooks or did ou write your own? Do you think that a unit on this topic shoUld be-included in regular textbooks-for 7th and 8th grades?: -Plea8e-make-ANY-additionAl-Cotment --atour-sitAIFT-tewohitigejt;77---petience with this unit which you think would be helpful to he PaneIT-responsible for preparin =mod -experimenting with textbook materials for grades 7 and .53 of adhere. l levels ,but relatively few were described as low level -teaOhing days varied from 8 to 26. aleiViing table -shows the topics which were reported easiest h-and the 108 which were reported most difficult.' to teach. ;pOL at S.,_how ,Many teachelArePorted these 'topics. Easiest to Most difficult ----Ttipic------3 Teach to Teach Transversal -4 Informal, proofs 0 rineiple 3 Angles, of a triangle Principlda of parallel E4leriment. Applying-principles -Converses -------------:- -----,:------____ Isoscelea triangle 2. -It was apparent the teachers thought thht the' vodab _ ='.vtks----difficult -at times , Some of them stated -thaf ere:RI:Jae much discOvery; for example: cutting out triangle and reminded the student of 'cutting- out paper One teacher remarkect.that this unit did not go as well as 't early units in her seventh. grade class. One. stated that for her eighth grade class it was -an excellent Several teachers- suggested measuring angles by. u ing a:pro-tractor.. They .proposed. that the idea- of approximate measurement be introduced at tie, same time. It was suggested- by several that principles stated. in final form should be in the guide for teaehers but not in the Atudents UNIT -IX I 0 L -GEOMETRY II . _ riingles, perpendicular-- bisectorS, parallelogr _ ri i t t ogle principl erg a a ge011etty:. In-this sequence = informal Geomery is intended-follow the units on Non--Metric geometry and Informal If- -the-eacher does not wish to give_ this -muCh-time geometry and prefers to try this unit rather than some of the oth this could be done providing,the students have had some exPerie4Pq, 4 ------with-the measure of length,, area, and angle:and .also_some.ex7. POrience with parallel: lines and transversals. In most schools Informal Geometry II.should notloe taught. befdre the eighth grade The writing committee believes that would be appropriate to 'teach some of the geometry units in 't seventh grade and some In the eighth. Congruent tri gles idea of -congruence should be illustrated with-ntmerous--. sueh 'd two, Teets. of paper; teitoOkd,: two .z . and two footballs 'Then the pupils ,shouidbe led to see' . : _ the importance of knowing whether- two obje'qtsare the saMe-' and shape} especially in business and manufacturing. In manufacturing an automobile or a. radii); she company' wisheb to make repairs as easy as possible by producing star -inter-changeable -parts such as -nuts-,:-and_tubez , -these --must--:be-the same size and sap,. = ' certain measurements of a part are taken, and if too many sped-MenS-deviate from the standard by more than:,.a ertain -,ampunttaa4td,' the' tolerance, the engineer knows that sOmeth ng has gone wrong in the 4 manufact'uring process. T-/X-2 Similarly, manufactu ing .a complitated-Machifte like an4Lirplane-engine, it is necessary to fittogether _ -many:parts which wereAma_e Should Work properly the-.-parte.mus e-be made in Certain sizes and shapes-to within very narrow tolerances.: en.attention-ls turned to the congruence :of o.triangles., an excellent opportunity is :provided. to emphasize how the mathe- matician-aPProaches a problem by considering a simple and 'quite 11-ideal", situation first. It can alSo be -Tainted out-that the solution of a problem- iri such an "ideal" situation often prOVes be very useful in solving a similar problem-I/Tr life;aituatiOns. Some prefer to.think of the movement of one triangle .in a lane_ so-that it coincides with a second triangle to which 0 ruent,,rather than the cutting out process selected in the chclice of-the-outtingout process was madesince it seemed,' y t&'be.more cledr.to students. It_is entirely: satieactory 111 'intuitive treatment 'of:-ge retry. is, 110pre ext. out that solid figures in 3 dimensions could no 'be . milar process. In the text, we suggest a procedure 1.0.6p1ds larComparing solid figures.. ndept' of a rigid motion should be emphasized; a.clidean e-sdefintdaS the study ofthose -properties of figures -igid.motions. .prl,sAges . 24 we L-ri the pupils. we can see whether two figures are, congrUen moving them, by imagining how they wul_d fitf. ;theywere :moo. -.if the figure which is moved is cut 56 out of tracing paper, then corresponding points .a. points which lie'overeach, other, can be marked; and. aAer the figUreS are, ga Xw the were anged° during yth motion. , Then Ask how the dIStancesibetvieen corresponding. points of the two figures compare Then adk \Whether theconverse_is e,whether if the two- ;figures ,can-be matehed-------point for point so that distances between corresponding points are equal, the figures will be -congruent. On page 31we to get the', childre hat theyl,can _ acv with a comPass the figure formed by all points in-ja plane. whose distande from a given point is 4 -ihah0. -Thus they should discover that there -are two podsible positions for the point 'B, I one on each side of the segment AC. On page 4 the :children may join he pdifit n to -the vertipes A and _C and extend the lines IANCand- CM until 'they meet he opl5osite, sides in points' P. and. R respectively: The point -the distan lengths equa and may be, at.e.d and nom C to P-and A __ng_the _corregponding Then A' P' and C' R'. intersect at the -'.point M'. As studepts,begin to look for "tests" for congruence Of two--triangles, .the teacher should emphasize the advantage of finding the tests as simp, as possible, that is tests involving e comparison 57 triangles possible. .Since it is stUdying-this unit will be familiar with the.,;-g---probablY a good idea t0 start with-the-.,case of-iizen:,angleand a side, even thOugh this casemay actually ult-forthe pupils tO 'see than some_of the Other Page 5 practical application of tests for congruence of gles_is:'suggesied measurement of certain parts of w o triangles is'sUfficient to shpw that they are-cOngruent then the measures of .these parts determine.uniquelY the-measures-of all . . otheripartsi , Appther practical applibation is the determination., inacceshible-distances. Suppose a man or -'Shore. wishes to find 11(514 far he ,is from a certain ship: -N He may pick another nearby location N on the shore. Then with a sextant, he can measure the angles SMN and. SNM and the-dibtance Be can: then make a drawing to Scale and measure the distance when-he...studies. itrigonometry,,,_ he_ will _ie_arn On page 5, the distance's of the airplane to Honoiulu is 1710 miles, -o three significant figures. In the sCale-drawing the leng 2620 100 3.28 inches 11-n75 '5/16711_nche . in this unit it is assumed that fi ruler and protractor . The teacher can ilse_hisi ud ent -abOubt e4uiring' construciion with ruler , and. compasd; The ex reises' reveal that In: some situation s there :is -actually an, "'advantage -'in the Use of compassed. would be well- for the teache'r to use the conventional terms in connection With these struments.-en-the figureip:to bemade: by-measuringzwl_th a _ and protractor, it is conventional to use the phrase -- , Y. aik the, figure% " When the d compass', is to be used it is' convention7'_ -say ,,i_eonstruct the figure . ntended to pro vrab the 4 155Adhity-fm4, student explore the; situations and discover the -pripciplea upon -, r .. i ... which there , can. agreement, rather than to have these -principled . teacher and then verified V -the pupil . Thls a pointed- _.out by process may .take. longer bilt it- would' -be better tp provie for this h material than t cover a great terial by tea her lecturing. On page 10, special' attention should be -balled t the dase . deal more ma-in which twc sides and. an angle included angle) are given. ppoae- that sides AB and BC- and the-: angle vert- x I 59 Where BC is lc -be drawn. two different angle one triangle ust long ,enou to be Perpendicular to side AC. BC is too short. to reach. _t4 . _ ____ In _trigonometry this case of given parts of a triangle; called the._ambiguous case. . -S ecial emphasis should be placed on the list .pOssibilities the top of page 10, again emphasizing, the advantage of the 'discover approach, rather than the memorizing of conclusions of others. It should.be made entirely clear that we have depe .intuitisin in developing the fOur principles on congruence. They are S6-olleted --rh- "-this 'unit- by' eXperiment and-observation.- The-use-the word "prove" shoul avoided throUghout kince we have not :developed in the units on intutkive geometry a logical structure : 'oh.,whAch- proof can be based. We might, however, calk what._we do An the parts'following the treatment of congruent triangles,-"informal 'deduction prOcessn. eetio berpier:tdicialar bisectors parallelogrAmS, or condU 1 ent ,may be Form e . i he a is .on cbncurrent _lines is ettidied, .it would -be- desirable to precede it by .the SectiOn. on perpergidicular bisectors Otherwise any of the'-_, remaining sections may be studied in any order. :These three sections' on perpendi ular biseCtoi.s parallelor grams, d concurrent -lines--provide a; nice opportunity to .show certain- properties of ,triangles d,quadriluterals _using the :sin7.-al deductive process In which most of -the, cOnclusions are ,-based On :the prindipies accepted for congruent triangles. These top os also provide opportunity-for further:diacciveily on the __par -i e. pupils boff properties' not- ncluded in the text:. The better ents- should be to try this. "well In this section as -well as In thoAeyhidh follov6 ,great. , ess- should be laid on the -discovery-or verif :-ication. of pro-N perties held by making use of preViously accepted:-principles in the _ _ chain of reasoning. The ability to -identify the reasons for con- olubibrib, and -Steps used in readhing a given conclusiOn Is more important than any one of the conclusiOns elf #07 T comments on'perpendicular bisectors apply equally well in this section. Consideration of special figures like squares, rectangles, and rhombuses will prov de a rich opportu discovery on the part Of the betjer pupil. may_usetheyOrd converse" if he chooses. material may, be adjuatedfor-diyferent classes'or ate:tents= na' OrepaUtion :Ould be me d Great Care,shd-,_1 . 4 e-exercistd-in th'a Class to See that no conclusions are based on a-converse-A) _=a-statement, when.thel atetent_hasbeen ihown",bu not the converse. Thit: is a common error in-everiday reasoning.: Students ?-in the unior high school can ,-cAre to avoid this. On page 18 exercise 5c 'Chalk he extensions of the.segme and -.1H. If the profitably observe great' children stir' do ndt see that a transversal intersecting two parallel line- HI-and JK, erase every hint' in'the ',figure ext6Ot these three lines and ask again. If the pupils 's 1 don't See it,' 'ask them- tp skim through Informal Geometry I agai an tri4. to find a figure similar to this dne If necessary, you shoUld then ask themto reaUoVer the ,-statement Of..princiPles.fer,.that unit --again'td bee._whetherny ofthemApplIeS4,.to this-figure' and implies-that two angles in the figure are equal. Concurrent lines Some of the most interesti .Concerned with the concurPency'of lines nary geometry eats- assg dted: , pis-like to wor h-the,dra s r-consj--ruct=--- ---ions; This material also 16/10.'itselrnicelyrto -tl approah, and, pupils, in worki li wi -'this materiti come to sense some ore to orrliness ,..and;1 , in th14-Sec Nmuch case af, -geometric relationships made of measurement., and in the tors use should be made of the under-the If ,z the e puPils T -IX -9, er wishes to, Spend Ire time on this material . 'be encouraged t" experplent with. the bisecebrs. of the anglstand with the meaia neither of whiphSre men-deed in the text. .The concurrency of the btsectors of two the: exter 'angles other (interior) a triangle and 'the bisecter the. -angle of the triangle would provide for further-.study of this idea on the part of pupils who ga'y find it so -that-wish to give further time to-the..topic. A Right Triangle Principle' Most eighth grade texts include work- on the Theoi-em br Pythagoras and it is intended that the teacher use his own judgment in adding to the materials of this unit from what he' . . . may want to teach from his own text,. This comment ppli aril .ularly to the applications of this principle to which nO att tloh . is given here. In other words this topic 'is. treated more from the point of view of-mathematical Interest than of pradtical interest. While the steps used In r.aching the principle n nO.sense be Considered a proof, they do suggest a way, of leading up the principle by relating Lhe new experience to. previously learned principleT'which have been supported by infortal deduction from, other principles or w p- Amental evidence The teacher ka y Hnii,,o that the purpose of the comparison of the wo iquares is to arrive at the conclusion; --that ve'ueen accepted on the bases oT ex-.63 T -IX-lO where' r and axe the lengths-of the shorter sides of a righ igle and-t.iS'.-theXengttI!of ths_hyPotenuse. An -appeal i made' --11--AY ) 41 "iie,oim areas of squares to do this, which incidentally ggeats 2.. , .., 4 ,,, , , -% whyllig-M1 -t read r -r-squar0 The unit is writttn,.so. that the 4. student zwill arrive at this conOlusi,on through a r6ther than having A-poin d out to hIpa by the tea e and-. theexerclsks 4. '1 -proved. The order of should fi in gu e is iinportan,, draw the square uf7 units on a side, and A . ex then 4ach corner measure off the triangles as suggi?ated theme ,e t. It then temdips Lu argue 1.,11 aquae, This quadrilateral, call thp "square on In writing Lill., unit grayal4 the I equation: this point it cif using letters qz All right ta. as measwe-a. f the 2 a-o1L, A i"L 1.,0ers. where m and n are aLl'at.e,,,, ao Lh04 144 a. and b' may be interchanged ) 1,.01 oladrilateral repres EFGR, is a s what we usu ly 4r, , oduce are fu -ult in Lhe.foin,ot'an LLU students mater tie det )a, page z:), with _-ntegersv rvdm the,. formulas . obta.ir case t- on page ?5, L tic), an a L d n , T-IX-11. The Babylbnian tablet referred to on the previous page is Plimpton -% on plate American 322 Columbia,University Collection, and app ars 25-in Neugebauer and Sachs, Mathematical Cuneiform Text_ 4; Oriental-Society, New Haven, donnecticut, Pythagoras was born on the island of 'Samos about 572 B.C. In 6 532 he- migrated to, southern Italy and founded a semireligious N brotherhood at Croton. This society became mixed up in politic Rich lad -_to attack6 by enemies. ilthagaras_theni_moved to Meta-._ poritcpmwherelledied'abol..4495 B.C. You can obtain mere inform-atop abOdt his Work from Heath, Lk Hist-of Greek Mathematics Oxfordunivarsitypress.Plutarchls Lives is a very famous classic. It is also assumed that the pupil will not have had much. experiende with . the squares of numbers and the square roots of numbers. usually. The teacher should feel perfeOtly free to itVe a more ale-1 Ttle use the symbol, for square root has been avoided vete treatment if the pupils are prepared for it -- that is if they th the ideas of _square and square roots and areoalready familiar ,,theduse the symbol for square root. In ohder to avolu othe complications while the emphasis oV ttentlbn-i ,gip geometric Adeas=, it 'l suggested in t4p unit that a table of square runts here to enter Into a teacb .the q ist students, shothld have has -xptrinee w1 Lh read1ngw a table of square uld ullicJ about, the most appropriate method to it does Aeqm clear that all While it is not the place aft oxiMate It as read. mom a table ov 0 40 square root of 5, for example, cannot be emphasized too much. TIX-12 If this is the first experience 6f the pupil witti..square root he should be given some exercises in which he finds such products as 1.4 x 1.4-, 1.41 x 1.41, and 1.414 x 1:414. He should see that while none 'of these. numbers 1.4; 1.41, d 1.414 is a square root of 2, that each Of these gives an appruAtmatlon of the square root f 2, which may-be satisfactory fc, oome purposes. it would be well also to compare the product 1.41 x 1.41, which is less t6ah 2, with A --the product 1.42 x 1.42, wh shows that the square rout The pupil should also r-a number even, though he may n decimals or fractions, 2. This comparison (itriwhere between 1.41 and 1.42. he "square root f 2" is express it exactly using 4 Exercises 11 -1 d t- show that we can by geo metric considerstiLl., quare root of be omitted if thc o6 'h 10 the 1ic len(h 4, luny, course, Ansvters to the exercises -Part A. Congruent triangles. 6 B 5 L The measure A and L5 wigleo A, El, and C are, respectively, B-T 82 29' C T 41 241 5 cos A = 9/1b ,..)b B, 1 .125, cos C = 3/4 .--. 7 5_ The various dioLa,;4-4 mc.,y L. Led by the law of cosines: if a: b, and.c are the 16,0 ,i khc 61de6 of the trianglerABC opposite-' 1. the angle A, B, a1,0 u. vu.,1,..,iluly, then 0. a---p f 2bc eQS A, etc In order to Catleulai t. mI L4L r1 A, we have solved for cos' ,-, ki 2 ,; d LIO t lb 25 27 9 .,_ COS A -If we wish to calehIoLL 11, then we t)ply thio 10,V4 ,e f_ou x Lu E, F, G, and 11, t1 ui °e 67 T -IX -14 and obtain theArela ion 2 2 2 y c 16 +, x2 Which iield6 the table 2e x cos A x 2 3 11 4 414-5 18.5 3.534 3.317 3.391 3.742 = 4.301 The distances in the rest of the table may be comput6C1 in the same way. Of course, the children are supposed to obtain these distances ", _y measurement, Instead of c.dmputation. Ve have included here :this additional inf ornistic n for your benefit. There may by an occasional pupil who wouii pL f t from the discussion here. 68 T-IX 11 5 3.53 2 -4' 4.30 5.50 5.15 4.9 2 . 57 2.50 1 7 32 3 39 . 51-21 5.50 .2.57 3-67 5.1 l lA 7 2. 21' fig. 4.9 i .57 2. 5 0 2 8i _',4., . You the chlidren tri tie 'fIrst, Mw -, and.:, then 4 -. to find':_ easy ay , to ii 1n the c01 :,Try to see w loiig. -akcs the -pv.11.)114 tcidlicoVer tic --y-'_ , -1-1 reap i, the di agonai. Let the children eqrtiphr the,tribrigles JAL), -.-. , d :AG; the Lviangfes JAk -KAG, an , the,: gigs ,TAF uc alas trianglbs JAF .T6e triay not Oe that e famll 1 riangies- are similar,' a.nd the-of corresponding sides are proportional -IX-i6 are congruent eat,are congruent, but not congruent to.the le triang14'fs not congruent to those of numbers 1 and 3. files and the included side of one triangle are ,equal sute'tO2 angles and the included side of another triangles are congruent. and d , and f -.Cluded by the angles with the same. measure do not e measure are not necessarily congruent. EXERCISES IN EXPLORING TRIANGLES ri'ang1es appear to be congruent he triangles appear be Congruent on ba triangles -compared 6 tr nglev are congruent. All sides,are eqUal. All the trianglekare equilateral since; all theirangles have a measure :of'06,0egrees.--, DraW-a line segrner3t equal- to the ure,of one side, call At a thumb tack at <A and snot fit.. E. Tie knots in thetting-dividthg- the string into-lerigtbs'that have the same i4eaqtirep-as the -second and third sides of the triangle. Attach ihe'end:nots to the'thumb tacks at'A and B. Stretch the string tight-and'place a. thumb tack at the third Xnot-C'to form triangle AB C. Measutethent 6f-the angles of tie triangles is no CONDITIONS FOR CONGRUENCE OF TRIANGLES T-IX-17 tll:whther the triangles are congruent. If the corresponding angles of two triangles are equal in measure (case I), then the triangles similar. The qUotients of the lengths of correspond-, ing sidesdin the two triangles axe equal. For example, in the -triangles below, if a, then The sum Eit, and el are the length a' es of the' sides, Ad LIA4 al-161e0 of degrees. Thereiope, in ccAde V, it' equal in measure khe theh so are the Thus case V owl x,.;atAe,1 L-111. -Tiangle is 180 angles of one triangle are aing angles of another triangle, Jr Lhe triangles equal in measure. 1. Triangles acc triangle undei these The third angle C 9) Is only one way to linaw the co..A Clone t4.1 1 1 II, A' 11.0 44he_ tArt i..langleo are eon6ruent. 71 T -IX-18 .The possible' positionp,of the point C on a given side ,of the line AB such that BAC = 30 degrees form -a ray with endpoint at A. The set of all points C such that the distance from B to-C is 2 -inches is the'circle with center at B and radius 2 inches. The intersection of these two sets consists of 2 points, indicated by CI .and 0" above. The possible lengths of AC are AC 311S.-if 2 in case V, AC -÷r7 2 3.92. two ah61 e. ,41,0 opp.Jsite one of them one triangle are equal In meaL,u1-e L. Lhe correspondi_ parts of another triangle, then the triangles are congruent. In case VI, if L4q, oppoolte one or them one triangle 0.1'6. c.;t4k4,,I of another triagi no L ne essarily responding parts congruent. See ex Part D: II, (L) (g) (L) 1 C 614..1 ItiOUS E. No 3. No 4. 5-AO n2 d cnapt_r Ly eaal one principh, 4 principle 1 At nor = ACP - bur yes 11'11-'4'4111 L...) It (1) 4i 1G 14., 11 -IX-19 A54 by construction lesecongruent by PrAnciple 3. ABR and ABS are since th8y are ,equal in measure 0 sum of their measures is 180 is -the clppendicular bisector at RS. 10. No 12. .One is the converse of the other E and F lie on the perpendicular bisector at CO, by principle, There -is-no other line on E and F thus iS.trie pe'rPendicular bisector of CD. 7Pat 1, 2. One 4. The triangles e Congruent"-AB T.DC LA,T C PARALLELOGRAMS a. Principles 1 and 5 in this chapter b. Principle 1 if HJ is drawn c. HI and JK are parallel,. intersected by the transversal HJ. 11$110 HJK, since they are; alternate interior anglep (principle 7 in the previouphapter) Similarly, HK and IJ are parallel lines intersected by the transversal HJ, and FIJI M KHJ for thp same reason. d. -Triangle HJK is congrUent to triangle JHI, by Principle 2. a. Definition off' ral101ogram, b. PrinCiple 7 of he previous chapter c. Same reason d. Any number is ual Lo itself. e. 2 f. Principl_ 1 AB T CD. Alternat, angle at A and ,C B and D are equal in measure. HM = MC and DM = BM since triangles AMB and CMD are congruent by pi- inglple 2. , The sum of LI aleo.ouv.i 01 A' and B is 180'degree8. If AB is extended, then it !,6 peen to be a transversal,intersecting. the lines BC and AD in LAtch a way A,hatorresponding angles:have e tal measures. Therefore BC and AD are parallel, Similarly AB CD-are _parallelf and sc ABCD is a parallelogram, ..by finition. 73 T -IX -2O angles AMIQ.. and .CDA are congruent. Print 14)43 c. .ADB.-=-4'CBD and ABD = d. Principle 4 ©t .the ,previous -chapter. The quadrilateral is a parallelogram. If the diagonals' at a quadrilateral b quadrilateral is a parallelogram. ect each other", A 1 The statement.in number 8 is probably true . 1S Show.that triangles ABO an BCO are. congruent. Then angle AOB = angle.BOO,T -14 -False 15. Example: the diagon4ls Part G: I. (a) 4, a c and e The.perpendiculai The perpendicular The altitudes are ,Part H: 49 square inches 9, and 16 square inches '12 square inches.' then the' D-rectengle .are e in length. CONCURRENT LINES b) 1 u be uoncUrrent LI161,0 neurrent _y triangle concurrent COMPARISPV WP = DGP, AE = BF, .p.ngle Principle'l 74, B. Pi'inciple 4. es, Principle '4. X-21 180° goo 'The measures-of the area at squa BUM of ihe'MeaSure of the areas are the saMe.' e EFGH of figure 2 and' the the squares in figure 1 the area 'Of the-square 14.1tft length of sides EF is Measured by, the'sum bt the'-irea of the squares of the two shorter side the, triangle AFE. incries A RIGHT TRIANGLE PRINCIPLE b c 'e. f. g. h. i. J k. 1, M. 0. p% , This 7 8 20-12 9 28 11 6 16 48 13 36 9 8065 '52 20 '99 n .12 13 3 24 25 4 15 17 21 29 5 35 37 6 4,40 41 53 60 61 6 33 65 7 63( 65 8 55 73 .8 81 85 77 85 89 '97 101 lo 1 5 able was o.,14oLvuC 'are".ny two integeps 'pare the lengths 'of tni e the intity 9 2 a 1 2 1 4 2 5 4 1 3 6 2 5 4 d by using the fact that.if,m n, then m2 2 2 inn and m2 right triangle. You may' verify in Rpm i the habove mn and b m 2, right trite the length -s 2- 2 2 es, a- m n and b 2 mn, obtained in this'° -a-d in others is an interesting. theorem that all f Whose .sides dare integers, can be ON PYTHAGOREAN 'THEOREM 7".c The short, sides of the right triangle have tkle sane measure ,-,... .25-, 16 + 3 b . 169- = 25:.4.7:-14)4 d. LCOO . 256- '4-' 144 . -3.6 C. 13 3:61 -, if you know the measure of the area of a square:, yoll find i side byof the stquare root of , this measure., Measured o values f squdre roots should be approximately the s'aple as cOmpleted F UNIT ... Sampl_e!Test _Questions . PART I. jRug, E All isosceles triangles ngruent. Chao zias was -the- first man to- discover he-42-Theorem., All-parallelpgrams are qUadrilatera In ==trifle- -at --= the- side. RT is nude e a F 5 Three or more__ -2, p.otehus 'F. side of -the try right dRT.j. d_rilateral __ A an the- -four -sided figure a hagoragazi_ -point -are said' bre-!_concentric_. triangle always the !longest hown, at the right, if sides _and; CD are parallel .and a :ways be a arallelogram. sides 'T A triangle may_ never inallide , two right angles . : :F1 1 If eAtteasures of the four or --.-a h 4 re-, 0.--- al ayS-reotaugl . -ira 1elo ran are trifle= are_. coneurrent. = the measures "of :all _the angles of a-. - allelo are a: _.=. equaj4- the .figure IS always a_: square. a a F .1 re t the- rigYit' lines ,AB and If_ line parallel. is the perpen rdialar .77 A ." .ABs dlso bisects CD. he figure at right, if Iii is the:, pe ndicular _ tOr of lines-and D, --AB.Tehebri, are parallei. shown at the right Mark the following statementd true.Or falde-on the basis _ the inftirmation given 'above. DE 11-17.., CD $1 CF. F 1'8; .AE= F- 19. Angle T gle CE'., angle CFE. A T II. IviULTIPLE 'CHOICE the right-.is parallelogram. fihich of the ollowing-:StateMents cannot be proved? -d ACD are cangruen _ -le BDC .E3C 8.5 feet the measure of AC is ca tO the me sure of A. 5 0 feet. C. DC. 'y E; feet. How many ,concurrent there be on a poi 0 D. How m figures would need to draw to priove that 'the tildes of .a triangle are concurrent. Es. :You annet-prove.itstatement b g drawing -Inwhic one of the following does it ear that all -----7, 79 a es or a the f e-are-coneurrent. Iii 'th -6 -.Mown at the right --. BC...and-. AD 11-1 DC. It .BD is extended to meet AC on F, iE CE. BE a-DE. In the figure at the right, if CD is the perpendicular bisector and lines E[', CD, and GIB are parallel-, then... Art BG. AE T AC. AC BC BG. A o of tile above is correct right,c F; 4 L the perpendicular bisector of pi. FOin.VB W, X, Y, aid eZ repreiont pos.sib lOcation4 of a-:buried treasure. If the only cilia to the locatimi of the treasure ia- that it IS buried at. t a point which' measures an qual distance from points 0 and M Co above informatiOn, In the triangular MP r the base XY Z le:bora:triangle And t the. congruent,, is OW or the Wt eples triangles. tblidWihg statement is n0 correc .the 1r of tranlea at the =rl ht ang-, e OMIT- --rangle_ fz4and line MN M line YZ: What. is the least amount of igle NOM - angle Yati -_ and MO T xz, one-Of, the above will prove. the triangles.. congruent :15iieet di i1 intefteclion the sides °fa, rA . . 4baw'of the above maw eCorrect.--. . , chOf thgse-ia.ngleg a e71-1gbt.-tri es accordin t i..,._ . , ;nth of the sides given?. ',----; e righttrianglers. 61-1.owing'form-is Us04,VaLrePrise b Ans weils arR:both-. ff4 lqwin 'se is of thr :eastume sides of right triangidh? 4-90 10-Only and i#. , i 'of-.these ''are poisOle'%, n '..the''-tirith-igla at the whibh the kbilbwing stateMeh. atT true are, s Items 1. and -The fire are ebngrudn it g1-e t Is -the measure of -length. o (10 .ft.) as -is t measureof the--000 If on can place one Square ,on top' the four vertices of the first on the four vertices of the second that the two squares are congru IXZ is a telephone Pole; XY the length of a wire 'support and yz the distanCe along the gle ',Ant---A i,to 5 apply to the figures SHOWN 'BELOW. Decide whether the .17riantilesAABC and D will be congrue9t using Only, the infer-'nation- given. 4), each clues 1, Angle A-LI e. C DF , FE. le A B .amgfe E.City. ber of days,-given to the teaching. nclU .test of this Unit: State: -BAJTK OF THIS SHEET IF YOU NEED EXTRA SPACE TO ANSWER ANY OF THE:QUESTIONS BELOW -a-stateMent-about-th _lity -level of the -pupils in-the class arid state whether your school uses some :homo-geneoua, gxoupIngi. at parts of the unit probed to be the MD What -:parts of _teach?' -e unit proved to be the most' difficult . did you omit any part?-Did' you use any supplementary dev al Materials? -If so, what were they, and at what ,points were they used?-Did you find it necessary to pro al practice material? If so, was from textbooks or did yOu wr with addition-n? 16. Do you thirthat unit on this topic should 'be included in regular textbooks for 7th and 8th grades?-; Please Make vadditional comments about .your teaching ex-perience with this .unit. which you think would be helpful to the Panel-responsible, for preparing and experimenting with textbook materials for grades 7 and 8. fTeacher6-1 Coftment'S-,'. 'Four teachers -reported.On :Unit LX.. They gave the.,number;,6f: spent on the' unit-130 190 18, 18, and descrilled the .olaos"... A l ) e q.eteept,:for,onewhichwas, ma elipo 41 ted.P0 _The following ~topics were. reported as eazie Congruent'triang1e Perpendidular bisectors tight;triangles All of the .unit Concurrent lines-The folioliing topics were reported as most dill icult to-.te-achi COncurreno TheOrem-of thagor,s. MOst of the unit APPlications of cOngrueney Since only 3 teachers reported the comments are limited.. three seemed to -think that it was a.difficult _This was_ be expected-as the unit was intended to be for eighth'gradepuibils.,-1 "Proof didnot-well in any of the :classes'. e-A r CO ,1----prOdeisi of measurement plays such an _important -part -- ry -li fe that evetyone should have a =clear. under-. . stan4,ing of its nature. A, substantial part o sr ithmet Lc taug in the elementary -achool relates of the early=wow mes:surement h-lidrenirithOommozi-'-uirit a the 'ratios between them; Tile bails concept the ,process pf :.measur whiCh represents, the is des to nt. Moat ned to familiarize use;L, -ith contrast to= the proces oximate ntimber of count frig se parate-', Objects which -unit is that-yielddra number o1 units. to is .in ids an exact number iyhen.7the a .riii,lnded.tr estimated, the an 'approximation in \the Berle -measUrementa.,..are_ap.prcttima ments such as .areas unit Selected r cif separate objects ,numbOr is treated as sense as a measurement. Since _icalOttlaelone,_made with rrkeasur sults which are also- aproximsate. r, a -given measiiremeri t' should:' be suit--fOic-tjte-Ourbo8V-TOr.-/---4 able for the which the rfTiasuremen t aarily.a dard un sod. 016-un For,ex ne.,eibeys. the it hiiight of a perti on _ the neti!eit had '- inch-, that using_tb.e half' s h ae _ . 0 nit 'meaeuremet should be, etaed a° as ed. :A 'me tour ement reported:as--. 3/4- in. 111131 ies that thei.tn:iit Used was 1/4 inch, . _ the result. is atated" to the nearest'fburth-inch. The esuit, thus applies -1111a any measurement iyhidh is lithirk =thei, side- of-th-e-.-.2 -3 4inchlinerk t Wtfich'I.s.:: more ttiari.2 -aVe 8'anct'ress:than 2 718 inches We -the- term eateet possible -error" to refer Which the actual measurement may -vary from' the =. . The use of the- word "erica." may -Cause some .. difficultyd It shoupl -be emph.aSized that fhe word is used here to mean th-at a Msesur.ement such as 2,-3/4 repro7-. series of meamu_rements, ranging .frtizt 4-1 to'-'2 3 This Meeningt, shOuld be dist inguishe4 from e--more-familiar-uee- of--a-- term -to--mean mistakes -in--u the meitaiir in instrument 'mist ekes in reading the niista.kee result 4 g: Iran use. o .podrly neii4icea . ale .Or 7 a ;faulty' instrument, such ai. 1 -The two terms "precision" and accuracy" are sometimes . i colifusino. ...In comrrion ueage # preC5.sion rifers othe stz the nit. o use f-meesurement 'd.; the smaller. the unit the ---4 , so0 =-- 'Phis =means , of the .metti4emerit ,0 the small:er,- .. . ---. osaible 'error liriplied. kocuracy.-- of a raeasureMe -is = .__ he.,fier cent, the greatest possible ,error' le.- -OP: the' .-a resent;-L--'In O'.ider to, develop learly-.the' measurement and greatest .pOssible the students:be given cone 111h4S-- oATe- -nearest inch, . -rable dt Y inch, Tfiit k04 jt Ian inch, and so on. They. ould. priced:to-state. the= _ _ greatest poss ible error f measuremen -and , to ,indtOa e within; What range a statekmeksurement fall., The greatest possible error of the meat.Fement- of the area of a rei tazg1 computed from linear matsurements Es ship st by means of a ;drawing and then b .computation The dia.. _ -I ; tributive law-then ,used to analyie the .Vidduct of thsit.- two. :dimensions _in -order, to determi the-greatest pOssible error -the-oomputedarea --It would to -relate the develop ment :means of the distributive law to drawinj so. the areas represented by Vie various produoba (3 1/4 x 1 /B, and 6o-on) dan be identified in the drawing This especially Suggested in case7:the'stlidents. e.v =not already beoome.--tb.orOughly.' familiar With the distrib ;law!. T othe t imeip. have idif ic me to with the same significant reigardless of the unit of meas ng whY s have the same ac,curacy . dev loPment along the following Ines i .sdmet imed helpful. Significant- digit Szie. defined as the digits in the numeral which show the 'numbeiti! 611 nt unit , he -number of -unite and the .pos 3570 Unit : 10 fte. No. of rani 035.7 _.Unit .0001 ft. No Each these, meastir er nt To determine the a -curacy, cent( of 'error In the f %ea Unit si cent nine find the relative error1, or er have -the ratio J 5 50 ible : 357_ r Possible or: 5 f Poshible 'Ivor : 93095 ignifiCant. In the is cond case, ratio is 400050 which equals' Students also -m-urements cure .035'f eve difficulty In understanding why larger number of aignifica t . digits haVe y. An example imilar -to that %above can be 750 useds 57: ft. 'tnit : 01 ft.. Unmet ft. -Relative or. of -3.57 f R ela.t ive errer, of units 357. Possible 43 r or 005. -ft. . , . of units 3570 Possible or .3.57 ' - -35700. of 35.70 fft .. is -.005, or 5 35.70 5570 9, ,or .005it, eE sc.-ope by units of ms desirable -, ;this' -sonit might tob increased._ in he hi-tory cf standard reports 'on suoll. topics ,7 1, measu ,r varioui.messu ng devices used in industry and eaisncs :, ,c stall and,' MIS work of - the United_ States Butler, C FL and Wrsn, F. L. Teac in cf, econdary 'Maths-matte s 279-289. New York:. B o cik omp axr4rs;g5 9.2 iP The poi D. may and 1 7 (1_3 a). 178= b BetwOsn 2 9/32"_ 1/1 .= 1/16 on If._cf _the 1 ,1 " 1/24" 3 0 1 /Ui 2 rom be anywhere 'between -.1 5 /4 1/8) _.--) 1 1/16" and 1 /3.- 6" and 2 11/32" unit used .7/10 ± 1/20 3. inches_ 4.0 'inches 3.. - a) .1 ftl ; .05t. ,; b 1 ft. 05--0 .001 in., .0005 in.; 1 and 13 years old This rdepeThis, on when the the present ..mcinth(plus a) 4: 100 .ct,14.; 50 ft. b f ) ,a1 ft. ; :05 ft ,e is the most' precise;, a 4200 b )-244000 -1/4 1 /a ft 4-1! ,001 ln. .0004 iquegtion is as)ced, ,or minus three Motithil ,1 ft .; .5 'ft. ) .0001-ft. ; .110005 f is the least precise -,48000 000 ; .05ift -; -s more prec -e-mOre-'0°6-Pre, isior? 1, yes, d and 5 .005 cih .96 % .qC)01; 2 -.1 10- ft. :.91. in. _ p 1 0003. .01,-, i0001 .in ;,1 .' .001 f 10 f s 000 'rni . 001 .same accuracy; doe ste -have': same accuracy. . h a _e-Isignif 1-cant digits ; d ' 'the \ igrifiet -di its . \, h) _ 4 c i d):: 4 .: .41 1)).; -.01 c) : 14 .. .. 014 .;More. -a-ificant:digliks, thesmallier the__I-.ter ben , Of e 7 '812 in has the greatest 1-scoUrscy' and -2 ri has the .!. . .act:Itti t , $ 1 1-I'Larges Area q -1 Smallest Area ..- -2 eq. I "diiferencs 4,1, R sq. ip. Area aroduCt:-,wcula be 1 t -1/613. .1/4 1 we ,'spe ithat,(Athek"error:-Iri- -the number of aqua :unite the ,computed area is the Same (except- for sis -ems The Lama xi: tayfrhe .they a might '13P:43 ller b t sq.-_. y 4 -differ' by. as ,much' 8.S .eqUare inches inc + 741 le e ob ,considered be exa . 4 u--en er:the unit, the lore :Preelhe is the Tie: gresteSt possible error n :Would be oe-sixteen i k if .tile length, ox 8' m- cured' -to the neare -,T11.1 tiller ifie per cent6, er , the greater 4C-eili4ac the xneasiirement temeasuremerit of a line is stated to be 10 plies _th t the 1 rie was measured- to the nearest inch. nTUre-pr9clise-ti-le_Meaiurement; the greater: -is the pos sible error. ____Aimeas_urem_ent _of 300:4I s has tYie ssme greatedt pcissibe -error as a meastiremeht There are -four significant digits in the measurement 7,003 miles. The term."-greatest. pos ible error" of a mea u emen not-refeir to a mista.ke made in t measi The greatedt.possible_e,rror of the sum of several approk-imatp measurements is :th6 same. as the ,kreatest Possible error of t PART I The. Moat ,precise A. 261 inches. ifiches C 260 inches iththe greatest, accuracy 1 the one with he easte' possible error .per cent of error. n b of significant. digits., The gr test possible :error in the sure of 115.5 6,05. 5.1 is: B. i.f05- in. .0 int.. .055 -.005 in. ART 00FLETION. The meaiureineht, a line seignent-waa-, State-dtd-be-11---ineh 1 The measuPement 'might be sated= as 11- + 2 The eatedt, Rossibie error In a measurement is -always the' -wilt' used, v A measurement of 341 ineh.es Phasifs the measurement 6 Mea5u T MIS e- pred2sion as 'a.' -tr.,LANEQUS. length -oX-% the line te'gmen. at eigh h of an inchi A Suppose you have .iryeapured. th h 'f.Y a rectangle eaiiest nth of au inch. Explalm hole; would . proxim t§ greatedt- popa417e, err 1 ea. c;its he uparest ar Comput e the per cent of error in the meaeureMen-25iinghe:' cOmparewith'the-lier cent off error Pa-the measurement 25 miles? 14 ( .02/4.27-per gent 7
4626	https://mathstoshare.com/2023/02/13/concavity-of-the-squared-sum-of-square-roots/	Skip to content Concavity of the squared sum of square roots The -norm of a vector is defined to be: If , then the -norm is convex. When , this function is not convex and actually concave when all the entries of are non-negative. On a recent exam for the course Convex Optimization, we were asked to prove this when . In this special case, the mathematics simplifies nicely. When , the -norm can be described as the squared sum of square roots. Specifically, Note that we can expand the square and rewrite the function as follows If we restrict to with for all , then this function simplifies to which is a sum of geometric means. The geometric mean function is concave when . This can be proved by calculating the Hessian of this function and verifying that it is negative semi-definite. From this, we can conclude that each function is also concave. This is because is a linear function followed by a concave function. Finally, any sum of concave functions is also concave and thus is concave. Hellinger distance A similar argument can be used to show that Hellinger distance is a convex function. Hellinger distance, is defined on pairs of probability distributions and on a common set . For such a pair, which certainly doesn’t look convex. However, we can expand the square and use the fact that and are probability distributions. This shows us that Helligener distance can be written as Again, each function is concave and so the negative sum of such functions is convex. Thus, is convex. The course As a final comment, I’d just like to say how much I am enjoying the class. Prof. Stephen Boyd is a great lecturer and we’ve seen a wide variety of applications in the class. I’ve recently been reading a bit of John D Cook’s blog and agree with all he says about the course here. Share this: Click to share on X (Opens in new window) X Click to share on Facebook (Opens in new window) Facebook Like Loading... Related Solving a system of equations vs inverting a matrixIn "numerical-methods" The sample size required for importance samplingIn "numerical-methods" Sampling from the non-central chi-squared distributionIn "statistics" One thought on “Concavity of the squared sum of square roots” Pingback: Solving a system of equations vs inverting a matrix – Maths to Share Leave a comment Cancel reply Comment Reblog Subscribe Subscribed Maths to Share Already have a WordPress.com account? Log in now. Maths to Share Subscribe Subscribed Sign up Log in Copy shortlink Report this content View post in Reader Manage subscriptions Collapse this bar
4627	https://pmc.ncbi.nlm.nih.gov/articles/PMC7230878/	Follicle-Stimulating Hormone (FSH) Action on Spermatogenesis: A Focus on Physiological and Therapeutic Roles - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. 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Learn more: PMC Disclaimer \| PMC Copyright Notice J Clin Med . 2020 Apr 3;9(4):1014. doi: 10.3390/jcm9041014 Search in PMC Search in PubMed View in NLM Catalog Add to search Follicle-Stimulating Hormone (FSH) Action on Spermatogenesis: A Focus on Physiological and Therapeutic Roles Daniele Santi Daniele Santi 1 Department of Biomedical, Metabolic and Neural Sciences, University of Modena and Reggio Emilia, 41126 Modena, Italy; santi.daniele@gmail.com (D.S.); giulia.brigante@unimore.it (G.B.); livio.casarini@unimore.it (L.C.); vincenzo.rochira@unimore.it (V.R.) 2 Unit of Endocrinology, Department of Medical Specialties, Azienda Ospedaliero-Universitaria of Modena, 41126 Modena, Italy; giorgia.spaggiari87@gmail.com Find articles by Daniele Santi 1,2, Pascale Crépieux Pascale Crépieux 3 Physiologie de la Reproduction et des Comportements (PRC), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Centre National de la Recherche Scientifique (CNRS), Institut Français du Cheval et de l’Equitation (IFCE), Université de Tours, 37380 Nouzilly, France; pascale.crepieux@inrae.fr (P.C.); eric.reiter@inra.fr (E.R.) Find articles by Pascale Crépieux 3, Eric Reiter Eric Reiter 3 Physiologie de la Reproduction et des Comportements (PRC), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Centre National de la Recherche Scientifique (CNRS), Institut Français du Cheval et de l’Equitation (IFCE), Université de Tours, 37380 Nouzilly, France; pascale.crepieux@inrae.fr (P.C.); eric.reiter@inra.fr (E.R.) Find articles by Eric Reiter 3, Giorgia Spaggiari Giorgia Spaggiari 2 Unit of Endocrinology, Department of Medical Specialties, Azienda Ospedaliero-Universitaria of Modena, 41126 Modena, Italy; giorgia.spaggiari87@gmail.com Find articles by Giorgia Spaggiari 2, Giulia Brigante Giulia Brigante 1 Department of Biomedical, Metabolic and Neural Sciences, University of Modena and Reggio Emilia, 41126 Modena, Italy; santi.daniele@gmail.com (D.S.); giulia.brigante@unimore.it (G.B.); livio.casarini@unimore.it (L.C.); vincenzo.rochira@unimore.it (V.R.) 2 Unit of Endocrinology, Department of Medical Specialties, Azienda Ospedaliero-Universitaria of Modena, 41126 Modena, Italy; giorgia.spaggiari87@gmail.com Find articles by Giulia Brigante 1,2, Livio Casarini Livio Casarini 1 Department of Biomedical, Metabolic and Neural Sciences, University of Modena and Reggio Emilia, 41126 Modena, Italy; santi.daniele@gmail.com (D.S.); giulia.brigante@unimore.it (G.B.); livio.casarini@unimore.it (L.C.); vincenzo.rochira@unimore.it (V.R.) Find articles by Livio Casarini 1, Vincenzo Rochira Vincenzo Rochira 1 Department of Biomedical, Metabolic and Neural Sciences, University of Modena and Reggio Emilia, 41126 Modena, Italy; santi.daniele@gmail.com (D.S.); giulia.brigante@unimore.it (G.B.); livio.casarini@unimore.it (L.C.); vincenzo.rochira@unimore.it (V.R.) 2 Unit of Endocrinology, Department of Medical Specialties, Azienda Ospedaliero-Universitaria of Modena, 41126 Modena, Italy; giorgia.spaggiari87@gmail.com Find articles by Vincenzo Rochira 1,2, Manuela Simoni Manuela Simoni 1 Department of Biomedical, Metabolic and Neural Sciences, University of Modena and Reggio Emilia, 41126 Modena, Italy; santi.daniele@gmail.com (D.S.); giulia.brigante@unimore.it (G.B.); livio.casarini@unimore.it (L.C.); vincenzo.rochira@unimore.it (V.R.) 2 Unit of Endocrinology, Department of Medical Specialties, Azienda Ospedaliero-Universitaria of Modena, 41126 Modena, Italy; giorgia.spaggiari87@gmail.com 3 Physiologie de la Reproduction et des Comportements (PRC), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Centre National de la Recherche Scientifique (CNRS), Institut Français du Cheval et de l’Equitation (IFCE), Université de Tours, 37380 Nouzilly, France; pascale.crepieux@inrae.fr (P.C.); eric.reiter@inra.fr (E.R.) Find articles by Manuela Simoni 1,2,3, Author information Article notes Copyright and License information 1 Department of Biomedical, Metabolic and Neural Sciences, University of Modena and Reggio Emilia, 41126 Modena, Italy; santi.daniele@gmail.com (D.S.); giulia.brigante@unimore.it (G.B.); livio.casarini@unimore.it (L.C.); vincenzo.rochira@unimore.it (V.R.) 2 Unit of Endocrinology, Department of Medical Specialties, Azienda Ospedaliero-Universitaria of Modena, 41126 Modena, Italy; giorgia.spaggiari87@gmail.com 3 Physiologie de la Reproduction et des Comportements (PRC), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Centre National de la Recherche Scientifique (CNRS), Institut Français du Cheval et de l’Equitation (IFCE), Université de Tours, 37380 Nouzilly, France; pascale.crepieux@inrae.fr (P.C.); eric.reiter@inra.fr (E.R.) Correspondence: manuela.simoni@unimore.it Received 2020 Mar 10; Accepted 2020 Apr 2; Collection date 2020 Apr. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( PMC Copyright notice PMCID: PMC7230878 PMID: 32260182 Abstract Background: Human reproduction is regulated by the combined action of the follicle-stimulating hormone (FSH) and the luteinizing hormone (LH) on the gonads. Although FSH is largely used in female reproduction, in particular in women attending assisted reproductive techniques to stimulate multi-follicular growth, its efficacy in men with idiopathic infertility is not clearly demonstrated. Indeed, whether FSH administration improves fertility in patients with hypogonadotropic hypogonadism, the therapeutic benefit in men presenting alterations in sperm production despite normal FSH serum levels is still unclear. In the present review, we evaluate the potential pharmacological benefits of FSH administration in clinical practice. Methods: This is a narrative review, describing the FSH physiological role in spermatogenesis and its potential therapeutic action in men. Results: The FSH role on male fertility is reviewed starting from the physiological control of spermatogenesis, throughout its mechanism of action in Sertoli cells, the genetic regulation of its action on spermatogenesis, until the therapeutic options available to improve sperm production. Conclusion: FSH administration in infertile men has potential benefits, although its action should be considered by evaluating its synergic action with testosterone, and well-controlled, powerful trials are required. Prospective studies and new compounds could be developed in the near future. Keywords: FSH, spermatogenesis, male infertility 1. Introduction The pituitary gland regulates human reproduction through the refined, combined action of the follicle-stimulating hormone (FSH) and the luteinizing hormone (LH) on the gonads. Upon transport in the bloodstream, FSH reaches the gonads, stimulating follicle development in females and spermatogenesis in males. While the FSH role on reproductive physiology in both sexes is clear, its therapeutic use for the treatment of infertility remains doubtful, especially in men. FSH is mostly used in the therapeutic context of assisted reproduction techniques (ART) for inducing multi-follicular growth in women, while the efficacy of FSH administration in infertile men is unclear. In clinical pictures characterized by deficient levels of FSH, such as male hypogonadotropic hypogonadism (HH), restoration of FSH action improves quantitative and qualitative sperm parameters and, therefore, fertility . The therapeutic approach to male HH aims at restoring the physiological functioning of the testis and can be obtained by the administration of gonadotropin-releasing hormone (GnRH), which mediates the endogenous, pituitary production of both FSH and LH. Proper serum gonadotropin levels may also be achieved by exogenous administration of these hormones, typically human chorionic gonadotropin (hCG), which, acting on the same membrane receptor for LH, the LHCGR, was assumed to be identical to LH for long time . hCG administration may be supported by adding FSH activity with either urinary-derived or recombinant follitropins . In HH patients, the increase of serum gonadotropin levels is suggested to be beneficial for male fertility, since the improvement of sperm parameters would have a positive impact on pregnancy rate. Although the full restoration of normal semen parameters is generally not achieved in HH men (depending on HH causes) and not even necessary to obtain a pregnancy, gonadotropin treatment is sufficient to stimulate spermatogenesis and provides a rationale for using, even if empirically, FSH, with or without hCG, in male idiopathic infertility, which is considered a form of “functional” hypogonadism. By analogy with the therapeutic benefits observed in HH males, exogenous FSH administration has been proposed in men with altered sperm production and FSH levels within the normal range, assuming its capacity to increase the spermatogenic output . In the present review, we evaluate the physiological role of FSH in spermatogenesis, highlighting its synergic action with testosterone. These concepts are fundamental to understand the potential pharmacological benefits of FSH administration in clinical practice. 2. Physiological Control of Spermatogenesis Spermatogenesis occurs within testicular seminiferous tubules in a stepwise fashion, requiring autocrine, paracrine, and endocrine stimuli that are controlled by both FSH and LH actions [6,7]. FSH is a glycoprotein formed by two subunits: alpha, shared with other glycoprotein hormones, and beta (FSHβ). The human FSHβ is encoded by the FSHB gene located on chromosome 11p21 . Through interaction with its receptor (FSHR) , FSH acts on its unique target in male cells, namely, the Sertoli cells, located at the basis of the seminiferous tubules of the testis [10,11]. These cells create a niche in which spermatogonia proliferate and mature [10,11]. Sertoli cells are connected together and to neighboring germ cells by gap junctions, permitting metabolite exchanges. Importantly, tight junctions, located at their basis, form the blood–testis barrier, isolating meiotic and post-meiotic germ cells from the bloodstream. In Leydig cells, LH stimulates testosterone production through the interaction with its specific receptor, the LHCGR . Testosterone achieves 50–100-fold higher concentrations within the testis than in peripheral circulation . All these aspects point out the importance of the testis environment for the support and maintenance of the spermatogenetic function [13,14]. The physiological role of FSH in spermatogenesis regulation was evaluated in different animal models. In rodents, FSH determines the final Sertoli cell number at puberty, by stimulating cell proliferation during fetal and neonatal life, whereas, in primates, this mitotic function is observed during neonatal and peri-pubertal stages . Consistently, early in life, FSH stimulates the transcription of genes involved in both DNA replication and cell cycle regulation . In humans, the FSHR is first expressed during the second half of gestation, but its activation occurs after the onset of FSH secretion in the newborn . Then, the peri-pubertal rise of FSH stimulates Sertoli cell proliferation . In adulthood, FSH drives Sertoli cells to produce regulatory molecules and nutrients required for spermatogenesis . In particular, FSH activates the transcription of genes involved in metabolic homeostasis and supports germ cell functions , with the synthesis of retinoic acid, lactate, type 2 plasminogen activator, as well as fatty acid metabolism and mitochondrial biogenesis [19,20]. FSH circulating levels correlate directly with Sertoli cell number and testicular volume in adults . Beyond the known FSH action on Sertoli cell proliferation, the precise role of this gonadotropin in spermatogenesis remains largely unclear. Genetically modified mouse models have been useful to better understand how FSH regulates spermatogenesis. In particular, in the adult mouse testis, FSH stimulates Sertoli cells to produce anti-apoptotic survival factors and adhesion molecules, facilitating germ cell maturation [22,23]. However, the absolute lack of FSH or FSHR, despite a reduction in Sertoli cell number, does not lead to azoospermia nor sterility [24,25]. In fact, Fshb or Fshr knockout (KO)-mice have reduced, yet persisting, sperm production. More precisely, the lack of FSH action leads to reduced Sertoli cell numbers but germ cell maturation persists, despite a decreased number of spermatogonia and spermatocytes. With this in mind, animal models suggest that FSH is required to elicit Sertoli cell proliferation and to maintain germ cell numbers, probably through the ability of Sertoli cells to nurture germ cells, whereas its action is dispensable to complete spermatogenesis. Since the endocrine regulation of gonadal functions in mice could differ from humans, data from rodent models should be considered in their context. In some cases, inactivating homozygous FSHR mutations in humans were associated with infertile male phenotypes , although these mutations are very rare, and the issue is a matter of debate [25,27] and merits additional investigations. However, it is also worth noting that the A189V inactivating mutation is related to male oligozoospermia/subfertility, not necessarily azoospermia . Wide evidence is available in the scientific literature, demonstrating the central role of LH in supporting spermatogenesis via induction of intratesticular testosterone production. Indeed, LH receptor (Lhr)-KO mice models have no testosterone production and exhibit an impaired progression of spermatogenesis that is restored after testosterone replacement . Thus, unlike FSH, the absence of LH-dependent testosterone production leads to azoospermia, suggesting that testosterone is strictly required for sperm production. These experimental models suggest that a quantitatively and qualitatively normal spermatogenesis requires the action of intratesticular testosterone in synergy with FSH. The existence of synergism between testosterone and FSH is also suggested by in vitro studies, since both testosterone and FSH regulate the expression of genes involved in blood–testis barrier development and functions, while FSH supports the organization of tight junctions and ectoplasmic specializations [16,31,32], suggesting that the synergistic action of FSH and testosterone regulate different levels of the spermatogenic processes . New insights on the nature of this synergy came from the Fshr-D580H mutation in Lhr-KO background mutant mice . The Fshr-D580H mutation, resulting in constitutive Fshr activation, reversed the azoospermia due to missing LH action obtained by the deletion of the Lhr gene, combined with the blockade of the residual testosterone activity by the antiandrogen flutamide . These data demonstrate that the constitutive activation of Fshr-mediated signaling may overcome the absence of both LH- and testosterone-mediated pathways. Actually, the continuous hyperactivation of FSH-like signals sustains the transcription of androgen-dependent genes and activates the spermatogenic pathway in a LH- and testosterone-independent way . Overall, the existing literature in other animal models suggests that signals induced by gonadotropins may be non-specific and redundant since in some species spermatogenesis occurs in the absence of FSH [24,25], indicating that the action of intratesticular testosterone is sufficient for supporting sperm production in physiological conditions. Although these effects might occur only in specific experimental settings [35,36], FSH is supposed to play pleiotropic roles on gametogenesis. This FSH-dependent gonadal regulation would be sex-specific, since the hormone is instead indispensable for follicular growth beyond the pre-antral stage in females . In humans, the physiological framework in which FSH acts is even more complex , and the effective action of the hormone on spermatogenesis remains poorly understood. The few data describing the effects of impaired FSH action in the human are provided by FSHR and FSHB inactivating mutations . The rare FSHR mutations described in men suggest they lead to subfertility with testosterone levels within the physiological range and reduced spermatogenesis, but not necessarily to azoospermia. This result reflects the clinical picture in Fshr-KO mice models, suggesting that spermatogenesis may occur in the absence of proper FSH action also in humans, while azoospermia occurs in all cases of inactivating FSHB mutations described so far . However, considering the paucity of FSHB and FSHR mutations described in men, we are currently unable to clarify the exact role of FSH for human spermatogenesis. Human inactivating LHCGR or LH beta (LHB) mutations are linked to 46, XY disorders of sexual development (DSD) phenotypically ranging from the complete feminization to less severe or no sexual abnormalities . In certain cases, LH signaling defects not fully inhibiting the hormone action lead to oligozoospermia and low intratesticular testosterone levels, but not azoospermia [40,41]. Thus, in the presence of physiological FSH production, very low levels of LH could be sufficient for sustaining intratesticular testosterone and sperm production in humans. These data may be suggestive of a possible FSH and LH synergistic action based on the redundancy of the gonadotropins’ signaling machinery [42,43]. On the other hand, human activating LHCGR mutations lead to increased testosterone levels associated with continuous production of spermatozoa, despite reduced FSH levels . It may be speculated that, during evolution, spermatogenesis has been maintained under partial dependence of both FSH and LH, at least in certain species , ensuring sperm production despite the loss of the action of one gonadotropin. Information about the physiological role of FSH in spermatogenesis comes from mammalian models of hemicastration, including primates, where a significantly increased volume of the contralateral testis occurs [45,46,47,48,49]. Considering that the adult Sertoli cell numbers do not change, the increase in testicular size is reasonably due to the growing amount of germ cells . Unilateral orchiectomy results in inhibin B decrease, resulting in endogenous FSH rise and support of sperm production through increased stimulation of B spermatogonia proliferation . Similar data were described in men after removal of a testis with malignant cells, where contralateral testicular hypertrophy occurred . In this setting, testicular hormonal secretion and sperm production returned to physiological levels within 120 days after hemicastration, confirming the compensatory increase of spermatogenesis in the remaining testis . These data are corroborated by studies of patients affected by pituitary FSH-secreting adenomas, which showed elevated testicular enlargement likely due to FSH over-stimulation, even when occurring after puberty . In summary, human spermatogenesis (Figure 1) is physiologically regulated both by FSH action and, synergistically, by LH-dependent intra-testicular testosterone, which, in turn, acts in Sertoli cells through the androgen nuclear receptor (AR). These mechanisms are exerted through the activation of partially overlapping intracellular signaling pathways since the absence of either gonadotropin does not necessarily determine azoospermia. However, qualitatively and quantitatively adequate spermatogenesis requires FSH action and high intratesticular testosterone levels, acting in additive and synergistic ways . These data provide new perspectives in the therapeutic approaches to male infertility, similar to what is currently done in women undergoing ART . Figure 1. Open in a new tab Schematic representation of a hemi-section of a seminiferous tubule. From the basal membrane to the lumen, spermatogenesis stages are reported up to the mature spermatozoa. This complex process is regulated synergistically by follicle-stimulating hormone (FSH) acting on Sertoli cells and by intra-testicular testosterone, produced by Leydig cells under luteinizing hormone (LH) stimulus. (FSH = follicle-stimulating hormone; n = haploid cell; 2n = diploid cell) 3. Mechanism of Action of FSH in the Sertoli Cell Many studies have aimed at deciphering the signaling pathways induced by FSH from birth to puberty in order to gain insights into the dynamics that govern the switch between the hormone mitotic and differentiating functions. Coupling of the FSHR to the G protein α subunit (Gα s)/cyclic adenosine monophosphate (cAMP)/cAMP-dependent protein kinase A (PKA)/cAMP-responsive elements binding protein (CREB) signaling pathway has been acknowledged as the sole effector mechanism of FSH for more than forty years . In Sertoli cells, FSH regulates the intracellular concentration of cAMP, resulting from the equilibrium between its synthesis by adenylate cyclase and degradation rate by phosphodiesterases (PDE) [54,55,56]. The intracellular concentration of second messengers is also fine-tuned by desensitization of the FSHR, which occurs within minutes upon FSH exposure [57,58], and it results in uncoupling of the ligand-bound receptor from G proteins. In vitro treatment of Sertoli cells by FSH leads to the recruitment at the receptor of kinases that target G protein-coupled receptor (GPCR), namely the G protein-coupled receptor kinases (GRKs) [57,58]. β-arrestins are then recruited to the FSHR as the result of both phosphorylation and agonist-induced conformational changes of the receptor. β-arrestins promote internalization by virtue of their ability to interact with components of the clathrin-coated pits. This desensitization process is of clear physiological relevance because, in the living organism, Sertoli cells are permanently exposed to FSH, being the receptor localized at the basal pole of the cell, in contact with the basal lamina and blood capillaries (Figure 2). Figure 2. Open in a new tab Early FSH-dependent signaling and receptor internalization. (A) FSHR extracellular domain is exposed to FSH binding in the cell membrane. (B) FSHRs may form multimers in the cell surface and, upon FSH binding, undergo conformational changes inducing activation of the Gα s protein and subsequent ATP to cAMP conversion by the adenylyl cyclase enzyme. The downstream PKA, pERK1/2, and pCREB activation target CRE sequences, triggering gene expression. (C) The GRK enzyme is recruited via a mechanism involving the βγ dimer of the G protein and mediates FSHR phosphorylation at specific intracellular sites. (D) β-arrestins form a signaling module associated with FSHR and ERK1/2, mediating the assembly of clathrin-coated pits and internalization of the receptor. Besides this β-arrestin/clathrin-related mechanism, the GTPase dynamin may participate in the internalization of FSHR. (E) FSH complexed with the receptor is internalized and routed to endosomal compartments sustaining the prolonged activation of signaling cascades. In the early 2000s, a number of discoveries have highlighted that FSH actually induces a complex network of molecular interactions within the cell, which largely exceeds G protein-dependent signaling. The coupling of the FSHR to the Gα s/cAMP/PKA signaling pathway is now viewed as one among several mechanisms contributing to the activation of the whole hormone-induced signaling network, where cAMP-dependent and cAMP-independent signaling co-exist. In Sertoli cells, promiscuity of G protein coupling has been demonstrated so far as the only way to promote cAMP-independent signaling. For example, the FSHR couples not only to Gα s but also to Gα i, which is involved in the post-natal mitotic response of Sertoli cells to FSH via extracellular-regulated kinase (ERK)/mitogen-activated protein kinase (MAPK)-dependent signaling . These data are supported by recent evidence that FSH limits the expression of Gα s, which fails to downregulate Gα i activity in infant primates . In this line, a dynamic computational model predicted that PKA inhibition does not compromise phosphatidylinositide(3,4,5) trisphosphate (PIP3) production in pre-pubertal rats . PKA-independency of this pathway has been validated experimentally by studying the activation of protein kinase B (PKB/Akt), a target of PIP3 kinase (PI3K) in Sertoli cells . ERK and Akt are regulated by FSH in an opposite manner during Sertoli cell post-natal development [59,61]. The tyrosine kinase receptor phosphatase Src homology 2-containing phosphotyrosine phosphatase (Shp2) is likely involved in this dual regulation of ERK and Akt by FSH, and disruption of this pathway in Shp2 knock-out mice leads to infertility because of abnormalities in the blood–testis barrier and premature exhaustion of the germ cell population . Numerous studies employing immortalized cell lines have opened the possibility that β-arrestins could be a major scaffold for many signaling modules of GPCRs in general , and of the FSHR in particular [65,66,67]. Their action is important because they sequentially dictate the kinetics of ERK activation profile, which is more sustained than the transient Gs-mediated ERK activation . In contrast, β-arrestins and Gα s act cooperatively to stimulate the ribosomal protein S6 kinase beta-1 (p70S6K) in response to FSH in a complex including one of the enzyme substrates (ribosomal protein S6) . Although p70S6K and β-arrestin 1 are identified in common protein complexes in Sertoli cells, to date, the role that β-arrestins could play as regulators of the kinetics and spatial organization of FSH-dependent signalling network has not been conclusively proven in these cells. However, the fact that oligozoospermic men bearing the inactivating Ala189Val mutation in the FSHR retain β-arrestin-dependent but cAMP-independent signaling suggests that β-arrestins may play a role in supporting fertility. Strikingly, in Sertoli cells, second messenger levels appear to be developmentally regulated, with the efficacy of FSH-induced cAMP production rising from birth to puberty , whereas the sensitivity to FSH of PIP3 production decreases over time . For example, in FSH-stimulated Sertoli cells, p70S6K is regulated by a subtle interplay between PKA- and PI3K-dependent signaling that operates in a developmentally-regulated manner . PIP3-dependent molecular events, such as the mammalian target of rapamycin (mTOR) and proline-rich Akt substrate of 40 kDa (PRAS-40) phosphorylation, appear to be involved in the mitogenic response to FSH, whereas 5′ AMP-activated protein kinase (AMPK) and phosphatase and tensin homolog (PTEN) counteract this effect [71,72]. However, the mechanisms whereby the FSHR couples to PIP3 regulation is still unclear. The FSH signaling network dictates the protein content of gonadal cells through long-term transcriptional regulations directly affected by transcriptional factors or chromatin remodeling, and through short-term post-transcriptional regulations at the level of RNA messenger (mRNA) translation and micro RNA (miRNA) turn-over. DNA microarray analyses of FSH-treated prepubertal rat Sertoli cells in vitro revealed effects on the steady-state level of many target mRNAs, as early as after 2 h of treatment . Suppressing FSH by a four-day passive immunization with neutralizing Abs led to the identification of hormone-regulated transcripts, among which some are well-known FSHR-responsive genes in Sertoli cells, including genes involved in cell cycle and survival regulation, such as cyclin D1. Comparable results were obtained in hypogonadal (hpg) mice, a convenient model to analyze the effect of exogenously administrated FSH on gene regulation in vivo [74,75]. Recently, the FSH-regulated gene expression landscape has been described not only in Sertoli cells from neonate or prepubertal rats but also in adults, when these cells fully provide metabolic and physical support of the germline . As expected, in immature Sertoli cells, genes involved in cell growth and proliferation, metabolism, and the MAPK and Wingless-related integration site (Wnt) pathways were upregulated, whereas in mature cells, genes relevant to the differentiated function of Sertoli cells, involved in phagocytosis, cytoskeleton remodeling, glucose metabolism, and insulin signaling, were expressed. In this study, FSH was delivered with testosterone in a pulsatile fashion, expected to amplify the gene responses, as previously described . This observation is very interesting because it suggests that hormone target cells decode a pulsatile signal vs. a monotonous signal in a quantitatively different manner. These transcriptomic analyses have provided an atlas of genes regulated by FSH signaling in vitro/ex vivo, but they indicate neither the transcription factors that could recognize the gene regulatory regions nor the upstream signaling pathways that could be involved. Systemic analyses of the global FSH-induced network from the receptor to the target genes will upgrade our understanding at the molecular level of FSH-induced biological responses. For example, the requirement of CREB in transcriptional regulation has to be reconsidered because not so many FSHR-responsive genes include a cAMP-responsive element (CRE) in their promoter regions. Hence, several PKA-mediated transcriptional responses could be mediated by other transcription factors, such as retinoic acid receptor α , a well-known regulator of germ cell development in the testis. The anabolic role of FSH in Sertoli cells is well established; however, the whole FSH-induced proteome is not available yet. However, several reports have suggested that FSH regulates the translation efficacy of pre-existing mRNA in Sertoli cells from pre-pubertal rats. For example, FSH stimulates the mTOR/p70S6K pathway, inducing the phosphorylation of regulators of translation initiation, such as the scaffold proteins eukaryotic initiation factor 4B and G (eIF4B and eIF4G, respectively). These molecular rearrangements are linked to recruitment of specific mRNAs to the polysomes, such as the c-fos and vascular endothelial growth factor (VEGF) encoding mRNA . In agreement with this trophic role, in transfected HEK293 cells expressing FSHR, the hormone enhances the translation of a particular subclass of mRNA, mainly encoding proteins of the translational machinery, the 5′TOP encoding mRNAs. However, polysome-bound mRNA identified by the riboTag technology in adult testis in vivo detected negligible changes in the amount of FSH-induced mRNAs . MiRNAs constitute a bona fide network, intertwined with cell signaling networks in the cell. In this line, Sertoli cell-selective knock-out in mice of the gene encoding Dicer, an enzyme involved in miRNA processing, has unraveled the role of miRNAs in regulating the expression of genes essential for meiosis and spermiogenesis . A rat model, where FSH and testosterone action was suppressed in vivo, has been created to identify the miRNA network at spermiation , a stage that is particularly sensitive to hormone regulation. Four of the regulated miRNAs that came out from a miRNA microarray analysis were complementary to the PTEN mRNA, and the hormonal input would lead to the degradation or synthesis inhibition of these miRNAs, then stabilizing PTEN at spermiogenesis . Interestingly, the PTEN protein level is massively enhanced following FSH cell stimulation in vitro, leading Sertoli cells to achieve terminal differentiation . Since FSH enhances PTEN protein level within minutes, the mechanisms involved probably occur post-transcriptionally, an assumption consistent with the hormone-induced degradation of miRNA that prevents the accumulation of PTEN locally in Sertoli cells. Hence, miRNA networks might regulate the compartmentalization of FSH signaling components in Sertoli cells and control the kinetics of these intracellular reactions. Sertoli cells represent a paradigmatic model of transition between a proliferative state and commitment to differentiation that is controlled by FSH [81,82] and by other factors like thyroid hormones , which both act antagonistically. Seminal knock-out experiments in mice have demonstrated that thyroid hormones are the master signal that arrests Sertoli growth via the α isoform of the thyroid hormone receptor [84,85,86]. Local autocrine/paracrine secretion of other factors, such as glial cell-derived neurotrophic factor (GDNF) , activin A, and insulin-like growth factor (IGF)-I also synergize, or at least complement, FSH action. In vivo ablation of the IGF system (both IGF-R and Ins-R knock-out) have convincingly entrenched that it is required for FSH-mediated mitogenic action in the pre-pubertal mouse . Based on previous observations, it can be assumed that FSH, IGF-I, and triiodothyronine (T3) directly impact on cell cycle regulators. For instance, both FSH and IGF-I induce the expression of cyclin D1 and D2 [59,88], promoting the cell cycle progression through the G1 phase, and downregulating the transcription of two cell cycle inhibitors, namely, the p15 Ink4 gene and, presumably, the p21 Cip gene , via p53 dephosphorylation . This action on the positive regulators of the cell cycle is counteracted by T3, which downregulates the expression of cyclin-dependent kinase 4 , whose expression is upregulated by FSH in granulosa cells . T3 likely inhibits molecular events occurring in early G1, and not later on, because neither the E, A and cyclins B nor cyclin-dependent kinase 2 (Cdk2) are modulated by T3 (Figure 3) . T3 acts not only through thyroid hormone receptor (TR)α1 but also through mitochondrial p43 receptors [84,85,91]. Figure 3. Open in a new tab Triiodothyronine (T3) counteracts the mitotic response to FSH, by inhibiting the transcription of Cdk4, via the JunD transcriptional regulator. Thus, the association of Cdk4 to FSH-induced CycD1 is hampered, and Sertoli cell mitoses cease. As indicated above, a number of studies established that FSH and testosterone are both mandatory for spermatogenesis to proceed appropriately. Interestingly, recent investigations on the respective role of FSH and testosterone by cell-specific knock-out of their respective receptor have shown that the extent of the Sertoli cell population is primarily determined by FSH, enabling the progression of germ cells through meiosis in concert with testosterone [33,93]. The steroid is also involved in suppression of Sertoli cell proliferation by targeting inhibitors of cell cycle progression . 4. Genetic Regulation of FSH Action on Spermatogenesis The gene encoding the FSHR is located on chromosome 2p21, including ten exons and nine introns . The first nine exons encode the extracellular receptor domain, mainly involved in ligand binding. Exon 10 encodes the hinge region, the seven transmembrane-spanning domains and the intracellular, carboxy-terminal tail . Once FSH binds the extracellular FSHR domain, the receptor undergoes conformational changes, activating several intracellular signaling pathways , as described above. Thus, spatial conformations of FSHR specifically linked to single nucleotide polymorphisms (SNPs) may modulate the receptor functionality. More than two thousand SNPs have been detected within coding and non-coding regions of the FSHR gene, and two of them have been repeatedly evaluated. The adenosine to guanine change at position 919 from the start codon (c.919A > G; rs6165) and the c.2039A > G (rs6166) SNPs are located in exon 10 of the FSHR gene, leading to the amino acid changes p.T307A and p.N680S, respectively, in the receptor protein chain . The physiological impact of FSHR SNPs was first described in men , although it was extensively studied in women undergoing assisted reproduction. The FSHR p.N680S homozygous S (S/S) genotype was related to higher FSH basal serum levels and to a higher number of FSH ampoules required to achieve similar estradiol levels during ovarian stimulation compared to the p.N680S homozygous N (N/N) genotype in 161 ovulatory women . This was the first demonstration of a possible role of SNPs on female reproduction, suggesting that the presence of the S/S variant renders the FSHR less sensitive to FSH . In vitro studies on human primary granulosa cells highlighted that the polymorphic variant of the receptor mediates different kinetics of the response to FSH stimulation . Indeed, the homozygous FSHR S variant leads to a slower increase of intracellular cAMP than the p.N680S homozygous N receptor variant, although similar plateau levels were achieved after one hour of treatment . Moreover, the FSHR genotype-specific cAMP kinetics impacts the activation of other molecules, such as ERK 1/2 and CREB, the expression of target genes, and progesterone production , reflecting clinical data demonstrating that the FSHR SNP p.N680S is a marker of ovarian response to FSH . Although the role of this SNP on female reproduction was further confirmed in other studies , several years passed before its potential role in man was demonstrated. In 2012, the first demonstration of a modulatory activity of the FSHR p.N680S SNP on male fertility was produced, reporting a lower testicular volume in two large cohorts of the p.N680S homozygous S carriers compared to the N/N and heterozygous (N/S) patients [103,104]. Since a relatively high sample size was required for consistently detecting the physiological effect in men, it may be concluded that the p.N680S SNP has a weak impact on male fertility . The FSHR promoter SNP, falling 29 nucleotides upstream of the transcriptional start codon, −29G > A (rs1394205), is another common polymorphism potentially impacting FSHR function [106,107] through the modulation of the gene transcriptional activity [108,109]. Interestingly, other SNPs involved in the gonadal response to FSH were identified within the FSHβ-encoding gene promoter. The FSHB −211G > T (rs10835638) is linked to reduced gene transcription and serum FSH levels . Many authors have evaluated the FSHB c.−211G > T effect in both sexes, finding that, in men, it is associated with lower testicular volume, sperm count, testosterone, and LH serum levels . The gonadal response to FSH would depend on FSHR functioning and number in the cell membrane, together with the amount of ligand available. Therefore, the combination of SNPs modulating the FSHR and FSHB gene transcription and receptor functioning may be predictive of the gonadal response to the hormone in vivo. A recent case–control study evaluated three FSHR SNPs (c.919A > G, c.2039A > G, and c.−29G > A) in 255 infertile men and 340 healthy controls . Although the frequency of allelic variants was similar between the two groups, a specific haplotype was detected more frequently in fertile men (i.e., −29G > A G allele, c.919A > G A allele, and c.2039A > G A allele) . Moreover, a meta-analysis of twelve studies available in the literature confirmed that the presence of either c.919A > G G (p.T307A A) or c.2039A > G G (p.N680S S) alleles is associated to an increased risk of male infertility . This result suggests that different genetic models should be considered for inferring the combinatorial contribution of FSHR SNPs in male infertility, including dominant, co-dominant, and recessive models. Previous studies examining potential associations between FSHR and FSHB SNPs and male fertility parameters have produced contradictory results, although they suggest that the FSH action on spermatogenesis is likely regulated by a combined effect of many SNPs . Further evidence could be obtained by new studies in which complex haplotypes calculated using several SNPs falling even within genes not involved in the control of the FSH action. Moreover, available studies did not collect all fertility-related clinical parameters , limiting our current understanding of the impact of SNP combinations on male reproduction. The association between spermatogenetic phenotypes and genetic background could be necessary for developing patient-targeted therapeutic approaches. 5. Therapeutic Options to Improve Sperm Production HH represents the main context of FSH application in males in clinical practice. This disease is due to congenital or acquired impairment of hypothalamic GnRH production and/or pituitary gonadotropin secretion [111,112,113,114]. The reduced gonadotropin stimulation results in low sex hormone secretion and sperm production. There is compelling evidence that HH patients benefit from either GnRH or gonadotropin replacement therapy, which results in an improvement of both spermatogenesis and androgenisation [115,116]. GnRH can be administered in a pulsatile fashion, subcutaneously via a portable infusion pump. This represents the most physiological approach to replacing gonadotropin stimulation of the testis. Pulsatile GnRH administration can restore the development of adult secondary sex characteristics, normal testosterone serum levels, and spermatogenesis [117,118,119,120]. However, in spite of being the most physiological option, this treatment is not often used outside highly specialized centers, mainly in the United States, due to the prolonged use of an external pump that can be inconvenient and costly to maintained. Instead of GnRH administration, gonadotropins could be directly used in HH. In particular, clinical benefits have been reported using hCG alone or associated with FSH . In this setting, a recent meta-analysis evaluated the effects of gonadotropin administration on sperm concentration in HH subjects . Ten studies provided hCG alone, fourteen provided concomitant hCG and FSH treatment, and twenty-three introduced FSH at a variable time point after hCG monotherapy . The comprehensive result showed a beneficial effect of all treatment regimens in almost 75% of patients, considering the appearance of at least one sperm at conventional semen analysis . This effect was detected by evaluating a total of 897 HH patients and obtained with either hCG alone or hCG plus FSH, although a slightly significantly better result came from the combined action of the two gonadotropins . This result confirms that the intra-testicular testosterone action is required for spermatogenesis, and FSH improves this clinical outcome synergistically. However, the relatively low HH prevalence represents the main obstacle in organizing large trials, which would allow us to establish the best gonadotropin replacement therapy for recovering spermatogenesis in this context. Moreover, HH treatment, either with GnRH or with gonadotropins, requires the use of injections that burden the patient, possibly limiting compliance and efficacy. FSH is sometimes empirically proposed in men presenting idiopathic infertility, where the pathogenesis of infertility is not clearly identified. In such cases, a specific treatment is not applicable, and the exogenous stimulation of sperm production may be only empirical. Historically, first attempts to stimulate spermatogenesis were made by the administration of estrogen receptor antagonists (i.e., tamoxifen and clomiphene citrate). Estrogen receptor antagonists are used off-label in male infertility in some countries, especially in the United States. The pituitary release of endogenous FSH and LH may be increased by inhibiting the negative feedback mechanism using estrogen receptor modulators (SERMs). A recent meta-analysis of eleven randomized trials, enrolling men with idiopathic infertility, showed a higher pregnancy rate in couples in which the male partner was treated with SERMs with antagonistic activity at the estrogen receptor compared to couples with untreated men . This result is reflected by a significant rise in sperm concentration and motility after SERM administration, confirming that sperm production may be improved by increasing gonadotropin stimulation . However, this result has been obtained in a limited number of patients with heterogeneous clinical conditions, limiting the conclusion in favor of or against the application of this treatment. Accordingly, the Cochrane meta-analysis concluded that there is not enough evidence in favor of SERM application in male infertility . The potential benefit of estrogen receptor antagonists on sperm parameters could be explained by endogenous FSH “overstimulation” and LH-mediated increase in intra-testicular testosterone concentration, increasing the synergic stimulation of FSH and LH on sperm production. Exogenous FSH administration was suggested as a possible therapeutic option in infertile men with FSH serum levels within the normal ranges (i.e., below 8 IU/L), despite possible on-label only in few countries [124,125]. The rationale of this approach relies on the attempt to overstimulate spermatogenesis, an approach that was investigated in non-human primates. Indeed, FSH administration to monkeys significantly increased the number of type A pale and B spermatogonia . These studies indicated that spermatogenesis could be enhanced above the physiological rate and can be further stimulated by FSH administration. However, eventual benefits from supra-physiological FSH levels in humans were never clearly demonstrated. Interestingly, hCG did not have any effect on monkeys [126,127], suggesting the existence of species-specific regulation of the gonadal response among primates. Seventeen clinical trials (twelve prospective randomized clinical trials and five non-randomized studies) and four meta-analyses have been published so far, aiming at determining FSH efficacy in male idiopathic infertility. Interestingly, heterogeneous type and dosage of gonadotropin preparations were used (Table 1). Moreover, fourteen out of seventeen trials considered pregnancy rate as a primary endpoint, which is not exempt from many potential biases (i.e., the fertility status of the female partner). The efficacy of FSH in supporting spermatogenesis should be evaluated considering sperm parameters as the primary endpoint. Finally, most of these trials exhibit many biases, limiting the evidence in favor of the efficacy of FSH administration in male idiopathic infertility. The major limitations of these studies are the primary outcome chosen, heterogeneity of FSH dosages and duration (Table 1), as well as the inclusion criteria. Table 1. Clinical trials available in the literature evaluating the efficacy of follicle-stimulating hormone (FSH) in male idiopathic infertility. Studies are classified considering the study design and the therapeutic dosage applied. \| FSH Doses \| Treatment Duration \| FSH Total Amount \| Authors \| FSH Doses \| Treatment Duration \| FSH Total Amount \| Authors \| :---: :---: :---: :---: \| \| RCT \| \| Recombinant FSH \| Urinary-derived FSH \| \| 50 IU on alternate days \| 12 weeks \| 2250 IU \| Foresta et al. (2002) \| 50 IU on alternate days \| 12 weeks \| 2250 IU \| Ding et al. (2015) \| \| 100 IU on alternate days \| 12 weeks \| 4500 IU \| Foresta et al. (2002), Foresta et al. (2005) [128,130] \| 100 IU on alternate days \| 12 weeks \| 4500 IU \| Ding et al. (2015) \| \| 6750 IU \| Colacurci et al. (2012) \| \| 150 IU on alternate days \| 12 weeks \| 6750 IU \| Simoni et al. (2016), Foresta et al. (2009) [105,132] \| 75 IU daily (150 IU on alternate days) \| Not reported \| Not reported \| Ben-Rafael et al. (2000), Matorras et al. (1997) [133,134] \| \| 5400 IU \| Selice et al. (2011) \| 13 weeks \| 6825 IU \| Knuth et al. (1987)§ \| \| 150 IU daily (300 IU on alternate days) \| 12 weeks \| 12600 IU \| Kamischke et al. (1998)§ \| 200 IU on alternate days \| 12 weeks \| 9000 IU \| Ding et al. (2015) \| \| 16 weeks \| 18000 IU \| Paradisi et al. (2006)§ \| 150 IU daily (300 IU on alternate days) \| 12 weeks \| 12600 IU \| Ding et al. (2015), Ben-Rafael et al. (2000) [129,133] \| \| Non randomized, retrospective trials \| \| Recombinant FSH \| Urinary-derived FSH \| \| 150 IU on alternate days \| 12 weeks \| 5400 IU \| Caroppo et al. (2003) \| 75IU on alternate days \| 12 weeks \| 3375 IU \| Foresta et al. (2000) \| \| 75 IU daily \| 12 weeks \| 6300 IU \| Ashkenazi et al. (1999) \| \| 4 weeks \| 2100 IU \| Bartoov et al. (1994) \| \| 150 IU daily \| 12 weeks \| 12600 IU \| Baccetti et al. (2004) \| Open in a new tab FSH: follicle-stimulating hormone; RCT: randomized controlled clinical trial. This study applied human chorionic gonadotropin (hCG) plus human menopausal gonadotropin (hMG). § In these studies, the control group was treated with placebos in a double-blind study design. Meta-analyses provide an overview of studies published so far, overcoming the limited sample size in each trial. However, even meta-analyses are sometimes based on very limited numbers of patients. The Cochrane collaboration combined seven studies, suggesting a final spontaneous pregnancy rate increase in couples in which the male partner was treated with FSH [144,145]. However, these analyses did not highlight a significant pregnancy rate increase after ART [144,145]. In 2015, a second meta-analysis was performed, enlarging inclusion criteria and combining fifteen trials (both randomized and non-randomized) in which FSH or hMG was administered to idiopathic infertile male partners of couples attending ART . This study suggested the benefit of this treatment on pregnancy rate, even outside the context of ART, due to a rise in total sperm numbers . However, these works, combined together, have only a maximum of 482 patients treated by FSH/hMG, limiting the strength of these conclusions. Recently, the clinical efficacy of FSH administration in male idiopathic infertility has been reported, highlighting that gonadotropin dosage and sperm number increase are positively correlated . Only one study with a pharmacogenetic design aimed at a priori identification of those infertile men expected to respond to FSH administration . In this trial, FSH treatment improved sperm quality established by considering the sperm DNA fragmentation index in FSHR p.N680S homozygous N carrier men . However, the FSH beneficial effect was not detected after a three-month therapy but rather after three more months of treatment interruption, suggesting that the current three-month-scheme is probably too short to obtain a sustained effect . Similarly, a post-hoc analysis of a different clinical trial confirmed that pharmacogenomic of FSHB could be useful to predict FSH efficacy in idiopathic male infertility [130,148]. In summary, the few properly designed clinical trials have contrasting outcomes and results obtained by meta-analyses do not overcome the heterogeneity of these studies, preventing conclusions useful for clinical practice. A recent Italian nation-wide survey evaluated the idiopathic male infertility management, detecting an FSH prescription rate of 55% . Although this study was not designed to detect FSH efficacy in male infertility, it showed a slight increase in semen parameters in about half of the treated men, confirming the need for properly designed larger clinical trials in this setting [5,52]. A further, interesting field of investigation is the role of FSH/gonadotropin treatment in cases of non-obstructive azoospermia undergoing sperm retrieval by testicular sperm extraction (TESE) and micro-TESE. The vast majority of observational trials and reviews suggest that elevated FSH serum levels should be considered negative predictive markers of sperm retrieval after TESE/micro-TESE. Given the (presumed) primary testicular inability to produce sperms in such cases, there is no clear rationale in proposing a further FSH stimulation in the presence of serum FSH levels above the normal range. However, some studies reported an increased rate of sperms extracted after TESE/micro-TESE in men with non-obstructive azoospermia treated with clomiphene citrate , hCG , hMG , and FSH [151,152]. These findings would suggest that the minimal, residual spermatogenic activity could be stimulated even in case of elevated serum FSH levels, at least in some patients. However, the role of FSH administration in men with non-obstructive azoospermia undergoing surgical procedures is not supported by sufficient evidence and remains experimental. 6. Future Perspectives Future perspectives in the clinical use of gonadotropins in the male should consider the challenges still present in treating both HH and idiopathic infertility. Considering gonadotropin administration in HH men, several questions remain unsolved. First, many studies have demonstrated that hCG alone is sufficient to restore spermatogenesis and that FSH addition could only raise the sperm number [1,153]. Thus, intra-testicular testosterone seems able to stimulate spermatogenesis alone, overwhelming the FSH action on Sertoli cells. Moreover, no clear evidence is available on the right timing in which FSH should be added to hCG stimulation or which FSH dosage is required. Thus, specific trials aiming at dissecting the best therapeutic approach to HH men are still outstanding, focusing on the need, the dose and the timing of FSH addition to hCG. Of note, all available results on this topic used hCG to stimulate intra-testicular testosterone production rather than the physiological hormone LH. Although LH and hCG act on the same receptor, recent studies demonstrated different signal transduction pathways at the cellular level and different clinical outcomes in vivo, at least in women undergoing assisted reproduction [2,154]. The reason for using hCG instead of LH is more related to historical and practical reasons (availability, half-life, and costs) than to scientific evidence. Indeed, the first gonadotropin isolation dates back to the 1930s to 1950s and the preparation obtained, called human menopausal gonadotropin (hMG), was a 50/50 mixture of FSH and LH. Methodological improvements to obtain pure hormones have resulted in the effective production of highly purified FSH from the urine of postmenopausal women but LH is lost in the chromatographic procedure. Therefore, there is the need to add hCG (from the urine of pregnant women) to modern hMG preparations. Considering hCG potency and half-life, no attempts were made to obtain pituitary LH from urine in the certainty that the two gonadotropins had the same biological effect . Now that recombinant gonadotropins are available, the new evidence on the differences between LH and hCG action should be considered and could open further perspectives in the potential therapeutic role of LH in stimulating testicular function. Gonadotropin administration to HH men leads to a sperm concentration increase up to a mean value of 5.92 million per mL until pregnancy is obtained, after which this regimen is stopped . Although this increase is not sufficient to reach the normozoospermic range according to the World Health Organization (WHO) (i.e., >15 million per mL) , it is largely compatible with fertility in these men [156,157,158,159,160,161]. Thus, the reason why pharmacological gonadotropin stimulation does not completely restore spermatogenesis in such patients, even after many months, sometimes years (insufficient duration? non-optimal stimulation protocols? hCG vs. LH?), is unclear, intriguing, and remains to be fully discovered. Similarly, future perspectives in this area should consider which is the best schedule of gonadotropin stimulation. Indeed, both FSH and LH are physiologically secreted by the pituitary gland in a pulsatile fashion, and this secretory rhythm is considered essential for fertility in all mammals, including in human . Whether the current, non-pulsatile regimen of administration of exogenous gonadotropins is the most effective to stimulate the human testis in clinical practice remains unclear. Another important issue to be developed is how to treat HH boys before puberty, as they are currently diagnosed very early. Indeed, the standard management of male HH adolescents remains testosterone administration in order to induce virilization and psycho-sexual maturation . However, testosterone replacement therapy does not induce spermatogenesis, which, on the contrary, requires the administration of gonadotropins . It is currently unclear whether puberty induction should be performed first with testosterone, followed by gonadotropins, or vice versa . Recently, Zacharin et al. described complete pubertal development in 19 HH boys, treated with both hCG and FSH, with the acquisition of fertility and a marked improvement of quality of life, regardless of previous testosterone replacement . Thus, it seems that gonadotropin administration is required to stimulate HH testicular development and this could be performed both before and after testosterone administration. However, what the right timing is to start gonadotropins’ stimulation to induce puberty remains controversial and requires properly designed trials. In the setting of male idiopathic infertility, further pharmacogenomics studies are needed to prospectively evaluate how to select patients to be treated and to tailor FSH administration. To this purpose, pharmacogenomics evaluation of the FSHR and FSHB SNP role on male fertility could be useful. Indeed, since genetic haplotypes are associated with a reduced physiological stimulation of FSH on spermatogenesis, these patients could benefit from higher FSH dosages or longer treatment duration. On the other hand, most favorable genetic haplotypes could be sufficiently stimulated with the current empirical FSH therapeutic regimens. Thus, new, comprehensive approaches, connecting the genetic background to spermatogenesis and treatment response, are required . Finally, new perspectives deal with the application of new compounds with gonadotropin activity. For example, corifollitropin alfa is a recombinant hormone obtained by combining the gonadotropin α subunit with a chimeric FSHβ subunit fused to the carboxy-terminal peptide (CTP) of the hCGβ-subunit. This compound has a longer half-life than FSH due to additional glycosylation sites in the CTP portion . Since, in women undergoing ART, corifollitropin alfa produces a similar therapeutic response as recombinant FSH formulations , it is claimed that one injection of corifollitropin alfa could replace seven FSH daily administrations. In the male context, since gonadotropin regimens commonly impose long-term treatment with frequent injections, the use of corifollitropin alfa may result in fewer medication errors and improved compliance . Replacement therapy with corifollitropin alfa has been evaluated in HH men, demonstrating similar effectiveness and safety to FSH . On the other hand, this new compound has never been tested in the context of male idiopathic infertility. Thus, future research on the potential application of corifollitropin alfa in idiopathic male infertility should be conducted. Further attempts have been made to develop long-lasting gonadotropins. Efforts made along these lines included the attachment of polyethylene glycol (PEG) to increase half-life, bioavailability, and biological activity. Covalent addition of PEG results in increased solubility and increased size, hence, reduced renal clearance and protection from proteolytic degradation. FSH conjugated to PEG retained FSH activity in vitro on bovine cumulus cells . Interestingly, PEGylated FSH displayed improved bioavailability compared to FSH in rats . Another strategy for prolonging hormone half-life consists in the development of sustained-release formulations using aluminum hydroxide gel suspension. This approach led to prolonged bioactivities in rabbits and cattle [172,173]. Single-chain gonadotropins with improved pharmacokinetics have also been reported (see for a recent review). A pioneer study demonstrated that a single-chain recombinant analog, in which the hCGβ-subunit was fused to the N-terminus of the gonadotropin α-subunit, displayed improved in vivo biopotency compared to heterodimeric hCG. This initial success led to the development of single-chain analogs of LH and FSH. They were engineered with the β-subunits oriented at the N-terminus of the α subunit and used the hCGβ CTP sequence as a linker, resulting in increased half-life in vivo. Interestingly, they had similar or even higher secretion in vitro and biological activity in vivo than their native hormone counterpart [176,177,178]. For instance, it was demonstrated that a single injection of single-chain FSH analog could stimulate estrogen production for 5 to 7 days in Rhesus monkeys . As expected, the O-linked glycosylation sites of the carboxyl-terminal peptide (CTP) reduced hepatic clearance rates, contributing to increased serum half-life and increased biopotency of the single-chain analogs. Some analogs presenting dual FSH and LH activities were later developed. Such dual-active analogs could be relevant for the treatment of HH and infertile men. These constructs included FSHβ-CTP-CGβ-α and FSHβ-CTP-LHβ-CTP-α, and, when injected in sheep, elicited increased serum estradiol concentration, ovarian weight, and formation of corpora lutea [177,180,181]. The clearance rate of the dual-active FSHβ-CTP-LHβ-CTP-α was also significantly improved compared with endogenous ovine gonadotropin (1 to 2 h). Despite these promising indications, to date, no single-chain or dual-active gonadotropin analog has reached the clinical phase of development. More recently, a fusion protein consisting of FSHα and β-subunits fused to immunoglobulin Fc fragments was constructed in order to benefit from the advantageous pharmacokinetic properties of immunoglobulins [182,183]. Female rats injected this FSH-Fc analog displayed significantly increased ovarian weight compared to FSH-treated animals. In a radically different approach, low molecular weight (LMW) chemicals exerting agonistic actions at the FSHR also represent an intriguing alternative for the treatment of men in HH and infertility. Although such compounds do not present relatively long half-life in vivo, they could be developed for oral administration. Another advantage is the relatively limited cost of this type of molecule compared to biologicals such as recombinant gonadotropins. Hence, long term treatments with a high dosage could be envisioned without compromising the patient’s compliance or health system financial sustainability. Although developing LMW agonists to gonadotropin receptors has been very challenging compared to some other GPCRs, this is likely due to the peculiar nature of FSH interaction with the extracellular domain of FSHR and the subsequent chain of molecular events that lead to signal transduction. Various LMW modulators acting at FSHR have been reported over the years. Amongst them, the thiazolidinone (TDZ) series was identified by combinatorial screening . Some TDZ derivatives exhibit agonistic/positive allosteric modulation (PAM) at the FSHR . Indeed, some TDZs are able to stimulate FSH-induced signaling in the absence of FSH and are able to induce folliculogenesis in immature rats. Despite such promising results, these compounds never reached the market due to unfavorable pharmacokinetic parameters . Other compounds from the TDZ series showed the ability to switch FSHR coupling from Gα s to Gα i without the functional consequences for reproduction being clarified to date . Another class of LMW ligands capable of binding at the FSHR is the benzamide series. These compounds show selectivity to FSHR compared to TSHR and LHCGR. In the presence of a physiological concentration of FSH (EC 20), some benzamide derivatives enhanced the FSH-mediated activity on the receptor, behaving as PAM . Similarly, the dihydropyridine compound Org 24444-0 was also reported to increase FSH-induced cAMP production in vitro through a PAM mechanism of action. Interestingly, Org 24444-0 reproduces the effects of FSH on follicle maturation in vivo . Other LMW ligands have also been identified for their ability to turn off FSHR signaling. Among them, tetrahydroquinolines showed potent inhibitory effects on cAMP production without impairing FSH binding. Unfortunately, no effect was observed in vivo . More recently, the ADX series included negative allosteric modulators (NAM) acting at the FSHR. In particular, ADX61623 shows decreased FSHR-induced cAMP production and improved FSH binding affinity. Furthermore, when applied to primary granulosa cells, ADX61623 decreased progesterone but not estradiol production . Two other compounds from the ADX series were later reported as biased NAMs at the FSHR: ADX68692 and ADX68693. ADX68692 decreases cAMP, progesterone, and estradiol production in vitro in rat granulosa cells. However, ADX68693 shows similar inhibitory action on progesterone while it does not impair estradiol production in the same model system. Interestingly, in vivo, ADX68692 efficiently reduces the number of oocytes recovered in mature female rats whereas ADX68693 does not . It is important to note, however, that these two compounds also affect LHCGR signaling, highlighting promiscuity among the gonadotropin receptors for this ADX series . Despite these promising developments, no LMW compound has reached clinical phases so far. Further development will be needed to develop an orally active LMW FSHR agonist. Whereas none of these different strategies to improve bioactivity/bioavailability/administration has been tested clinically yet, they do open promising research avenues for the development of gonadotropin therapeutics that are better adapted to the treatment of men in HH and infertility. 7. Conclusions In conclusion, here we highlight how FSH acts in the sophisticated and not completely understood regulatory process resulting in male fertility. Human spermatogenesis is regulated by FSH and LH-dependent intra-testicular testosterone, with synergistic, partially overlapping mechanisms. Starting from these notions, FSH administration is proposed as a potentially effective therapeutic approach to male infertility. The exogenous FSH administration seems to improve sperm production, although it remains unclear whether it acts as replacement therapy or overstimulating treatment. This latter concept is supported by experimental demonstrations that spermatogenesis could be boosted beyond its physiological rate. However, the FSH-dependent overstimulation on spermatogenesis must still be verified in infertile men with properly designed clinical trials. Similarly, new scientific evidence is needed to confirm the efficacy of FSH administration in male infertility. A really powerful, phase 2 clinical trial is urgently required to produce evidence about FSH therapeutic potential. In this setting, the pharmacogenomic basis of FSH response, as well as the most effective dose and duration, must be addressed. Acknowledgments M.S. is a LE STUDIUM RESEARCH FELLOW, Loire Valley Institute for Advanced Studies, Orléans & Tours, France— INRAE, CNRS, Université de Tours, IFCE, PRC, Nouzilly, 37380, France, receiving funding from the European Union′s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 665790. 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[DOI] [PubMed] [Google Scholar] Articles from Journal of Clinical Medicine are provided here courtesy of Multidisciplinary Digital Publishing Institute (MDPI) ACTIONS View on publisher site PDF (2.3 MB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract 1. Introduction 2. Physiological Control of Spermatogenesis 3. Mechanism of Action of FSH in the Sertoli Cell 4. Genetic Regulation of FSH Action on Spermatogenesis 5. Therapeutic Options to Improve Sperm Production 6. Future Perspectives 7. 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4628	https://www.unirioja.es/cu/jvarona/aliquot.html	Aliquot sequences Welcome to Juan L. Varona WWW page about aliquot sequences. First of all, sorry for the English in this page; I am sure it will have some spelling and grammar mistakes. Also, note that the notation for some mathematical formulas follows the one of TeX. For a positive integer n, let sigma(n) denote the sum of the divisors of n, and s(n) = sigma(n) - n. A perfect number is a number n such that s(n) = n, and an amicable pair of numbers is (n,m) satisfying s(n) = m, s(m) = n. In a similar way, cycles of numbers (a_1, a_2, ... , a_p) such that s(a_i) = a_{i+1} for i = 1, ... , p-1 and s(a_p) = a_1 are known as aliquot cycles or sociable numbers. Iterating with the function s, it appears a aliqout sequence. For instance, starting from n = 20, s(20) = 1+2+4+5+10 = 22, s(22) = 1+2+11 = 14, s(14) = 1+2+7 = 10, s(10) = ... and so on. For one of such sequences, there are four possibilities: (i) it terminates at 1 (the previous term being a prime), (ii) it reaches a perfect number, (iii) it reaches an amicable pair or a cycle, (iv) it is unbounded. Catalan-Dickson conjecture says that (iv) does not actually happen. But other researchers disagree with this conjecture and they think that there are unbounded sequences; in fact, the alternative conjecture from Guy-Selfridge states that there are a lot of sequences that go to infinity, perhaps almost all those that start at an even number. The smallest n for which there was ever doubt was 138, but Lehmer showed that the sequence terminates at s^{177}(138) = 1. Since then, the first number whose behavior is not known is 276. Many researchers have investigated the behavior of this sequence, but it is not yet known if its end exists. But there are many other sequences whose end is in doubt. For instance, there are five main sequences starting in a number smaller than 1000 whose behavior is unknown (Lehmer five: 276, 552, 564, 660 and 966). Before the year 1980, there were fourteen main sequences starting between 1000 and 2000 (Godwin fourteen), but at the present time only twelve remain unknown (Godwin showed that the 1848 sequence terminates, and so did Dickerman for the 1248 one). In 1994, A. W. P. Guy and R. K. Guy published a article with a table showing the status of the sequences starting with numbers less than or equal to 7044. In the same year, the book Unsolved problems in Number Theory (2nd ed.), from R. K. Guy, updates the information of the article. In this book, a extensive bibliography on this subject can be found. Since then, Manuel Benito and myself have been checking the aliquot sequences for numbers under 10000. For some starting values, we have shown for the first time that the sequence terminates. In December 22, 1996, we found the record for the maximum of a terminating sequence: the one starting at 4170 converges to 1 after 869 iterations getting a maximum of 84 decimal digits at iteration 289. In Octuber 1999, Wieb Bosma beated this record: he found that the aliquot sequence starting with 44922 terminates after 1689 iterations (at 1) after reaching a maximum of 85 digits at step 1167. Later, in December 3, 1999, he beated again the record finding that the sequence starting at 43230 finished: it terminates (at 1) after 4357 steps after reaching a maximum of 91 digits at step 967. On June 10, 2001, Manuel Benito and myself got again the record for the highest known terminating sequence. The sequence starting at 3630 reaches a maximum of 100 digits at index 1263, and ends (at 1) at step 2624 (with the prime 59 being the previous term). Later, in December 25, 2001, we found that the sequence starting at 6160 finished: it terminates (at 1) after 3027 steps (with the prime 601 being the previous term) after reaching a maximum of 96 digits at step 1631. In our web pages, we will deal only with sequences starting under 10000. At present time, the aliquot sequences starting in a number under 10000, and whose end is yet unknown are the following: 276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, 2232, 2340, 2360, 2484, 2514, 2664, 2712, 2982, 3270, 3366, 3408, 3432, 3564, 3678, 3774, 3876, 3906, 4116, 4224, 4290, 4350, 4380, 4788, 4800, 4842, 5148, 5208, 5250, 5352, 5400, 5448, 5736, 5748, 5778, 6396, 6552, 6680, 6822, 6832, 6984, 7044, 7392, 7560, 7890, 7920, 8040, 8154, 8184, 8288, 8352, 8760, 8844, 8904, 9120, 9282, 9336, 9378, 9436, 9462, 9480, 9588, 9684, 9708, 9852. These sequences have have pursued to at least 100 digits. To get more information about these sequences: Table that summarize the present state of these sequences. Occurrences of numbers of 80 digits, 90 digits, and 100 digits in these sequences. The sequences themselves in the ftp server. All our work has been done by using free packages available on internet. We have run the programs on many computers from the authors and some colleages, and their respective institutions. We have used the following packages, that are available at their corresponding web pages (or anonymous ftp sites): UBASIC PARI-GP KANT-KASH MIRACL If you are interested in this subject, you can found pre-print with our results (in tex, dvi and pdf formats), the last term for any doubtful sequence, and the present state of the calculus by using anonymous ftp from mat.unirioja.es/pub/aliquot. Here, you can download a simple PARI-GP program or KASH program that allow to compute an aliquot sequence. In any case, feel free to contact us; perhaps the results in these web pages and/or the ftp server are not up to date. Other people that is involved in this subject: Wolfgang Creyaufmueller, from Aachen, is cumputing aliquot sequences starting a in number between 10^5 and 10^6 up to a minimum of 60 digits. This work is documented in his website, both in English and in German. Paul Zimmermann is computing iterates of the sequences starting in the numbers 276, 552, 564, 660, 966, 1074, 1134 and 204828. Also, he is working with aliquot sequences starting between 50000 and 100000. He maintains a web page. Jim Howell is working in the aliquot sequence starting in 966. The status can be seen in his aliquot page Wieb Bosma is working with aliquot sequences starting between 10000 and 50000. He also maintains a web page. Christophe Clavier is continuing with sequences starting in a number under 10000. See his web page. Mitchell Dickerman (who completed the 1248 sequence) is working with unitary aliquot sequences (one takes only the unitary divisors, i.e. those divisors d which do not divide n/d). Last modification of this page: September 16, 2004. Personal home page (in Spanish): Personal home page (in English):
4629	https://www.quora.com/When-do-you-use-the-verb-see-in-progressive-form-Some-might-say-its-a-stative-verb-so-dont-use-the-progressive-form-But-I-wonder-how-they-are-going-to-explain-the-following-sentence-are-you-seeing-this	When do you use the verb 'see' in progressive form? Some might say: 'it's a stative verb so don't use the progressive form.' But I wonder how they are going to explain the following sentence: 'are you seeing this?'? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In English Language and Gram... Stative Verb Grammar Present Continuous Tense Subject Verb Agreement Progressive Form (grammar... English Grammar Verb Tenses English Grammar Rules 5 When do you use the verb "see" in progressive form? Some might say: "it's a stative verb so don't use the progressive form." But I wonder how they are going to explain the following sentence: "are you seeing this?"? All related (37) Sort Recommended Jo Canfield EFL/ESL Instructor · Author has 16.6K answers and 10.9M answer views ·2y A speaker chooses to use a certain aspect (simple, continuous, perfect, or perfect continuous) because that aspect conveys the information he wishes to communicate. The continuous aspect communicates the information “takes or took place over a period of time" or, in other words, "the action had a beginning, a middle, and an end". We don't usually use the continuous aspect with stative verbs or verbs of perception because the concept of “took place over a period of time" is intrinsic to the meaning of the verb. Since the meaning “took place over a period of time" is part of the meaning of the verb Continue Reading A speaker chooses to use a certain aspect (simple, continuous, perfect, or perfect continuous) because that aspect conveys the information he wishes to communicate. The continuous aspect communicates the information “takes or took place over a period of time" or, in other words, "the action had a beginning, a middle, and an end". We don't usually use the continuous aspect with stative verbs or verbs of perception because the concept of “took place over a period of time" is intrinsic to the meaning of the verb. Since the meaning “took place over a period of time" is part of the meaning of the verb, we have no need to use the continuous aspect to tell the listener that the verb took place over a period of time. This is a far too complicated idea to explain to beginning language learners, so to stop them getting into the habit of using the present continuous inappropriately, we simply tell them “Don’t use stative verbs in the continuous aspect.” You can use stative verbs and verbs of perception in the present continuous when they express the idea you want to express. They usually communicate the idea of “temporarily true" or “right now, at the moment of speaking" (as opposed to always) true". Now, you've taken the words “Are you seeing this?" out of context, so I could only guess as to why the speaker chose to use the continuous aspect. My suggestion would be to go back and analyze the text, asking yourself “What function does the continuous aspect perform here?” and "Why did the speaker choose to use the continuous aspect here?” Because, frankly, with no context, I have no way of knowing. If you're serious about learning English, then you might want to get yourself a copy of Practical English Usage, by Michael Swan and The English Verb, by Michael Lewis. (They're expensive. Sometimes you can find used copies.) Upvote · 9 1 Promoted by Grammarly Grammarly Great Writing, Simplified ·Aug 18 Which are the best AI tools for students? There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you? That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners. Here are five AI tools inside Grammarly’s document editor that are worth checking out: Do Continue Reading There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you? That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners. Here are five AI tools inside Grammarly’s document editor that are worth checking out: Docs – Your all-in-one writing surface Think of docs as your smart notebook meets your favorite editor. It’s a writing surface where you can brainstorm, draft, organize your thoughts, and edit—all in one place. It comes with a panel of smart tools to help you refine your work at every step of the writing process and even includes AI Chat to help you get started or unstuck. Expert Review – Your built-in subject expert Need to make sure your ideas land with credibility? Expert Review gives you tailored, discipline-aware feedback grounded in your field—whether you're writing about a specific topic, looking for historical context, or looking for some extra back-up on a point. It’s like having the leading expert on the topic read your paper before you submit it. AI Grader – Your predictive professor preview Curious what your instructor might think? Now, you can get a better idea before you hit send. AI Grader simulates feedback based on your rubric and course context, so you can get a realistic sense of how your paper measures up. It helps you catch weak points and revise with confidence before the official grade rolls in. Citation Finder – Your research sidekick Not sure if you’ve backed up your claims properly? Citation Finder scans your paper and identifies where you need sources—then suggests credible ones to help you tighten your argument. Think fact-checker and librarian rolled into one, working alongside your draft. Reader Reactions – Your clarity compass Writing well is one thing. Writing that resonates with the person reading it is another. Reader Reactions helps you predict how your audience (whether that’s your professor, a TA, recruiter, or classmate) will respond to your writing. With this tool, easily identify what’s clear, what might confuse your reader, and what’s most likely to be remembered. All five tools work together inside Grammarly’s document editor to help you grow your skills and get your writing across the finish line—whether you’re just starting out or fine-tuning your final draft. The best part? It’s built for school, and it’s ready when you are. Try these features and more for free at Grammarly.com and get started today! Upvote · 999 201 99 34 9 3 Related questions More answers below I know that stative verbs are not used in progressive or in their continuous form, but how do we know that verbs such as "like and prefer" are stative verb and not action verb? Is "begin" a stative verb? What is the present progressive verb form of “go”? In English grammar, why are some verbs followed by an infinitive and others by a gerund? What is the difference between, "I like to run in the morning" and "I like running in the morning"? Why do some verbs only allow one or the other form? Can we use a 'feel' stative verb in present continuous? R. Frawley taught English grammar for 40+ years, not a purist · Author has 30K answers and 11.5M answer views ·2y Most so-called “stative verbs” can be and are pressed into service in progressive form, sometimes with a specific meaning (She’s seeing a doctor about her rash. She’s been seeing someone recently.) or just as an interesting quirk: I’m loving this book. I’m liking the new music I’m hearing. Upvote · 9 2 Elizabeth Henderson Former English Teacher. · Author has 49.9K answers and 60M answer views ·2y Usage of “stative verbs” is complex. To simplify things for beginners, we tell them, “Don’t use a stative verb in the continuous form”. But it’s only a rule of thumb, not intended to be a rigid rule that must never be broken. I admit I’m having difficulty thinking of a convincing context for “Are you seeing this?” But I have no difficulty with “Are you seeing what I’m seeing?” or (talking about a colour-blind person) “Is she seeing that blue dress the same way I’m seeing it?” Upvote · 9 1 Christina Biava PhD in Linguistics, University of Illinois at Urbana-Champaign (Graduated 1991) · Author has 4K answers and 3.6M answer views ·Updated 2y The rule about stative verbs not being used with progressive aspect is just a general rule of thumb—i.e., it is not 100%. Specifically, in situations/sentences when an action verb uses the progressive, a stative verb oftens doesn’t. For instance, “have” is usually not used in the progressive even when the action is right now when the meaning of “have” is “possess/own”: “Right now I am having three cars.” (instead of the correct “Right now I have/own three cars.”) But “Right now I am having a hard time staying awake.” (“have” = “experiencing”) “See” is a verb that is not often used in the progressi Continue Reading The rule about stative verbs not being used with progressive aspect is just a general rule of thumb—i.e., it is not 100%. Specifically, in situations/sentences when an action verb uses the progressive, a stative verb oftens doesn’t. For instance, “have” is usually not used in the progressive even when the action is right now when the meaning of “have” is “possess/own”: “Right now I am having three cars.” (instead of the correct “Right now I have/own three cars.”) But “Right now I am having a hard time staying awake.” (“have” = “experiencing”) “See” is a verb that is not often used in the progressive. However, the example given shows that it can be used in conversation when a speaker is asking if the listener understands what has just been said/explained. The progressive gives the question a hint of tentativeness, not as direct/forceful as the simple present would be, i.e. “Do you see this?” Upvote · 9 2 Sponsored by USA Corporate Incorporate in the US correctly and avoid costly mistakes later. Download our free e-book to learn about visas, LLCs, and more from a leading US incorporation service. Download 999 116 Related questions More answers below What is the verb form of "needy"? Is “feel” a stative verb? What form of verb do we use after "have to, has to, and had to"? When should we use stative verbs in "ing" form in a sentence with examples? What is the definition of a verb ending in -ing? Richard Lueger Former editor, ESL teacher (Parliament & Gov't of Canada) · Author has 13.5K answers and 19.2M answer views ·Sep 22 Related Can you explain why some verbs can't be used in the progressive form, like "I was being at home"? The main reason that you shouldn’t say that you were being at home is that the vast majority of native English speakers would never say that. They know it’s wrong. But there is a logical basis to how we use these verb tenses. The whole point of the progressive or continuous tenses (or ‘aspect’ as some would prefer) is that they describe some action that progresses and that involves some change through time. This action may be short-term and temporary, as in “I was drinking coffee when you phoned,” or even very long-term, as “The sun is slowly burning out.” The stative or state of being verbs des Continue Reading The main reason that you shouldn’t say that you were being at home is that the vast majority of native English speakers would never say that. They know it’s wrong. But there is a logical basis to how we use these verb tenses. The whole point of the progressive or continuous tenses (or ‘aspect’ as some would prefer) is that they describe some action that progresses and that involves some change through time. This action may be short-term and temporary, as in “I was drinking coffee when you phoned,” or even very long-term, as “The sun is slowly burning out.” The stative or state of being verbs describe unchanging states that just are, as in “I am my father’s son,” “I know how to calculate the area of a circle” and “I owe my success to my wife.” In regard to your sentence, there is no action involved or change in your status during the time you’re at home. When ‘to be’ is used in continuous tenses, it has an idiomatic sense: it describes temporary, atypical behaviour, as in “I was being silly when I said I was a teakettle.” In such sentences ‘to be’ means ‘to act in a certain way.’ That is, it describes an action, not a state of being. Of course, ‘to be’ is often used in continuous tenses when it’s part of a passive voice verb phrase, as in “I’m being sued by my neighbour,” but then it’s just an auxiliary verb. Upvote · 9 1 Raja Sekhar Telagathoty Educator 117M + Content Views & 24.7M + Upvotes (2016–present) · Author has 16.7K answers and 95.4M answer views ·Updated 4y Related What is the full detail of stative verbs? STATIVE VERBS 6 October 2018 In English grammar Stative verb describes a state rather than an action. This particular state lasts for some time ⏰ only. The state of being as opposed to an action. The stative verb usually can not be used in the present progressive tense form. English can be a global language as well as an eccentric one, with full of exceptions in terms of grammatical conclusions. As such, never arrive at a grammatical conclusion immediately Verbs of Sensual Perception verbs such as see,hear, smell, taste, feel can not be used in the present continues state. Despite the fact some exe Continue Reading STATIVE VERBS 6 October 2018 In English grammar Stative verb describes a state rather than an action. This particular state lasts for some time ⏰ only. The state of being as opposed to an action. The stative verb usually can not be used in the present progressive tense form. English can be a global language as well as an eccentric one, with full of exceptions in terms of grammatical conclusions. As such, never arrive at a grammatical conclusion immediately Verbs of Sensual Perception verbs such as see,hear, smell, taste, feel can not be used in the present continues state. Despite the fact some exemptions have been offered by grammarians I see you. I can see your car. I have just seen your car . But it is forbidden to use in continues form. I am SEEING you . in correct usage. Yet some exceptions : I am SEEING her means you have a relationship with a woman . I am SEEING the doctor tonight means you have already taken an appointment with the doctor concerned. But not exactly a date . SEEING the door locked I returned home. You can use it in a simple sentence as it has a mono clause. The pizza pronounced ( p i : t s a ) TASTES good . I SMELL something bad. May be bad odour or something unusual as well. Verbs of Cognition verbs such as understand, know, forget, can not be used in progressive form. I understand what you said. I am UNDERSTANDING , can be incorrect. I KNOW where you are. I FORGET the past always. Verbs of Possession verbs such as have,own posses and belong can not be used in present continues form. HAVE : exceptions. I HAVE two cars. Here have is taken to be a possessive verb. I am HAVING two wives can be incorrect and you are likely to be taken to the book. When Have is taken to be an ordinary verb : Here HAVE means take a meal or give a party. Have can also be used to mean : take a meal, drink, a bath, a lesson. Give a party to entertain guests. They are HAVING a party tomorrow. I am HAVING dinner is accepted as it refers to eating only. I belong to AP state. But no belonging . I OWN this apartment. Verbs of Mental Activity verbs such as remember and agree. I always REMEMBER you, but not I am remembering you. I AGREE with you always. Verbs of Feeling and Emotion Verbs like love like,hate and fear can not be used in present progressive form. I LOVE my mother. I LIKE my father. LOVING LIKING can not be used in the present progressive form. I HATE them for their bad deportment. Means I always HATE them. She HATES visiting him : means she once gets acquainted with him, now she does not want to visit him owing to diverse reasons. Verbs of State verbs like contain, consist, depend, deserve and involve can not be used in present form. This book CONTAINS 560 pages. You DESERVE the treat. The team CONSISTS of good allrounders. But COMPRISE can not be followed by OF. The team COMPRISES ABCD etc. Despite the fact the above mentioned verbs can be used in the progressive form as per the mood of the verb . Noticing the query I could not wait for anything except typing the clarification. I always welcome qualitative queries from readers those who want to excel in greater proficiency in English language. Upvote · 99 74 9 3 999 149 Promoted by Webflow Metis Chan Works at Webflow ·Feb 4 Which HTML editor is the best for web development? With the various HTML editors available today there are several considerations to keep in mind when deciding on which is the right fit for your company including ease of use, SEO controls, high performance hosting, flexible content management tools and scalability. Webflow offers all those features and allows you to experience the power of code – without writing it. You can take control of HTML5, CSS3, and JavaScript in a completely visual canvas — and let Webflow translate your design into clean, semantic code that’s ready to publish to the web, or hand off to developers. If you prefer more cus Continue Reading With the various HTML editors available today there are several considerations to keep in mind when deciding on which is the right fit for your company including ease of use, SEO controls, high performance hosting, flexible content management tools and scalability. Webflow offers all those features and allows you to experience the power of code – without writing it. You can take control of HTML5, CSS3, and JavaScript in a completely visual canvas — and let Webflow translate your design into clean, semantic code that’s ready to publish to the web, or hand off to developers. If you prefer more customization you can also expand the power of Webflow by adding custom code on the page, in the , or before the of any page. Trusted by 200,000+ leading organizations – Get started for free today! Get started for free today! Upvote · 99 37 9 6 Glyn Hughes Former Teacher (1963–1991) · Author has 2.7K answers and 1.9M answer views ·2y The present progressive form is commonly used along with the future progressive in preference to the standard future form “I shall see”, meaning “I shall meet”. I shall be seeing him tomorrow/I’ll be seeing/I’m seeing him tomorrow Both forms are also used with many other verbs, but by no means all. I’m going/working/’playing /I’ll be going/working/playing there tomorrow He’s leaving/arriving/He’ll be leaving/arriving on Thursday But, whilst you would say “She’ll be living in Germany at Christmas/We’ll be waiting for you at the entrance”, you would never say “”She is living/will live in Germany at C Continue Reading The present progressive form is commonly used along with the future progressive in preference to the standard future form “I shall see”, meaning “I shall meet”. I shall be seeing him tomorrow/I’ll be seeing/I’m seeing him tomorrow Both forms are also used with many other verbs, but by no means all. I’m going/working/’playing /I’ll be going/working/playing there tomorrow He’s leaving/arriving/He’ll be leaving/arriving on Thursday But, whilst you would say “She’ll be living in Germany at Christmas/We’ll be waiting for you at the entrance”, you would never say “”She is living/will live in Germany at Christmas” or “We are waiting for you at the entrance”, with reference to the future, but you could say “We will wait for you at the entrance”. There are no easy rules to follow. Usage varies depending on which verb you’re using, so you need to be very fluent in the language to determine which form(s) are acceptable. I’m finding it difficult to imagine when “Are you seeing this?” - or, even, “Do/Can you see this?” -would be used, but I can just about envisage situations when “Do/Can you see that” and “Are you seeing/Will you be seeing that tomorrow”. Are you enjoying this? Are buying this? and Are you watching this? sound fine to me, but not Are you seeing this?. In what context have you heard it? Upvote · 9 1 Claire Bush Academic Writing Tutor at Mercy College (2015–present) · Author has 480 answers and 454.7K answer views ·5y Related Why we use the sentences like, " you are looking good" and " I am feeling better", " I am seeing a bird" knowing that " look, see, feel, etc." are stative verbs or we can't use stative verb in continuous form, except some situations…? Generally speaking, we are beginning to see progressive aspect being coupled with stative verbs. Some of your examples do make me slightly nauseated - I am seeing a bird is one of them. This is new usage which I hope dies off but who knows? One exception in your list is “I’m feeling better. This one I can accept as it does communicate a difference in meaning than the simple aspect. I feel better is conclusive. I am done being sick, or upset. I’m fine. On the other hand, I am feeling better does not conclude. It says I am progressing away from what was bothering me but I can’t say I am fine. I Continue Reading Generally speaking, we are beginning to see progressive aspect being coupled with stative verbs. Some of your examples do make me slightly nauseated - I am seeing a bird is one of them. This is new usage which I hope dies off but who knows? One exception in your list is “I’m feeling better. This one I can accept as it does communicate a difference in meaning than the simple aspect. I feel better is conclusive. I am done being sick, or upset. I’m fine. On the other hand, I am feeling better does not conclude. It says I am progressing away from what was bothering me but I can’t say I am fine. I am getting there but not conclusively well. but yeah. This awful I am believing thing has got to stop. I hope. But you know the rule in language: Usage rules. Upvote · 9 1 9 1 Sponsored by Reliex Looking to Improve Capacity Planning and Time Tracking in Jira? ActivityTimeline: your ultimate tool for resource planning and time tracking in Jira. Free 30-day trial! Learn More 99 24 Frank Charles Former ly Frank Dauenhauer. New account since Feb. 2023. (1960–1991) · Author has 360 answers and 132.3K answer views ·Updated 2y When do you use the verb "see" in progressive form? You use it when you want to say things like these: “Seeing is believing.” “We must stop him from seeing her.” “Seeing him, she seemed to brighten a little.” “Are you seeing anyone now?” “I'm looking forward to seeing you.” “I’ll be seeing you.” Why are you seeing this kind of usage as a problem? Apparently, I’m not seeing it as you do. Upvote · 9 4 9 1 Lynn E. Lived in Greater Boston Area (1952–1977) · Author has 38.2K answers and 49.7M answer views ·6y Related Is natural to use "have" in the progressive form when experiencing some sort of pain? For example, " I am having a terrible headache", meaning I am experiencing it right now. Have is considered a non-action, or stative verb in this sort of sentence. But like any stative verb, it can, in fact, sometimes be used in the progressive, if we want particularly to emphasize its on-going or recurring nature. The plain, ordinary way to say it is: I have a headache. But I can also say, I’ve been having a lot of headaches recently, or yes: Please don’t bother me. I am having a terrible headache at the moment. It’s odd, in that it seems to describe the headache as an on-going attack of some sort, rather than just your run-of-the-mill headache. So it is not the basic, normative wa Continue Reading Have is considered a non-action, or stative verb in this sort of sentence. But like any stative verb, it can, in fact, sometimes be used in the progressive, if we want particularly to emphasize its on-going or recurring nature. The plain, ordinary way to say it is: I have a headache. But I can also say, I’ve been having a lot of headaches recently, or yes: Please don’t bother me. I am having a terrible headache at the moment. It’s odd, in that it seems to describe the headache as an on-going attack of some sort, rather than just your run-of-the-mill headache. So it is not the basic, normative way to express it. Like McDonald’s “I’m lovin’ it” it draws attention to itself by being a bit unusual way of expressing the point. Upvote · 9 2 Sophie Former Teacher · Author has 5.9K answers and 6.9M answer views ·Sep 20 Related Can you explain why some verbs can't be used in the progressive form, like "I was being at home"? You can say “I was being”, though “being at home” sounds weird. You can say, “You were being stupid.” I am racking my brain to think of any verbs that absolutely can’t be used in the progressive form. I mean, there are some (like “mean”) that do tend to be used in the simple form, but it would be OK to say, “That’s not what I was meaning.” The ad that goes, “I’m loving it” destroys the rule that you can’t put “love” in the progressive form. English is a flexible language in many ways, and this is one of them. Sorry I’m not being more helpful here… Upvote · 9 1 Sue Ann Keller retired teacher, published writer, freethinker (2000–present) · Author has 1.2K answers and 562K answer views ·6y Related What kind of verb is 'to wonder'? Can it be used in the progressive? It’s an intransitive verb, and yes, it can be used in the progressive tenses. I am wondering whether I should tell my wife about my indiscretion. I was wondering what would happen if I missed my first day of work. I had been wondering about my investments when my broker called to advise me to sell. I have been wondering all day whether it will snow. I will be wondering about his loyalty as long as he continues to work here. Upvote · 9 2 Related questions I know that stative verbs are not used in progressive or in their continuous form, but how do we know that verbs such as "like and prefer" are stative verb and not action verb? Is "begin" a stative verb? What is the present progressive verb form of “go”? In English grammar, why are some verbs followed by an infinitive and others by a gerund? What is the difference between, "I like to run in the morning" and "I like running in the morning"? Why do some verbs only allow one or the other form? Can we use a 'feel' stative verb in present continuous? What is the verb form of "needy"? Is “feel” a stative verb? What form of verb do we use after "have to, has to, and had to"? When should we use stative verbs in "ing" form in a sentence with examples? What is the definition of a verb ending in -ing? What are "durative", "punctual" and "stative" verbs? Can you explain why some verbs, such as "know," can be used in both progressive and non-progressive forms? Are these forms interchangeable or do they convey different meanings? Which verb is to be used with 'everyone'? Can we use a third-person singular in stative verbs? Is “see” an action verb? Related questions I know that stative verbs are not used in progressive or in their continuous form, but how do we know that verbs such as "like and prefer" are stative verb and not action verb? Is "begin" a stative verb? What is the present progressive verb form of “go”? In English grammar, why are some verbs followed by an infinitive and others by a gerund? What is the difference between, "I like to run in the morning" and "I like running in the morning"? Why do some verbs only allow one or the other form? Can we use a 'feel' stative verb in present continuous? What is the verb form of "needy"? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
4630	http://anakamura.weebly.com/uploads/4/9/1/4/4914438/alg1x_3-9_lecture_notes.pdf	3.9: Weighted Averages [Algebra 1(X)] Hawaii Content and Performance Standards III  Standard 10: Patterns, Functions, and Algebra: SYMBOLIC REPRESENTATION: Use symbolic forms to represent, model, and analyze mathematical situations.  Benchmark MA.AI.10.1: Solve linear equations and inequalities in one variable using a variety of strategies (e.g., algebraically, by graphing, by using a graphing calculator).  Benchmark MA.AI.10.2: Translate between verbal mathematical situations and algebraic expressions. Common Core State Standards for High School Mathematics:  A.CED.1: Create equations that describe numbers or relationships. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Objectives:  To solve mixture problems.  To solve uniform motion problems. Mixture Problems A weighted average (weighted mean) combines two or more different averages into one average for a set of data. In a weighted average, some data points will contribute to the average more than others. The weighted average can be calculate by finding the sum of the product of the number of units and the value per unit divided by the sum of the number of units. ܅܍ܑ܏ܐܜ܍܌ ۯܞ܍ܚ܉܏܍ൌValueሺ# unitsሻ൅Value ሺ# unitsሻ Sum of # units A mixture problem is situation in which two or more parts are combined into a whole. Example 1: Solve a Mixture Problem with Prices TRAIL MIX. Assorted dried fruit sells for $5.50 per pound. How many pounds of mixed nuts selling for $4.75 per pound should be mixed with 10 pounds of dried fruit to obtain a trail mix that sells for $4.95 per pound? Step 1: Categorize Information Item Value # of Units Weighted Average = Step 2: Use Weighted Average Formula and Solve Weighted Average ൌValueሺ# unitsሻ൅Value ሺ# unitsሻ Sum of # units Step 3: Answer the Question Example 2: Solve a Mixture Problem with Percents SCIENCE. A chemistry experiment calls for a 30% solution of copper sulfate. Kendra has 40 milliliters of 25% solution. How many milliliters of 60% solution should she add to obtain the required 30% solution? Step 1: Categorize Information Item Value # of Units Weighted Average = Step 2: Use Weighted Average Formula and Solve Weighted Average ൌValueሺ# unitsሻ൅Value ሺ# unitsሻ Sum of # units Step 3: Answer the Question Uniform Motion Problems Motion problems are another application of weighted averages. Uniform motion problems are problems where an object moves at a certain speed, or rate. The formula ݀ൌݎݐ is used to solve these problems, where d represents distance, r represents rate, and t represents time. Weighted Rate ൌRateሺtimeሻ൅Rate ሺtimeሻ Total Time Example 3: Solve for Average Speed TRAVEL. On Richard’s drive to his aunt’s house, the traffic was light, and he drove the 45‐mile trip in one hour. However, the return trip took him two hours. What was his average speed for the round trip? Example 4: Solve a Problem Involving Speeds of Two Vehicles SAFETY. Under ideal conditions, a siren can be heard up to 440 feet. However, under normal conditions, a siren can be heard from only 125 feet. In one particular situation, a car and an emergency vehicle are heading toward each other. The car is traveling at a speed of 30 miles per hour or about 44 feet per second. The emergency vehicle is traveling at a speed of 50 miles per hour or about 74 feet per second. If the vehicles are 1000 feet apart and the conditions are ideal, in how many seconds will the river of the car first hear the siren?
4631	https://artofproblemsolving.com/wiki/index.php/AM-GM_Inequality?srsltid=AfmBOoqdk3E2n93NQLyul-00h-J4NH9q6L_tZOy_RCQUx8mRWhYiJ-w5	Art of Problem Solving AM-GM Inequality - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki AM-GM Inequality Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search AM-GM Inequality In algebra, the AM-GM Inequality, also known formally as the Inequality of Arithmetic and Geometric Means or informally as AM-GM, is an inequality that states that any list of nonnegative reals' arithmetic mean is greater than or equal to its geometric mean. Furthermore, the two means are equal if and only if every number in the list is the same. In symbols, the inequality states that for any real numbers , with equality if and only if . The AM-GM Inequality is among the most famous inequalities in algebra and has cemented itself as ubiquitous across almost all competitions. Applications exist at introductory, intermediate, and olympiad level problems, with AM-GM being particularly crucial in proof-based contests. Contents [hide] 1 Proofs 2 Generalizations 2.1 Weighted AM-GM Inequality 2.2 Mean Inequality Chain 2.3 Power Mean Inequality 3 Problems 3.1 Introductory 3.2 Intermediate 3.3 Olympiad 4 See Also Proofs Main article: Proofs of AM-GM All known proofs of AM-GM use induction or other, more advanced inequalities. Furthermore, they are all more complex than their usage in introductory and most intermediate competitions. AM-GM's most elementary proof utilizes Cauchy Induction, a variant of induction where one proves a result for , uses induction to extend this to all powers of , and then shows that assuming the result for implies it holds for . Generalizations The AM-GM Inequality has been generalized into several other inequalities. In addition to those listed, the Minkowski Inequality and Muirhead's Inequality are also generalizations of AM-GM. Weighted AM-GM Inequality The Weighted AM-GM Inequality relates the weighted arithmetic and geometric means. It states that for any list of weights such that , with equality if and only if . When , the weighted form is reduced to the AM-GM Inequality. Several proofs of the Weighted AM-GM Inequality can be found in the proofs of AM-GM article. Mean Inequality Chain Main article: Mean Inequality Chain The Mean Inequality Chain, also called the RMS-AM-GM-HM Inequality, relates the root mean square, arithmetic mean, geometric mean, and harmonic mean of a list of nonnegative reals. In particular, it states that with equality if and only if . As with AM-GM, there also exists a weighted version of the Mean Inequality Chain. Power Mean Inequality Main article: Power Mean Inequality The Power Mean Inequality relates all the different power means of a list of nonnegative reals. The power mean is defined as follows: The Power Mean inequality then states that if , then , with equality holding if and only if Plugging into this inequality reduces it to AM-GM, and gives the Mean Inequality Chain. As with AM-GM, there also exists a weighted version of the Power Mean Inequality. Problems Introductory For nonnegative real numbers , demonstrate that if then . (Solution) Find the maximum of for all positive . (Solution) Intermediate Find the minimum value of for . (Source) Olympiad Let , , and be positive real numbers. Prove that (Source) See Also Proofs of AM-GM Mean Inequality Chain Power Mean Inequality Cauchy-Schwarz Inequality Inequality Retrieved from " Categories: Algebra Inequalities Definition Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
4632	https://www.annchildneurol.org/m/journal/view.php?number=1163	Annals of Child Neurology ABOUT Aims and scope About the journal Editorial board Best practice Subscription information Management team Open access ARTICLE CATEGORY Editorial Review Original article Letter to the editor Others BROWSE ARTICLES Current issue All issues Ahead-of print Most view Most download Most cited EDITORIAL POLICY Research and publication ethics Advertising policies Preprint policy AUTHOR INFORMATION Instructions to authors Author’s checklist E-submission Copyright transfer agreement Search Close PDF Share Facebook Twitter Google LinkedIn Metrics Park and Kim: Isolated Unilateral Sixth Nerve Palsy as the First Manifestation of Multiple Sclerosis Letter to the editor Published online: August 26, 2019 DOI: ### Isolated Unilateral Sixth Nerve Palsy as the First Manifestation of Multiple SclerosisHyo Yun Park, MD, Hyo Jeong Kim, MD Department of Pediatrics, Gachon University Gil Medical Center, Gachon University College of Medicine, Incheon, Korea Corresponding author: Hyo Jeong Kim, MD Department of Pediatrics, Gachon University Gil Medical Center, Gachon University College of Medicine, 21 Namdong-daero 774beon-gil, Namdong-gu, Incheon 21565, Korea Tel: +82-32-460-3224, Fax: +82-32-460-2362, E-mail: greatelena@gilhospital.com Received July 1, 2019 Revised July 23, 2019 Accepted August 7, 2019 Copyright © 2019 Korean Child Neurology Society This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Clinical features of multiple sclerosis (MS) vary widely depending on the area of the lesions. Although brainstem lesions are common in MS, isolated cranial nerve palsies are rare, especially as an initial manifestation . Among the cranial nerves, the fifth nerve is most commonly involved, followed by the 7th, 6th, 3rd, and 8th nerves at MS . Sixth nerve palsy results in ipsilateral horizontal gaze palsy. Common ocular motor symptoms in MS include nystagmus and internuclear ophthalmoplegia (INO) . Unusual ocular motor findings including bilateral 3rd nerve palsy, opsoclonus, and isolated 6th nerve palsy had been reported in MS patients . Here we report the case of a 15-year-old patient who presented with diplopia due to unilateral 6th nerve palsy as the first manifestation of MS. A 15-year-old man presented with diplopia for 6 days. Physical examination showed limited abduction of the right eye on right lateral gaze. Adduction of the left eye on right lateral gaze was normal (Fig. 1). Other neurological and blood test findings were normal. Magnetic resonance imaging (MRI) of the brain demonstrated multiple ovoid lesions in the periventricular white matter, cerebellum, corpus callosum, basal ganglia, thalamus, pons, and midbrain as a T2 high-signal intensity (Fig. 2). Both gadolinium-enhancing (Fig. 2I) and non-enhancing lesions were present. The lesion in the right pons, which might be in the right sixth cranial nerve pathway, innervated the ipsilateral lateral rectus muscle and might have caused the limited abduction of the ipsilateral eye (Fig. 2B, arrow). MRI of the spine demonstrated T2 hyperintensity and heterogeneous enhancement of the lesions at C4 and C6 (Fig. 2G, H, and J). These findings demonstrated dissemination in time and space. Cerebrospinal fluid (CSF) showed a white blood cell count of 12/mm 3 (lymphocyte 99%), 27.9 mg/dL protein, 65 mg/dL glucose, and no bacteria, malignant cells, or oligoclonal bands. CSF immunoglobulin G was elevated (1.95), and antibodies to aquaporin 4 were absent. This patient was not tested antibodies against myelin oligodendrocyte glycoprotein. Ophthalmologic examinations including fundus examination, optical coherence tomography, visual evoked potential, and visual acuity showed normal findings. Brainstem auditory evoked potential and upper and lower extremity somatosensory evoked potentials were normal. Based on the 2017 McDonald criteria, the patient was diagnosed with MS; one attack and two or more lesions on MRI demonstrating dissemination in time and space. He was treated with intravenous (IV) methylprednisolone (1 g/day) for 5 days, which he did not respond to, followed by oral prednisone (1 mg/kg/day, tapered over 5 weeks). Therefore, IV immunoglobulin (2 g/kg divided over 4 days) was administered, but the lateral rectus muscle dysfunction did not improve. Subsequently, he received a plasma exchange five times in 10 days and had marked improvement in diplopia and the lateral rectus muscle function; Grading of abduction was improved from –4 (no movement beyond the midline) to –1 (75% of movement remains). He has been receiving interferon-β 1b subcutaneously since the 5 days after the last plasma exchange. Interferon-β 1b was titrated to 0.25 mg (8 million IU) every other day. At the 7-month follow-up, the 6th cranial nerve palsy had completely resolved. No neurological symptoms had occurred until the 14-month follow-up. Follow-up MRIs of the brain and spine revealed a slight decrease in the lesion size. INO is a common ocular motor symptom in MS, affecting 30% of cases . INO is an eye movement disorder caused by a lesion of the medial longitudinal fasciculus (MLF), which affects conjugate eye movement by connecting the paramedian pontine reticular formation (PPRF)—abducens nucleus complex of the contralateral side to the oculomotor nucleus of the ipsilateral side . The MLF lesion impairs the adduction of the affected eye during contralateral gaze, while the contralateral eye abducts but is accompanied by nystagmus. If the lesion is located in the PPRF, abducens nucleus, or contralateral MLF, conjugate horizontal gaze palsy to the ipsilateral side occurs instead of ipsilateral lateral gaze palsy. Our patient presented with limited abduction of the unilateral eye and intact conjugate adduction of the contralateral eye. This indicated that the pathologic lesion involved the 6th nerve alone, and not the abducens nucleus, PPRF, or MLF . Isolated 6th nerve palsies are rare, seen only in 0.4% to 1.0% of MS . There have been no reports of isolated sixth nerve palsies in children, so they might be even rarer. After MS diagnosis, aggressive acute management including steroid, immunoglobulin, and plasma exchange was provided in a timely manner, leading to complete resolution of the 6th nerve palsy. Interferon-β was initiated in the early period and has been maintained without complications and relapses. Incomplete recovery indicates poor prognosis, and early, effective treatment of MS prevents irreversible long-term complications . Although isolated 6th nerve palsies are rarely observed in MS, especially in children, MS should be a differential diagnosis. Conflict of interestConflict of interest No potential conflicts of interest relevant to this article was reported. Fig.1. Ocular motility photography. Test of ocular motility demonstrated limited abduction of the right eye at right gaze (A), esotropia of the right eye at primary position (B), and normal ocular motility at left gaze (C). The consent was obtained from the patient and the guardians regarding the publication of the patient’s images. Fig.2. Magnetic resonance imaging (MRI) of the brain demonstrates multiple ovoid lesions in the cerebellum (A), pons (B, arrow), midbrain (C), basal ganglia, thalamus (D), and periventricular white matter (E, F), as a T2 high-signal intensity. MRI of the spine demonstrates T2 hyperintense and heterogeneous enhancing lesions at C3, C4, C6 (G) and T7 (H). MRI of gadolinium-enhanced T1-weighted images demonstrates several lesions in the frontal lobe (I) and C4 (J). REFERENCESREFERENCES References Zadro I, Barun B, Habek M, Brinar VV. Isolated cranial nerve palsies in multiple sclerosis. Clin Neurol Neurosurg 2008;110:886-8. [Article][PubMed] de Seze J, Vukusic S, Viallet-Marcel M, Tilikete C, Zephir H, Delalande S, et al. Unusual ocular motor findings in multiple sclerosis. J Neurol Sci 2006;243:91-5. [Article][PubMed] Rufa A, Cerase A, De Santi L, Mandala M, Nuti D, Giorgio A, et al. Impairment of vertical saccades from an acute pontine lesion in multiple sclerosis. J Neuroophthalmol 2008;28:305-7. [Article][PubMed] Bae YJ, Kim JH, Choi BS, Jung C, Kim E. Brainstem pathways for horizontal eye movement: pathologic correlation with MR imaging. Radiographics 2013;33:47-59. [Article][PubMed] Naldi P, Collimedaglia L, Vecchio D, Rosso MG, Perl F, Stecco A, et al. Predictors of attack severity and duration in multiple sclerosis: a prospective study. Open Neurol J 2011;5:75-82. [Article][PubMed][PMC] Copyright © Korean Child Neurology Society. View Full Site Go to Top
4633	https://www.youtube.com/watch?v=smfTfd80UbQ	Balancing the Equation for the Combustion of Octane (C8H18) Wayne Breslyn (Dr. B.) 890000 subscribers 64 likes Description 20049 views Posted: 6 Apr 2022 To balance the chemical equation for the combustion of Octane (C8H18 + O2 = CO2 + H2O) you first must correctly count all of atoms on each side of the chemical equation. When we balance an equation like Combustion of Octane our goal is to provide the correct coefficients for each compound. The coefficient is the number that goes in front of the compound. To balance C8H18 + O2 = CO2 + H2O you'll need to be sure to count all of atoms on each side of the chemical equation. Once you know how many of each type of atom you can only change the coefficients (the numbers in front of atoms or compounds) to balance the equation for Combustion of Octane. Important tips for balancing chemical equations: Only change the numbers in front of compounds (the coefficients). Never change the numbers after atoms (the subscripts). The number of each atom on both sides of the equation must be the same for the equation to be balanced. For a complete tutorial on balancing all types of chemical equations, watch my video: • Balancing Equations in 5 Easy Steps: • More Practice Balancing: For help with moles to grams conversions and more: • More Moles to Grams Practice: • Molar Mass in Three Easy Steps: • Understanding the Mole: • Moles - Gram Conversions: • How to Balance Chemical Equations: • Mole Ratio: • Reaction Stoichiometry: Drawing/writing done in InkScape ( Screen capture done with Camtasia Studio 4.0. Created on a Dell Dimension laptop computer with a Wacom digital tablet (Bamboo). 7 comments Transcript: let's balance the equation for the complete combustion of octane that's C8 h18 plus O2 this is combustion because we have a hydrocarbon so it's organic combustion we're adding oxygen gas that reacts to form carbon dioxide and water that makes this a combustion reaction let's balance the equation so I've counted the atoms up on both sides do note that in the products we have the two oxygen here and we have one with the water so two + 1 that gives us three we'll leave the O2 the oxygen gas here till last because when we change the coefficient in front of the O2 it won't change the number of carbons or hydrogens so in combustion reactions leave the O2 until last let's put an eight in front of the carbon dioxide 1 time 8 that gives us 8 this8 it goes to everything so we've got to update the oxygen so 8 2 is 16 plus we have that one oxygen there with the water 17 oxygen atoms let's do the hydrogens we could put a nine here 2 9 that would give us 18 that would balance the hydrogens let's update the oxygen so 8 2 that's 16 + 9 1 that gives us 25 oxygen atoms that's a bit of a problem but here's how to think about it 2 times some number will give us 25 2 25 over 2 that would give us 25 the twos would cancel out and we can just put a 25 over 2 here in front of the oxygen this equation it's balanced when we're balancing equations like combustion reactions here what we're really interested in is the coefficients those are the ratios of each substance so some teachers like to have this as a whole number that's pretty easy to fix we just multiply everything by two so these coefficients they're correct as well again it's the ratio that we're really interested in if we wanted to write the states for the combustion of octane that would look like this octane would be a liquid we have oxygen gas carbon dioxide gas and then water because this is exothermic combustion reactions are exothermic it'd get hot and water that would be a gas at least initially this is Dr B with the equation for the combustion of octane thanks for watching
4634	https://notes.mathforge.org/notes/published/hexagon	Notes hexagon Skip the Navigation Links \| Home Page \| Feeds Hexagon A hexagon is a polygon with six sides. Regular Hexagon In a regular hexagon, the interior angle is and the exterior angle is . A regular hexagon can be decomposed into six equilateral triangles. This shows that the long diameter is twice the side length, while the short diameter is times the side length. The area is times the square of the side length. Revised on October 3, 2021 15:41:22 by Andrew Stacey Views: Print \| TeX \| Source \| Linked from: two regular hexagons iii solution, two regular hexagons iii techniques, hexagon and a square techniques, two hexagons and a triangle solution, six semi-circles inside a hexagon solution, six semi-circles inside a hexagon techniques, a circle and two semi-circles in two hexagons solution, three circles in a hexagon in a circle solution, three circles in a hexagon in a circle techniques, seven hexagons solution, seven hexagons techniques, three hexagons and a rectangle solution, three hexagons and a rectangle techniques, two overlapping hexagons and a rectangle solution, two overlapping hexagons and a rectangle techniques, three regular hexagons iii solution, three circles in a semi-circle solution, nested circles and polygons solution, nested circles and polygons techniques, crossed hexagon solution, stacked hexagons solution, stacked hexagons techniques, divided hexagon solution, regular hexagon and rectangle solution, semi-circle on a hexagon solution, shaded spikes in a hexagon solution, two hexagons and a square solution, divided hexagon ii solution, divided hexagon ii techniques, two hexagons and a square techniques, three regular hexagons iii techniques, three circles in a semi-circle techniques, three hexagons in a trapezium solution, three hexagons in a trapezium techniques, two regular hexagons ii solution, two regular hexagons ii techniques, overlapping triangle and hexagon solution, triangle overlapping a hexagon solution, six semi-circles in a circle solution, six semi-circles in a circle techniques, circle in a hexagon in a semi-circle solution, circle in a hexagon in a semi-circle techniques, three regular hexagons solution, three regular hexagons techniques, nested circles and polygons ii solution, polygons with equal perimeter solution, polygons with equal perimeter techniques, nested circles and polygons ii techniques, dissected hexagon solution, dissected hexagon techniques, divided hexagon techniques, equilateral triangle in a hexagon solution, equilateral triangle in a hexagon techniques, two squares on a hexagon solution, two squares on a hexagon techniques, hexagon in a quarter circle solution, a square and a hexagon solution, a square and a hexagon techniques, quartered hexagon solution, rectangle on a hexagon solution, divided hexagon iii solution, divided hexagon iii techniques, hexagon and concentric circles solution, hexagon and concentric circles techniques, semi-circle on a hexagon techniques, shaded spikes in a hexagon techniques, a circle and two semi-circles in two hexagons techniques, overlapping triangle and hexagon techniques, quartered hexagon techniques, triangle overlapping a hexagon techniques, two hexagons in a rectangle solution, two hexagons in a rectangle techniques This site is running on Instiki 0.30.2(MML+) Powered by Ruby on Rails 2.3.18
4635	https://com-emergency.sites.medinfo.ufl.edu/files/2013/02/peritonsillar-abscess.pdf	Peritonsillar Abscess NICHOLAS J. GALIOTO, MD, Broadlawns Medical Center, Des Moines, Iowa P eritonsillar abscess is the most com-mon deep infection of the head and neck in young adults, despite the widespread use of antibiotics for treating tonsillitis and pharyngitis. This infection can occur in all age groups, but the highest incidence is in adults 20 to 40 years of age.1,2 Peritonsillar abscess most commonly occurs during November to December and April to May, which coincides with the high-est incidence rates of streptococcal pharyn-gitis and exudative tonsillitis.3,4 Peritonsillar abscesses are almost always first encountered by the family physician, and those with appro-priate training and experience can diagnose and treat most patients. Prompt recogni-tion and initiation of therapy is important to avoid potential serious complications. Anatomy The two palatine tonsils lie on the lateral walls of the oropharynx in the depression between the anterior tonsillar pillar (palatoglossal arch) and the posterior tonsillar pillar (pala-topharyngeal arch). The tonsils form during the last months of gestation and grow irregu-larly, reaching their largest size at approxi-mately six or seven years of age. The tonsils begin to gradually involute at puberty, and by older age little tonsillar tissue remains.5 When healthy, the tonsils do not project beyond the tonsillar pillars medially.2 Each tonsil has a number of crypts on its surface and is surrounded by a capsule that provides a pathway for blood vessels and nerves. Peri-tonsillar abscesses form in the area between the palatine tonsil and its capsule.1 Etiology Peritonsillar abscess has traditionally been regarded as the end point of a continuum that begins as acute exudative tonsillitis, pro-gresses to cellulitis, and eventually forms an abscess. A recent review implicates Weber’s glands as playing a key role in the formation of peritonsillar abscesses.6,7 This group of 20 to 25 mucous salivary glands are located in the space just superior to the tonsil in the soft palate and are connected to the surface of the tonsil by a duct.7 The glands clear the tonsillar area of debris and assist with the digestion of food particles trapped in the tonsillar crypts. If Weber’s glands become inflamed, local cellulitis can develop. As the infection progresses, the duct to the surface of the tonsil becomes progressively more obstructed from surrounding inflam-mation. The resulting tissue necrosis and pus formation produce the classic signs and symp-toms of peritonsillar abscess.8 These abscesses generally form in the area of the soft palate, just above the superior pole of the tonsil, in the location of Weber’s glands.7 The occurrence of peritonsillar abscesses in patients who have undergone tonsillectomy further supports the theory that Weber’s glands have a role in the pathogenesis. Other clinical variables include significant periodontal disease and smoking.6 Peritonsillar abscess remains the most common deep infection of the head and neck. The condition occurs primarily in young adults, most often during November to December and April to May, coinciding with the highest incidence of streptococcal pharyngitis and exudative tonsillitis. A peritonsillar abscess is a polymicrobial infection, but Group A streptococcus is the predominate organism. Symptoms generally include fever, malaise, sore throat, dysphagia, and otalgia. Physical findings may include trismus and a muffled voice (also called “hot potato voice”). Drainage of the abscess, antibiotics, and supportive therapy for maintaining hydration and pain control are the foundation of treat-ment. Antibiotics effective against Group A streptococcus and oral anaerobes should be first-line therapy. Steroids may be helpful in reducing symptoms and speeding recovery. To avoid potential serious complications, prompt rec-ognition and initiation of therapy is important. Family physicians with appropriate training and experience can diag-nose and treat most patients with peritonsillar abscess. (Am Fam Physician. 2008;77(2):199-202, 209. Copyright © 2008 American Academy of Family Physicians.) This article exempli-fies the AAFP 2008 Annual Clinical Focus on infectious disease: prevention, diag-nosis, and management. ▲ Patient information: A handout on peritonsillar abscess, written by the author of this article, is provided on page 209. Downloaded from the American Family Physician Web site at www.aafp.org/afp. Copyright © 2008 American Academy of Family Physicians. For the private, noncommercial use of one individual user of the Web site. All other rights reserved. Contact copyrights@aafp.org for copyright questions and/or permission requests. Peritonsillar Abscess 200 American Family Physician www.aafp.org/afp Volume 77, Number 2 ◆ January 15, 2008 Clinical Manifestations Patients with peritonsillar abscess appear ill and present with fever, malaise, sore throat, dysphagia, or otalgia. The throat pain is mark-edly more severe on the affected side and is often referred to the ear on the same side. Phys-ical examination usually reveals trismus, with the patient having difficulty opening his or her mouth because of pain from inflammation and spasm of masticator muscles.9 Swallowing is also highly painful, resulting in pooling of saliva or drooling.9 Patients often speak in a muffled voice (also called “hot potato voice”). Markedly tender cervical lymphadenitis may be palpated on the affected side. Inspection of the oropharynx reveals tense swelling and erythema of the anterior tonsillar pillar and the soft palate overlying the infected tonsil. The tonsil is generally displaced inferiorly and medially with contralateral deviation of the uvula (Figure 1). The most common symp-toms and physical findings are summarized in Table 1. Potential complications of peritonsil-lar abscess are outlined in Table 2. Death can occur from airway obstruction, aspiration, or hemorrhage from erosion or septic necrosis into the carotid sheath. Diagnosis The diagnosis of peritonsillar abscess is often made on the basis of a thorough his-tory and physical examination. Differential diagnosis includes infectious mononucleosis, lymphoma, peritonsillar cellulitis, and ret-romolar or retropharyngeal abscess. Patients often present with peritonsillar cellulitis with the potential to progress to abscess formation. In peritonsillar cellulitis, the area between the tonsil and its capsule is edematous and ery-thematous, but pus has not yet formed.1 On occasions when the diagnosis of peri-tonsillar abscess is in question, the presence of pus on needle aspiration or radiologic test-ing may help confirm the diagnosis. Trans-cutaneous or intraoral ultrasonography also can be helpful in identifying an abscess and in distinguishing peritonsillar abscess from peritonsillar cellulitis.1,6 If spread of the infection beyond the peritonsillar space or complications involving the lateral neck space are suspected, computed tomography (CT) or SORT: KEY RECOMMENDATIONS FOR PRACTICE Clinical recommendation Evidence rating References Treatment for peritonsillar abscess should include drainage and antibiotic therapy. C 1, 3, 6, 12 Initial empiric antibiotic therapy for peritonsillar abscess should include antimicrobials effective against Group A streptococcus and oral anaerobes. C 8, 13, 14 Steroids may be useful in reducing symptoms and in speeding recovery in patients with peritonsillar abscess. B 17 A = consistent, good-quality patient-oriented evidence; B = inconsistent or limited-quality patient-oriented evidence; C = consensus, disease-oriented evidence, usual practice, expert opinion, or case series. For information about the SORT evidence rating system, see page 131 or Figure 1. Patient with right peritonsillar abscess. Table 1. Common Symptoms and Physical Findings in Patients with Peritonsillar Abscess Symptoms Fever Malaise Severe sore throat (worse on one side) Dysphagia Otalgia (ipsilateral) Physical findings Erythematous, swollen soft palate with uvula deviation to contralateral side and enlarged tonsil Trismus Drooling Muffled voice (“hot potato voice”) Rancid or fetor breath Cervical lymphadenitis Tonsil Soft palate swelling Peritonsillar Abscess January 15, 2008 ◆ Volume 77, Number 2 www.aafp.org/afp American Family Physician 201 magnetic resonance imaging (MRI) is indicated. CT can distinguish between peritonsillar cellulitis and peritonsillar abscess, as well as demonstrate the spread of the infection to any contiguous spaces in the deep neck region (Figure 2). MRI has the advantage of improved soft-tissue definition over CT without exposure to radiation. Additionally, MRI is superior to CT in detecting complications from deep neck infections such as internal jugular vein thrombosis or erosion of the abscess into the carotid sheath. Disadvan-tages of MRI include longer scanning times, higher cost, lack of availability, and the potential for claustrophobia.10 Treatment Drainage of the abscess, antibiotics, and supportive ther-apy to maintain hydration and pain control are the foun-dation of treatment for peritonsillar abscess. Because peritonsillar cellulitis represents a transitional stage in the development of peritonsillar abscess, its treatment is similar to that of a peritonsillar abscess, excluding the need for surgical drainage. The main procedures for the drainage of peritonsillar abscess are needle aspiration, incision and drainage, and immediate tonsillectomy. Drainage using any of these methods combined with antibiotic therapy will result in resolution of the peritonsillar abscess in more than 90 percent of cases.6 The acute surgical management of peritonsillar abscess has evolved over time from routine immediate tonsillectomy to increased use of incision and drainage or needle aspiration.11 Immediate abscess tonsil-lectomy has not been proven to be any more effective than needle aspiration or incision and drainage, and it is consid-ered to be less cost-effective.12 Several studies comparing needle aspiration with incision and drainage have found no significant statistical differences in outcomes.11,12 Although it is not routinely performed for the treat-ment of peritonsillar abscess, immediate tonsillectomy should be considered for patients who have strong indi-cations for tonsillectomy, including those who have symptoms of sleep apnea, a history of recurrent tonsil-litis (four or more infections per year despite adequate medical therapy), or a recurrent or nonresolving peri-tonsillar abscess.6 Initial empiric antibiotic therapy should include antimicrobials effective against Group A streptococcus and oral anaerobes.13 The most com-mon organisms associated with peritonsillar abscess are listed in Table 3.8,14 Although peritonsillar abscesses are polymicrobial infections, several studies have shown intravenous penicillin alone to be as clinically effective as broader-spectrum antibiotics, provided the abscess has been adequately drained.12,14 In these studies, inad-equate clinical response following 24 hours of antibiotic therapy played a significant role in the decision to use broad-spectrum antibiotics. Several other studies have reported that more than 50 percent of culture results demonstrated the presence of beta-lactamase produc-ing anaerobes, leading many physicians to use broader- spectrum antibiotics as first-line therapy.8,14,15 Table 4 shows suggested antimicrobial regimens.16 Although steroids have been used to treat edema and inflammation in other otolaryngologic diseases, their role in the treatment of peritonsillar abscess has not Table 2. Complications of Peritonsillar Abscess Airway obstruction Aspiration pneumonitis or lung abscess secondary to peritonsillar abscess rupture Death secondary to hemorrhage from erosion or septic necrosis into carotid sheath Extension of the infection into the tissues of the deep neck or posterior mediastinum Poststreptococcal sequelae (e.g., glomerulonephritis, rheumatic fever) when infection is caused by Group A streptococcus Table 3. Common Organisms Associated with Peritonsillar Abscess Aerobic bacteria Group A streptococcus Staphylococcus aureus Haemophilus influenzae Information from references 8 and 14. Anaerobic bacteria Fusobacterium Peptostreptococcus Pigmented Prevotella Figure 2. Computed tomography of a right peritonsillar abscess. Area of abscess Right tonsil Uvula Peritonsillar Abscess 202 American Family Physician www.aafp.org/afp Volume 77, Number 2 ◆ January 15, 2008 been extensively studied. A recent study reported that 32 patients who received a single high dose of steroids (methylprednisolone [Depo-Medrol] 2 to 3 mg per kg up to 250 mg) intravenously plus antibiotics responded much more quickly to treatment than 28 patients who received antibiotics plus placebo.17 The use of steroids in the treatment of peritonsillar abscess appears to help speed recovery, but additional studies are needed before making a recommendation for their routine use.6,17 When the family physician is inexperienced in treat-ing peritonsillar abscess or when complications or ques-tions arise during treatment, an otolaryngologist should be consulted. Once the diagnosis has been established, drainage or aspiration of the abscess should be per-formed in a setting where possible airway complications can be managed.3 Peritonsillar aspiration is a technique well suited for the family physician who has had appro-priate training. The patient should be observed for a few hours after aspiration to ensure he or she can tolerate oral antibiotics and pain medications. Outpatient follow-up should occur in 24 to 36 hours.18 Oral antibiotics are con-tinued for 10 days. Most patients with a peritonsillar abscess can be treated in an outpatient setting, but a small percentage (e.g., 14 percent in one study) may require hospitaliza-tion.12 Hospital stays usually do not exceed two days and are required for pain control and hydration. The overall risk of developing a second peritonsillar abscess is approximately 10 to 15 percent.11,12 Up to 30 per-cent of patients with a peritonsillar abscess meet the crite-ria for tonsillectomy.12 This operation may be performed immediately or delayed until the abscess has resolved. The Author NICHOLAS J. GALIOTO, MD, is associate director of the Family Medicine Residency Program and director of the Transitional Year Residency Pro-gram for Broadlawns Medical Center in Des Moines, Iowa. He also has a clinical teaching appointment in the Department of Family Medicine at the University of Iowa Carver College of Medicine in Iowa City. Dr. Galioto received his medical degree from Creighton University in Omaha, Neb., and completed a family medicine residency at Broadlawns Medical Center. Address correspondence to Nicholas J. Galioto, MD, Broadlawns Medi-cal Center, 1801 Hickman Rd., Des Moines, IA 53104 (e-mail: ngalioto@ broadlawns.org). Reprints are not available from the author. Author disclosure: Nothing to disclose. REFERENCES 1. Steyer TE. Peritonsillar abscess: diagnosis and treatment [Published cor-rection appears in Am Fam Physician. 2002;66(1):30]. Am Fam Physi-cian. 2002;65(1):93-96. 2. Khayr W, Taepke J. Management of peritonsillar abscess: needle aspi-ration versus incision and drainage versus tonsillectomy. Am J Ther. 2005;12(4):344-350. 3. Belleza WG, Kalman S. Otolaryngologic emergencies in the outpatient setting. Med Clin North Am. 2006;90(2):329-353. 4. Bisno AL, Gerber MA, Gwaltney JM, Kaplan EL, Schwartz RH, for the Infectious Diseases Society of America. Practice guidelines for the diag-nosis and management of group A streptococcal pharyngitis. Infectious Diseases Society of America. Clin Infect Dis. 2002;35(2):113-125. 5. Berkovitz BK, ed. Pharynx. In: Standring S, ed. Grays Anatomy. The Anatomical Basis of Clinical Practice. 39th ed. New York, NY: Churchill Livingstone, 2005:623-625. 6. Herzon FS, Martin AD. Medical and surgical treatment of peritonsillar, retropharyngeal, and parapharyngeal abscesses. Curr Infect Dis Rep. 2006;8(3):196-202. 7. Passy V. Pathogenesis of peritonsillar abscess. Laryngoscope. 1994; 104(2):185-190. 8. Brook I. Microbiology and management of peritonsillar, retropharyn-geal, and parapharyngeal abscesses. J Oral Maxillofac Surg. 2004; 62(12):1545-1550. 9. Nwe TT, Singh B. Management of pain in peritonsillar abscess. J Laryn-gol Otol. 2000;114(10):765-767. 10. Gidley PW, Ghorayeb BY, Stiernberg CM. Contemporary manage-ment of deep neck space infections. Otolaryngol Head Neck Surg. 1997;116(1):16-22. 11. Johnson RF, Stewart MG, Wright CG. An evidence-based review of the treatment of peritonsillar abscess. Otolaryngol Head Neck Surg. 2003;128(3):332-343. 12. Herzon FS, Harris P. Mosher Award thesis. Peritonsillar abscess: inci-dence, current management practices, and a proposal for treatment guidelines. Laryngoscope. 1995;105(8 pt 3 suppl 74):1-17. 13. Brook I. The role of beta lactamase producing bacteria and bacte-rial interference in streptococcal tonsillitis. Int J Antimicrob Agents. 2001;17(6):439-442. 14. Kieff DA, Bhattacharyya N, Siegel NS, Salman SD. Selection of antibiot-ics after incision and drainage of peritonsillar abscesses. Otolaryngol Head Neck Surg. 1999;120(1):57-61. 15. Ozbek C, Aygenc E, Unsal E, Ozdem C. Peritonsillar abscess: a compari-son of outpatient IM clindamycin and inpatient IV ampicillin/sulbactam following needle aspiration. Ear Nose Throat J. 2005;84(6):366-368. 16. Fairbanks DN, ed. Pocket Guide to Antimicrobial Therapy in Otolaryn-gology—Head and Neck Surgery. 12th ed. Alexandria, Va.: American Academy of Otolaryngology—Head and Neck Surgery Foundation, Inc., 2005:40, 86-90. 17. Ozbek C, Aygenc E, Tuna EU, Selcuk A, Ozdem C. Use of ste-roids in the treatment of peritonsillar abscess. J Laryngol Otol. 2004;118(6):439-442. 18. Roberts, JR. ED considerations in the diagnosis and treatment of peri-tonsillar abscess. Emerg Med News. 2001;23(3):6,9-10. Table 4. Antimicrobial Regimens for Peritonsillar Abscess Intravenous therapy Ampicillin/sulbactam (Unasyn) 3 g every six hours Penicillin G 10 million units every six hours plus metronidazole (Flagyl) 500 mg every six hours If allergic to penicillin, clindamycin (Cleocin) 900 mg every eight hours Oral therapy Amoxicillin/clavulanic acid (Augmentin) 875 mg twice daily Penicillin VK 500 mg four times daily plus metronidazole 500 mg four times daily Clindamycin 600 mg twice daily or 300 mg four times daily Information from reference 16.
4636	https://www.isbtweb.org/asset/CDF9E27E-DD96-482A-B578BA864FA08B46/	Anti -D Immunoglobulin (Ig) Administration to avoid sensitisation in pregnancy - an aide memoire Potentially sensitising events (PSE) during pregnancy (see Appendix 1 on next page) Gestation LESS than 12 weeks Vaginal bleeding associated with abdominal pain Administer at least 500 IU anti -D Ig within 72 hours of event. Confirm patient details and administration details Evacuation of retained products of conception (ERPC) / instrumentation of uterus Medical or surgical termination of pregnancy (See updated information from NICE guidance NG126 and NG140 in the shaded text box on the next page) Ectopic / molar pregnancy Gestation 12 to 20 weeks For any PSE Administer at least 500 IU anti -D Ig within 72 hours of event. Confirm patient details and administration details For continuous uterine bleeding (see key points above) Gestation 20 weeks to term For any PSE (Irrespective of whether RAADP has been given) Request a test for FMH (e.g. Kleihauer Test) and immediately administer at least 500 IU anti -D Ig within 72 hours of event. Confirm patient details and administration details Does the test for FMH (e.g. Kleihauer test) indicate that further anti -D Ig is required? Administer more anti -D Ig following discussion with laboratory and ensure woman is followed up until all fetal cells have been cleared Key points to note:  Women who are confirmed to have immune (allo) anti -D do not need (or should not receive) anti D Ig  Where the results of the cffDNA (cell free fetal DNA) test are available and show that the fetus/baby is D -negative, anti -D Ig does not need to be given  Confirm that the cffDNA result relates to the current pregnancy  Person administering anti -D Ig should confirm the woman’s’ identity, discuss risk/benefits, gain informed consent and record in patient’s notes . Confirm product dose and expiry date  Following potentially sensitising events ( PSE - see appendix 1) , anti -D Ig should be administered as soon as possible and always within 72 h ours of the event. If, exception ally, this deadline has not been met some protection may be offered if anti -D Ig is given up to 10 days after the sensitising event . After 10 days refer to local policy  Each new sensitising event should be managed with an appropriate additional dose of ant i-D Ig regardless of timing or dose of anti -D Ig administered for a previous event  In the event of continual uterine bleeding which is clinically judged to represent the same sensitising event, with no features suggestive of a new presentation or a signif icant change in pattern or severity of bleeding, a minimum dose of 500 IU anti -D Ig should be given at 6 weekly intervals with 2 weekly estimation s of fetomaternal haemorrhage ( FMH )  Appropriate tests for FMH should be carried out for all D-negative pregnancies when a PSE has occurred after 20 weeks of gestation and additional dose(s) of anti -D Ig should be administered as necessary  Routine Antenatal Ant i-D Ig Prophylaxis (RAADP) is a separate entity and should be always be given at the appropriate time in the second trimester, even if one or more doses of anti -D Ig for PSE ha ve been administered  Diagnosis AND delivery of intrauterine death ( IUD ) are 2 separate sensitising event sAnti -D Immunoglobulin (Ig) Administration - an aide memoire Routine antenatal anti -D Ig p rophylaxis (RAADP) Sampling for Routine Antenatal Anti -D Prophylaxis (Irrespective of whether anti -D Ig has already been given for PSE) Administration of Routine Antenatal Anti -D Prophylaxis (Irrespective of whether anti -D Ig has already been given for PSE) Take a blood sample to confirm group & check antibody screen prior to anti -D Ig administration – do not wait for results before administering anti -D Ig Administer 1500 IU anti -D Ig at 28 – 30 weeks OR Administer at least 500 IU anti -D Ig at 28 weeks and then administer at least 500 IU anti -D Ig at 34 weeks Confirm patient details and administration details At delivery (or Intra uterine death (IUD ) >20 weeks) Is the baby’s group confirmed as D -positive? OR Are cord samples not available following IUD? Request a test for FMH (e.g. Kleihauer Test) Administer at least 500 IU anti -D Ig within 72 hours of delivery Confirm patient details and administration details Does the test for FMH (for e.g. Kleihauer Test) indicate that further anti -D is required? Administer further dose(s) of anti -D Ig as advised by your laboratory and ensure woman is follow ed up until all fetal cells have been cleared Confirm correct product batch number & expiry date, IU dose and patient’s ID with the authorisation/product label. Document d ate & time given Updated recommendations as per NICE guidance NG140: Offer anti -D prophylaxis to women who are rhesus D negative and are having an abortion after 10 +0 weeks' gestation. Do not offer anti - D prophylaxis to women who are having a medical abortion up to and including 10 +0 weeks' gestation. Consider anti -D prophylaxis for women who are rhesus D negative and are having a surgical abortion up to and including 10 +0 weeks' gestation. Providers should ensure that: rhesus status testing and anti -D prophylaxis supply does not cause any delays to women having an abortion and anti -D prophylaxis is available at the time of the abortion. NG 126: Offer anti -D rhesus prophylaxis at a dose of 250 IU (50 micrograms) to all rhesus -negative women who have a surgical procedure to manage an ectopic pregnancy or a miscarriage. Do not offer anti -D rhesus prophylaxis to women who: receive solely medical management for an ectopi c pregnancy or miscarriage or have a threatened miscarriage or have a complete miscarriage or have a pregnancy of unknown location. Do not use a Kleihauer test for quantifying feto -maternal haemorrhage. British Society for Haematology have issued an update to their guidelines on 3 rd Feb 2020 and state that the more recent NICE guidance , noted above, should be referred to in relation to ectopic pregnancy and also abortion care pending an update the BSH guidelines ( -s-h.org.uk/guidelines/guidelines/use -of -anti -d-immunoglobin -for -the -prevention -of -haemolytic -disease -of -the - fetus -and -newborn/ ) Appendix 1 - Potentially sensitising events in pregnancy (From the ‘BCSH guideline for the use of anti -D immu noglobulin for the prevention of haemolytic disease of the fetus and newborn 2014 ‘ ) Amniocentesis, chorionic villus biopsy and cordocentesis Evacuation of molar pregnancy Ectopic pregnancy Antepartum haemorrhage/Uterine (PV) bleeding in pregnancy Ectopic pregnancy Miscarriage, threatened miscarriag e External cephalic version Intrauterine death and stillbirth Therapeutic termination of pregnancy Abdominal trauma (sharp/blunt, open/closed) In -utero therapeutic interventions (transfusion, surgery, i nsertion of shunts, laser) Delivery – normal, instrumental or Caesarean sectio n
4637	https://www.grc.nasa.gov/www/k-12/Sample_Projects/Ohms_Law/ohmslaw.html	\| \| \| \| \| \| \| \| \| --- --- --- --- \| \| \| \| \| --- \| \| \| National Aeronautics andSpace Administration Glenn Research Center \| \| \| \| \| \| Educational Product \| \| Teachers \| Grades 9-12 \| \| OHM'S LAW V = I x R Where: V = Voltage I = Current R = Resistance Example Problem: A nine volt battery supplies power to a cordless curling iron with a resistance of 18 ohms. How much current is flowing through the curling iron? Sketch: Solution: 1.) Since V(Voltage) and R(Resistance) are known, solve for I(Current) by dividing both sides of the equation by R. 2.) The R's on the right hand side of the equation cancel. 3.) I is then left in terms of V and R. 4.) Substitute in the values for V(Voltage) and R(Resistance). 5.) Solve for I(Current). Problem #1 A 110 volt wall outlet supplies power to a strobe light with a resistance of 2200 ohms. How much current is flowing through the strobe light? Sketch: Choose your answer below 0.5 amps 2.0 amps 0.05 amps 1.0 amps Problem #2 A CD player with a resistance of 40 ohms has a current of 0.1 amps flowing through it. Sketch the circuit diagram and calculate how many volts supply the CD player? Choose your answer below 0.0025 volts 4.0 volts 10.0 volts 400.0 volts This activity was created by Steven Gutierrez, for the Learning Technologies Project, at the NASA Glenn Research Center, in 1996. Ohm's Law Web Page Navigation .. Beginner's Guide Home Page Please direct any comments to: Thomas.J.Benson@nasa.gov
4638	https://math.stackexchange.com/questions/5018958/proving-that-if-f-has-a-local-extremum-at-a-then-fa-0	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. current community your communities more stack exchange communities Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Proving that if $f$ has a local extremum at $a$, then $f'(a)=0$ Knowing that $a$ is in the interior of $\mathcal{D}_f$ and $f$ is differentiable in $a$. Prove that if $f$ has a local extremum at $a$, then $f'(a)=0$. Here is what I have done: $\exists\delta>0:a-\delta<x<a+\delta,x\neq a \implies f(x) \leq f(a) \implies f(x)-f(a)\leq0$ If $a-\delta<x<a$ then $$x-a < 0 \implies \frac{f(x)-f(a)}{x-a} \geq0 \implies \lim_{x\to a-}\frac{f(x)-f(a)}{x-a} \geq0$$ Similarly, for $a<x<a+\delta$ it follows that $\lim_{x\to a+}\frac{f(x)-f(a)}{x-a} \leq 0$ Thus $\lim_{x\to a}\frac{f(x)-f(a)}{x-a} = 0 \implies f'(a)=0$ Have I done this correctly? 1 Answer 1 Yes, you did it correctly, but for your proof, you must assume - without loss of generality - that a is a local maximum for f. Extremum means: minimum or maximum. You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Explore related questions See similar questions with these tags. Related Hot Network Questions Subscribe to RSS To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Mathematics Company Stack Exchange Network Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA . rev 2025.9.26.34547
4639	https://pressbooks.bccampus.ca/intermediatealgebrakpu/chapter/2-5-absolute-value-equations/	2.5 Absolute Value Equations – Intermediate Algebra Skip to content Menu Primary Navigation Home Read Sign in Search in book: Search Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices. Book Contents Navigation Contents Preface Chapter 1: Algebra Review 1.1 Integers 1.2 Fractions (Review) 1.3 Order of Operations (Review) 1.4 Properties of Algebra (Review) 1.5 Terms and Definitions 1.6 Unit Conversion Word Problems 1.7 Puzzles for Homework Chapter 2: Linear Equations 2.1 Elementary Linear Equations 2.2 Solving Linear Equations 2.3 Intermediate Linear Equations 2.4 Fractional Linear Equations 2.5 Absolute Value Equations 2.6 Working With Formulas 2.7 Variation Word Problems 2.8 The Mystery X Puzzle Chapter 3: Graphing 3.1 Points and Coordinates 3.2 Midpoint and Distance Between Points 3.3 Slopes and Their Graphs 3.4 Graphing Linear Equations 3.5 Constructing Linear Equations 3.6 Perpendicular and Parallel Lines 3.7 Numeric Word Problems 3.8 The Newspaper Delivery Puzzle Chapter 4: Inequalities 4.1 Solve and Graph Linear Inequalities 4.2 Compound Inequalities 4.3 Linear Absolute Value Inequalities 4.4 2D Inequality and Absolute Value Graphs 4.5 Geometric Word Problems 4.6 The Cook Oil Sharing Puzzle 4.7 Mathematics in Life: The Eiffel Tower Midterm 1 Preparation Midterm 1: Version A Midterm 1: Version B Midterm 1: Version C Midterm 1: Version D Midterm 1: Version E Chapter 5: Systems of Equations 5.1 Graphed Solutions 5.2 Substitution Solutions 5.3 Addition and Subtraction Solutions 5.4 Solving for Three Variables 5.5 Monetary Word Problems 5.6 Solving for Four Variables 5.7 Solving Internet Game Puzzles Chapter 6: Polynomials 6.1 Working with Exponents 6.2 Negative Exponents 6.3 Scientific Notation (Homework Assignment) 6.4 Basic Operations Using Polynomials 6.5 Multiplication of Polynomials 6.6 Special Products 6.7 Dividing Polynomials 6.8 Mixture and Solution Word Problems 6.9 Pascal's Triangle and Binomial Expansion Chapter 7: Factoring 7.1 Greatest Common Factor 7.2 Factoring by Grouping 7.3 Factoring Trinomials where a = 1 7.4 Factoring Trinomials where a ≠ 1 7.5 Factoring Special Products 7.6 Factoring Quadratics of Increasing Difficulty 7.7 Choosing the Correct Factoring Strategy 7.8 Solving Quadriatic Equations by Factoring 7.9 Age Word Problems 7.10 The New Committee Member's Age Midterm 2 Preparation Midterm 2: Version A Midterm 2: Version B Midterm 2: Version C Midterm 2: Version D Midterm 2: Version E Chapter 8: Rational Expressions 8.1 Reducing Rational Expressions 8.2 Multiplication and Division of Rational Expressions 8.3 Least Common Denominators 8.4 Addition and Subtraction of Rational Expressions 8.5 Reducing Complex Fractions 8.6 Solving Complex Fractions 8.7 Solving Rational Equations 8.8 Rate Word Problems: Speed, Distance and Time Chapter 9: Radicals 9.1 Reducing Square Roots 9.2 Reducing Higher Power Roots 9.3 Adding and Subtracting Radicals 9.4 Multiplication and Division of Radicals 9.5 Rationalizing Denominators 9.6 Radicals and Rational Exponents 9.7 Rational Exponents (Increased Difficulty) 9.8 Radicals of Mixed Index 9.9 Complex Numbers (Optional) 9.10 Rate Word Problems: Work and Time 9.11 Radical Pattern Puzzle Chapter 10: Quadratics 10.1 Solving Radical Equations 10.2 Solving Exponential Equations 10.3 Completing the Square 10.4 The Quadratic Equation 10.5 Solving Quadratic Equations Using Substitution 10.6 Graphing Quadratic Equations—Vertex and Intercept Method 10.7 Quadratic Word Problems: Age and Numbers 10.8 Construct a Quadratic Equation from its Roots Midterm 3 Preparation and Sample Questions Midterm 3: Version A Midterm 3: Version B Midterm 3: Version C Midterm 3: Version D Midterm 3: Version E Chapter 11: Functions 11.1 Function Notation 11.2 Operations on Functions 11.3 Inverse Functions 11.4 Exponential Functions 11.5 Logarithmic Functions 11.6 Compound Interest 11.7 Trigonometric Functions 11.8 Sine and Cosine Laws 11.9 Your Next Year's Valentine? Final Exam Preparation Final Exam: Version A Final Exam: Version B Appendix Reference Section Answer Key 1.1 Answer Key 1.2 Answer Key 1.3 Answer Key 1.4 Answer Key 1.6 Answer Key 1.7 Answer Key 2.1 Answer Key 2.2 Answer Key 2.3 Answer Key 2.4 Answer Key 2.5 Answer Key 2.6 Answer Key 2.7 Answer Key 2.8 Answer Key 3.1 Answer Key 3.2 Answer Key 3.3 Answer Key 3.4 Answer Key 3.5 Answer Key 3.6 Answer Key 3.7 Answer Key 4.1 Answer Key 4.2 Answer Key 4.3 Answer Key 4.4 Answer Key 4.5 Answer Key 4.6 Mid Term 1: Review Questions Answer Key Midterm 1: Version A Answer Key Midterm 1: Version B Answer Key Midterm 1: Version C Answer Key Midterm 1: Version D Answer Key Midterm 1: Version E Answer Key Answer Key 5.1 Answer Key 5.2 Answer Key 5.3 Answer Key 5.4 Answer Key 5.5 Answer Key 5.6 Answer Key 5.7 Answer Key 6.1 Answer Key 6.2 Answer Key 6.3 Answer Key 6.4 Answer Key 6.5 Answer Key 6.6 Answer Key 6.7 Answer Key 6.8 Answer Key 6.9 Answer Key 7.1 Answer Key 7.2 Answer Key 7.3 Answer Key 7.4 Answer Key 7.5 Answer Key 7.6 Answer Key 7.7 Answer Key 7.8 Answer Key 7.9 Answer Key 7.10 Midterm 2: Prep Answer Key Midterm 2: Version A Answer Key Midterm 2: Version B Answer Key Midterm 2: Version C Answer Key Midterm 2: Version D Answer Key Midterm 2: Version E Answer Key Answer Key 8.1 Answer Key 8.2 Answer Key 8.3 Answer Key 8.4 Answer Key 8.5 Answer Key 8.6 Answer Key 8.7 Answer Key 8.8 Answer Key 9.1 Answer Key 9.2 Answer Key 9.3 Answer Key 9.4 Answer Key 9.5 Answer Key 9.6 Answer Key 9.7 Answer Key 9.8 Answer Key 9.9 Answer Key 9.10 Answer Key 9.11 Answer Key 10.1 Answer Key 10.2 Answer Key 10.3 Answer Key 10.4 Answer Key 10.5 Answer Key 10.6 Answer Key 10.7 Answer Key 10.8 Midterm 3 Prep Answer Key Midterm 3: Version A Answer Key Midterm 3: Version B Answer Key Midterm 3: Version C Answer Key Midterm 3: Version D Answer Key Midterm 3: Version E Answer Key Answer Key 11.1 Answer Key 11.2 Answer Key 11.3 Answer Key 11.4 Answer Key 11.5 Answer Key 11.6 Answer Key 11.7 Answer Key 11.8 Answer Key 11.9 Final Exam: Version A Answer Key Final Exam: Version B Answer Key Intermediate Algebra Chapter 2: Linear Equations 2.5 Absolute Value Equations When solving equations with absolute values, there can be more than one possible answer. This is because the variable whose absolute value is being taken can be either negative or positive, and both possibilities must be accounted for when solving equations. Example 2.5.1 Solve When there are absolute values in a problem, it is important to first isolate the absolute value, then remove the absolute value by considering both the positive and negative solutions. Notice that, in the next two examples, all the numbers outside of the absolute value are moved to one side first before the absolute value bars are removed and both positive and negative solutions are considered. Example 2.5.2 Solve Example 2.5.3 Solve Note: the objective in solving for absolute values is to isolate the absolute value to yield a solution. Example 2.5.4 Solve Often, the absolute value of more than just a variable is being taken. In these cases, it is necessary to solve the resulting equations before considering positive and negative possibilities. This is shown in the next example. Example 2.5.5 Solve Since absolute value can be positive or negative, this means that there are two equations to solve. Remember: the absolute value must be isolated first before solving. Example 2.5.6 Solve Now, solve two equations to get the positive and negative solutions: There exist two other possible results from solving an absolute value besides what has been shown in the above six examples. Example 2.5.7 Consider the equation from Example 2.5.5. What happens if, instead, the equation to solve is or For there is no so there will be just one solution instead of two. Solving this equation yields: For the result will be “no solution” or , since an absolute value is never negative. One final type of absolute value problem covered in this chapter is when two absolute values are equal to each other. There will still be both a positive and a negative result—the difference is that a negative must be distributed into the second absolute value for the negative possibility. Example 2.5.8 Solve Solving this means solving: Removing the absolute value signs leaves: Questions For questions 1 to 24, solve each absolute value equation. Answer Key 2.5 Previous/next navigation Previous: 2.4 Fractional Linear Equations Next: 2.6 Working With Formulas Back to top License Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted. Share This Book Powered by Pressbooks Pressbooks User Guide \|Pressbooks Directory \|Contact Pressbooks on YouTubePressbooks on LinkedIn
4640	https://www.healthline.com/nutrition/vegan-vs-vegetarian	#### Nutrition Vegan vs. Vegetarian: What’s the Difference? Medically reviewed by Kim Chin, RD — Written by Alina Petre, MS, RD (NL) — Updated on July 24, 2024 A vegetarian does not eat any animal flesh such as meat, poultry, or fish. A vegan is a stricter vegetarian who also avoids consuming dairy, eggs, and any other ingredients derived from animals. Vegetarian diets have reportedly been around since as early as 700 B.C. Several types exist, and individuals may practice them for a variety of reasons. These include health, ethics, environmentalism, and religion. Vegan diets appeared a little more recently, but have been getting a good amount of press. This article takes a look at the similarities and differences between these two diets. It also discusses how they affect your health and the environment. What is a vegetarian diet? According to the Vegetarian Society, a vegetarian is someone who does not eat any meat, poultry, game, fish, shellfish, or by-products of animal slaughter. Vegetarian diets contain various levels of fruits, vegetables, grains, pulses, nuts, and seeds. The inclusion of dairy, honey, and eggs depends on the type of diet you follow. The most common types of vegetarians include: Lacto-ovo vegetarians: vegetarians who avoid all animal flesh, but do consume dairy and egg products Lacto vegetarians: vegetarians who avoid animal flesh and eggs, but do consume dairy products Ovo vegetarians: vegetarians who avoid all animal products except eggs Vegans: vegetarians who avoid all animal and animal-derived products People who do not eat meat or poultry but do consume fish are considered pescatarians, whereas part-time vegetarians are often referred to as flexitarians. Although they’re sometimes considered vegetarians, pescatarians and flexitarians do eat animal flesh. So, they do not technically fall under the definition of vegetarianism. ADVERTISEMENT Save on GLP-1 treatments with LifeMD Find GLP-1 meds and expert care through LifeMD’s comprehensive platform. Take a short quiz to see if you qualify for help with insurance reimbursement. As low as $0–$25 copay for GLP-1 weight loss treatment. TAKE THE QUIZ 4.9 Star Reviews 745k+ members Medicare Accepted What is a vegan diet? A vegan diet can be viewed as the strictest form of vegetarianism. Veganism is currently defined by the Vegan Society as a way of living that attempts to exclude all forms of animal exploitation and cruelty as much as possible. This includes exploitation for food and any other purpose. So, a vegan diet not only excludes animal flesh, but also dairy, eggs, and other ingredients that come from animals. These include: gelatin honey carmine pepsin shellac albumin whey casein some forms of vitamin D3 Vegetarians and vegans often avoid eating animal products for similar reasons. The largest difference is the degree to which they consider animal products acceptable. For instance, both vegans and vegetarians may exclude meat from their diets for health or environmental reasons. Vegans also choose to avoid all animal by-products because they believe this has the largest impact on their health and the environment. In terms of ethics, vegetarians are opposed to killing animals for food, but generally consider it acceptable to consume animal by-products such as milk and eggs, as long as the animals are kept in adequate conditions. On the other hand, vegans believe that animals have a right to be free from human use, whether it’s for food, clothing, science, or entertainment. As a result, they seek to avoid all animal by-products, regardless of the conditions in which animals are bred or housed. The desire to avoid all forms of animal exploitation is why vegans choose to forgo dairy and eggs — products that many vegetarians have no problem consuming. Nutrition considerations for vegetarian and vegan diets ResearchTrusted Source shows vegetarian and vegan diets tend to be low in saturated fat and cholesterol. They also tend to contain high amounts of vitamins, minerals, fiber, and healthy plant compounds. What’s more, both diets contain a high amount of nutrient-dense foods. These may include fruit, vegetables, whole grains, nuts, seeds, and soy products. On the other hand, poorly planned vegetarian and vegan diets could result in low intakes of some nutrients, particularly iron, calcium, zinc, and vitamin D. Both diets also tend to contain limited amounts of vitamin B12 and long-chain omega-3 fatty acids, although levels of these nutrients are generally lower in vegans than vegetarians. While vegetarian and vegan diets tend to lean heavily on fruits, legumes, and vegetables, some items might be diary- and meat-free but are still: highly processed high in added sugars cooked using methods that can add excess fat Cookies, french fries, candies, and even nut based ice creams may fall into the vegan and vegetarian category yet still contain refined carbohydrates, are highly processed, are high in added sugar, or are deep fried. These items should be consumed in moderation. ADVERTISEMENT Save on GLP-1 treatments with LifeMD Find GLP-1 meds and expert care through LifeMD’s comprehensive platform. Take a short quiz to see if you qualify for help with insurance reimbursement. As low as $0–$25 copay for GLP-1 weight loss treatment. TAKE THE QUIZ 4.9 Star Reviews 745k+ members Medicare Accepted Which is healthier? According to a report from the Academy of Nutrition and DieteticsTrusted Source, both vegetarian and vegan diets can be considered appropriate for all stages of life, as long as the diet is planned well. An insufficient intake of nutrients such as omega-3 fatty acids, calcium, and vitamins D and B12 can negatively impact various aspects of health, including mental and physical health. Both vegetarians and vegans may have lower intakes of these nutrients. However, studiesTrusted Source show that vegetarians tend to consume slightly more calcium and vitamin B12 than vegans. Nonetheless, both vegetarians and vegans should pay special attention to nutrition strategies meant to increase the absorption of nutrients from plant foods. It may also be necessary to consume fortified foods and supplements, especially for nutrients such as iron, calcium, omega-3, and vitamins D and B12. Vegetarians and vegans should strongly consider: analyzing their daily nutrient intake getting their blood nutrient levels measured taking supplements accordingly The few studiesTrusted Source directly comparing vegetarian to vegan diets report that vegans may have a somewhat lower risk of developing type 2 diabetes, heart disease, and various types of cancer than vegetarians. That said, most studies so far have been observational in nature. This means that it’s impossible to say exactly which aspect of the vegan diet produces these effects and to confirm that diet is the only determining factor. Veganism is about more than what you eat Although vegetarians and vegans may choose to avoid animal products for similar purposes, this choice often extends beyond diet for vegans. In fact, veganism is often considered a lifestyle strongly anchored in animal rights. For this reason, many vegans also avoid purchasing clothing items containing silk, wool, leather, or suede. What’s more, many vegans boycott companies that test on animals and purchase only cosmetics that are free of animal by-products. People known as “ethical vegans” also tend to steer clear of circuses, zoos, rodeos, horse races, and any other activities involving the use of animals for entertainment. Finally, many environmentalists adopt a vegan diet for its reduced impact on the earth’s resources and the benefits it has against climate change. The bottom line Vegetarians and vegans may avoid consuming animal products for similar reasons, but do so to various extents. Several types of vegetarians exist, and vegans are at the strictest end of the vegetarian spectrum. Both types of diet can be considered safe for all stages of life, but vegan diets may even offer additional health benefits. However, it’s important for both vegetarians and vegans to plan their diets well in order to avoid health complications over the long term. ADVERTISEMENT Explore these resources for weight loss Great for sustainable weight loss plans Prioritizes a balanced relationship with food Uses a quiz to create a custom plan and estimated timeline Now pairs GLP-1 treatments with psychological support $209 per year with other plans available 21-Day Free Trial TAKE QUIZ Great for a modern approach to healthcare $9/month with your Prime membership Accessible and affordable care designed around your needs Offers 24/7 care, in-app health records, messaging, and more Accepts insurance for in-office and virtual visits START 14-DAY FREE TRIAL Great for tailored GLP-1 support Access weight-loss medication, if qualified Meet 1-on-1 with doctors and registered dietitians Get insurance support to maximize your coverage Keep the weight off with our trusted Points® program Save up to 48% per month with select plans TAKE QUIZ Great for finding help with GLP-1 medication cost Access to GLP-1 treatments, a dedicated doctor, and care team As low as $0 copay for Wegovy, Zepbound through LifeMD In-app care, insurance, primary care, and more included Start for just $75 today, with medication rush shipped TAKE QUIZ How we reviewed this article: Healthline has strict sourcing guidelines and relies on peer-reviewed studies, academic research institutions, and medical journals and associations. We only use quality, credible sources to ensure content accuracy and integrity. You can learn more about how we ensure our content is accurate and current by reading our editorial policy. Definition of veganism. (n.d.). Dinu M, et al. (2017). Vegetarian, vegan diets and multiple health outcomes: A systematic review with meta-analysis of observational studies. Eating veggie. (n.d.). Fernandes S, et al. (2024). Exploring vitamin B12 supplementation in the vegan population: A scoping review of the evidence. Łuszczki E, et al. (2023). Vegan diet: nutritional components, implementation, and effects on adults' health. Mekina V, et al. (2016). Position of the Academy of Nutrition and Dietetics: Vegetarian diets. Rose D, et al. (2019). Carbon footprint of self-selected US diets: nutritional, demographic, and behavioral correlates. van de Kamp ME, et al. (2018). Reducing GHG emissions while improving diet quality: exploring the potential of reduced meat, cheese and alcoholic and soft drinks consumption at specific moments during the day. Our experts continually monitor the health and wellness space, and we update our articles when new information becomes available. Current Version Jul 24, 2024 Written By Alina Petre Edited By John Bassham Medically Reviewed By Kim Chin, RD Copy Edited By Copy Editors Aug 23, 2021 Medically Reviewed By Kimberley Rose-Francis RDN, CDCES, LD VIEW ALL HISTORY Share this article Evidence Based This article is based on scientific evidence, written by experts and fact checked by experts. Our team of licensed nutritionists and dietitians strive to be objective, unbiased, honest and to present both sides of the argument. This article contains scientific references. The numbers in the parentheses (1, 2, 3) are clickable links to peer-reviewed scientific papers. Medically reviewed by Kim Chin, RD — Written by Alina Petre, MS, RD (NL) — Updated on July 24, 2024 related stories The Best Diabetes-Friendly Diets for Weight Loss The Paleo Diet — A Beginner's Guide Plus Meal Plan Can the Paleo Diet Help You Lose Weight? The Nordic Diet: An Evidence-Based Review 14 Health Foods That Taste Better Than Junk Foods Was this article helpful? Yes No
4641	https://www.cuemath.com/semicircle-formulas/	Semicircle Formulas Half a portion of any circle is known as a semicircle and is formed by cutting a whole circle along the diameter. Various parameters related to a semi-circle like a diameter, area, the perimeter can be calculated using semicircle formulas. The diameter of a circle divides the circle into two equal semicircles. The area of any semicircle is half of the area of a circle. Let us understand the semicircle formulas using solved examples in the following sections. What are Semicircle Formulas? The semicircle formulas include the formulas to calculate the area, perimeter, and circumference of a semicircle. These formulas are based on the fact that a semicircle is half a portion of a full circle. Area of a Semicircle Formula Since we know that a semicircle is half a circle, the semicircle area will be half of the area of a circle. So, the area of a circle is πR2 where R is the radius of the circle. The area of a semi-circle refers to the region or inner space of the semi-circle. Hence, Area of a semicircle formula = πR2/ 2 square units. where, Perimeter of a Semicircle Formula The perimeter of a semicircle formula is used to calculate the perimeter of a semicircle. We must know either the diameter or radius of a circle along with the length of the arc. To evaluate the length of the arc of the semicircle, we must calculate the circumference of a circle. where, Circumference of a Semicircle Formula The circumference of a semicircle is defined as the length of the arc around the semicircle. It is half of the circumference of a circle. The difference between the circumference and perimeter of a semicircle is that the circumference is only the length of the arc which is the curved portion on the boundary, while the perimeter of the semicircle includes circumference as well as diameter. Circumference of semicircle formula = πR units where, Let us see how to use these semicircle formulas in the following section. Book a Free Trial Class Examples on Semicircle Formula Let us take a look at a few examples to better understand the formulas of the semicircle. Example 1: Using the semicircle formula calculate the area of the semicircle whose diameter is 12 in. Express your answer in terms of π. Solution: To find: The area of the semicircle, Given: Diameter of the semicircle = 12 in Radius of semicircle = 12/2 = 6 in Using semicircle formulas, Area of Semicircle = 1/2 × (π r2) Area = (π × 62)/2 = 36 π/2 = 18 π in2 Answer: The area of the semicircle is 18π in2. Example 2: If the radius of a semicircle is 7 units, then using the semicircle formula find its perimeter. Solution: To find: The perimeter of a semicircle Given: The radius of semicircle = 7 units Using the perimeter of a semicircle formula, Perimeter of a semicircle = πr + d = πr + 2r = (7 × 22/7 + 14) units = (22 + 14) units Answer: The perimeter of the semicircle is 36 units. Example 3:Using the semicircle formulas, calculate the circumference of a semi-circle whose diameter is 8 units. Solution: To find: The circumference of a semicircle Given: The diameter of semicircle = 8 units Radius = 8/2 = 4 units Using the circumference of a semicircle formula, Circumference of a semicircle = πr units C = (3.14) × (4) C = 12.56 units Answer: The circumference of the semicircle is 12.56 units. FAQs on Semicircle Formulas What is the Area of a Semicircle Formula? The area of a semi-circle refers to the total amount of inner space of the semi-circle. It is expressed as 'Area of a semicircle formula = πR2/ 2 square units', where, What is the Perimeter of a Semicircle Formula? The perimeter of a semicircle formula is defined as the sum of half of the circumference of the circle and the diameter of a circle. It is expressed as, where, What are the Formulas of Semicircle in Geometry? There are two important formulas of semicircle in geometry which are given below: What is the Circumference of a Semicircle Formula? The formula of the circumference of a semicircle is πR units. It is the length of the curved edge of the semicircle which is exactly half of the circumference of a circle. What is the Formula for Finding the Diameter of a Semicircle? The diameter of a semicircle can be easily found if any of the following values are known: Area, perimeter, or circumference. If A is the area of the semicircle, then the diameter (d) = √(8A/π) units. If P is the perimeter, then diameter (d) of semicircle formula = 2P/(π + 2) units. And, if C is the circumference, then the diameter (d) = 2C/π units. What is the Difference Between the Area of a Circle and Semicircle Formulas? The area of the circle formula is expressed as A = πr2. Whereas, the area of the semicircle is half the area of the circle. Therefore, the area of the semicircle is πr2/2, where 'r' is the radius. The area of a semicircle is expressed in square units like in2, cm2, m2, yd2, ft2, etc.
4642	https://www.youtube.com/watch?v=nb-_bs6eSYo	Max product of 3 real numbers (KristaKingMath) Krista King 272000 subscribers 418 likes Description 30884 views Posted: 2 Jun 2014 ► My Partial Derivatives course: Learn how to find three positive numbers with sum 100 and maximum possible product. To do this, write two equations that define the relationship between the numbers, and then use the equations together to reduce the equation of the product to just two variables. Take first-order and second-order partial derivatives. Use the first-order partial derivatives to find critical points of the function, and eliminate all critical points that don't match the criteria of the question. Then use the second-order partial derivatives in the second derivative test to verify that the remaining critical point is a maximum of the product function. ● ● ● GET EXTRA HELP ● ● ● If you could use some extra help with your math class, then check out Krista’s website // ● ● ● CONNECT WITH KRISTA ● ● ● Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;) Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!” So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student—from basic middle school classes to advanced college calculus—figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: FACEBOOK // TWITTER // INSTAGRAM // PINTEREST // GOOGLE+ // QUORA // 32 comments Transcript: today we're going to be talking about how to find three positive numbers when those three positive numbers have a sum of 100 and when we're trying to find the maximum product of those three positive numbers So in other words what we're looking to do is find three positive numbers which we'll call x y and z who have a sum of 100 So if we say x + y + z = 100 that would satisfy the first part of the question We're also looking for these three numbers x y and z to have the maximum product So we're looking to maximize the product of those three numbers So if we say product is equal to x y and z where the product is we're multiplying those three numbers together this is the equation that we're looking to maximize Since we want to maximize this second equation here for product we need to reduce it to at most two variables And the way that we're going to do that is by solving this top equation for one of these variables and then plugging it in to the second equation So if we solve this first equation here for Z we'll get Z is equal to 100 - X - Y We just subtract X and Y from both sides Now we have a value for Z that we can plug into our product formula So we'll say product is equal to X Y Here's where we plug in for Z 100 - x - y Now notice that our product equation is in terms of two variables only x and y At this point instead of using p for product let's just go ahead and call this a function in terms of x and y And we'll go ahead and distribute the xy across the 100 - x - y When we do that we'll get 100 xy -x xy is a -x^2 y and xy a - y is a -x y^2 Now if you haven't realized it by now essentially we're dealing with an optimization problem And this is a multivariable optimization problem If you'll remember from single variable optimization problems once we got the equation together that we were going to maximize we needed to take the derivative of the equation Well in this case since we have a multivariable function we're going to need to take partial derivatives of the function So we'll take the partial derivative of f with respect to x When we do that we'll get 100 yus 2xy - y^ 2 If we take the partial derivative of f with respect to y we'll get 100x - x^2 - 2x y So these are our partial derivatives We're also going to need second order partial derivatives You're going to need those every time to verify that you do in fact have a maximum or a minimum depending on what the optimization problem is asking you for So let's go ahead and find our second order partial derivatives We'll do that up here We'll say the second order partial derivative of f with respect to x We'll start remember with the first order partial derivative with respect to x and take the partial derivative with respect to x Again 100 y will drop away because there's no x variable involved there We'll just be left with here -2 y and our y^2 will drop away because there's no x variable involved Taking the partial derivative of f with respect to y is going to give us 100x here will drop away There's no y variable involved.x^2 drops away and here we're just left with -2x Now if I take the mixed second order partial derivative and I have x y like this I'll start with my first order partial derivative for x but this time I'm going to take the partial derivative with respect to y That's going to leave me with 100 minus 2x - 2 y I'll come back to these in a second I'm going to need them later But for now what we're looking to do is use our first order partial derivatives to find critical points of our function f ofxy The way that we're going to do that is by solving this as a system of simultaneous equations So notice this first equation the partial derivative of f with respect to x We can factor a y out of this equation because we have y in each term Here's what that looks like We'll get y 100 - 2x - y We'll go ahead and set that equal to zero Our second equation we can factor out an x because we have an x in every term When we do that we'll get x 100 - x - 2 y and that's going to be equal to zero Now here's how we're going to find critical points We're going to find values of y that satisfy this first equation Then we're going to plug those into our second equation to find corresponding values of x So here's what that looks like Because we have this factored we're going to set each of these factors equal to zero on the right hand side Take them as separate equations and solve for values of y So this equation here this first equation is only true if either y is equal to 0 or 100 - 2x - y is equal to z If y is equal to zero then we have the value y equals 0 No problem If we have this second factor here 100 - 2x - y = 0 If we take that equation and we solve that for y we would add y to both sides and we'd get y is equal to 100 - 2x These are the two values of y that come out of our first equation What we want to do now is plug those values of y into our second equation to find corresponding values of x So if we plug in y = 0 this -2 y term will go away and we'll be left with x 100 - x is equal to 0 When that happens we have again two factors here that we need to solve for individually So either x= 0 and in that case we have the critical point 0 0 because we said that x was equal to 0 and that came from plugging yals 0 into this equation So y would be zero there or we have 100 - x is equal to 0 And in that case we would add x to both sides and we'd get x= 100 And we'd have the critical point x is 100 and y is zero because we got to that equation by plugging in 0 for y So we've generated two critical points from the value y= 0 by plugging that into our second equation What happens when we plug in y = 100 - 2x well remember we're going to plug it into this equation here we would get x 100 - x - 2 here's where we plug it in for y 100 - 2x and that's going to be equal to zero Now if we want to simplify this here we get x 100 - x - 200 + 4x and that's going to be equal to zero simplifying here we'll get x -100gativex plus a 4x is + 3x and that's equal to zero So again here we have two factors that we have to solve for individually when we plugged in 100 - 2x either x is equal to 0 So in that case we would have the critical point 0 then we would want to plug that x value back into 100 because we don't want to call the critical point 0 100 - 2x we want two real numbers So we plug x= 0 back into this equation for y and we get y = 100 - 2 0 This term goes away We get y= 100 So that point is 0 100 Or we have this second factor here -100 + 3x is equal to 0 So solving that for x we'd add 100 to both sides and we'd get 3x is equal to 100 or x = 100 over 3 So we'd have the point here 100 over 3 To find the corresponding y value we'd plug this x value back into y = 100 - 2x and we get y is equal to 100 - 2 100 over 3 or 200 over 3 If we want to find a common denominator here we'd multiply this 100 by 3 over3 and get 300 over 3 So you can see here that our result is 300 over3 - 200 over3 which is 100 over3 So then our corresponding y value is 100 over3 So we've exhausted all of our critical point options We generated four potential critical points But what we need to realize at this point is that the problem has asked for three positive numbers And if we look here at our critical points what we notice is that this second critical point is out here because zero is not a positive number It's just zero So that can't be a potential critical point These points are both zero And we have a zero value here for y So none of these three critical points here can satisfy the question we've been asked That means the only critical point that we can investigate is this first one here 100 over3 100 over3 Before we go on to find our third positive number we want to verify that this critical point does in fact generate a maximum Remember that we were looking to find the maximum product of these three positive numbers So what we have to do is use the second derivative test to verify that this is in fact a maximum Remember our second derivative test we use the equation for d And what we do is we plug this critical point into the second order partial derivative with respect to x So we get -2 100 over3 So -2 100 / 3 We multiply that by whatever we get when we plug the critical point into the second order partial derivative with respect to y So again we'd get -2 100 over3 here -2 100 over 3 And then we subtract whatever we get when we plug our critical point into the mixed second order partial derivative and we square that result So we subtract whatever we get when we plug in there which is going to be 100 - 2 100 over 3 - 2 100 over 3 I'm plugging it into this equation right here And we want to square that result Squaring it is part of our second derivative test So now we just need to calculate the value here We get here -200 over 3 -200 over 3 minus here we'll get 100 - 200 over 3 - 200 over 3 and we're squaring that whole thing Continuing to simplify here -200 -200 gives us a positive 40,000 divided by 9 Then we have here minus if we find a common denominator here with this 100 we multiply it by 3 over3 and we call this 300 over 3 300 - 200 is 100 - 200 again is a -00 So this whole thing here is -100 over3 when we square it we get a positive 10,000 over 9 like this Now all we have to do is subtract We can see that that result for D is going to be a positive 30,000 / 9 And we could simplify further but we don't need to because all that we care about is whether this answer right here is positive or negative or equal to zero In this case it's greater than zero When the second derivative test is greater than zero we have to look at the second order partial derivative with respect to x evaluated at the critical point that we found Well when we evaluate this at the critical point that we found we get -200 over 3 This value is less than zero When the second derivative test is greater than zero and the second order partial derivative with respect to x is less than zero that tells us that we have a maximum value at the critical point we found which was 100 over 3 100 over3 That verifies that this critical point does in fact represent a maximum of our function f ofxy So we verify that this is a maximum All that's left to do is find the value of z because we only have here values of x and y To find a value for z we can plug into this equation right here We get z is equal to 100us 100 over 3 - 100 over 3 If we find a common denominator by multiplying this 100 by 3 over3 we get 300 over 3 And we can see that Z is equal to 100 over 3 Because we get 300 - 100 - 100 is a result of 100 And that tells us that our three positive numbers with a sum of 100 and maximum possible product are the numbers 100 over 3 100 over 3 and 100 over 3
4643	https://chamberland.math.grinnell.edu/papers/arctangent.pdf	Mathematical Assoc. of America American Mathematical Monthly 121:1 August 4, 2018 2:23 p.m. arctan˙2.tex page 1 Arctangent Formulas and Pi Marc Chamberland and Eugene A. Herman Abstract. Using both geometrical and analytical approaches, new multivariable formulas con-necting the arctangent function and the number π are produced. 1. INTRODUCTION. Since the discovery of Machin’s formula π 4 = 4 arctan 1 5 −arctan 1 239 , (1) the arctangent function has been ubiquitous in calculations of π. While formulas like (1) have been heavily explored , we seek formulas that link π with a linear combi-nation of arctangents of general arguments. The simplest example is the well-known equation π 2 = arctan(x) + arctan 1 x (2) for all x > 0. Another example, a variant of an equation due to Euler, states π 2 = arctan(x) −arctan(x −y) + arctan x2 −xy + 1 y for all x and when y > 0. The goal of this note is to develop arctangent formulas with several variables. 2. GEOMETRY OF TRIANGLES AND TETRAHEDRA. This study started serendipidously by considering the inscribed circle in a general triangle: see Figure 1. The area of the triangle can be computed in two ways. By dissecting the triangle into r r r b a a c c b Figure 1. Inscribed circle in a triangle. three subtriangles, we ﬁnd that its total area A satisﬁes A = 1 2(a + c)r + 1 2(a + b)r + 1 2(b + c)r = (a + b + c)r, January 2014] ARCTANGENT FORMULAS AND PI 1 Mathematical Assoc. of America American Mathematical Monthly 121:1 August 4, 2018 2:23 p.m. arctan˙2.tex page 2 where r is the radius of the inscribed circle. Alternatively, applying Heron’s formula to the original triangle yields A = q abc(a + b + c). Setting the two expressions equal produces r = s abc a + b + c. Since the six angles surrounding the center of the inscribed circle sum to 2π, this produces π = arctan a r a + b + c abc ! + arctan b r a + b + c abc ! (3) + arctan c r a + b + c abc ! for all a, b, c > 0. To generalize this geometric approach, one could consider an (n −1)-sphere in-scribed in a simplex in n dimensions. The volume of the simplex can be calculated with the Cayley–Menger determinant. More challenging is the generalization of the angles around the sphere’s center, sometimes called “solid angles”; see [3, 4]. The complexity of this approach, particularly in higher dimensions, suggests an analytic approach for ﬁnding formulas similar to equation (3). 3. ARCTANGENT AND SYMMETRIC POLYNOMIALS. Some beautiful iden-tities connect the tangent function with symmetric polynomials. Let xi = tan(θi) for i = 1, 2, 3, . . . and let ek(x) denote the kth elementary symmetric polynomial in the variables x1, x2, x3, . . .. The ﬁrst few examples are e0(x) = 1, e1(x) = X i xi, e2(x) = X i 0 for all i. This produces π 2 = arctan y1 √y1y2 + arctan y2 √y1y2 (8) = arctan ry1 y2 + arctan ry2 y1 . Of course this is equivalent to equation (2). This approach can be generalized by letting xi = yif(y1, y2, . . . , yn) for i = 1, 2, . . . , n. Setting the numerator of equation (6) equal to zero produces π = arctan y1 s y1 + y2 + y3 y1y2y3 ! + arctan y2 s y1 + y2 + y3 y1y2y3 ! + arctan y3 s y1 + y2 + y3 y1y2y3 ! , (9) which is the same as equation (3), while setting the denominator of equation (6) equal to zero gives π 2 = arctan y1 √y1y2 + y2y3 + y3y1 + arctan y2 √y1y2 + y2y3 + y3y1 + arctan y3 √y1y2 + y2y3 + y3y1 . (10) Equation (7), with its numerator equal to zero, generates π = arctan y1 s y1 + y2 + y3 + y4 y1y2y3 + y1y2y4 + y1y3y4 + y2y3y4 ! + arctan y2 s y1 + y2 + y3 + y4 y1y2y3 + y1y2y4 + y1y3y4 + y2y3y4 ! + arctan y3 s y1 + y2 + y3 + y4 y1y2y3 + y1y2y4 + y1y3y4 + y2y3y4 ! + arctan y4 s y1 + y2 + y3 + y4 y1y2y3 + y1y2y4 + y1y3y4 + y2y3y4 ! . (11) January 2014] ARCTANGENT FORMULAS AND PI 3 Mathematical Assoc. of America American Mathematical Monthly 121:1 August 4, 2018 2:23 p.m. arctan˙2.tex page 4 It is natural to wonder whether there are other “simple” choices of f that produce equations of the form C = arctan (y1f(y1, y2, . . . , yn)) + arctan (y2f(y1, y2, . . . , yn)) + · · · + arctan (ynf(y1, y2, . . . , yn)) (12) for some constant C and all y1, y2, . . . , yn > 0. Theorem 1. The only equations of the form (12) satisﬁed by a nonzero function f = q p q, where p and q are polynomials, are the equations (8)–(11). Proof. Letting xi = yif(y1, y2, . . . , yn), use equations (4) and (12) to write D := tan(C) = tan(arctan(x1) + · · · + arctan(xn)) = e1(x) −e3(x) + e5(x) −· · · e0(x) −e2(x) + e4(x) −· · · = fe1(y) −f 3e3(y) + f 5e5(y) −· · · e0(y) −f 2e2(y) + f 4e4(y) −· · · . (13) For the rest of the proof, the y will be suppressed from ei. Now we break the proof into two cases. n is even. Let n = 2m. Equation (13) can be written as D e0 −f 2e2 + · · · + (−1)mf 2me2m = fe1 −f 3e3 + · · · + (−1)m−1f 2m−1e2m−1. (14) First, consider the case where f = p/q and p and q are polynomials with no common factors. Equation (14) can be written as D e0q2m −p2q2m−2e2 + · · · + (−1)m−1p2me2m = pq2m−1e1 −p3q2m−3e3 + · · · + (−1)m−1p2m−1qe2m−1. Since p is a factor of all but one of the terms, we must have that p divides e0q2m. This forces p = α for some constant α. Similarly, q is a factor of all but one term, so because the elementary symmetric polynomials are irreducible, either q = β or q = βe2m for some constant β. The latter option will not work because this produces the term De0q2m = D(βe2m)2m, the only term with degree (2m)2. This implies that f is a constant, necessarily the trivial case f = 0. Now suppose that f = p p/q, f is not rational, and p and q are polynomials with no common factors. Equation (14) can be written as D e0qm −pqm−1e2 + · · · + (−1)mpme2m = rp q qme1 −pqm−1e3 + · · · + (−1)m−1pm−1qe2m−1. This equation only holds in two cases: D = 0 or D = ∞. 4 c ⃝THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 Mathematical Assoc. of America American Mathematical Monthly 121:1 August 4, 2018 2:23 p.m. arctan˙2.tex page 5 When D = 0, we obtain 0 = qm−1e1 −pqm−2e3 + · · · + (−1)m−1pm−1e2m−1. (15) As before, the irreducibility of ek implies there is a constant α such that p = α or p = αe1, and subsequently, there exists a constant β and k ∈N ∪{0} such that q = βek 1e2m−1. By replacing p and q in equation (15) and considering the degree of each term, we see that the only possible option (up to scaling) is m = 2 (n = 4), p = e1, and q = e3. This is equation (11). The case D = ∞is similar, and implies 0 = e0qm −pqm−1e2 + · · · + (−1)mpme2m. This forces p = α for some constant α and q = β or q = βe2m for some constant β. The only feasible option (up to scaling) is m = 1 (n = 2), p = 1, and q = e2. This is equation (8). n is odd. Let n = 2m + 1. Equation (13) can be written as D e0 −f 2e2 + · · · + (−1)mf 2me2m = fe1 −f 3e3 + · · · + (−1)mf 2m+1e2m+1. Paralleling the even case, one can show that there is no nontrivial rational function f satisfying this equation. If f takes the form p p/q, one again needs to consider two cases, D = 0 and D = ∞. If D = 0, this forces p = α or p = αe1 for some constant α, and q = β or q = βe2m+1 for some constant β. The only feasible solution (up to scaling) is m = 1 (n = 3), p = e1, and q = e3. This is equation (9). If D = ∞, one has p = α for some constant α and either q = β or q = βe2m for some constant β. The only feasible solution (up to scaling) is m = 1 (n = 3), p = 1, and q = e2. This is equation (10). There are other choices of f, but they get more complicated. A simple example uses equation (5) with θ1 + θ2 = π/4, producing π 4 = arctan p y2 1 + 6y1y2 + y2 2 −y1 −y2 2y1 ! + arctan p y2 1 + 6y1y2 + y2 2 −y1 −y2 2y2 ! for y1, y2 > 0. REFERENCES 1. Arndt, J., Haenel, C. (2001). Pi — Unleashed. New York: Springer. 2. Bronstein, M. (1989). Simpliﬁcation of real elementary functions. In: Gonnet, G. H., ed. Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation. ISSAC ’89 (Portland US-OR, 1989-07). New York: ACM, pp. 207211. 3. Eriksson, F. (1990). On the Measure of Solid Angles. Mathematics Magazine. 63(3): 184–187. 4. Wikipedia. (2018). Solid Angle. en.wikipedia.org/wiki/Solid_angle 5. Wikipedia. (2018). List of trigonometric identities. en.wikipedia.org/wiki/List_of_ trigonometric_identities Department of Mathematics and Statistics, Grinnell College, Grinnell IA 50112 chamberl@grinnell.edu January 2014] ARCTANGENT FORMULAS AND PI 5 Mathematical Assoc. of America American Mathematical Monthly 121:1 August 4, 2018 2:23 p.m. arctan˙2.tex page 6 Department of Mathematics and Statistics, Grinnell College, Grinnell IA 50112 eaherman@gmail.com 6 c ⃝THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121
4644	https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_1e_(OpenStax)/04%3A_Exponential_and_Logarithmic_Functions/4.08%3A_Fitting_Exponential_Models_to_Data	Published Time: 2015-10-31T20:48:03Z 4.8: Fitting Exponential Models to Data - Mathematics LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This 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Home 2. Bookshelves 3. Precalculus & Trigonometry 4. Precalculus 1e (OpenStax) 5. 4: Exponential and Logarithmic Functions 6. 4.8: Fitting Exponential Models to Data Expand/collapse global location Precalculus 1e (OpenStax) Front Matter 1: Functions 2: Linear Functions 3: Polynomial and Rational Functions 4: Exponential and Logarithmic Functions 5: Trigonometric Functions 6: Periodic Functions 7: Trigonometric Identities and Equations 8: Further Applications of Trigonometry 9: Systems of Equations and Inequalities 10: Analytic Geometry 11: Sequences, Probability and Counting Theory 12: Introduction to Calculus 13: Trigonometric Functions 14: Appendix Back Matter 4.8: Fitting Exponential Models to Data Last updated Jan 2, 2021 Save as PDF 4.7: Exponential and Logarithmic Models 4.E: Exponential and Logarithmic Functions (Exercises) Page ID 1359 OpenStax OpenStax ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Learning Objectives 2. Building an Exponential Model from Data 1. EXPONENTIAL REGRESSION 2. How to: Given a set of data, perform exponential regression using a graphing utility 3. Solution 4. Exercise 4.8.1 4.8.1 5. Q&A: Is it reasonable to assume that an exponential regression model will represent a situation indefinitely? Building a Logarithmic Model from Data LOGARITHMIC REGRESSION How to: Given a set of data, perform logarithmic regression using a graphing utility Example 4.8.2 4.8.2: Using Logarithmic Regression to Fit a Model to Data Solution Exercise 4.8.2 4.8.2 Building a Logistic Model from Data LOGISTIC REGRESSION How to: Given a set of data, perform logistic regression using a graphing utility Example 4.8.3 4.8.3: Using Logistic Regression to Fit a Model to Data Solution Exercise 4.8.3 4.8.3 Media Key Concepts Contributors and Attributions Learning Objectives Build an exponential model from data. Build a logarithmic model from data. Build a logistic model from data. In previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points exactly. In this section, we use a modeling technique called regression analysis to find a curve that models data collected from real-world observations. With regression analysis, we don’t expect all the points to lie perfectly on the curve. The idea is to find a model that best fits the data. Then we use the model to make predictions about future events. Do not be confused by the word model. In mathematics, we often use the terms function, equation, and model interchangeably, even though they each have their own formal definition. The term model is typically used to indicate that the equation or function approximates a real-world situation. We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review. Take a moment to rethink each of these functions, reflect on the work we’ve done so far, and then explore the ways regression is used to model real-world phenomena. Building an Exponential Model from Data As we’ve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that’s not the whole story. It’s the way data increase or decrease that helps us determine whether it is best modeled by an exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let’s review exponential growth and decay. Recall that exponential functions have the form y=a b x y=a b x or y=A 0 e k x y=A 0 e k x. When performing regression analysis, we use the form most commonly used on graphing utilities, y=a b x y=a b x. Take a moment to reflect on the characteristics we’ve already learned about the exponential function y=a b x y=a b x (assume a>0 a>0): b b must be greater than zero and not equal to one. The initial value of the model is y=a y=a. If b>1 b>1, the function models exponential growth. As x x increases, the outputs of the model increase slowly at first, but then increase more and more rapidly, without bound. If 0<b<1 0<b<1, the function models exponential decay. As x x increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the x-axis. In other words, the outputs never become equal to or less than zero. As part of the results, your calculator will display a number known as the correlation coefficient, labeled by the variable r r, or r 2 r 2. (You may have to change the calculator’s settings for these to be shown.) The values are an indication of the “goodness of fit” of the regression equation to the data. We more commonly use the value of r 2 r 2 instead of r r, but the closer either value is to 1 1, the better the regression equation approximates the data. EXPONENTIAL REGRESSION Exponential regression is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. We use the command “ExpReg” on a graphing utility to fit an exponential function to a set of data points. This returns an equation of the form, y=a b x(4.8.1)(4.8.1)y=a b x Note that: b b must be non-negative. when b>1 b>1, we have an exponential growth model. when 0<b<1 0<b<1, we have an exponential decay model. How to: Given a set of data, perform exponential regression using a graphing utility Use the STAT then EDIT menu to enter given data. Clear any existing data from the lists. List the input values in the L1 column. List the output values in the L2 column. Graph and observe a scatter plot of the data using the STATPLOT feature. Use ZOOM to adjust axes to fit the data. Verify the data follow an exponential pattern. Find the equation that models the data. Select “ExpReg” from the STAT then CALC menu. Use the values returned for a and b to record the model, y=a b x y=a b x. Graph the model in the same window as the scatterplot to verify it is a good fit for the data. Example 4.8.1 4.8.1: Using Exponential Regression to Fit a Model to Data In 2007, a university study was published investigating the crash risk of alcohol impaired driving. Data from 2,871 2,871 crashes were used to measure the association of a person’s blood alcohol level (BAC) with the risk of being in an accident. Table 4.8.1 4.8.1 shows results from the study. The relative risk is a measure of how many times more likely a person is to crash. So, for example, a person with a BAC of 0.09 0.09 is 3.54 3.54 times as likely to crash as a person who has not been drinking alcohol. Table 4.8.1 4.8.1\| BAC \| 0 \| 0.01 \| 0.03 \| 0.05 \| 0.07 \| 0.09 \| \| Relative Risk of Crashing \| 1 \| 1.03 \| 1.06 \| 1.38 \| 2.09 \| 3.54 \| \| BAC \| 0.11 \| 0.13 \| 0.15 \| 0.17 \| 0.19 \| 0.21 \| \| Relative Risk of Crashing \| 6.41 \| 12.6 \| 22.1 \| 39.05 \| 65.32 \| 99.78 \| Let x x represent the BAC level, and let y y represent the corresponding relative risk. Use exponential regression to fit a model to these data. After 6 6 drinks, a person weighing 160 160 pounds will have a BAC of about 0.16 0.16. How many times more likely is a person with this weight to crash if they drive after having a 6 6-pack of beer? Round to the nearest hundredth. Solution Using the STAT then EDIT menu on a graphing utility, list the BAC values in L1 and the relative risk values in L2. Then use the STATPLOT feature to verify that the scatterplot follows the exponential pattern shown in Figure 4.8.1 4.8.1: Figure 4.8.1 4.8.1 Use the “ExpReg” command from the STAT then CALC menu to obtain the exponential model, y=0.58304829(2.20720213 E 10)x y=0.58304829(2.20720213 E 10)x Converting from scientific notation, we have: y=0.58304829(22,072,021,300)x y=0.58304829(22,072,021,300)x Notice that r 2≈0.97 r 2≈0.97 which indicates the model is a good fit to the data. To see this, graph the model in the same window as the scatterplot to verify it is a good fit as shown in Figure 4.8.2 4.8.2: Figure 4.8.2 4.8.2 Use the model to estimate the risk associated with a BAC of 0.16 0.16. Substitute 0.16 0.16 for x x in the model and solve for y y. y=0.58304829(22,072,021,300)x Use the regression model found in part(a)=0.58304829(22,072,021,300)0.16 Substitute 0.16 for x≈26.35 Round to the nearest hundredth y=0.58304829(22,072,021,300)x Use the regression model found in part(a)=0.58304829(22,072,021,300)0.16 Substitute 0.16 for x≈26.35 Round to the nearest hundredth If a 160 160-pound person drives after having 6 6 drinks, he or she is about 26.35 26.35 times more likely to crash than if driving while sober. Exercise 4.8.1 4.8.1 Table 4.8.2 4.8.2 shows a recent graduate’s credit card balance each month after graduation. Table 4.8.2 4.8.2\| Month \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| \| Debt ($) \| 620.00 \| 761.88 \| 899.80 \| 1039.93 \| 1270.63 \| 1589.04 \| 1851.31 \| 2154.92 \| Use exponential regression to fit a model to these data. If spending continues at this rate, what will the graduate’s credit card debt be one year after graduating? Answer a The exponential regression model that fits these data is y=522.88585984(1.19645256)x y=522.88585984(1.19645256)x. Answer b If spending continues at this rate, the graduate’s credit card debt will be $4,499.38$4,499.38 after one year. Q&A: Is it reasonable to assume that an exponential regression model will represent a situation indefinitely? No. Remember that models are formed by real-world data gathered for regression. It is usually reasonable to make estimates within the interval of original observation (interpolation). However, when a model is used to make predictions, it is important to use reasoning skills to determine whether the model makes sense for inputs far beyond the original observation interval (extrapolation). Building a Logarithmic Model from Data Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best. Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we’ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic regression analysis, we use the form of the logarithmic function most commonly used on graphing utilities, y=a+b ln(x)y=a+b ln⁡(x). For this function All input values, x x,must be greater than zero. The point (1,a)(1,a) is on the graph of the model. If b>0 b>0,the model is increasing. Growth increases rapidly at first and then steadily slows over time. If b<0 b<0,the model is decreasing. Decay occurs rapidly at first and then steadily slows over time. LOGARITHMIC REGRESSION Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. We use the command “LnReg” on a graphing utility to fit a logarithmic function to a set of data points. This returns an equation of the form, y=a+b ln(x)(4.8.2)(4.8.2)y=a+b ln⁡(x) Note that all input values, x x,must be non-negative. when b>0 b>0, the model is increasing. when b<0 b<0, the model is decreasing. How to: Given a set of data, perform logarithmic regression using a graphing utility Use the STAT then EDIT menu to enter given data. Clear any existing data from the lists. List the input values in the L1 column. List the output values in the L2 column. Graph and observe a scatter plot of the data using the STATPLOT feature. Use ZOOM to adjust axes to fit the data. Verify the data follow a logarithmic pattern. Find the equation that models the data. Select “LnReg” from the STAT then CALC menu. Use the values returned for a and b to record the model, y=a+b ln(x)y=a+b ln⁡(x). Graph the model in the same window as the scatterplot to verify it is a good fit for the data. Example 4.8.2 4.8.2: Using Logarithmic Regression to Fit a Model to Data Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed countries since the beginning of the 20 th century. Table 4.8.3 4.8.3 shows the average life expectancies, in years, of Americans from 1900–2010. Table4.8.3 4.8.3\| Year \| 1900 \| 1910 \| 1920 \| 1930 \| 1940 \| 1950 \| \| Life Expectancy(Years) \| 47.3 \| 50.0 \| 54.1 \| 59.7 \| 62.9 \| 68.2 \| \| Year \| 1960 \| 1970 \| 1980 \| 1990 \| 2000 \| 2010 \| \| Life Expectancy(Years) \| 69.7 \| 70.8 \| 73.7 \| 75.4 \| 76.8 \| 78.7 \| Let x x represent time in decades starting with x=1 x=1 for the year 1900, x=2 x=2 for the year 1910, and so on. Let y y represent the corresponding life expectancy. Use logarithmic regression to fit a model to these data. Use the model to predict the average American life expectancy for the year 2030. Solution Using the STAT then EDIT menu on a graphing utility, list the years using values 1–12 1–12 in L1 and the corresponding life expectancy in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logarithmic pattern as shown in Figure 4.8.3 4.8.3: Figure 4.8.3 4.8.3 Use the “LnReg” command from the STAT then CALC menu to obtain the logarithmic model, y=42.52722583+13.85752327 ln(x)y=42.52722583+13.85752327 ln⁡(x) Next, graph the model in the same window as the scatterplot to verify it is a good fit as shown in Figure 4.8.4 4.8.4: Figure 4.8.4 4.8.4 To predict the life expectancy of an American in the year 2030 2030, substitute x=14 x=14 for the in the model and solve for y y: y=42.52722583+13.85752327 ln(x)Use the regression model found in part(a)=42.52722583+13.85752327 ln(14)Substitute 14 for x≈79.1 Round to the nearest tenth y=42.52722583+13.85752327 ln⁡(x)Use the regression model found in part(a)=42.52722583+13.85752327 ln⁡(14)Substitute 14 for x≈79.1 Round to the nearest tenth If life expectancy continues to increase at this pace, the average life expectancy of an American will be 79.1 79.1 by the year 2030 2030. Exercise 4.8.2 4.8.2 Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. Table 4.8.4 4.8.4 shows the number of games sold, in thousands, from the years 2000–2010. Table 4.8.4 4.8.4\| Year \| 2000 \| 2001 \| 2002 \| 2003 \| 2004 \| 2005 \| \| Number Sold (thousands) \| 142 \| 149 \| 154 \| 155 \| 159 \| 161 \| \| Year \| 2006 \| 2007 \| 2008 \| 2009 \| 2010 \| Number Sold (thousands) \| 163 \| 164 \| 164 \| 166 \| 167 Let x x represent time in years starting with x=1 x=1 for the year 2000. Let y y represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data. If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand. Answer a The logarithmic regression model that fits these data is y=141.91242949+10.45366573 ln(x)y=141.91242949+10.45366573 ln⁡(x) Answer b If sales continue at this rate, about 171,000 171,000 games will be sold in the year 2015 2015. Building a Logistic Model from Data Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or limiting value. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients. It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic regression analysis, we use the form most commonly used on graphing utilities: y=c 1+a e−b x y=c 1+a e−b x Recall that: c 1+a c 1+a is the initial value of the model. when b>0 b>0, the model increases rapidly at first until it reaches its point of maximum growth rate, (ln(a)b,c 2)(ln⁡(a)b,c 2). At that point, growth steadily slows and the function becomes asymptotic to the upper bound y=c y=c. c c is the limiting value, sometimes called the carrying capacity, of the model. LOGISTIC REGRESSION Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command “Logistic” on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form y=c 1+a e−b x(4.8.3)(4.8.3)y=c 1+a e−b x Note that The initial value of the model is c 1+a c 1+a. Output values for the model grow closer and closer to y=c y=c as time increases. How to: Given a set of data, perform logistic regression using a graphing utility Use the STAT then EDIT menu to enter given data. Clear any existing data from the lists. List the input values in the L1 column. List the output values in the L2 column. Graph and observe a scatter plot of the data using the STATPLOT feature. Use ZOOM to adjust axes to fit the data. Verify the data follow a logistic pattern. Find the equation that models the data. Select “Logistic” from the STAT then CALC menu. Use the values returned for a a, b b, and c c to record the model, y=c 1+a e−b x y=c 1+a e−b x. Graph the model in the same window as the scatterplot to verify it is a good fit for the data. Example 4.8.3 4.8.3: Using Logistic Regression to Fit a Model to Data Mobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular service. Table 4.8.5 4.8.5 shows the percentage of Americans with cellular service between the years 1995 and 2012. Table 4.8.5 4.8.5\| Year \| Americans with Cellular Service (%) \| Year \| Americans with Cellular Service (%) \| --- --- \| \| 1995 \| 12.69 \| 2004 \| 62.852 \| \| 1996 \| 16.35 \| 2005 \| 68.63 \| \| 1997 \| 20.29 \| 2006 \| 76.64 \| \| 1998 \| 25.08 \| 2007 \| 82.47 \| \| 1999 \| 30.81 \| 2008 \| 85.68 \| \| 2000 \| 38.75 \| 2009 \| 89.14 \| \| 2001 \| 45.00 \| 2010 \| 91.86 \| \| 2002 \| 49.16 \| 2011 \| 95.28 \| \| 2003 \| 55.15 \| 2012 \| 98.17 \| Let x x represent time in years starting with x=0 x=0 for the year 1995. Let y y represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data. Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent. Discuss the value returned for the upper limit, c c. What does this tell you about the model? What would the limiting value be if the model were exact? Solution Using the STAT then EDIT menu on a graphing utility, list the years using values 0–15 0–15 in L1 and the corresponding percentage in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logistic pattern as shown in Figure 4.8.5 4.8.5: Figure 4.8.5 4.8.5 Use the “Logistic” command from the STAT then CALC menu to obtain the logistic model, y=105.73795261+6.88328979 e−0.2595440013 x(4.8.4)(4.8.4)y=105.73795261+6.88328979 e−0.2595440013 x Next, graph the model in the same window as shown in Figure 4.8.6 4.8.6 the scatterplot to verify it is a good fit: Figure 4.8.6 4.8.6 To approximate the percentage of Americans with cellular service in the year 2013, substitute x=18 x=18 for the in the model and solve for y y: y=105.7379526 1+6.88328979 e−0.2595440013 x Use the regression model found in part(a)=105.7379526 1+6.88328979 e−0.2595440013(18)Substitute 18 for x≈99.3 Round to the nearest tenth y=105.7379526 1+6.88328979 e−0.2595440013 x Use the regression model found in part(a)=105.7379526 1+6.88328979 e−0.2595440013(18)Substitute 18 for x≈99.3 Round to the nearest tenth According to the model, about 98.8% of Americans had cellular service in 2013. The model gives a limiting value of about 105 105. This means that the maximum possible percentage of Americans with cellular service would be 105 105, which is impossible. (How could over 100 100 of a population have cellular service?) If the model were exact, the limiting value would be c=100 c=100 and the model’s outputs would get very close to, but never actually reach 100 100. After all, there will always be someone out there without cellular service! Exercise 4.8.3 4.8.3 Table 4.8.6 4.8.6 shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012. Table 4.8.6 4.8.6\| Year \| Seal Population (Thousands) \| Year \| Seal Population (Thousands) \| --- --- \| \| 1997 \| 3.493 \| 2005 \| 19.590 \| \| 1998 \| 5.282 \| 2006 \| 21.955 \| \| 1999 \| 6.357 \| 2007 \| 22.862 \| \| 2000 \| 9.201 \| 2008 \| 23.869 \| \| 2001 \| 11.224 \| 2009 \| 24.243 \| \| 2002 \| 12.964 \| 2010 \| 24.344 \| \| 2003 \| 16.226 \| 2011 \| 24.919 \| \| 2004 \| 18.137 \| 2012 \| 25.108 \| Let x x represent time in years starting with x=0 x=0 for the year 1997. Let y y represent the number of seals in thousands. Use logistic regression to fit a model to these data. Use the model to predict the seal population for the year 2020. To the nearest whole number, what is the limiting value of this model? Answer a The logistic regression model that fits these data is y=25.65665979 1+6.113686306 e−0.3852149008 x y=25.65665979 1+6.113686306 e−0.3852149008 x. Answer b If the population continues to grow at this rate, there will be about 25,634 25,634 seals in 2020. Answer c To the nearest whole number, the carrying capacity is 25,657 25,657. Media Access this online resource for additional instruction and practice with exponential function models. Exponential Regression on a Calculator Visit this website for additional practice questions from Learningpod. Key Concepts Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. We use the command “ExpReg” on a graphing utility to fit function of the form y=a b x y=a b x to a set of data points. See Example 4.8.1 4.8.1. Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. We use the command “LnReg” on a graphing utility to fit a function of the form y=a+b ln(x)y=a+b ln⁡(x) to a set of data points. See Example 4.8.2 4.8.2. Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit. We use the command “Logistic” on a graphing utility to fit a function of the form y=c 1+a e−b x y=c 1+a e−b x to a set of data points. See Example 4.8.3 4.8.3. Contributors and Attributions Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under aCreative Commons Attribution License 4.0license.Download for free at This page titled 4.8: Fitting Exponential Models to Data is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform. Back to top 4.7: Exponential and Logarithmic Models 4.E: Exponential and Logarithmic Functions (Exercises) Was this article helpful? 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4645	https://cnxus.org/wp-content/uploads/2022/04/OECD20DAC_Evaluating20Peacebuilding.pdf	DAC Guidelines and Reference Series Evaluating Peacebuilding Activities in Settings of Conflict and Fragility Improving Learning for Results DAC GUIDELINES AND REFERENCE SERIES Evaluating Peacebuilding Activities in Settings of Conflict and Fragility IMPROVING LEARNING FOR RESULTS This work is published on the responsibility of the Secretary-General of the OECD. The opinions expressed and arguments employed herein do not necessarily reflect the official views of the Organisation or of the governments of its member countries. This document and any map included herein are without prejudice to the status of or sovereignty over any territory, to the delimitation of international frontiers and boundaries and to the name of any territory, city or area. ISBN 978-92-64-10679-6 (print) ISBN 978-92-64-10682-6 (PDF) Series: DAC Guidelines and References Series ISBN 1990-0996 (print) ISBN 1990-0988 (PDF) The statistical data for Israel are supplied by and under the responsibility of the relevant Israeli authorities. The use of such data by the OECD is without prejudice to the status of the Golan Heights, East Jerusalem and Israeli settlements in the West Bank under the terms of international law. Photo credits: Cover: © Holly Kuchera/Dreamstime.com. Corrigenda to OECD publications may be found on line at: www.oecd.org/publishing/corrigenda. © OECD 2012 You can copy, download or print OECD content for your own use, and you can include excerpts from OECD publications, databases and multimedia products in your own documents, presentations, blogs, websites and teaching materials, provided that suitable acknowledgement of OECD as source and copyright owner is given. All requests for public or commercial use and translation rights should be submitted to rights@oecd.org. Requests for permission to photocopy portions of this material for public or commercial use shall be addressed directly to the Copyright Clearance Center (CCC) at info@copyright.com or the Centre français d’exploitation du droit de copie (CFC) at contact@cfcopies.com. Please cite this publication as: OECD (2012), Evaluating Peacebuilding Activities in Settings of Conflict and Fragility: Improving Learning for Results, DAC Guidelines and References Series, OECD Publishing. FOREWORD EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 3 Foreword The international community, including members of the Development Assistance Committee (DAC), has paid increasing attention to situations of conflict and fragility, acknowledging that these settings represent some of the great development challenges of our time. Rising levels of resources go into these contexts, but the fact that no fragile state has yet to reach any of the Millennium Development Goals is a stark reminder to us all that results are difficult to achieve and sustain in these situations. Finding answers to improve delivery is urgent, not least for the populations suffering from conflict and poverty. To deliver better results in situations of conflict and fragility we need to improve our understanding of the impacts and effectiveness both of programmes and projects aimed at supporting peace and of development and humanitarian activities operating in conflict settings. While the use of evaluation has become widespread in development and methods continue to evolve, it has grown clear that a special approach is needed to support learning and accountability in the context of conflict and fragility. How can evaluations provide strong evidence and lessons about what works and why in complex conflict settings – where change processes are non-linear and engagement politicised, and, where data are often missing or unreliable? How can the findings of these evaluations be used to inform policy making, programme design and implementation? It is against this backdrop that I am pleased to present this OECD-DAC guidance on Evaluating Peacebuilding Activities in Settings of Conflict and Fragility and to highlight the timely and relevant contribution it makes to international policy debates. Whereas a few years ago it was thought that evaluation in situations of conflict and fragility was impossible – or even objectionable – the process of developing and testing this guidance has proven otherwise. Importantly, it has stimulated critical thinking and shown how an evaluative perspective can be useful to policy makers and practitioners – not just for commissioning evaluations, but throughout the programme cycle. It has also demonstrated that more rigorous assessment of the theories underlying donor action in settings of conflict and fragility can help debunk outdated myths about the role of aid in preventing violent conflict and supporting long-term development processes. Evaluating Peacebuilding Activities in Settings of Conflict and Fragility was born out of a collaborative effort between evaluation and the conflict-and-fragile-state communities. By establishing a common understanding of key concepts and encouraging more and better evaluation, particularly of development co-operation activities in settings of conflict and fragility, it has created a community of practice bridging these diverse disciplines. I think it safe to say that the process itself has shown the value-added of cross-DAC work. I encourage all those concerned with supporting positive change and sustainable development in today’s fragile and conflict-affected regions to utilise this guidance – not just for commissioning evaluations, but as an input to learning and accountability throughout government. J. Brian Atwood, Chair OECD Development Assistance Committee ACKNOWLEDGEMENTS EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 4 Acknowledgements This guidance document has benefited from the valuable contributions of policy makers, practitioners, researchers, and conflict and evaluation experts, in development agencies and elsewhere, who want to see a strengthened evidence base and better results in settings of fragility and conflict. The OECD Development Assistance Committee (DAC) International Network on Conflict and Fragility (INCAF) and the DAC Network on Development Evaluation members and respective secretariats have led this collaborative process. The list of contributors over the years is long but a few should be specially mentioned for their role in completing the work. Following the application phase, revision of the guidance was led by a joint task force made up of: Beate Bull of Norad's Evaluation Department; Ivo Hooghe from the Office of the Special Evaluator of International Co-operation in Belgium; Megan G. Kennedy-Chouane, of the OECD DAC Network for Development Evaluation Secretariat; and Asbjørn Wee of the OECD DAC INCAF Secretariat. Megan G. Kennedy-Chouane managed the process for the OECD. Special thanks are due to the Peer Review Panel for their review of early drafts: Ted Kliest from the Policy and Operations Evaluation Department (IOB) in the Netherlands Ministry of Foreign Affairs and Monika Egger Kissling from the Human Security Division, Directorate of Political Affairs, in Switzerland's Federal Department of Foreign Affairs. Special thanks also go to Norway and Norad's Evaluation Department for strong support and engagement throughout the process of developing the guidance, and for hosting the feedback workshop in February 2011. Mariska van Beijnum of the Clingendael Institute carried out useful revisions based on lessons from the application phase. Special appreciation goes to all those who guided the application phase and provided substantive input to this work over the years – especially Asbjørn Eidhammer, Cristina Hoyos, Dominique de Crombrugghe, Gørild Mathisen, Peter Woodrow, Diana Chigas, Anna Hellström, Hans Lundgren, Lisa Williams, Annette Brown, Marie Gaarder and Thania Paffenholz. And finally, thanks to Ken Kincaid for editing, and to Jenny Gallelli, Alexandra Chevalier, Stephanie Coic and the OECD Public Affairs and Communication directorate for preparing this publication. TABLE OF CONTENTS EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 5 Table of contents Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Why guidance on evaluating peacebuilding activities in settings of conflict and fragility? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Who will benefit from this guidance and how should it be used? . . . . . . . . . . . . . . 18 Scope and structure of the guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Chapter 1. Conceptual background and the need for improved approaches in situations of conflict and fragility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 The need to better understand and adapt to conflict and fragility . . . . . . . . . . . . . 22 Principles and objectives of peacebuilding and statebuilding support . . . . . . . . . . 23 Aid that does harm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Improving programme design and strategic planning . . . . . . . . . . . . . . . . . . . . . . . 29 Chapter 2. Addressing challenges of evaluation in situations of conflict and fragility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Challenges to evaluations in situations of conflict and fragility . . . . . . . . . . . . . . . 32 Overcoming challenges to evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Chapter 3. Preparing an evaluation in situations of conflict and fragility . . . . . . . . . . . 39 Defining the purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Analysing conflict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Deciding the scope of the evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Selecting evaluation criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Outlining key evaluation questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Incorporating gender equality and women’s empowerment . . . . . . . . . . . . . . . . . . 48 Looking at the big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Selecting the best-fit evaluation methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Dealing with timing and logistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Co-ordinating with other actors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Working with local and country stakeholders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Considering a joint evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Writing terms of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Setting up evaluation management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Selecting the evaluation team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Controlling quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 TABLE OF CONTENTS EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 6 Chapter 4. Conducting an evaluation in situations of conflict and fragility . . . . . . . . . 57 Allow an inception phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Identify and assess the theory of change and implementation logic . . . . . . . . . . . 58 Gather data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Criteria for evaluating interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Draw conclusions and make recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Management response and follow-up action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Disseminate findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Feed back into programming and engage in learning . . . . . . . . . . . . . . . . . . . . . . . . 75 Annex A. Conflict analysis and its use in evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Annex B. Understanding and evaluating theories of change . . . . . . . . . . . . . . . . . . . . . 80 Annex C. Sample terms of reference for a conflict evaluation . . . . . . . . . . . . . . . . . . . . . 87 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Tables 3.1. Some key questions for conflict analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2. Examples of evaluation scopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 A.1. Summary of selected conflict analysis tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 B.1. Common theories of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Figure 4.1. Assumptions underlying the theory of change in a fictitious peace journalism programme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Evaluating Peacebuilding Activities in Settings of Conflict and Fragility Improving Learning for Results © OECD 2012 7 Executive summary The high human, economic, political and social costs of violent conflict – coupled with a growing sense that such suffering and devastation could be avoided or at least mitigated – have led to increasing shares of development and humanitarian assistance being spent in settings of violent conflict and state fragility. In the decade to 2009 the share of overseas development assistance (ODA) to fragile, conflict-afflicted countries doubled to USD 46 billion and 37% of total available ODA. International actors now recognise the centrality of these challenges for global development. Yet the scale of that effort is not reflected in its results. Findings from evaluations in these fields show that there are substantial weaknesses in programme design, effectiveness, and management. Peacebuilding and statebuilding support is often not based on a clear, strategic understanding of the conflict and (potential) role of international support in transforming key conflict drivers. Programmes lack basic conflict sensitivity and are not well adapted to the context in which they operate. The logic and assumptions underlying many activities in these fields are untested and objectives are unclear. Sketchy understanding of a conflict and unchecked assumptions can produce interventions that actually worsen tensions and fuel the conflicts they seek to mitigate. The need for more and better evaluation in conflict settings Furthermore, a persistent evaluation gap (few or weak evaluations of peacebuilding and conflict prevention activities), and little to no evaluation activity in settings of violent conflict, has meant that there is often very little credible information about the effectiveness and results of such endeavours. Learning and accountability have been weak. Research and experience, including the testing of the draft guidance, have shown that evaluations in these fields tend to be weak in terms of data, methods and validity of findings. Fewer rigorous methods are used and questions of causality are often inadequately addressed. Many evaluations in this field focus on process and mapping the context. Both internal and external validity tend to be quite low – meaning it is hard to draw broader lessons that can be applied to other contexts and it is difficult to draw credible conclusions about effectiveness and what works. How and why guidance was developed Evaluating Peacebuilding Activities in Settings of Conflict and Fragility (hereafter referred to as “the Guidance”) was developed by the OECD’s Development Assistance Committee (DAC) through the collaboration of its subsidiary bodies working on conflict and fragility and on evaluation. In 2008, the OECD produced a draft guidance, (Guidance on Evaluating Conflict Prevention and Peacebuilding Activities – Working Draft for Application Period) that was used to evaluate a range of activities. The findings from that application phase led to the 2008 draft EXECUTIVE SUMMARY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 8 being revised. The result is the present guidance, Evaluating Peacebuilding Activities in Settings of Conflict and Fragility. This guidance aims to help improve programme design and management and strengthen the use of evaluation in order to enhance the quality of conflict prevention and peacebuilding work. It seeks to guide policy makers and country partners, field and programme officers, evaluators and other stakeholders engaged in settings of conflict and fragility by supporting a better, shared understanding of the role and utility of evaluations, outlining key dimensions of planning for them, setting them up, and carrying them out. This guidance is to be used for assessing activities (policies, programmes, strategies or projects) in settings of violent conflict or state fragility, such as peacebuilding and conflict prevention work and development and humanitarian activities that may or may not have specific peace-related objectives. This encompasses the work of local, national, regional and non-governmental actors, in addition to development co-operation activities. The central principles and concepts in this guidance, including conflict sensitivity and the importance of understanding and testing underlying theories about what is being done and why, are applicable to a range of actors. Understanding key concepts The document begins with a discussion of key concepts and provides an overview of current policy debates. It describes the convergence of the concepts of peacebuilding, statebuilding and conflict prevention and addresses the emerging international consensus that such contexts require specific, adapted approaches. It considers the principles for engagement in fragile states as the backdrop to evaluating such engagement and outlines the preconditions for evaluability, which should be handled by those designing and managing such programmes. Such conditions include setting clear, measurable objectives for peace-related activities, collecting baselines data and monitoring activities. Challenges of evaluating in fragile, conflicted-affected settings While no two situations of conflict and fragility are alike – all are specific to a place and time – they share some characteristics, many of which make evaluation particularly challenging. Evaluations in the field of conflict prevention and peacebuilding expose both evaluators and evaluated to violence. International engagement in settings of violent conflict is often highly politicised, complex and multifaceted, encompassing not only development activities, but ones that are humanitarian, diplomatic, and even military in nature. Evaluators may struggle to maintain safe “evaluation space” where they can produce credible, defensible findings. It can be particularly difficult to establish clear attribution and causality in settings that are complex and where changes for peace (or renewed violence) are often non-linear and unpredictable. Further complicating evaluation in these settings are the relatively weak programme designs and the lack of agreed upon, proven strategies for effectively working towards peace. Baseline and monitoring data, including information on implementation, is often lacking. These weaknesses in design and programme management can make peacebuilding and statebuilding activities less effective and particularly difficult to evaluate. EXECUTIVE SUMMARY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 9 Overcoming challenges by understanding the conflict context Moving from the discussion of challenges, the guidance outlines a number of basic principles to help overcome them, setting the stage for a description of key elements of planning for, preparing, implementing, and learning from an evaluation. The guidance argues that conflict analysis – which includes understanding conflict dynamics and actors as well as the economic and political context – is essential for designing and implementing strategies and programmes, as well as for evaluating such work. A clear analysis of the causes, drivers and dynamics of conflict and fragility sets the analytical framework for evaluation, and should also be used to ensure conflict sensitivity. Adjusting the evaluation to be sensitive to the conflict context may have implications for methodology, data collection and findings. Conflict sensitivity and theories of change Conflict sensitivity is an overarching principle highlighted in this guidance. All engagement (including evaluation) in such settings should be sensitive to conflict and avoid doing harm. However, conflict sensitivity does not of itself build peace and being sensitive to the conflict is not synonymous with being effective. Questions of conflict sensitivity will therefore be evaluated alongside an assessment of effectiveness and other criteria. The concept of theory of change is presented as a way of encouraging critical thinking about the assumptions and strategies of peacebuilding and statebuilding. A theory of change is the understanding of how a specific activity will result in achieving desired changes in a particular context – it is the logic that underlies action. Developing better founded, more clearly stated theories about how peacebuilding and statebuilding can be achieved and supported is a key message from this guidance for decision makers, managers, and programme staff. Policies and programmes should use theories and assumptions that are tested and evidence based, which set out in clear cause-effect terms how they intend to produce outputs, outcomes and impacts. Doing so will not only help in the assessment of effectiveness and impact, but also contribute to knowledge about violence, peace and development. Untested or incorrect theories of change are often one reason why development assistance is failing to produce peacebuilding results. Evaluation contributes to testing theories of change and to building up the evidence base on peacebuilding and statebuilding. Evaluations at work Analysis against the DAC criteria constitutes the main substance of the evaluation study. Evaluators examine relevance, sustainability, effectiveness, efficiency and impact of activities in relation to the specific conflict context in order to answer the main evaluation questions posed in the terms of reference. The guidance describes how these criteria might be adjusted and offers examples of conflict-related lines of query. Data availability and other challenges may affect analysis, particularly when assessing impact. The last phase of an evaluation is to draw the conclusions and feed the findings into relevant planning, management, learning, research, or accountability processes. Dissemination strategies should be tailored to the target audiences, reaching them with timely, relevant information backed up by sound evidence. Actionable recommendations based on the conclusions should be presented as opportunities for learning and commissioning institutions should EXECUTIVE SUMMARY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 10 ensure systematic response to the findings. Such an approach will increase receptivity and the chances that findings will be fed back into programme design and decision-making. In these ways, more and better evaluation will contribute to identifying strategies and programmes that progress towards “peace writ large”. GLOSSARY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 11 Glossary Activity Actions taken or work performed through which inputs, such as funds, technical assistance and other types of resources, are mobilised to produce specific outputs (OECD, 2002). In this guidance, “activity” may include projects, programmes, policies and country assistance strategies. See also intervention. Attribution The ascribing of a causal link to observed (or expected to be observed) changes and a specific intervention (OECD, 2002). Baseline study An analysis describing the situation prior to a development intervention, against which progress can be assessed or comparisons made (OECD, 2002). Conflict analysis A systematic study of the political, economic, social, historical and cultural factors that directly influence the shape, dynamics and direction of existing or potential conflicts. It includes an analysis of conflict causes and dynamics as well as assessments of the profiles, motivations, objectives and resources of conflict protagonists (CDA, 2007; Conflict Sensitivity Consortium, 2004). Conflict mapping A representation of the main aspects of a conflict analysis, illustrating relationships between actors, causes, causal relationships, etc. Conflict prevention Actions undertaken to reduce tensions and to prevent the outbreak or recurrence of violent conflict. Beyond short-term actions, it includes the notion of long-term engagement. It consists of operational prevention, (i.e. immediate measures applicable in the face of crisis), and structural prevention, i.e. measures to ensure that crises do not arise in the first place or, if they do, that they do not recur (OECD, 2001b; United Nations, 2001a). Conflict sensitivity Systematically taking into account both the positive and negative impacts of interventions, in terms of conflict or peace dynamics, on the contexts in which they are undertaken, and, conversely, the implications of these contexts for the design and implementation of interventions (Conflict Sensitivity Consortium, 2004). Counterfactual The situation or condition which hypothetically may prevail for individuals, organisations, or groups were there no intervention, e.g. the war that would have occurred had a peacebuilding intervention not taken place. GLOSSARY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 12 Country programme One or more donor’s or agency’s portfolio of interventions and the assistance strategy behind them, in a partner (recipient) country. Driving factors of conflict The trends, currents, causes or fundamental influences that affect a conflict and help determine its characteristics, direction and ultimate outcome. Do no harm Ways in which international humanitarian and development assistance given in conflict settings may be provided so that, rather than exacerbating and worsening the conflict, it helps local people disengage from fighting and develop systems for settling the problems which prompt conflict within their societies (CDA, 2004; OECD 2010d). Evaluability Extent to which an activity or programme can be evaluated in a reliable, credible fashion. Evaluability assessments call for the early review of a proposed activity or programme in order to ascertain whether its objectives are adequately defined and its results verifiable (OECD, 2002). Evaluation Evaluation refers to the process of determining merit, worth or value of an activity, policy or programme. It consists of the systematic and objective assessment of an on-going or completed project, programme or policy, its design, implementation and results. The aim is to determine the relevance and fulfilment of objectives, development efficiency, effectiveness, impact and sustainability. An evaluation should provide information that is credible and useful, enabling the incorporation of lessons learned into the decision-making process of both recipients and donors (Scriven, 1991; OECD, 2002). Ex ante evaluation Evaluation performed before the implementation phase of an intervention (OECD, 2002). Ex post evaluation Evaluation of an intervention after it has been completed (OECD, 2002). Formative evaluation Evaluation intended to improve performance, most often conducted during the implementation phase of projects or programmes. Fragility, fragile state, fragile situation National, regional and local territories where the state (including central and local authorities) has weak capacity to carry out basic governance functions, and lacks the ability to develop mutually constructive relations with society. Fragile states are also more vulnerable to internal or external shocks such as economic crises or natural disasters (OECD 2011b). These situations tend to be characterised by poor governance, to be prone to violent conflict, and to show limited progress towards development. An aggregate of governance and security criteria, or of capacity, accountability and legitimacy criteria are usually used as measures of fragility. GLOSSARY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 13 Goal The higher-order objective to which a development intervention is intended to contribute (OECD, 2002). Impacts Positive or negative, primary and secondary effects produced by an intervention, directly or indirectly, intended or unintended (OECD, 2002). Results that lie beyond immediate outcomes or sphere of an intervention and influence the intensity, shape or likelihood of a conflict. Indicator Quantitative or qualitative factor or variable that provides a simple and reliable means to measure achievement, to reflect the changes connected to an intervention, or to help assess the performance of a development actor (OECD, 2002). Inputs The financial, human, and material resources used for the development intervention (OECD, 2002). Intervention A general term that refers to the subject of the evaluation and may refer to an activity, project, programme, strategy, policy, topic, sector, operational area, country strategy, institutional performance, etc. Examples are policy advice, projects, programmes (OECD, 2010c). Joint evaluation An evaluation involving more than one donor agency and/or country partner (OECD, 2002). Logical framework (log frame) Management tool used to improve the design of interventions, most often at the project level. It involves identifying strategic elements (inputs, outputs, outcomes, impact) and their causal relationships, indicators, and the assumptions or risks that may influence success and failure. It thus facilitates planning, execution and evaluation of development interventions. Monitoring A continuing function that uses systematic collection of data on specified indicators to provide management and the main stakeholders of an intervention with information regarding the use of allocated funds, the extent of progress, the likely achievement of objectives and the obstacles that stand in the way of improved performance (OECD, 2002). Objective (project or programme objective) The intended physical, financial, institutional, social, environmental, or other results to which a project or programme is expected to contribute (OECD, 2002). Outcome The likely or achieved short-term and medium-term effects of an intervention’s outputs (OECD, 2002). Outputs The products, capital goods, and services which result from a conflict prevention and peacebuilding intervention (OECD, 2002). Participatory evaluation Evaluation method in which representatives of agencies and stakeholders (including beneficiaries) work together in designing, carrying out, and interpreting an evaluation (OECD, 2002). GLOSSARY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 14 Peace analysis An assessment of the peacebuilding environment, including existing peace efforts, actors, de-escalating factors (reduce armed conflict or tensions), and connectors (Paffenholz and Reychler, 2007). Peacebuilding Actions and policies “aimed at preventing the outbreak, the recurrence or continuation of armed conflict”, encompassing “a wide range of political, developmental, humanitarian and human rights programmes and mechanisms”, including “short and long term actions tailored to address the particular needs of societies sliding into conflict or emerging from it” (UN, 2001b). Includes long-term support to, and establishment of, viable political and socio-economic and cultural institutions capable of addressing the proximate and root causes of conflicts, as well as other initiatives aimed at creating the necessary conditions for sustained peace and stability (OECD, 2001b). Policy coherence The systematic promotion of mutually reinforcing policy actions across government departments and agencies, creating synergies towards achieving the agreed objectives (OECD, 2001c). There are four dimensions of coherence: a) consistency between ends and means of a policy; b) consistency of policies and activities across government departments; c) consistency of policies and activities pursued by different actors; and d) alignment of policies, activities and processes between external actors and conflict affected or conflict prone countries (Picciotto and Weaving, 2006). Product An evaluation product is the output of an evaluation process and may take different forms, including written or oral reports, visual presentations, community meeting or videos. Programme theory (see also logical framework and theory of change) A programme theory is a hypothesis or model of how a programme is intended to produce intended results and the factors affecting or determining its success. A programme theory often combines a theory of change and an implementation model (Bamberger et al., 2006). Project/programme cycle management (PCM) Management approach that systematically follows the cycle of planning, implementing, monitoring and evaluating the intervention. Reliability Reliability refers to the consistency or dependability of data and evaluation judgements, with reference to the quality of the instruments, procedures and analyses used to collect and analyse data. Evaluation information is reliable when repeated observations under similar conditions produce similar results. Reliability contributes to credibility that can be additionally enhanced through a transparent evaluation process (OECD, 1991). GLOSSARY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 15 Results based management (RBM) A management strategy focusing on performance and achievement of outputs, outcomes and impacts. Result The output, outcome or impact (intended or unintended, positive and/or negative) of a development intervention. Risk assessment/risk analysis An analysis or an assessment of factors (called assumptions in the log frame) that affect or are likely to affect the successful achievement of an intervention’s objectives. A detailed examination of the potential unwanted and negative consequences to human life, health, property, or the environment posed by an intervention; a systematic process for providing information regarding such undesirable consequences; the process of quantification of the probabilities and expected impacts for (OECD, 2002). Stakeholders Agencies, organisations, groups or individuals who have a direct or indirect interest in the intervention or its evaluation (OECD, 2002). Statebuilding Statebuilding is a term used to describe the construction of a legitimate, functioning state. The OECD/DAC has defined statebuilding as an internal process to enhance capacity, institutions and legitimacy of the state, driven by state-society relations (OECD, 2008a). Summative evaluation A study conducted at the end of an intervention (or a phase of that intervention) to determine the extent to which anticipated outcomes were produced. Summative evaluation is intended to provide information about the worth of the programme (OECD, 2002). Terms of reference (TOR) A written document presenting the purpose and scope of the evaluation, the methods to be used, the standard against which performance is to be assessed or analyses are to be conducted, the resource and time allocated, and reporting requirements (OECD, 2002). Theory of change The assumptions that link a programme’s inputs and activities to the attainment of desired ends. A set of beliefs about how and why an initiative will work to change the conflict. It includes both implementation theory and programme theory (Weiss, 1995; Church and Rogers, 2006). Theory-based evaluation An evaluation that tracks the anticipated sequence of linkages from inputs and activities to outcomes and impacts (Weiss, 1995). Triangulation The use of multiple theories, methods and/or data sources to verify and substantiate an assessment. It is used to overcome the biases that come from unitary disciplines, single observers, self-interested informants, and partial methods (OECD, 2002; Weiss, 1995). Validity The extent to which data collection strategies and instruments measure what they purport to measure (OECD, 2002). “External validity” refers to the extent to which findings or conclusions from one evaluation/context are applicable and valid in another. Evaluating Peacebuilding Activities in Settings of Conflict and Fragility Improving Learning for Results © OECD 2012 17 Introduction Why guidance on evaluating peacebuilding activities in settings of conflict and fragility? In recent years, the international community has paid increasing attention to situations of conflict and fragility, acknowledging that they are one of the great development challenges of our time. As growing shares of resources, time and energy are devoted to projects, programmes, and policy strategies for countries affected by conflict and fragility, more evidence of the effectiveness of these endeavours is essential. Donors, practitioners and developing country governments show mounting interest in learning more about what does and does not work, and why, and in improving understanding of what contributes positively to sustainable peace and development. The project of developing guidance to strengthen evaluation and learning in these contexts began with the identification of a persistent evaluation gap (too few or weak evaluations of peacebuilding and conflict prevention activities). Development actors undertake little to no evaluation activity in settings of violent conflict and the peacebuilding and conflict prevention fields have been under-evaluated (OECD, 2007a). Part of the explanation for the lack of evaluation activity is that evaluating in these contexts presents unique challenges. This guidance considers that the main challenge specific to evaluations in fragile and conflict-affected settings is the threat of violence. Other challenges covered in this guidance are: complexity, weak theoretical foundations, data collection, attribution, a highly political environment and multiple actors and multiple agendas. Challenges are further discussed in Chapter 2. The lack of attention to evaluation and the challenges described above have meant that there is little credible evidence of the effectiveness and results of peacebuilding and conflict prevention endeavours. Research and experience, including the testing of the draft guidance, have shown that evaluations in these fields tend to be weak in terms of data, methods and validity of findings. Fewer rigorous methods are used and questions of causality are often inadequately addressed. Many evaluations in this field focus on process and mapping the context (FAFO 2006). Validity, both internal and external, tends to be low-meaning it is hard to draw broader lessons that can be applied to other contexts and difficult to draw credible conclusions about effectiveness and what approaches work. The process of developing Evaluating Peacebuilding Activities in Settings of Conflict and Fragility (also referred to as “the Guidance”) was spurred by a recognition in the peace and conflict prevention community of the lack of solid information about the actual results of peacebuilding efforts. Recognising the need for better, more tailored approaches to evaluation in conflict settings, the OECD’s Development Assistance Committee (DAC) launched an initiative to develop guidance on evaluating conflict prevention and INTRODUCTION EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 18 peacebuilding activities. The initiative brought together practitioners and policy makers from the International Network on Conflict and Fragility (then the DAC Network on Conflict, Peace and Development Co-operation) with evaluation experts from the DAC Network on Development Evaluation. The OECD (2008a) produced draft guidance in 2008 which was used to evaluate various conflict prevention activities and external peacebuilding and statebuilding support in a number of major conflict settings including the Democratic Republic of Congo, Sri Lanka, Southern Sudan, and Afghanistan. The guidance has been revised on the basis of the substantive and methodological findings from this application phase. The goal of this guidance is to promote critical reflection. It aims to help fill the learning and accountability gap in settings of conflict and fragility by providing direction to those undertaking or commissioning evaluations and helping them better understand the sensitivities and challenges that apply in such contexts. At the same time, it aims to assist policy makers and practitioners working on peacebuilding and statebuilding to better understand the role and utility of evaluation and grasp how an evaluation lens can help strengthen programme design and management. With these objectives in mind, the Guidance offers advice on aspects of evaluating donor engagement in conflict-affected and fragile situations that differ from evaluation in more stable environments. To provide a complete picture it also covers some steps that apply to all development evaluations. Who will benefit from this guidance and how should it be used? Different target audiences will benefit in different ways from this guidance. The primary audience includes policy staff, donors, field and desk officers in foreign service offices and development agencies, partner country governments, non-governmental and international organisations (NGOs), and United Nations (UN) organisations involved in commissioning or supporting evaluations in situations of conflict and fragility. Secondly, it targets evaluators and evaluation managers, including the evaluation departments of developing countries and development agencies. Evaluators will benefit by gaining a clearer view of what commissioners expect from their work. Given the diversity of the intended audience, some sections may be more relevant than others to individual readers. Evaluating Peacebuilding Activities in Settings of Conflict and Fragility provides an overview of key concepts relevant to evaluation in conflict situations and fragile states. It can be read while designing a programme or developing a strategic policy, while commissioning or programming an evaluation, and during the planning and carrying out of a specific evaluation. This is not a prescriptive instruction manual. Rather, it seeks to contribute to fostering thoughtful, critical approaches by highlighting and clarifying specific challenges for evaluation. It should be viewed as a living guidance that will continue to evolve as evaluation methodologies and peacebuilding practices improve. It outlines key steps and main points to consider at each stage in the evaluation process and suggests tools that may support that process. The information and advice it volunteers should be applied carefully, based on an evaluation’s context and intended purpose. To that end, this guidance is designed to be practical and to respond to the particular challenges that characterise fragile, conflict-affected situations and which evaluations must address. INTRODUCTION EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 19 Scope and structure of the guidance Evaluating Peacebuilding Activities in Settings of Conflict and Fragility builds on existing literature and experience in development agencies and countries affected by conflict and fragility. This includes the lessons learned during the two-year application phase of a draft version of this guidance, when the suggested approach was tested in evaluations of external support in conflict settings. The draft guidance was employed for evaluations of multi-donor engagement in Southern Sudan (Bennett et al., 2010), Sri Lanka (Chapman et al., 2009), and the Democratic Republic of Congo (Brusset et al., 2011), as well as single-donor evaluations of the Norwegian contribution to peace in Haiti (Norad, 2009), the Swedish Provincial Reconstruction Teams in Afghanistan (unpublished), the German Civil Peace Service Programme (Paffenholz, 2011), and the European Commission’s peacebuilding portfolio (EC, 2011). Chapter 1 outlines the conceptual background of international engagement in settings of conflict and fragility, including main donor policy commitments, and examines why better understanding of conflict and fragility matters in today’s development context. It is of particular relevance to those with limited experience in the conflict and peace domains and presents the overarching concepts that guide and inform decision making and evaluation. If Chapter 1 is the theory, Chapters 2-4 are the practice. They form the “hands-on” core of the guidance and will be useful for all readers, particularly those with limited evaluation background. These chapters also provide seasoned evaluators with further ideas drawn from experience. They are guidance for planning, managing, implementing, and learning from evaluation. Underlying the chapters is the importance of understanding that each evaluation differs in its scope and purpose. Methodologies can and should be tailored accordingly. Chapter 2 describes challenges to evaluation in settings of conflict and fragility. It then considers the principles that should guide evaluation and help it rise to the challenges of a fragile, conflict-affected setting. It emphasises the importance of a conflict analysis for assessing an intervention and for ensuring that the evaluation itself is conflict sensitive. Evaluations should also seek to be ethically responsible and transparent about strengths and weaknesses. Chapter 3 considers the key steps in preparing an evaluation. It looks at the stages of defining an evaluation’s purpose and scope and conducting a conflict analysis. It then examines timing and logistics, co-ordination with other actors, management methods, and hiring evaluation teams. Chapter 4 deals with conducting an evaluation – from performing initial research to identifying the logic behind the development intervention, plugging gaps in data, and using OECD evaluation criteria of relevance, effectiveness, impact, sustainability and efficiency to assess the activity. Finally, it gives advice on follow-up, learning from evaluations, and feeding the lessons back into programming. The annexes provide additional detail to complement Chapters 1-4. Annex A goes into further detail on conflict analysis, looking at different approaches and the use of the analysis in evaluation. Annex B provides further detail on the concept and use of theories of change. Annex C considers how to draw up a terms of reference document, using the example of an imaginary peace journalism training course. An extensive bibliography provides references and resources for further reading. Evaluating Peacebuilding Activities in Settings of Conflict and Fragility Improving Learning for Results © OECD 2012 21 Chapter 1 Conceptual background and the need for improved approaches in situations of conflict and fragility Chapter 1 outlines the conceptual background to working in settings of conflict and fragility. Arguing that such settings require a deep understanding of context and conflict, the chapter first seeks to characterise fragile and conflict-affected situations. It then looks at the purpose and goals of external engagement and describes, based on recent evaluations, how development assistance sometimes misses its targets and can even “do harm” when international partners have not sufficiently understood and adapted to the real context-specific drivers of peace and conflict. It is suggested that better conflict analysis and clearer targeting, together with more explicit and tested theories of change and results-based management can contribute to improving the knowledge base for development assistance programmes and facilitate evaluation. 1. CONCEPTUAL BACKGROUND AND THE NEED FOR IMPROVED APPROACHES IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 22 The need to better understand and adapt to conflict and fragility Armed conflict has devastating effects on human life. People in fragile and conflict-affected situations are more than twice as likely to be undernourished and lack clean water as those in other developing countries (World Bank, 2011). Children are affected particularly badly: a child in a fragile state is twice as likely to die before the age of five and also less likely to be able to attend school (ibid.) Violence and state fragility are often characterised by systematic violations of fundamental human rights. The impacts of conflict on political, social and economic development are also profound. When violent conflict breaks out, development is derailed. Acknowledging the fact that countries affected by repeated cycles of political and criminal violence represent a central challenge for global development, donors provide them with substantial amounts of aid. Official development assistance (ODA) to fragile and conflict-affected states has doubled over the past decade, reaching USD 46 billion in 2009 and accounting for 37% of the total available ODA (OECD, 2011d). There is, however, an increasing body of evidence to suggest that aid aimed at achieving sustainable peace and development is not making a lasting contribution to peace and development. In 2005, a review of more than 75 evaluations in the conflict fragility field pointed to substantial weaknesses in programme effectiveness, design, and management (Fafo Institute, 2006). These findings were confirmed during the application of the earlier draft of this guidance (Kennedy-Chouane, 2011). In Southern Sudan for example, it was found the support provided by multiple donors in 2005-2010 was often mistargeted. Because donors did not fully take into account key drivers of violence, there was an overemphasis on basic services and a relative neglect of security, policing, and the rule of law, which were found to be essential in the process of state formation for the future South Sudan and therefore, critical to preventing future conflict (Bennett et al., 2010). Similarly, in the Democratic Republic of Congo, it emerged that one of the principal conflict drivers is that the Congolese justice system lacks credibility, political commitment, and competence and maintains a delicate relationship with customary law. The justice system is particularly inept in dealing with complex land ownership conflicts, which fuels violence and human rights violations, particularly where populations have been displaced and in the context of the unregulated exploitation of natural resources. Several large-scale multi-donor projects have targeted the restoration of justice and the rule of law. However, issues relating specifically to property titles, rent, and land rights have not been treated adequately within these programmes, according to local government and community groups surveyed (Brusset et al., 2011). There is no universal definition of fragile and conflict-affected situations – analysts and donors still have different notions of exactly what is meant. The OECD Principles for Good Engagement in Fragile States (2007a) – known as the Fragile States Principles – outline that in fragile situations, governments lack the political will and/or capacity to fulfil the basic conditions for poverty reduction, development, security, and human rights. In other words, vicious cycles of conflict commence when political and economic stresses and pressures 1. CONCEPTUAL BACKGROUND AND THE NEED FOR IMPROVED APPROACHES IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 23 on justice and security meet weak institutions (World Bank, 2011). The OECD (2011b) also states that fragile and conflict-affected states are those that have weak capacity for carrying out the basic functions of governing their populations and territory and lack the ability to develop mutually constructive and reinforcing relations with society. As pointed out in the Fragile States Principles, fragile and conflict-affected situations cover a broad spectrum. However, they do share some common features: ●They are inherently high-risk environments – for the people who live there, for their governments, and for those who provide and implement humanitarian and development assistance. The risks are not only related to the security situation (e.g. threats to staff, difficulty of movement, lack of access to information), but to the achievement of development objectives: risk of programme and strategic failure, fiduciary risks (corruption), and risks to the reputations of donors and implementing agencies. ●They are characterised by complex political economies and state-society relations, in which development partners can be parties to on-going conflicts and contested peace processes complicate efforts to prioritise needs and identify a strategic vision for sustainable peace. ●They are most often characterised by weak or non-existent national and local capacities and institutions. They are thus incapable of identifying or building consensus on peacebuilding priorities, developing strategies, implementing programmes, or monitoring progress. ●They are exposed to a combination of internal and external stresses that heighten the risk of violent conflict. Internal causes of conflict arise from political, economic, social and security-related dynamics (e.g. political exclusion, legacies of violence, crime, low GDP per capita, unemployment, identity-based conflict, and inequality). External stresses and regional conflicts can further exacerbate internal stresses (price shocks, for example, impact on inequality and unemployment) and some, like drug trafficking, can even cause them (World Bank, 2011). ●External humanitarian and development action is often part of donor’s broader geopolitical and economic agendas – such as combating international terrorism, stabilising access to scarce resources like oil, fighting transnational organised crime, opening markets for domestic firms and curbing immigration flows. As such, aid is at a higher risk of being politicised in fragile, conflicted situations than in more stable ones, and development actors may not be in the lead in setting the agendas for engagement. Principles and objectives of peacebuilding and statebuilding support The past few years have seen an increase not only in international engagement in situations of conflict and fragility, but also in the convergence of development, security, human rights, humanitarian assistance, peacebuilding, statebuilding, and related agendas. It is internationally acknowledged that sustainable peace and development are critically linked to the capacity and legitimacy of the state. Donors need to base their interventions not only on the need to support short-term stability (or the cessation of hostilities) or on the provision of humanitarian aid, but on a broader understanding of how their interventions affect state-society relations and longer term prospects for the development of a functioning, legitimate state. Donor engagement in fragile and conflict-affected states is largely guided by the overarching aims of preventing conflict, peacebuilding, and statebuilding (see Box 1.1). Box 1.1. Key donor agendas for situations of conflict and fragility Conflict Prevention Conflict prevention refers not only to actions undertaken in the short term to reduce manifest tensions and to prevent the outbreak or recurrence of violent conflict (OECD; 1997, 2001a). It also includes long-term engagement that addresses the built-in capacities of societies to deal with conflicting interests without resorting to violence (Menkhaus, 2006), and extends to the management of disputes with destabilising potential. Such work helps de-legitimise the belief that violence is an inevitable or acceptable way of resolving disputes, making nonviolent alternatives known and more attractive, addressing structural and immediate causes, and reducing vulnerability to triggers. The goal is not to prevent all conflict. Some conflict is natural, inevitable, and a positive part of development and other change processes. Instead, the emphasis is on preventing harmful violent responses to the inevitably diverging interests and conflicting objectives that exist in all societies. Peacebuilding Although most peacebuilding focuses on the transition from war to peace, the concept and practices of peacebuilding are, in principle, about supporting sustainable peace, regardless of whether or not political conflicts have recently produced violence. Indeed, the mere threat of violence occurring is sometimes enough to kick-start a peacebuilding process. Peacebuilding, in other words, is undertaken because violent conflict is looming, is going on, or has recently ceased (OECD, 2011b). The emerging UN consensus (2007) is that: “Peacebuilding involves a range of measures aimed at reducing the risk of lapsing or relapsing into conflict, by strengthening national capacities for conflict management and laying the foundations for sustainable peace. It is a complex, long-term process aimed at creating the necessary conditions for positive and sustainable peace by addressing the deep-rooted structural causes of violent conflict in a comprehensive manner. Peacebuilding measures address core issues that affect the functioning of society and the state.” Such wording points to a preventive, as well as a post-conflict, role for the concept and practice of peacebuilding. Statebuilding Statebuilding has been defined by the OECD DAC as “an endogenous process to enhance capacity, institutions and legitimacy of the state driven by state-society relations” (OECD, 2008a). The process must be understood against a background of long-term historical and structural factors that contribute to shaping the contours of state formation and the nature of state-society relations. And it must be understood within the exigencies of current circumstances in the country concerned. These may include the risk of violent conflict or effects of previous conflict – either internally or in the region – or the impact of economic pressures generated by global recession, debt, limited trade opportunities, financial imbalances and commodity prices (OECD, 2011b). 1. CONCEPTUAL BACKGROUND AND THE NEED FOR IMPROVED APPROACHES IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 24 The objectives of conflict prevention, peacebuilding, and statebuilding are inextricably linked. Efforts to support and achieve them essentially address the same underlying problems. Their aims, too, are consistent: to help societies move in directions conducive to nonviolent resolution of conflict, address grievances and injustice, and move towards 1. CONCEPTUAL BACKGROUND AND THE NEED FOR IMPROVED APPROACHES IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 25 sustained peace and development (OECD, 2011b). The wished-for end result of donor engagement in situations of conflict and fragility is not simply the absence of open conflict but a deeper peace, often referred to as “peace writ large”, i.e. societal-level peace or the bigger peace beyond the micro level of a single project (CDA Collaborative Learning Projects, 2004). The complicating factor is that peace and conflict are context-specific: there is no one blueprint either for the end state or the means of achieving it that can be applied to all situations. Donors recognise that much remains to be done to improve their engagement. Working with partner countries, they have committed themselves to a number of principles and guidance documents that underline what differentiates engagement in fragile and conflict-affected situations (Box 1.2) from development co-operation in other settings. Box 1.2. Principles for donor engagement in situations of conflict and fragility ●Principles for Good International Engagement in Fragile States and Situations (OECD, 2007a). ●Guidelines for actors involved in development co-operation, peacebuilding, statebuilding and security in fragile and conflict-affected states. ●Supporting Statebuilding in Situations of Conflict and Fragility: Policy Guidance (OECD, 2011b). ●Actionable guidance on the way development actors provide support to statebuilding in fragile and conflict-affected situations with a focus on strengthening state-society engagement. ●The New Deal for International Engagement in Fragile States (OECD, 2011c). ●Agreed at the 4th High Level Forum on Aid Effectiveness in Busan, South Korea, the New Deal aims to improve the effectiveness of aid in contexts of conflict and fragility. It sets out five peacebuilding and statebuilding goals and outlines how partners will work towards achieving them. ●International Support to Post-Conflict Transition: Rethinking Policy, Changing Practice (OECD, 2012). ●Presents recommendations for better practice in order to improve the speed, flexibility, predictability and risk management of aid during transition. ●Paris Declaration on Aid Effectiveness (2005), Accra Agenda for Action (2008) and the Busan Declaration on Effective Development Co-operation (2011). ●International commitments to improve effectiveness of development co-operation including by increasing co-ordination and country ownership, adapting to differing country environments and giving increased attention to fragile and conflict-affected countries. ●Managing Risk in Fragile and Transitional Contexts (OECD, 2011e). Provides information to help donors understand how to balance risks and opportunities in order to protect the integrity of their institutions while delivering better results to those who need it most. ●Principles and Good Practice of Humanitarian Donorship (GHD, 2003). ●United Nations Security Council Resolution 1325 (United Nations Security Council, 2000). Establishes that equality between men and women is essential to achieving and sustaining peace. Calls for the protection of women and girls and for equal participation in peace processes and post-conflict reconstruction efforts. 1. CONCEPTUAL BACKGROUND AND THE NEED FOR IMPROVED APPROACHES IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 26 One particularly important international commitment came in 2007, when OECD country ministers approved the Fragile States Principles as a guide to donor engagement in fragile states. The Principles highlight the importance of viewing countries in their particular contexts and thinking carefully about the objectives and likely impact of specific activities. They also underline the peculiarities of fragile states, which call for well sequenced and prioritised action across political, economic, administrative, and security-related domains. Such an effort entails shared analysis, objectives, strategies and resources. The ten principles are: 1. Take context as the starting point. 2. Ensure all activities do no harm. 3. Focus on statebuilding as the central objective. 4. Prioritise prevention. 5. Recognise the links between political, security and development objectives. 6. Promote non-discrimination as a basis for inclusive, stable societies. 7. Align with local priorities in different ways and in different contexts. 8. Agree on practical co-ordination mechanisms between international actors. 9. Act fast… but stay engaged long enough to give success a chance. 10. Avoid pockets of exclusion (“aid orphans”). Recent work by the OECD (2011e) has also helped to build consensus among development partners on the need to better manage and mitigate risks. Experience shows that donors tend to focus on fiduciary risks and ones that jeopardise the reputations of development agencies, using them as a reason not to engage in high-risk conflict or post-conflict countries. Development partners should, however, think more about contextual risks – the re-emergence of violent conflict, humanitarian disasters, etc. – and accept that the risks of non-engagement are often higher than those of engagement (OECD, 2011c). Risk mitigation does not mean eliminating risk, but, rather, finding appropriate ways of dealing with it. Shared risk assessments can be one such way. The international community has made much progress towards understanding – and improving – the role of external partners in settings of conflict and fragility. These Fragile States Principles have contributed to changing donor policy and, to some extent, donor behaviour (OECD, 2011d). However, the effects and results of applying them have not been rigorously evaluated. Aid that does harm There is an emerging understanding that ill-designed, poorly implemented, or badly co-ordinated interventions in fragile and conflict-affected situations can increase tensions and undermine capacities for peace. They can, in other words, “do harm” (Anderson, 1999a). Conflict sensitivity is needed to mitigate such harm by systematically taking into account both the positive and negative impacts of interventions (International Alert, 2007a). The implication for approaches used to deliver aid is that they need to be tailored to the demands of high-risk environments. Donors need to be realistic about what they can achieve as external partners in limited timeframes with limited capacities. Too often they underestimate the challenges of engaging in fragile and conflict-affected situations and draw up plans and schedules that 1. CONCEPTUAL BACKGROUND AND THE NEED FOR IMPROVED APPROACHES IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 27 have little grounding in reality. This tendency was confirmed by findings of the pilot evaluations carried out during the development of this guidance. Evaluations have also found that donors fail to prioritise their engagement and lack clear strategies to address core peace drivers and conflict-mitigating factors. A focus on providing humanitarian aid or basic services – and a neglect of key priorities aimed at building and sustaining peace – tends to be driven by the untested assumption that all development activities will somehow contribute to peace (Kennedy-Chouane, 2011). The illuminating example of a well-meaning intervention in Tajikistan (Box 1.3) demonstrates the critical importance of understanding the real post-war peace drivers of a particular context. Box 1.3. Do no harm – an example from Tajikistan At the end of the civil war in Tajikistan, one international NGO undertook massive housing reconstruction in a southern province. The intent of the effort was to i) to encourage people displaced during the fighting to return to the region and ii) to support reconciliation between the former foes by getting them to work together in rebuilding the destroyed villages. Priority for reconstruction went to the villages that had suffered the most damage. In these, the NGO worked with local people to decide which houses would be rebuilt and organise work crews to do the construction. They agreed that “anyone from the village who wanted a job” would be hired in these crews. A few months later, they had successfully sponsored the reconstruction of almost 60% of the damaged housing in the region. However, one day a local man came into the NGO’s compound with a Kalashnikov and threatened the staff, saying, “Why are you favouring that group that we defeated in the war? If you don't start building some houses for my clan, I will kill you.” The NGO staff members were astounded. They had meant to be completely inclusive and to ensure that everyone who suffered in the conflict received equal attention. What they had not known until this moment was that during the conflict, the greatest damage had been done in villages occupied by only one (rather than both) of the local warring groups. By focusing their assistance on the areas of greatest damage, and by hiring people from the villages in those areas to work on the rebuilding, they had inadvertently provided almost all their assistance to one side of the conflict – and the “losing side” at that. Their project design had unintentionally reinforced existing inter-group divisions by focusing on mono-ethnic villages and channelling all their support to one group. With a project redesign the NGO was able to supply building materials and support to multi-ethnic villages, to the damaged homes of the other ethnicity, and to community buildings that both groups shared such as schools, clinics and mosques. Source: CDA Collaborative for Development Projects (2000). Another widespread assumption is that being “conflict sensitive” is, ipso facto, doing peacebuilding work (OECD and CDA, 2007). As a result, much of the ODA aimed at fragility and conflict does not affect their driving factors. And, even when it is better targeted, it often remains ineffective. A case in point described by Bennett et al. (2010) is Southern Sudan. Between 65% and 85% of the total aid, including humanitarian assistance, from multiple donors in 2005-2010 targeted traditional socio-economic aid sectors. While this aid was provided in a way that many partners described as “conflict sensitive” (avoiding exacerbating ethnic tensions and trying to right historic inequalities), the conflict analysis Box 1.4. Weaknesses around conflict analysis Experience shows a number of recurrent challenges to the production and use of conflict analysis. ●Partial analysis. Due to time or resource constraints, it is often tempting to limit the focus of a conflict analysis to a donor’s particular programme or strategy and how it might fit the context. Such an approach can lead practitioners to miss important aspects of the context or to develop misguided or irrelevant programmes. ●Many people carry out context analysis, believing it to be conflict analysis. A context analysis seeks a broad understanding of the entire political, economic and social scene. A conflict analysis is more narrowly focused on the specific elements of that broader picture that may cause, trigger, or propel incompatible interests or violence. Conflict analysis focuses on those political, economic, social, historical and other factors that directly influence the shape and dynamics of the situation of conflict and fragility. ●Analysis is not updated. Analyses are performed only at the front end of a programme. There are seldom efforts at ongoing in-depth analysis or monitoring and adjusting over time. ●Programming is not linked to analysis. Even when practitioners do perform an analysis, they often fail to link their programme strategy to it or adjust activities and strategies to changing dynamics over time. Many implementing agencies and field staff work on the basis of an implicit analysis, often based on their own experience. Some programmes – frequently effective ones – are grounded in an informal analysis that draws on the long experience of local people or long-time observers of a conflict. These analyses can be quite sophisticated and may be constantly updated as individuals move about and talk with many different people. However, when analysis is done in this way, different members of the same project team or organisation sometimes operate on the basis of quite different understandings of the situation and their programme’s role in it. This undermines the development of coherent strategies, weakens sustainability (when staff leave, so does their analysis) and significant assumptions often remain undiscussed and untested. Therefore, efforts to make the implicit analysis more explicit and to share observations are usually valuable. Source: CDA Collaborative Learning Projects (2009). 1. CONCEPTUAL BACKGROUND AND THE NEED FOR IMPROVED APPROACHES IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 28 conducted by evaluators showed that lack of social services could be cited as neither the sole nor a significant cause of conflict. The aid, while avoiding doing harm, was clearly not addressing the sectors most likely to be factors in sustainable peace. To contribute to sustainable peace, donors should work on different priorities across humanitarian, development, conflict prevention, stabilisation, and peacebuilding activities. Priorities may need to be adjusted over time as a conflict evolves or political context shifts. Such an approach often involves a combination of working “in” and “on” conflicts. There is a widely held belief that traditional development activities (in areas such as health and education) can have a positive impact on conflict dynamics. However, this assumption needs to be critically examined and there is a growing consensus that development work should be complemented by activities that focus specifically on removing the causes and drivers of conflicts and strengthening the capacities, institutions and norms necessary for conflict management. Evaluation experience shows that the main issue in determining the effectiveness of donor engagement in situations of conflict and 1. CONCEPTUAL BACKGROUND AND THE NEED FOR IMPROVED APPROACHES IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 29 fragility is not the effect that activities labelled as “peacebuilding” have on peace. It is much more closely related to the influence that all forms of aid combined have on peace (Chapman et al., 2009; Bennett et al., 2010; Brusset et al., 2011). Development assistance should deliberately work in and on conflicts rather than simply endeavouring to get round them (OECD; 1998, 2001b). There is a real and growing need for thoughtful examination of development actors’ policy and practice in countries affected by violent conflict and state fragility. Consensus does exist that distinctive approaches are required to deliver effective support. Yet more work is needed to operationalise conflict-sensitivity concepts and achieve actual change at donor headquarters and country level, especially in terms of knowledge of conflict dynamics and how interventions relate to conflict, increased emphasis on outcomes and impact, and better understanding of the means of achieving these. Further work is also required to understand how individual donors should engage with national governments and align with country priorities and systems in situations where state legitimacy is weak or states are actors in violent conflict. More and better evaluation will contribute over time to helping practitioners better understand how to make their interventions more conflict sensitive, as well as more effective. Improving programme design and strategic planning Because good programme design is key to not only working effectively in support of peace and development, but also a prerequisite for good evaluation, it is important to consider the basics of planning, monitoring, and management. Evaluation findings in recent years have shed light on core dimensions of quality programming. When evaluation and its requirements are an integral part of programming activities from the outset, it contributes to more effective programming and facilitates better evaluation. As the OECD (1991) states: “clear identification of the objectives which an aid activity is to achieve is an essential prerequisite for objective evaluation”. Some basic components of good programming are baseline information on key indicators, conflict analysis, clear and measurable objectives, a testable programme logic or theory, and monitoring data. These elements of programme design and management may, for a variety of reasons, be lacking or missing from assistance activities in fragile and conflict-affected situations, especially ones where fragility is prolonged and there is open, armed conflict. ●Planning and conflict analysis entails identifying the most relevant contribution(s) that donors, governments and other actors can make to support peacebuilding and reduce fragility in a specific country or conflict setting. Concrete understanding of the political, economic and social dimensions of conflict and fragility is a necessity. It is good practice to conduct a context and conflict analysis as one of the first steps in planning. The understanding that is gained should influence strategy, policy, and programme design and implementation. (Box 1.4 describes some of the weaknesses of development strategies and activities related to conflict analysis.) ●Setting clear, realistic objectives with clear target outcomes related to the context deserve specific attention. Peacebuilding and statebuilding programmes and policy goals tend to be general, vague, and consequently difficult to manage and evaluate. ●Theory of change is a set of beliefs about how change happens and, as such, it explains why and how certain actions will produce the desired changes in a given context, at a given time (Weiss, 1995; Church and Rogers, 2006). Developing a sound, clear, evidence-based theory of 1. CONCEPTUAL BACKGROUND AND THE NEED FOR IMPROVED APPROACHES IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 30 change is one potentially useful way to improve design. Theory of change thinking is an approach that encourages critical thinking throughout the programme cycle. ●Results-based management is a management strategy that focuses on performance and the achievement of outputs, outcomes, and impacts. A key component is monitoring, which tracks how a programme is progressing and enables the adjustment of activities and strategies as the conflict setting shifts. Effective monitoring plays a crucial role in making programmes flexible and adaptable to changing contexts, particularly in complex situations of fragility and violence. In summary, reliable, comprehensive programme design, management, and monitoring can contribute to better policies and programmes and improve the effectiveness of interventions. They also set the stage for successfully evaluating conflict prevention and peacebuilding programmes, notably by creating a theoretical framework and setting up necessary data management systems. Programme planners, policy makers, implementing staff, and evaluators should work together to strategise about how best to develop a well designed, effectively monitored, evaluable intervention. Evaluating Peacebuilding Activities in Settings of Conflict and Fragility Improving Learning for Results © OECD 2012 31 Chapter 2 Addressing challenges of evaluation in situations of conflict and fragility This chapter is first of the three that form the main evaluation guidance. Building on the conceptual basis of Chapter 1, it outlines key challenges to evaluation in these settings and then describes core principles for addressing these challenges, including the OECD evaluation principles. The chapter considers the role of conflict analysis and the need to understand the particular context of the intervention. These principles should guide an evaluation in fragile, conflicted settings throughout the process described in Chapters 3 and 4. 2. ADDRESSING CHALLENGES OF EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 32 This chapter describes some of the key challenges to evaluating in settings of conflict and fragility and then sets out the core principles for meeting these challenges. Challenges to evaluations in situations of conflict and fragility This guidance considers that the main challenge specific to evaluations in fragile and conflict-affected settings is understanding and adapting to violent conflict, while mitigating the risk that evaluations themselves become part of the conflict or cause harm to those involved. Other challenges addressed are: complexity, weak theoretical foundations, challenges to data collection, attribution, a highly political environment, multiple actors and multiple agendas. The high risk of violence Evaluations of interventions in the field of conflict prevention and peacebuilding expose – in contrast to almost all forms of evaluation – both evaluators and evaluated to real risk. Potential implications are profound. First, the threat of violence may constrain the evaluators’ ability to raise issues, collect material and data, recruit and retain local staff, meet interlocutors, publish findings, and disclose sources. Defending the integrity of evaluation findings in highly politicised and even dangerous settings can pose problems for evaluation teams, particularly where evaluation findings may potentially be misused by different parties to a conflict or harm those involved. Second, the risk of harm may mean that the information obtained is biased, incomplete and/or (voluntarily or involuntarily) censored. Consequently, evaluations must address the operational and methodological consequences of the risk of violence. More specifically, in order to deal with this challenge, it is advisable that the evaluation itself include a conflict analysis in order to assess the intervention and to ensure that the evaluation process and product is conflict sensitive. Complex and unpredictable contexts and interventions Few would dispute that settings of conflict and fragility are complex, combining multifaceted, multi-directional change processes with high levels of unpredictability, a general lack of information, and potential strategic misinformation. The way programmes are implemented on the ground may differ widely from original plans, as practitioners change what they are doing to adapt to an evolving conflict. As a result, it may be difficult to identify what exactly should be evaluated. Although unpredictability and complexity may be inevitable, their frequently negative ramifications for evaluations need not be. Evaluators must prepare for risks, develop robust designs, and ensure sufficient flexibility to counter the challenges of unpredictability and complexity. They should select methods that help to capture complex social change processes and illuminate interactions between interventions and the context. 2. ADDRESSING CHALLENGES OF EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 33 Multiple actors Many players work in fragile and conflict-affected settings, seeking to effect change and influence the situation, which adds additional dimensions of complexity and uncertainty. Actors may be members of the diplomatic corps or the military; development and humanitarian agencies or government bodies; informal power structures or various local groups. These many actors have different cultures, loyalties, institutional features and interests, and do not always pull in the same direction. There are also differences in terminologies, planning cultures, and approaches between the different agencies on the donor side. Weak theoretical foundations and evidence base The theories underpinning international support to peacebuilding, conflict prevention and statebuilding are weak. There is a lack of agreed upon, proven strategies for effectively working towards peace. The logic underpinning donor activities is often unclear. Numerous strategies and programmes are poorly designed with ill-defined objectives and a lack of clearly stated, tested (or testable) theories of change (i.e. the implicit or explicit understandings of how it is hoped that what is being done will contribute to peace). Programme approaches are often contested and evolve rapidly to adapt to the changing context, meaning it may be difficult to establish what activities and strategies are actually being implemented. All of which makes programmes less easily “evaluable”. Challenges to data collection Challenges encompass scarcity of data, lack of monitoring, high personnel turnover, and erratic access to field data in certain regions at certain points in time. While the lack of timely, relevant, comparable data of high quality is not unique to situations of conflict and fragility, data problems tend to be compounded in these settings due, for example, to weak state statistical capacities and a multiplicity of international actors with incoherent data systems. However, this guidance suggests that more for data collection sources are available than currently used and that resources and institutions with special competence in this area exist and should be taken advantage of. Attribution Attribution is the ascribing of a causal link from a specific intervention to observed (or expected) changes. While attribution poses a problem in all areas of development work, attributing results to any particular policy or single intervention in conflict contexts is even more difficult. The difficulty arises principally from the fluidity and complexity of conflicts settings themselves and from frequently non-linear nature of change processes. For example, other activities (beyond the scope of the evaluation), such as military interventions, may actually be responsible for changes that are attributed to conflict prevention or peacebuilding activities. It can be very difficult for evaluators to control for these outside variables. Related challenges include the difficulty of creating a counter-factual or control group, especially when looking at country or regional conflicts, which is necessary to describe with reasonable certainty what would have happened had the activity in question not taken place 2. ADDRESSING CHALLENGES OF EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 34 Politicisation Fragile and conflict-affected settings are highly political environments. Due to the politicisation of international involvement and political sensitivities in national contexts, evaluators may find it difficult to maintain a safe, credible “evaluation space”. Box 2.1. Political constraints in conflict settings: lessons from Sri Lanka The evaluation took place in 2008-2009 in the complex political context of strained relations between the government and donors and significant security restrictions on travel outside Colombo. This limited the range of parties that could be interviewed and made donors hesitant to release sensitive strategy and programming material to the evaluation team. In response to the worsening security situation, the evaluation team decided to exclude from the sample the activities associated with the peace negotiations, as well as other aspects of diplomatic engagement, the security sector, and some donors’ internal analyses. This was a significant decision that led to agreement by most donors to support the evaluation, even though it meant important areas of donor engagement and the history of conflict prevention and peacebuilding work were not assessed. Source: Chapman et al. (2009). Overcoming challenges to evaluation This section outlines core principles for evaluation in settings of violent conflict and fragility. These principles should be carried throughout the evaluation process, informing each of the steps outlined in Chapters 3 and 4. The specific issues outlined here complement the general evaluation principles and standards which the OECD has set out. The application phase of this guidance showed that these general evaluation principles, as outlined below are also relevant and valid in conflict settings. There is no excuse not to apply them. When applied carefully, they enhance the credibility, use, and rigour of the evaluation process and its end results. The OECD DAC’s Quality Standards for Development Evaluation (2006a) provides a guide to good practice. This short document, built through international consensus, is a staple reference for all evaluations, including those in settings of conflict and fragility. The standards draw on the core principles that evaluation processes should be impartial, credible, transparent, and independent. They should also be useful and relevant, informing decision makers and contributing to learning. An evaluation report should describe transparently the data sources, data collection instruments and analytical methods used and identify their strengths and weaknesses. Evaluation teams should deal with attribution and causality in a credible way. Commissioning agencies should make the results of evaluation widely available and ensure that they are used systematically by decision makers and others to support learning and accountability. The standards also state that the collaboration of development partners is essential. This and other key references for evaluation are presented in OECD (2011d), Evaluating Development Co-operation: Summary of Key Norms and Standards. Context as the starting point: conflict analysis What is known about a situation of conflict and fragility, its causes, components, and dynamics? Conflict analysis – which includes analysis of the political economy, stakeholders, 2. ADDRESSING CHALLENGES OF EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 35 and conflict drivers and causes – is central to any evaluation of donor engagement in situations of conflict and fragility. Conflict analysis provides an analytical framework for assessing the relevance, effectiveness, and impact of peacebuilding activities, as explored in Chapter 4. Conflict analysis may be used as the basis for assessing whether activities have been sufficiently sensitive to the conflict setting, determining the scope of the evaluation (what will be evaluated), and identifying pertinent evaluation questions. Another of its functions is to ensure that the evaluation itself is conducted in a conflict-sensitive way. Conflict analysis, as the basis for evaluative analysis, is a key aspect of conflict prevention and peacebuilding evaluations, regardless of the design and methods used. It is as important for evaluations using randomised control groups, regression analyses, surveys, and large sample sizes, as it is for qualitative evaluations with in-depth case studies and focus groups. (The use of conflict analysis is covered in more detail in Chapter 3 and Annex A). Conflict sensitivity Conflict sensitivity refers to the ability of an organisation to a) understand the context in which it is operating, b) understand the interaction between the intervention and that context, and c) act upon that understanding in order to avoid negative impacts and maximise positive impacts on the conflict (CDA, 2009). All activities in a fragile and conflict-affected setting must be conflict sensitive. The principles of conflict sensitivity, adopted by the OECD in 2001, assert that international assistance must, at a minimum, avoid negative effects on conflict – “do no harm” – and, where possible, make a positive contribution to conflict prevention and peacebuilding. Fragile States Principle 2 (OECD, 2007) reiterates the commitment to conflict sensitivity, emphasising the importance of basing interventions on strong conflict and governance analysis in order to avoid inadvertently aggravating social tensions or exacerbating conflict. Conflict prevention and peacebuilding policies, projects and programmes, and development or humanitarian activities in conflict settings sometimes do cause harm, often unwittingly as in the example given Chapter 1 (Box 1.3). When assistance does cause harm in a situation of conflict and fragility, it produces direct or indirect effects that aggravate grievances, increase tension and vulnerabilities, and/or perpetuate conflict and fragility in some way. Such effects may be the result of a project or programme engagement – i.e. how its humanitarian or development outcomes contribute to peace or affect conflict. However, they may also spring from the operational aspects of an engagement (Uvin, 1999ab) – i.e. how, where, and when donors and agencies operate and how they implement and distribute aid. As a policy or programme should be conflict sensitive, so should the evaluation process itself. Evaluations carried out before, during, or after a violent conflict must be conflict sensitive because they are themselves interventions that may impact on the conflict. In this respect, it is important to understand that questions asked as part of an evaluation may shape people’s perception of a conflict. Evaluators should be aware that questions can be posed in ways that reinforce distrust and hostility towards the “other side”. Evaluators should keep in mind that the way they act, including both the explicit and implicit messages they transmit, may affect the degree of risk. Moreover, the evaluation process itself may actually put people in danger. A number of the evaluators who contributed to this guidance spoke of incidents where someone they 2. ADDRESSING CHALLENGES OF EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 36 had questioned in the course of their evaluation work had been arrested or otherwise threatened. Measures should be taken to avoid this. For example, in one evaluation the evaluation team leader decided that the names of its local members should not published in the report, because of possible repercussions they could face as a result. Their identities were protected and the local experts operated instead as external resource staff and key informants. In some cases, such as the real-time evaluation of Denmark’s humanitarian aid in south-eastern Somalia (Polastro et al., 2011), it may be considered more prudent and effective to rely on local staff or national teams that can more easily travel in dangerous zones – though their safety must also be protected. It is especially important to consider the safety of interpreters and other local staff, partners and beneficiaries, whom evaluators may inadvertently expose to greater risks than they themselves face. International evaluators leave after a short while, which may influence the risks they are prepared to take. Local people stay, however, and face possible reprisals. Such risks should be identified and addressed at the outset of the process and included in the planning and implementation of the evaluation. Doing so is the responsibility of evaluation commissioners and team leaders and a requirement of conflict-sensitive, ethical evaluation. Evaluators and commissioners should discuss and take appropriate measures to ensure conflict sensitivity, the ethical conduct of the evaluation and the protection of those involved. A thorough, up-to-date understanding of the conflict is the first step in a conflict-sensitive evaluation process. The evaluation report must explain what measures were or were not taken to ensure the conflict sensitivity of the evaluation itself and any impact that taking or not taking them may have had on the results of the evaluation. Evaluating conflict sensitivity (and effectiveness) It is important to understand that conflict sensitivity does not automatically deliver an effective peace programme or policy. A conflict-sensitive intervention is not necessarily effective in addressing drivers of conflict and fragility. Nor are explicit peacebuilding interventions necessarily conflict sensitive. For example, a reconstruction programme that rebuilds destroyed homes and provides small income-generation grants to returning refugees and internally displaced persons may avoid “doing harm” and try to rebuild relationships across conflict lines. It sponsors inter-ethnic dialogue between returnees and host community members, provides “balancing grants” to the host communities for priority community infrastructure or income-generation projects, and sponsors sports and cultural events for youth. It succeeds in ensuring that aid does not disproportionately benefit one group, and supports rebuilding of relationships among some community members. However, while it may be conflict sensitive, the reconstruction programme may not be effective peacebuilding as such, insofar as its activities do not address the drivers of the conflict, which could be, for example, impunity and injustice or conflicting visions of the future. In assessing conflict sensitivity, it is important to look at the extent to which the intervention aggravates or mitigates grievances, vulnerabilities or tensions. For interventions that do not have explicit peacebuilding goals, evaluators would assess the effects of the development or humanitarian outputs and outcomes (e.g. infrastructure development, a more operational police or judicial system, etc.) on the drivers of conflict or fragility. For example, a poverty reduction programme may have positive development results, but a thorough conflict analysis might reveal that, while the programme reduced levels of poverty overall, one group gained more than another, causing deeper resentment 2. ADDRESSING CHALLENGES OF EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 37 among excluded groups. If poverty reduction strategies helped achieve greater equity, they might contribute to peace. In addition, all activities, whether explicitly aimed at peacebuilding or not, should be examined to assess their conflict sensitivity. One of the more widely used conflict sensitivity tools, the Do No Harm Framework (Anderson, 1999), draws attention to the unintended consequences of aid planning and practice. Although it was originally developed for humanitarian aid it is also regularly applied to development and peacebuilding interventions. It identifies five ways in which operational components of an intervention may affect a conflict: ●theft/diversion: fuelling the conflict with stolen or diverted goods/funds; ●market effects: changing local markets with an influx of outside goods; ●distribution: distributing goods along the lines of the conflict; ●substitution effects: replacing existing functioning systems or structures; ●legitimisation: giving legitimacy to a group or leader by working with them. It also identifies four ways in which the behaviour of agencies, especially those implementing programmes, sends messages that reinforce the modes of warfare or, alternatively, non-conflictual relations. These include behaviour that: ●conveys respect or disrespect to people and communities, ●communicates an agency’s willingness or unwillingness to be held accountable, ●treats people in ways that are perceived as fair or unfair, ●demonstrates transparency or lack of transparency. Negative patterns can undermine an organisation’s efforts and put its staff in danger, lead to relationships that are antagonistic and untrusting, and make partners and communities feel humiliated. In extreme cases, violating the principles of respect, accountability, fairness and transparency can lead to violence against an organisation or within the community. Evaluators may need to examine the target agency’s own ways of working to determine whether the intervention is conflict sensitive. This would include examining inadvertent impacts of decisions about staffing, criteria for selection of beneficiaries, selection of local partners, relations with local authorities (including military actors and government), and processes and procedures for distributing aid (ibid.). Often, simple decisions about hiring – such as requirements regarding language – can result in staff that is disproportionately drawn from one conflict group. Similarly, seemingly objective criteria for the selection of beneficiaries (e.g. needs) can result in one group obtaining much more assistance than another and, consequently, contribute to escalating tensions. While the implication is not that donors or implementing staff abandon their criteria or redistribute aid, they must be aware of unintended conflict effects and develop options within the programme to mitigate them (ibid.). Being conflict sensitive and evaluating conflict sensitivity are two imperative dimensions of evaluating conflict prevention and peacebuilding work. A clear, critical assessment of an activity or a policy’s impacts will cover both intended and unintended consequences and thus offer insights into the sensitivity of the activity under evaluation. Evaluators can help assess whether or not the standard of conflict sensitivity has been achieved – as well as provide insights on how to improve sensitivity. Evaluating Peacebuilding Activities in Settings of Conflict and Fragility Improving Learning for Results © OECD 2012 39 Chapter 3 Preparing an evaluation in situations of conflict and fragility This chapter considers how evaluation commissioning agencies and planners may set up an evaluation. Its base premise is that effective preparation makes for effective evaluation. It thus examines each of the key preparatory steps, looking first at how to define the purpose of an evaluation and how to conduct (or commission) a conflict analysis. The chapter then goes onto discuss how to identify the key questions an evaluation must ask. It examines timing and logistics, co-ordination with other actors, selecting evaluation criteria, management, methods, the evaluation team, and the dissemination of evaluation results. 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 40 This chapter is designed to be of interest to those who commission evaluations (programme managers, staff or evaluation offices) and evaluation managers, as well as staff and decision makers involved in selecting evaluation topics and questions. It sets out some key steps in preparing an evaluation. The first three steps – purpose, conflict analysis, and scope – may usefully be considered together. Summary of key steps for preparing an evaluation 9 Define the purpose of the evaluation 9 Analyse the conflict context 9 Consider gender equality 9 Determine the scope of the evaluation 9 Decide on evaluation criteria 9 Outline key evaluation questions 9 Select evaluation approach and method to fulfil purpose 9 Take timing and logistical issues into consideration 9 Co-ordinate with other actors 9 Determine how the evaluation will be managed 9 Select and contract the evaluation team 9 Prepare to disseminate evaluation results 9 Control quality Develop terms of reference Defining the purpose Every evaluation, regardless of the context, should begin with the question: What is this evaluation meant to ascertain and how will this information be used? Defining the purpose and objectives of an evaluation is the most important planning step. If the purpose is not clear, the evaluation will not be. Commissioners should think about: Who is the intended audience? Who is to receive the findings and what will they do with the results? What kind of information is needed? Evaluations may have a number of different, sometimes concurrent, purposes. Accountability and learning are two of most frequent, and most evaluations combine them. ●Accountability seeks to find out whether an activity has been performed as intended and/or whether it has achieved the expected results. 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 41 ●Learning looks to provide evidence and improve knowledge of results and performance, which can help improve ongoing or future activities and increase understanding of what works, what does not, and why. Box 3.1 gives some examples of the accountability and learning objectives of evaluations, as described in evaluation reports. Box 3.1. Defining the purpose of an evaluation ●Accountability: “The purpose of the evaluation is to assess whether Norway has, with its transitional assistance, contributed to increased security (and stability) in Haiti, and whether gains achieved are likely to be sustained.” (Norad, 2009). ●Accountability: “This report was prepared to ascertain whether Asian Development Bank policy conditions had been met and whether they led to achievement of the Tajikistan Post Conflict Infrastructure Programme’s stated objectives or purpose.” (ADB, 2007). ●Learning: “The overall objectives of this project were, first, to develop a method for assessing the impact of development co-operation in conflict zones, and second, to apply this method in North East Afghanistan.” (Böhnke et al, 2010). ●Accountability and learning: “The joint evaluation of conflict prevention and peace building in the Democratic Republic of Congo has a double purpose: to provide accountability to the public and to decision makers in development co-operation, and to generate lessons for improvement. The emphasis is on the learning side, with a view to developing more strategic policies and programmes.” (Brusset et al, 2011). Source: Development Evaluation Resource Centre (DEReC) website, www.oecd.org/dac/evaluationnetwork/derec. The DAC Principles for Evaluation of Development Assistance (OECD, 1991) state that “to have an impact on decision-making evaluation findings must be perceived as relevant and useful”. Evaluations involve real costs, including the use of resources which could otherwise be deployed elsewhere, and should therefore be judged on the value of the information they provide. Usefulness is an important principle in evaluation. Use can take many different forms, before and during the implementation of an evaluation, or even many years after. In some cases decision makers use the findings to change or modify a programme directly, based on the recommendations presented. But in many cases use is less direct. An evaluation may contribute, along with other evaluations and research, to building up general knowledge over time on a particular topic, for instance. Behavioural or organisational changes may be caused by engaging in the evaluation process itself. Factors that may influence use of evaluations, and can be kept in mind when planning an evaluation, are the institutional environment (incentives and capacity for use), the relevance of the evaluation (timing, involvement of stakeholders, credibility), and the quality of dissemination (evaluation product, communication channels and mechanisms) (Feinstein, 2002). For evaluations involving multiple stakeholders, a shared understanding of the overall goal is crucial. In order to ensure such shared understanding and, later, the usefulness of the evaluation, involving stakeholders as part of the preparation and planning process is recommended. However, their degree of involvement depends on the evaluation’s design and purpose. An evaluation focused on learning is likely to be more participatory, whereas stakeholders would be less involved in an evaluation with an accountability purpose. 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 42 Analysing conflict Conflict analysis helps identify the causes, drivers and dynamics of conflict and fragility. It provides an analytical framework for understanding the complex, changing context in which an intervention is implemented. A conflict analysis identifies the key factors relating to conflict and fragility and the linkages between them, pointing to the sources and dynamics of violence as well as peace. A good analysis of conflict and fragility should include (or be linked to) an in-depth analysis of the political economy and broader development context (Kennedy-Chouane, 2011). Evaluators will always need to have an understanding of the conflict as the basis for their work, though they may not necessarily need to perform a conflict analysis themselves. The evaluation could be based on analysis provided by the evaluation target itself, the commissioning agency, an independent research institute or consultant, or a participatory process with stakeholders. An analysis may also be performed by the evaluation team as an early step in the evaluation process or used by commissioners to refine the evaluation scope and define key questions. Box 3.2. Two examples of the use of conflict analysis – Democratic Republic of Congo and Sri Lanka In the Democratic Republic of Congo, the conflict analysis – a combination of scientific research and workshops in the field – identified four important drivers of conflict during the inception phase. The four drivers – land ownership, weakness of the state, security sector, natural resources – helped delineate the scope of the evaluation. The drivers also helped evaluate the relevance of interventions (Did they target the right drivers of conflict?) and their impact (Did the conflict prevention and peacebuilding assistance have an effect on these drivers?) (Brusset et al., 2011). In Sri Lanka, the study used the existing comprehensive strategic conflict assessments conducted in 2001 and 2005 as a point of reference and background for the analysis. Though the team recognised that the conflict had deepened since 2005, it felt its root causes remained unchanged and were sufficiently covered by the earlier analyses (Chapman et al., 2009). Source: Brusset et al. (2011) and Chapman et al. (2009). As described in Chapter 1, a thorough understanding of the context of conflict and fragility should be part of the design and management of all interventions. If a conflict analysis has been carried out as part of developing a donor’s strategic engagement or programme design, the evaluation team will need to review the analysis and assess its quality and relevance at the outset of the programme and how it was adapted (or not) over time. Evaluators will need to consider whether the underlying analysis (explicit or implicit) was sufficient and accurate, whether it was effectively translated into relevant strategies and objectives, and whether it was adapted to the situation of conflict and fragility over time. Tips for reviewing a conflict analysis are provided in Box 3.3. There are many different models, tools and frameworks for conflict analysis used by development donors and others working in and on conflict and fragility. The aim is to gain a broad and deep understanding of the context in order to evaluate the intervention in 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 43 question. Notwithstanding the diversity of different models of conflict analysis, there is growing consensus on what is required in a good analysis: ●It distinguishes between the structural causes of conflict and fragility (both issues and people) and dynamic events and trends. ●It identifies positive as well as negative forces affecting the conflict (and chances for peace). ●It prioritises drivers of conflict and fragility and identifies which ones can be influenced by external action. ●It is operationally useful, and reflected in programme design, monitoring and evaluation. Nevertheless, in order to avoid the trap of becoming too comprehensive (and thus difficult to operationalise), it is important to distinguish those elements of the broader context that directly influence the conflict and how they do so. For instance, if poverty is identified as an important factor in the context, conflict analysis should identify which aspects of poverty influence tensions, resentments, and violence and in what way. Box 3.3 outlines more key elements to look for in a conflict analysis. Drawing on the outcomes of the conflict analysis, evaluators can assess the relevance and impacts of the activity or policy in question. For instance, the outcomes of the analysis will help to gauge whether or not an intervention addresses the relevant needs of the context, i.e. the causes of conflict and fragility. Additional information will, however, be needed to evaluate all dimensions of relevance and impact. For instance, the relevance of donor activities to overall country strategies or donor priorities may not be revealed through a standard conflict analysis and will have to be captured with other data, including programme documents and information on policies. One way of developing an analysis is to involve a range of stakeholders early on in the evaluation process. As it is not always possible to obtain all the competing perspectives from the different parties at the same time, it may be advisable to interview people separately to gain a deeper, wider understanding of the situation. However, evaluators should be aware that it will likely be difficult to gain consensus on the nature of the conflict as contending groups will not agree. This, of course, is a natural characteristic of conflict – and competing interpretations of history and causes may be an important dimension that the analysis captures. The outcomes of interviews, therefore, should be triangulated with secondary sources such as policy documents, programme/project notes, and grey literature such as reports from research institutes and think tanks. In interviews it can be particularly useful to engage in discussion to prioritise drivers of conflict and fragility, to understand which drivers are really important (and which less so), and to make general assertions more specific. Most evaluations look at an intervention or overall engagement in a single country or conflict region. In that case, the conflict analysis focuses on understanding that particular conflict, and also examines sub-regional or local conflict dynamics as relevant. For evaluations that involve analysis of activities across various conflict contexts – e.g. a thematic evaluation looking at women’s role in peace processes or an evaluation of a disarmament programme that operates in several different post-conflict countries – the analysis of conflict will be approached differently. Analysis could draw on existing research and empirical evidence about the (assumed) connections between the type of activities and violent conflict or state fragility in general. Case studies and comparative analysis can be Box 3.3. Checklist for reviewing a conflict analysis If a conflict analysis has been carried out as part of developing a donor’s strategic engagement or programme design, the team will need to review the analysis and assess its quality and relevance at the outset of the programme and how it was adapted (or not) over time. In this process, the evaluation team should pose the following questions: 1. Given the resources and capacities of the agency or organisation being evaluated, was the appropriate conflict analysis approach or tool chosen to guide the design and implementation of the programme(s) or policy(ies)? Did the analysis generate adequate information for determining the relevance of the intervention to the needs of the peacebuilding process; to the effectiveness of the programme designs and implementation; and to an assessment of the appropriateness of the theory of change? 2. Was the analysis kept up-to-date from the time the programme or policy was designed through the period of time under evaluation? Does it capture the evolution of the conflict in a way that can be used to look at relevance and longer term impacts? (If not, the evaluation team may need to update the analysis.) 3. Was the process of conflict analysis appropriate and effective? a) Was the analysis conducted by skilled people with an understanding of conflict and of the? b) Did the analysis gather information from a wide range of sources? Did it include perspectives from all the main stakeholders in the conflict? c) Was the analysis conducted in a conflict-sensitive manner? For example, did it ask questions in a way that avoided exacerbating divisions? If the analysis was conducted by convening stakeholder workshops, did the facilitators possess, or lack, sufficient skills to engage conflicting parties in productive discussion? Did the analysis process put researchers (and local partners) at risk by sending them into insecure areas? Did it put interviewees at risk by exposing them to retaliation? 4. Was the analysis done at the appropriate level? For example, if a programme was to be initiated at the provincial level, was a national analysis supplemented by an analysis of conflict dynamics within the province? 5. Were the conclusions reasonable? Were critical elements missing from the analysis? To what degree was the analysis shaped by the expertise of the agency or their general beliefs about how to bring about positive change? Was the analysis linked to strategy? Did it actually inform implementation and activities? 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 44 used to test these hypotheses and assess the intervention’s relevance, effectiveness and impact, as discussed in Chapter 4. Given the many different conflict analysis models and frameworks, evaluation commissioners and managers need to give conflict analysis careful thought. The analysis method selected should be well adapted to the context, the evaluation scope, and available resources. It follows that funders of evaluations should ensure that resources for the conflict analysis are proportional to the task envisaged and that the evaluation team has the necessary skills to analyse conflict. Broad questions that can be included are listed in Table 3.1, while Annex A provides further discussion on different approaches and links to conflict analysis resources. Table 3.1. Some key questions for conflict analysis Profile What is the political, economic, and socio-cultural context? What are the emergent political, economic and social issues? Are there important regional/international dynamics? What are the geographic dimensions? What areas that are prone to conflict and fragility, or affected by them, can be situated within the context? Is there a history of conflict? Conflict causes and potentials for peace What are the structural causes of conflict and fragility? What issues can be considered as proximate or dynamic causes of conflict and fragility? What triggers could contribute to the outbreak or further escalation of violence? What are the strategies or habits for dealing with conflict that contribute to violence? What new or emerging factors contribute to prolonging conflict and fragility dynamics? Have original causes shifted due to events during war and mass violence? What factors can contribute to peace and stability? What existing factors bring people together and can be built upon or reinforced? What are the most important drivers of conflict and peace? Which factors have the greatest influence on the situation? Actors Who are the main actors (people who perpetuate or mitigate the situation of conflict and fragility)? How do they contribute to or mitigate conflict? What are their interests, goals, positions, capacities and relationships? What capacities for peace and stability can be identified? Who can make a difference? What actors can be identified as “spoilers” (those who benefit from ongoing violence or who resist movement towards peace and stability)? Why? Are they inadvertent or intentional spoilers? Dynamics and future trends What are the relationships and dynamics among the key drivers of conflict and peace? What are the current conflict and fragility trends? What are the negative reinforcing cycles? What are the windows of opportunity? What scenarios can be developed from the analysis of the conflict and fragility profile, drivers and actors? How might different scenarios play out given likely future developments (in the short and long run)? Source: Adapted from International Alert (2007a) and Paffenholz and Reychler (2007). 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 45 Deciding the scope of the evaluation The scope of the evaluation should be clearly defined. The scope specifies the issues, funds, or types of interventions to be covered, including the time period and geographical coverage. When determining an evaluation’s scope, it is important to clarify and agree which types of aid it will cover and how. Evaluations may consider all or part of aid in a particular context, including explicit peacebuilding efforts, other forms of development assistance and humanitarian aid. Table 3.2 provides an overview of the hierarchy of evaluation scopes and some real-life examples for reference. Table 3.2. Examples of evaluation scopes Type of evaluation Definition Example System-wide or country-level Evaluation of the response by all (or most) international partners in a particular country or to a particular armed conflict or outbreak of violence. Multi-donor evaluation of Support to Conflict Prevention and Peacebuilding Activities in Southern Sudan 2005-2010 (Bennett et al., 2010). Partial system Evaluation of a part of a system (e.g. thematic or sector study), which may include cross-country or cross-conflict analysis. Joint Evaluation Programme on theme of Support to Displaced Persons (Borton et al., 2005). Evaluation of the German Civil Peace Service (Paffenholz , 2011). Single-agency response Evaluation of the overall response to a particular country or armed conflict by one international partner (funding, channelling, or implementing agency). Evaluation of Norwegian Support to Peacebuilding in Haiti 1998-2008 (Norad, 2009). Single project Evaluation of a single project, programme or policy undertaken by a single agency. Evaluation of the “Open Fun Football” School Programme in the Balkans region (Danida, 2011). Source: Examples drawn from the DAC Evaluation Resource Centre (DEReC), www.oecd.org/dac/evaluationnetwork/derec. 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 46 The scope should also spell out specific policies to be addressed in the evaluation – country memorandums of understanding or (joint) donor engagement strategies, for example. Conflict analysis can inform the process of determining the scope of the evaluation. Box 3.4. Using conflict analysis to inform the scope of an evaluation in the Democratic Republic of Congo A multi-donor evaluation was launched in 2008 to assess the role of external partners in supporting peacebuilding and conflict prevention in the eastern Democratic Republic of Congo. The use of a conflict analysis was very helpful for determining what key conflict factors – both the obvious and the less obvious ones – should be covered in the evaluation. At first, the evaluation focused on sexual and gender-based violence, child soldiers, and natural resources, which the commissioning evaluation departments considered to be key factors in the conflict. At that time (2008), these were generally accepted as important but the choice of those three factors was not based on a conflict analysis. Once the evaluation got underway, the team used conflict analysis to identify land issues and the weakness of the state as major conflict drivers, and these became part of the evaluation scope. Source: Brusset et al. (2011). To tailor an evaluation’s scope to its purpose and available resources, planners should ask these questions: What activities and policies will be covered? How far along the “results chain” will the evaluation go? Will it look for immediate impacts or on broad conflict dynamics? The answers to these questions will influence the selection of criteria and methods as described in the next two sections. Selecting evaluation criteria When planning an evaluation and drawing up its terms of reference, commissioners will need to determine which criteria will be analysed. The criteria to be examined are usually included in the key evaluation questions (see below) and will make up the main analytical content of the evaluation. The five OECD DAC criteria for evaluating development assistance – relevance, effectiveness, efficiency, sustainability and impact – are usually considered, though it may be more manageable to focus the evaluation on looking at a few criteria in-depth, depending on the evaluation purpose and intended use. Additional considerations that may be particularly relevant to situations of conflict and fragility, namely coherence and co-ordination, could also be subject to examination. Chapter 4 features a section, “Criteria for evaluating interventions”, that discusses use of the criteria in an evaluation. Outlining key evaluation questions Evaluation managers should develop a list of questions (or lines of inquiry) that an evaluation will answer. The type of intervention, the stage of implementation, and what the evaluation hopes to achieve determine the specific evaluation questions. In some cases, questions will be specific at the outset of an evaluation. In others, general questions will be refined through an iterative process during the evaluation. When considering 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 47 evaluation questions, evaluation managers should also think about which methods to apply for answering the questions and whether those methods are feasible in the available time and budget? When evaluating peacebuilding or statebuilding support and development interventions in fragile or conflict-prone contexts, evaluators might (in addition to assessing conflict sensitivity) pursue the following lines of inquiry: ●Is the intervention addressing the driving factors of the conflict? Does (or could) it address key tensions that have been identified as key factors in past, current, or possible future conflict? ●Has an analysis of conflict and fragility dynamics been undertaken and has it influenced programming and implementation choices? Humanitarian activities in conflict situations are guided by the core principles of neutrality and impartiality, and other Principles and Good Practices of Good Humanitarian Donorship agreed in Sweden in 2003 (GHD, 2003). Evaluations are likely to focus on assessing the extent to which a humanitarian intervention abided by the principles and the results it produced. And because humanitarian actions may have unintended (positive or negative) influence on conflict dynamics, evaluations must also consider conflict sensitivity. If the scope of an evaluation takes in all external engagement in a particular country, the relevance and effectiveness of humanitarian aid in relation to the conflict and fragility dynamics should also be considered. When examining the entire portfolio of assistance in a country affected by conflict, fragility, or prolonged humanitarian crises, evaluators might include humanitarian interventions in their analyses in order to assess the overall impacts on peace and conflict of interventions by external partners. They might also wish to consider the balance between humanitarian aid and other types of assistance. Some questions an evaluation might ask about humanitarian aid include: ●Does the intervention avoid creating tensions within the crisis-affected community; between displaced people and host communities; between agencies over the type and quantity of assistance? ●Does the provision of humanitarian aid impact the role and legitimacy of the state or have an influence on statebuilding processes? ●Is there coherence between humanitarian activities and other types of assistance? Because peacebuilding and statebuilding interventions may affect men and women differently, commissioners will often include questions on gender inequalities. Questions may focus on disparities in the family, the community, the marketplace, the state, and consider issues like gender-determined division of labour and role assignment; unequal access to and control over resources, benefits, and services; disparate participation in public and private spaces; and gender-specific practical or strategic needs such as protection from violence. A gender analysis can form the basis for studying gender dimensions in the evaluation. It is important to be realistic about what an evaluation can achieve, particularly when selecting evaluation questions. The testing phase of this guidance revealed a tendency for commissioning agencies and consultants to have overly ambitious expectations in respect to scope, content and timelines. For example, some evaluation terms of references contained dozens of evaluation questions, making it difficult for the evaluation team to answer all of the questions properly within the expected timeframe. One solution may be to provide a few broad evaluation questions at the outset, which can be examined in 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 48 greater depth, or added to, as part of the evaluation analysis. In an evaluation of the German Civil Peace Service (Paffenholz, 2011), the evaluation team first did a pilot of one of the eight case studies (Uganda) and used that experience to fine-tune the evaluation questions. They realised that several questions in the terms of reference could not be answered in the individual case studies and had to find other ways to address those questions. Incorporating gender equality and women’s empowerment Those planning an evaluation will need to determine how it will cover gender issues. Field experience and extensive research show that women and men and boys and girls experience, engage in, and are affected by violent conflict in different ways. Educational levels, responsibilities and mobility are some of the factors that can vary with gender and may affect the resources available to women and men in a conflict or post-conflict situation. Conflict itself can play a role in forming or changing a society’s understanding of gender roles – i.e. what it expects of different individuals in a given context. In many cases, conflict increases the burden placed on women. Systematic violations of women’s rights and their exclusion from economic, social and political spheres are barriers to development and may affect conflict dynamics. A clear, critical understanding of gender equality within a particular conflict context is, therefore, important for policy makers and practitioners, as well as for evaluators. United Nations Security Council Resolution 1325 (UNSC, 2000) was the first Security Council Resolution to link women’s experiences of conflict to the international and peace security agenda. It established that equality between men and women was essential to achieving and sustaining peace, and that equal participation in peace processes and post-conflict reconstruction efforts was critical to peacebuilding and statebuilding. It also called for the protection of women and girls and the inclusion of gender equality considerations in peacekeeping operations and training. In line with Resolution 1325, gender dynamics are part of conflict analysis. This requires differentiating between the roles, experiences and perspectives of men, women, girls and boys, as well as among women and men of different social, ethnic, religious or economic groups. Such analysis should not, however, fall into the trap of gender-based stereotypes. In the past, development agencies viewed women and girls primarily through the lens of victimhood. As a result they were too often left out of peacebuilding and statebuilding processes. In light of UNSCR 1325, there is growing consensus on two dimensions critical to understanding women’s roles in conflict situations: the targeting and victimisation of women and girls and the key part they play in peacemaking and rebuilding societies. This approach can help to ensure that peacebuilding processes take women’s needs into account and include making space for women in government and in key post-conflict decision-making processes. That being said, it should be acknowledged that women may also be perpetrators of violence, just as men may be victims. Those commissioning an evaluation should determine how it addresses gender-related issues in accordance with its focus and the activity or policy it is assessing. Planners may choose to make gender a cross-cutting theme or specific focus (some development agencies have particular requirements for the coverage of gender equality in evaluations). A programme that does not have adequate understanding of different gender needs and roles, or fails to adjust to them, may lack effectiveness, impact and relevance. Such issues might be included in the evaluation questions. “Criteria for evaluating interventions” in Chapter 4 gives several examples of questions on gender equality. 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 49 Evaluation managers might decide to include certain requirements on gender equality and women’s empowerment in the terms of reference as a way of ensuring that they are integrated into the evaluation’s objectives. For instance, commissioners may request that the evaluation team include a gender expert and use gender-disaggregated data and gender-sensitive indicators. An example of how gender could be incorporated into a conflict evaluation would be the evaluation of an infrastructure reconstruction project in a conflict setting that also looks at whether jobs created by the project have affected the livelihoods of working age men and women differently and whether this impacted on conflict drivers. As part of their efforts to ensure a conflict-sensitive approach, the evaluation team should examine how considerations of gender equality may affect their own work – in the make-up of their team or their engagement with stakeholders, for example. Looking at the big picture Experience in fragile and conflict-affected contexts has resulted in a growing emphasis on the need to look beyond individual projects, development assistance, and individual actors in order to understand peace and development processes more broadly. An individual activity may successfully achieve its short-term outcomes, such as training police officers or providing new livelihoods for former soldiers. However, these micro-level successes have widely been seen as not “adding up” to real progress towards peace. There is often a paradox between programme reports or self-evaluations that show successful programmes with good results and a simultaneous lack of progress towards peace – or even escalation of violence – at the macro level (sometimes called “peace write large”). The success of individual development or humanitarian activities and the outcome of overall peace and statebuilding processes generally depend not on the actions or strategies of a single funding or implementing agency, but on other factors. For instance, external shocks, government policy decisions or the diplomatic pressure that the international community exerts – or does not exert – on governments and warring parties might drive conflict and peace dynamics. Such factors are beyond the scope of the evaluated activity, but evaluators must nonetheless consider them in order to draw reasonable conclusions and attribute results. Evaluations that focus narrowly on donor engagement or look only at the programme or sector level may fail to identify important system-wide effects or constraints. It would not, of course, be realistic for every single evaluation to cover the entire policy arena or all dimensions of conflict and fragility in a given context. Activity or programme-level evaluations are also valuable. Nevertheless, development agencies should also plan evaluations that capture strategic issues or may ask evaluation teams to examine questions related to the broader context during an evaluation of a single programme or activity. Selecting the best-fit evaluation methodology There is no single blueprint methodology for evaluating donor engagement in fragile and conflict-affected situations. Rather, the golden rule is to apply the right tools and methods to the right questions. Methods should be chosen according to the evaluation purpose and key objectives and should involve a credible approach to attribution that avoids potential biases. The complex nature of interventions in fragile and conflict-affected 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 50 situations generally makes it necessary to combine different methodologies in order to answer the evaluation questions. Many favour a mixed-method approach, using both qualitative and quantitative methods and data. All the evaluations that tested this guidance used such mixed-method approaches. Planners should examine questions of methodology, deciding, for example, where there will be comparison groups (to inform discussion of counterfactuals), single case studies, and time or single series data. Extensive literature exists on the strengths and weaknesses of evaluation methods and their applicability to different contexts and purposes. Several types are listed in the glossary. An overview of methods and their suitability to conflict prevention and peacebuilding work may be found in Church and Rogers (2006) and in OECD (2008b). Encouraging participation of both women and men and knowing the informal rules of communication between men and women, is central to selecting a gender-sensitive approach. The incorporation of both women and men in the sample or study population should be ensured and potential obstacles to women’s participation in the evaluation addressed. For instance, it could be difficult for evaluators to speak directly with women and women may not express themselves freely in the presence of men. The methodological implications of these gender dynamics should be considered. Dealing with timing and logistics Schedules and evaluation plans are often decided well in advance. However, the timing for evaluating conflict prevention and peacebuilding interventions should be determined not only by the phase of the policy, programme, or project cycle, but also in relation to current conflict realities. The timing of the evaluation should be appropriate to the current dynamics of conflict and fragility and useful for informing policy discussion and/or programme adjustments (according to objectives). Commissioning organisations may have to adjust their expectations in the light of conflict and fragility-related constraints. The terms of reference should be clear about realistic time frames. To identify the right time and good entry points for an evaluation, the questions below should be considered, bearing in mind the outcomes of the conflict analysis. Clear terms of reference will help ensure the conflict sensitivity of the evaluation process itself, particularly how it proposes managing logistics and timing. ●What is happening in the situation of conflict and fragility? At what stage is the conflict cycle? Watch carefully for potential conflict triggers (elections, controversial celebrations, etc.). ●Would an evaluation at this moment be disruptive to the policy, project, or programme itself? ●Would an evaluation spark political reaction that could undermine the intervention by calling attention to it or by inadvertently feeding political forces in opposition to it? ●Would an evaluation put stakeholders at personal or political risk? Will there be sufficient access to stakeholders, or sufficient safeguards, to avoid bias that might endanger the policy, programme or project and the staff and stakeholders. ●Has the activity been in place long enough to provide useful experience and learning? Is the assessment of outputs, outcomes, and impacts based on a realistic time frame? 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 51 ●How long has it been since any previous evaluation or review was performed (either of a donor’s own activities or of relevant or similar activities of other donors)? ●Are there any logistical issues that must be taken into consideration (security restrictions, election process, weather patterns, major national holidays, access to transport, etc.)? Box 3.5. Conflict-related constraints on evaluations in Sri Lanka and the Democratic Republic of Congo In Norway’s evaluation of peace efforts in Sri Lanka during the period 1997-2009, the team faced a challenging task. The evaluators were granted full access to the Norwegian Ministry of Foreign Affairs’ archives and Norwegian individuals involved in the peace process. However, they could not gain access to a number of key people in Sri Lanka. They included senior figures in the Liberation Tigers of Tamil Eelam (dead), second-level cadres (in prison), and the incumbent government. The team sought to compensate by studying secondary sources, such as published research (including the team’s own), unpublished reports, and media coverage. In addition, international and national actors, experts and observers were interviewed. In the Democratic Republic of Congo, a first attempt in 2008 at evaluating peacebuilding support in Eastern DRC was cancelled due to a resurgence of violence. A rebel group, the National Congress for the Defence of the People, launched an offensive in North Kivu and the safety of the evaluators in the field could not be guaranteed. In 2009, there was a fresh attempt at conducting an evaluation. This time the set-up was different. Greater emphasis was placed on policy analysis and document-based study and interviews. Had there been a fresh upsurge in violence, the new set-up would have made it possible to continue the evaluation. However, without the field missions the evaluation would have also shifted in focus, looking at donor policy only and not assessing results in the country. Source: Norad (2011) and Brusset et al. (2011). Co-ordinating with other actors In line with the evaluation quality standards drawn up by DAC (OECD, 2010c) and the widely endorsed principles of aid effectiveness (OECD, 2005), evaluation work should be co-ordinated where possible and purposeful. To facilitate co-ordination, it is important to examine the institutional, organisational, and project-level context of the intervention in order to identify key stakeholders. Actors may include development agencies, bilateral donors, and multilateral institutions, providers of South-South assistance, implementing agencies, non-state actors, civil society, humanitarian actors and military forces. Commissioners should consider the interests different actors might have and contributions they could make in terms of data and decide if and how they might be involved in the evaluation. For example, non-governmental organisations that play a major role in the sector could be invited to serve on an evaluation reference group to lend the evaluation additional scope and reach. Or military actors that have access to data about the security situation may be asked to contribute this information without actually joining the evaluation process. For an evaluation looking at support to education, the Ministry of Education could be engaged and contribute to defining key evaluation questions and understanding potential links between education and conflict. Increasingly, development agencies, humanitarian organisations and security forces are working together in situations of conflict and fragility. The current emphasis among 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 52 many countries on “whole-of-government” approaches can often involve a great variety of actors from diverse backgrounds in any one evaluation. Co-ordinating such a diverse cross-section requires special consideration when setting up the management structure. If handled carefully, bringing together different development assistance players in a single evaluation can be a learning experience in that it broadens the scope of analysis and affords an opportunity for assessing differences in intervention methods and theories of change. However, a very large number of actors, ill-defined roles, or unclear objectives can make co-ordination a real problem. “Criteria for evaluating interventions” in Chapter 4, which discusses the importance of ensuring the coherence and co-ordination of an evaluated activity, also considers the involvement of different actors. Working with local and country stakeholders Evaluation managers may decide to invite local stakeholders (country offices or embassy staff, national governments, civil society organisations, beneficiaries, implementing partners) to take part in planning and conducting an evaluation – especially when learning and using the results are of the essence. It is generally accepted that there is a need for external partners to increase the involvement of local people and intended beneficiaries in evaluation. Local involvement may contribute to ensuring a more transparent, stronger relationship between external actors and local communities, in line with the Fragile States Principles and extensively borne out by evaluation experience. Engaging with knowledgeable local people and those targeted by programmes can provide critical input for understanding the context and conflict and carrying out the evaluation analysis. Involvement of people from different sides or with different perspectives on the conflict can be critical to understanding links between the intervention and conflict dynamics. Nevertheless, planners must take great care when deciding whom to involve, and how to involve them, in the context of fragility and violent conflict. The need to protect those involved and safeguard the objectivity and impartiality of the evaluation may influence such decisions. Development interventions that affect daily activities, resources, roles and responsibilities, opportunities and rights of the beneficiaries may have different implications (intended and unintended, positive and negative) for women and men and boys and girls. Evaluation managers must make sure that different points of view are included throughout the evaluation process. This may require considerations about what to analyse, which questions to ask stakeholders, what data collection procedures to adopt, what type of report to write and how to disseminate it. In some contexts it requires including female members in the evaluation team. Evaluation planners will also need to determine how to handle the participation of partner country government institutions and what level of involvement is appropriate and useful in a specific conflict context. This issue is of critical importance for the feasibility of the evaluation, for ownership of the process and its results, for transparency, and for potential interest in and use of the findings. Donors generally carry out their conflict prevention, peacebuilding and statebuilding actions in support of and in partnership with host governments. A logical extension of that co-operation is working together in evaluation. Such partnerships, however, may pose challenges where governments lack legitimacy or are primary actors in an ongoing conflict. The political context and its high stakes not only affect external partners, they are also likely to have very real impacts on 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 53 how and why partners engage in an evaluation process. Donors need to learn more about managing partnerships in evaluations in fragile, conflict-affected settings. Considering a joint evaluation Joint evaluations that bring together different actors (development agencies, partner countries, etc.) can contribute to harmonised approaches where analysis and follow-up are shared. They are also seen as promoting opportunities to generate additional learning about how a variety of activities “add up” to produce an overall development impact. Joint evaluations often have a broader scope, capturing a more complete picture of development co-operation in a particular context. Some will cover many – even all – interventions in a particular conflict and fragility zone to assess their combined impacts. For example, by pooling information from several partners, the multi-donor evaluation in Southern Sudan was able to cover some 85% of the entire donor portfolio (Box 3.6). The OECD DAC Evaluation Network’s Guidance for Managing Joint Evaluations (OECD, 2006) contains a number of practical suggestions and details as to why joint evaluations are conducted in development and how to conduct them. Box 3.6. A joint evaluation in Southern Sudan A major joint evaluation of peacebuilding efforts in Southern Sudan brought together a large number of development partners along with representatives of what would soon be the independent Government of South Sudan. A reference group in Southern Sudan was established and chaired by the transitional Ministry of Finance and Economic Planning to oversee and interact with the evaluation team during the evaluation. This group also involved representatives of government institutions, donors and agencies, the United Nations, the Joint Donor Team, and the NGO Forum Secretariat. The evaluation concluded that support to conflict prevention and peacebuilding had been only partially successful. Donor strategies did not fully incorporate key drivers of violence, which resulted in an overemphasis on basic services and a relative neglect of security, policing and the rule of law, all essential to state formation. Assistance in preparing Southern Sudan for secession was insufficient. There was an over-use of nominally “good” practice – particularly with respect to ownership and harmonisation – at the expense of much needed in-depth knowledge and field presence. While harmonisation, co-ordination and alignment do not run counter to conflict prevention and peacebuilding per se, they are not in themselves sufficient responses to state fragility. Source: Bennett et al. (2010). Writing terms of reference The preceding steps should inform the development of an evaluation’s terms of reference, which outline what is expected of the evaluator or evaluation team and from the evaluation process itself. As such, it is a key instrument for managing expectations of evaluations and helps guide the evaluation process. The terms of reference or scope of work is usually written by the person(s) commissioning or managing the evaluation. Terms of reference should specify whether or not a final report will be published and other requirements for completion of the process. It may be useful to describe in the terms of reference what dissemination is planned and who will be responsible for it. Annex C contains a sample terms of reference document. 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 54 Setting up evaluation management Commissioners and planners should set out the procedures for managing an evaluation. They should state clearly who is responsible for what (headquarters, field office, evaluators, partners, etc.), the team’s degree of independence, the role of the evaluation team leader, decision-making processes and procedure for the clearance of reports. Experience has demonstrated the importance of being clear about whom the evaluation team should report to and how, including criteria for acceptance of different reports (length, content, writing style, etc.). Additional critical questions to be addressed are: ●How will management of the process be handled and what will the role of the different parties be? How will relations with the partner government(s) and conflict parties be managed? ●Should there be a reference or steering group? A reference group is an advisory committee that serves as an intermediary between management and the evaluators. It may also provide independent oversight of the evaluation. The group is usually made up of a variety of stakeholders and experts and can be used as a way of including stakeholders who are not otherwise directly involved in the evaluation. ●Will there be a management group? Who will be involved? A management group is useful when conducting a joint evaluation – it is made up of a group of selected representatives from those donors and agencies commissioning the evaluation and is responsible for the management of the evaluation process, including contracting and overseeing consultants (on behalf of the other donors and agencies). Calendars, deadlines, and funding should be clearly and realistically determined and sufficiently flexible to adapt to a rapidly changing context. To the extent possible, the evaluation management should ensure a balance of genders in steering, management and reference groups. Selecting the evaluation team An evaluation team made up of members with complementary skills tailored to the task ahead is recommended. Planners and commissioners should specify the required competencies in the terms of reference and, if relevant, the tender document. They should spend time on this task, as it will be decisive in the evaluation process. People who are knowledgeable about conflict and peace – and the different priority areas within the field relevant to the evaluation subject – are critical to the quality of evaluations of donor engagement in situations of conflict and fragility. However, it is equally important to have knowledgeable evaluation experts in the team. As stated above, performing a gender-sensitive evaluation may well require gender experts in the evaluation team. Particular attention also needs to be given to the perception of bias in the team. When hiring staff from the conflict-affected region or conflicting groups, it is important to take into consideration and adjust to possible threats to them, the rest of the evaluation team, and the credibility of the evaluation itself, which their involvement may jeopardise. The risk of the team being perceived as biased, or not being given access to certain information, should also be weighed, and the report should address and describe any implications for data collection or analysis. Controlling quality Ensuring the quality of the evaluation process and products – including reports – is key. There are several ways to organise quality assurance. The DAC Quality Standards for 3. PREPARING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 55 Development Evaluation (OECD, 2010) can provide guidance as to what to emphasise. In some cases, often when dealing with challenging or complex multi-donor evaluations, internal or external experts review draft evaluation products for quality. They can do so prior to sharing the products with the wider stakeholder group, which can, in addition to improving quality, save time and money by reducing the number of rounds of revisions with the wider group. Product quality assurance and approval may also be tied to payment schedules. Evaluating Peacebuilding Activities in Settings of Conflict and Fragility Improving Learning for Results © OECD 2012 57 Chapter 4 Conducting an evaluation in situations of conflict and fragility Chapter 4 considers the business of conducting an evaluation. It begins with the inception phase, and then looks at how to identify theories of change and the implementation logic underpinning the activity being evaluated. The next step is the issue of gaps in baseline data and how and where to source data in order to plug the gaps. The chapter looks at the criteria evaluators should use, focusing in particular on the DAC criteria of relevance, effectiveness, impact, sustainability and efficiency. It then describes how to bring an evaluation to a close. The chapter looks at drawing conclusions and issuing recommendations and at the reports evaluators produce. The next step the chapter discusses is communicating the evaluation’s (positive or negative) results to stakeholders and disseminating the lessons learned. Finally, to close the loop, the chapter emphasises the importance of feeding findings back into programme design and management. 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 58 A number of core steps should be part of any evaluation. Evaluators and managers (and others involved in the actual process of evaluating) do not have to perform them in the order in which they are set out below. But they should bear them in mind. Key steps in the evaluation process 9 Allow for an inception phase 9 Identify the implementation logic and theory of change 9 Deal with missing baselines and other gaps 9 Gather data 9 Examine the effort using various criteria 9 Draw conclusions and make recommendations 9 Conduct reporting 9 Ensure quality 9 Feedback on the evaluation 9 Management response 9 Disseminate findings 9 Feedback into programming and learning Allow an inception phase Given the complexity of conducting evaluations of donor engagement in fragile and conflict-affected situations and the underlying weaknesses in programming, an inception phase can help identify issues that need to be addressed before proceeding with the evaluation. The scope of an inception phase can range from simple to complex. Usually it will involve desk study, document review, and the production of an inception report, but some preliminary field work may also be required. An inception phase may be useful for assessing or conducting conflict analyses. It could also be used to map the evaluation subject, conduct a donor policy and portfolio or country policy analysis, or carry out thematic studies as needed. Evaluators present the results of an inception phase in an inception report that can be the basis for discussing data availability, evaluability, and the feasibility of planned data collection strategies. Inception reports are also useful tools for adapting methods and for fine-tuning the approach chosen to address the key issues specified in the terms of reference. (See “Reporting” in this chapter). Identify and assess the theory of change and implementation logic A theory of change is a set of beliefs about how change happens. It explains why and how people think certain actions will produce the changes they desire in a given context, at a particular moment in time (Weiss, 1995; Church and Rogers, 2006). It is a term used to 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 59 describe the links between the context, the intervention inputs, the implementation strategy, and the intended outputs and outcomes. Implementation logic, also called programme logic, is a term that describes why an activity is doing what it is doing, usually at the project level. It is closely linked to the theory of change. Evaluation can reveal whether success, or lack of it, is attributable to programme design and theory or to implementation. Contributing to testing theories of change and implementation logic is one of the prime contributions evaluation can make to research and broader learning. In situations of conflict and fragility it can be especially important for evaluators to identify theories of change, since they are often implicit, unexamined, and untested. In Kosovo, for example, the international community operated for several years on the assumption (theory of change) that peace could be achieved by improving relations between the two main conflicting parties. On the basis of this theory, it funded numerous programmes to promote dialogue, exchanges, youth interaction, women’s groups, and so forth. All were aimed at building cross-community relationships. However, a study found that without intra-community work to bridge internal divides and create more responsible leadership, cross-community interventions had little effect. The activities were operating on an incomplete theory of change. The study also found that cross-community relationships did not help prevent violence or strengthen collective resistance to violence in 2004. Bonding social capital – intra-ethnic networks of trust and reciprocity – and mono-community leadership, motivated by the strategic desire to secure independence for Kosovo, led to more effective action (Chigas et al., 2006). Causation: Between theory and outcomes A theory of change can take different forms at different levels. Theories of change at the country and conflict levels may be quite general – for example, strengthening the capacity of the government will help improve governance and therefore reduce conflict and violence. In this broad vision, development actors may operate in different ways (e.g. through budget support or capacity support to individual ministries) or work in different sectors, such as social services or finance and planning. Evaluations of specific projects or programme would be likely to formulate more directly causal theories of change which link specific inputs and activities to desired micro-level outcomes and to broader peace dynamics. For example, in the Southern Sudan evaluation (Box 3.6), the team identified broad theories of change across the full portfolio of interventions they were examining. They did not drill down to the specific implementation logic level or look at individual projects. Instead, they grouped the multiple activities of different donors under broad theories of change, and then tested each one. Programme documents do not always state in explicit cause-effect terms how a programme aims to produce its intended outputs, outcomes, and impacts. Accurate, clearly worded theories of change are necessary for effective programming and should be evaluated. More importantly, the assumptions that underpin theories should be the subject of evaluation. By identifying how an intervention was expected to contribute to sustainable peace or address conflict and fragility factors, evaluators use the theory of change to assess relevance, effectiveness, and impact. An example of this thinking is the imaginary anti-bias peace journalism programme described in Annex C and illustrated in Figure 4.1. The programme could be understood to work on a theory of change whereby it will train journalists and increase their knowledge of conflict dynamics, so reducing bias in 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 60 reporting and contributing to a reduction in tensions and violence between warring groups. An evaluation of this programme would test the underlying theory that improved knowledge reduces reporting bias and the secondary theory that improved reporting lessens the tendency to resort to violence in the broader community. Key questions for analysis would be how workshops, awareness raising, and skills development actually change conflict reporting and what impact they have on critical conflict dynamics. The programme could track the language used in reporting before and after training. It might also survey public attitudes and, at the same time, the programme activities to see if they were achieving the expected results or whether unexpected obstacles had arisen. It might turn out, for instance, that individual journalists have very little influence over the use of inflammatory language and that editors determine the use of “colourful” language to boost sales, so reinforcing stereotypes. Such a finding would suggest that the “theory” of training journalists to influence public opinion was flawed. Figure 4.1. Assumptions underlying the theory of change in a fictitious peace journalism programme Inputs and activities Assumptions: activity to output Training for journalists 1. Training is relevant and effective. 2. Information is understood and accepted. Assumptions: outcome 1. The journalists trained have a role to play in influencing public opinion. 2. They are willing and knowledgeable enough to choose actions that will maximize the possibility of reducing bias. Outputs Improved understanding of conflict stereotypes Updated attitudes towards conflict actors Actions (changes in news coverage to influence public opinion) Reduced propensity towards violence against stereotyped groups Improved lives Intermediate outcomes Outcome Impact Source: Content from OECD (2008b); diagram adapted from Gaarder, 3iE (2011). Where do theories of change originate? One task related to identifying and evaluating theories of change is to identify where they originate. Are they empirical – based on solid evidence from past programmes? Are they drawn from programme designers’ personal and professional experience or from the experience of the stakeholders and beneficiaries consulted during programme design? Or are they research-based? When working theories are not explicit they will need to be uncovered through document review or interviews with implementing staff and other stakeholders. The evaluation process may also reveal that different stakeholders involved in a particular intervention operate on different assumptions (theories) about how their efforts will promote change for peace. By drawing out underlying theories and opening up a discussion of the validity of different theories, the evaluation process can be useful in helping those 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 61 involved in a programme reach consensus on what they are doing and why. This clarity can help improve programme design and implementation. Evaluations of donor engagement in fragile and conflict-affected situations very often cover multiple projects and – in some cases – multiple donors. Such donor engagement may involve multiple theories or multiple variations of a broad general notion, e.g. more aid for social services supports stability. Box 4.1. Evaluating success and failure in peacebuilding When evaluating it is important to distinguish between and analyse various types of success and failure as an input to learning and future programme design. In this field it helps to distinguish a failure of the theory (wrong assumption about how change will happen) from a failure in implementation. Theory failure indicates the failure of a conflict prevention or peacebuilding activity due to a flawed causal relationship – in other words, underlying assumptions about how to bring about change in this context are false. A faulty theory of change could be based on an inaccurate conflict analysis, or it could reflect misdirected priorities or mismatched objectives. Implementation failure denotes a problem with the execution of the activity itself (inputs/outputs, staff capability, timing, location, security environment or budget) or with management systems. Such problems could include sudden changes in the conflict that disrupt or reverse progress, despite an otherwise well designed activity. Source: OECD (1999). Gather data As mentioned in the introduction, evaluators and evaluation managers sometimes encounter weaknesses in policies, strategies, and interventions. There may be unclear or unstated objectives, a poorly articulated theory of change or programme logic (as described above), missing indicators, no monitoring data, or no baseline information. The security situation can have serious implications for data and may particularly affect the ability of evaluators to travel to certain regions or countries and to access affected communities, programme beneficiaries or other key informants. Evaluation teams, working in co-operation with the commissioning organisations, must address the issue of weak or missing data. They should consider ways to (re)construct or compensate for missing baselines and other data during the evaluation process, bearing in mind that nothing can ever fully replace solid planning. Tips on how to compensate for or work around such gaps without endangering the quality of the evaluation can be found in a number of publications (World Bank, 2005b; Bamberger et al., 2006; OECD, 1999). Box 4.2 considers an example of how a donor-supported project in Afghanistan is working to increase the availability and quality of data. A key phase in the evaluation process is collating data to trace the “story” of a programme or policy and its effects, which include how they affect stakeholders’ viewpoints. Evaluators gather information from programme documents and reports, any available monitoring data, and interviews with programme staff, partner organisations, local officials, target groups, participants, third parties (groups such as neighbouring communities not targeted by a programme), and international and national actors who Box 4.2. Building statistics for monitoring and evaluation in Helmand Province The Helmand Monitoring & Evaluation Programme (HMEP) is a programme funded jointly by the UK’s Department for International Development (DFID) and the Provincial Reconstruction Team (PRT). It is designed to assist improved delivery and effectiveness of stabilisation and development programming in Helmand, Afghanistan. HMEP collates data from third-party sources as well as drawing on a dedicated research capacity to establish baselines and monitor indicators of progress against the Helmand Plan. Data is presented graphically and geospatially and stored on an interactive database, accessible from an online website. HMEP produces quarterly monitoring and analytical reports and up to four ad hoc reactive reports per year aligned with PRT and DFID reporting requirements. In addition, HMEP is set up to support the development of programmatic capacity in the PRT and the integration of monitoring and evaluation into planning and programme implementation. Source: Coffey International Development, “Helmand Monitoring and Evaluation Programme”, our-projects/helmand-monitoring-and-evaluation-programme. 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 62 observe the intervention from a distance (civil society organisations, research institutions, donors, media, academia, think tanks). Data collection strategies depend on an evaluation’s design, its purpose, and the information sources available. Strategies that can be used include random and purposeful sampling. Quantitative and qualitative data can be collected through censuses, observation, household surveys, interviews, questionnaires, anthropological or ethnographic research, participatory workshops and discussion groups. National statistics systems and major NGOs will often have available demographic data, though these may not be suited to the required sample size or detail. Moreover, some private polling companies have begun to specialise in household surveys and data collection to professional standards in conflict-affected areas. Box 4.3 gives examples of data sources and the evaluation literature provides further detail on data collection techniques. Prolonged violence and situations of high tension pose significant data availability problems that often restrict evaluators’ work. However, there might be more data available than is first evident in the planning phase and in programme documents. High staff turnover often shortens institutional memory. Moreover, different external and national actors are frequently poorly co-ordinated and do not always share the information they collect with other actors to whom existing studies, assessments, data, evaluations and surveys may not be known or available. Valuable baseline data might be gathering dust in documentary archives somewhere with few aware of it. It is sometimes worthwhile investing time up front in an evaluation phase to map data already collected, either as part of the evaluation inception phase or as a pre-study. Evaluators and other researchers should share and use data collected by others, as too many are gathered and allowed to lie fallow. It is also advisable that those commissioning an evaluation and the evaluation team agree to contingency plans for sudden shifts in the situation on the ground that may impact collection of data that are necessary for the evaluation. Evaluation managers may include data management provisions in the terms of reference. Box 4.3. Examples of data sources from Somalia and Afghanistan Somalia: Using multiple data sources and accounting for data shortcomings A 2011 evaluation conducted by DARA (an independent organisation specialised in humanitarian evaluation) looked at the effectiveness of the humanitarian response in Somalia. It drew on the following data sources: ●Literature on past assistance to Somalia and the contextual variables, and 300 web pages and relevant publications, and assorted documents. ●Semi-structured group and individual interviews with 489 stakeholders. Of these, 189 (112 in Kenya and 77 in Somalia) were carried out with a range of individuals involved in the response, who included representatives from UN agencies, the Red Cross and Red Crescent, international and national NGOs, local government actors, and donors. Women accounted for 24% of interviewees. There were group interviews with over 300 people from the affected population. ●Field observations in Kenya and Somalia, in camps for the internally displaced and Somali refugees. ●Online survey of former staff and key informants involved in the humanitarian response (response was low with only ten respondents answering). In total, the evaluation team gathered 3 117 items of information but encountered the following challenges: high staff turnover, limited time for field work, insecurity, poor consistency of data, disaggregation gaps (limited availability of data disaggregated by age, gender, vulnerabilities and outcomes in the period under review). Safety concerns prevented international evaluators from visiting areas proposed in the terms of reference, although national evaluators were allowed to visit some. North East Afghanistan: Creating new data The more fragile a state is, the less data tend to be available. Evaluation teams often have to collect their own data. In the context of an evaluation of the impact of aid in North East Afghanistan, the evaluation team gathered data from various sources: The core of the data was collected with two surveys among 2 000 head of household respondents in 80 villages. Evaluators also collected data from other sources to create a profile for each community, containing information on the history, demography, ethnic composition, political and social organisation and resource endowment. In other settings, much of these data would be readily available from existing sources, such as a national census. In the Afghan context, however, evaluators had to collect these data themselves – which was possible only in the context of a multi-year research project. In order to collect information on major events and changes affecting the communities, a monitoring and reporting system was set-up covering 40 villages. Local correspondents completed semi-structured reports four times a year. The evaluation team collected additional qualitative data during field research visits. Finally, evaluators obtained data on aid flows from various international development organisations in the region. The data were then compiled and used for quantitative and qualitative analysis of the impact of aid on stability. Sources: Polastro et al. (2011) and Böhnke et al. (2010). 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 63 During conflict and in its immediate aftermath, when mistrust is rife and most intervention stakeholders also have a stake in the conflict, the reliability (not to mention availability) of data and information provided is often particularly problematic. Depending on their own position within the conflict, different actors may have diverse, or even 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 64 contradictory, interpretations of an intervention’s positive and negative impacts or relevance. To ensure reliability, evaluators should use multiple sources or types of information and a mix of sound quantitative and qualitative data. They should triangulate the data they use, ensure that sources are transparent, and verify the data’s validity before analysing them – by fact checking with key stakeholders and interviewees, for example. By combining multiple data sources, evaluators can offset the bias that comes from relying on a single type of information and single observers. In the evaluation report’s description of methods, any issues around data (including data gaps or problems with inconsistency) should be described along with the impact these data problems had on the reliability and validity of the evaluation’s conclusions. To avoid increasing tension between groups (in accordance with the “do no harm” principle), decisions about how to involve various groups in data collection should be based on a clear understanding of stakeholder roles and interests, drawing on the understanding of stakeholders developed in the conflict analysis. Where access or security concerns impinge on data gathering, other methods – including consulting with knowledgeable proxies able to provide representative views or information – could be explored. Consideration to coverage may be required – to include perspectives from both within and outside the capital city, for example – depending on what issues the evaluation intends to address. Evaluations should draw on accurate, relevant data about women, men and gender relations. This helps ensure evaluations are gender and conflict sensitive, makes gender disparities more visible, and assists in answering the evaluation questions. Gender-disaggregated statistical data and gender analysis (information on the position of women and men in a particular conflict context) should inform the evaluation. Where disaggregated data are not available, combinations of quantitative and qualitative approaches can be used to gain the necessary understanding of gender relations and differentiate results for men, women, boys and girls. Where it is deemed that inclusion of such data is not necessary or not feasible, this should be clearly justified in the evaluation report. For example, in the case of an evaluation of the impact of aid on local perceptions in rural North East Afghanistan (Böhnke et al., 2010) the evaluation team used a survey. Though the team wanted to include women and youth, in the end the survey respondents were all male heads of households. Their decision was based on an analysis of the local context and consultation with regional and gender experts, and on an earlier experiment using female researchers to survey Afghan women, which found that security threats and the social custom of having men accompany each female researcher made it nearly impossible for the teams to work effectively. The evaluators decided that the choice not to include women in the survey was justified in this case because the opinions of older men (as the political representative of their household) were most relevant to answering the key evaluation questions. They verified the decision with experienced local teams and tested it before doing the initial baseline. In the evaluation report the team explained this decision, stating that, “while we think that the tasks at hand warrant these choices and the resulting limitations, we are nevertheless conscious of the fact that our research design is not equipped to capture trends for all of Afghanistan, nor does it capture perceptions of women [...]” (Böhnke et al., 2010). Transparency over methods and data limitations is a requirement of high-quality evaluations. Many interventions work to build peace and prevent conflict by creating change in people's attitudes, thought processes, and relationships. In such cases, it may be necessary 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 65 to collect attitudinal data, conduct interviews, workshops, or focus group discussions with stakeholders, or carry out surveys to collect quantitative data. Measuring intangible changes in areas such as perceptions through interviews requires the same triangulation vetting as other types of data. Relevant interviewees (primary sources) may be missing, because they are either in prison, dead, or unavailable in other ways. The evaluation report must account for such data gaps and make clear how they have been compensated for. Investing time early in the preparation phase for researching the data situation should help prevent surprises like missing data at a later stage, which can derail or delay the evaluation process. Criteria for evaluating interventions When evaluating development co-operation programmes and projects it is useful to consider the criteria outlined in the DAC Principles for Evaluation of Development Assistance and additional criteria that may be valuable. The analysis of these criteria forms the main content of an evaluation report. This section suggests how each OECD DAC evaluation criterion might be adapted to the field of conflict and fragility. The specific questions outlined below are guiding examples, not a comprehensive or obligatory list. The evaluation criteria are interlinked, each criterion shedding light on the intervention being evaluated from a slightly different perspective in order to develop as comprehensive a picture as possible of the intervention. When read together, the criteria should assist the evaluation team in developing a clear understanding of the activity or policy being evaluated and its contribution to statebuilding and peacebuilding. This adaptation of the criteria is based on contributions from Paffenholz and Reychler (2007), the draft guidance (OECD, 2008b), OECD and CDA (2007), and Anderson and Olson (2003). Relevance The relevance criterion is used to assess the extent to which the objectives and activities of the intervention(s) respond to the needs of beneficiaries and the peacebuilding process – i.e. whether they address the key driving factors of conflict revealed through a conflict analysis. Relevance links the outcomes of the conflict analysis with the intervention’s objectives, although the relevance of the intervention might change over time as circumstances change. Understanding relevance may also involve an assessment of the extent to which an intervention ties in with overall strategies and policy frameworks of the country or external partners. Different conflict groups or actors may have different perspectives on the relevance of an intervention and its results. Women and men may also perceive the relevance of the intervention differently. Assessing an intervention in relation to the conflict is key to evaluating its relevance. If staff, managers, or others involved in design and implementation have already carried out some kind of conflict analysis, its accuracy and use should be assessed. Assessing whether or not the conflict analysis proves (or has proved) accurate will be an important aspect of the evaluation. Assessing the analysis will contribute to learning and to refining theories about why violence occurs and what the most important determinants of conflict dynamics and long term statebuilding are. If no process of systematic analysis has taken place previously, the evaluation team may either talk to staff and stakeholders to understand what underlying (unarticulated) conflict understanding is guiding their work, or facilitate a more formal exercise for conducting a conflict analysis. The analysis is 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 66 needed to identify the expected connections between the programme outcomes and peacebuilding goals and the relevance to key driving factors of conflict and fragility. Questions on relevance might include the following: ●Is the intervention based on a valid analysis of the situation of conflict and fragility? Has the intervention been flexibly adapted to updated analyses over time? ●In the light of the conflict analysis, is the intervention working on the right issues in this context at this time? Does the intervention appear to address relevant key causes and drivers of conflict and fragility? Or does it address the behaviour of key driving constituencies of the conflict? ●What is the relevance of the intervention as perceived by the local population, beneficiaries and external observers? Are there any gender differences with regard to the perception of relevance? ●Are the stated goals and objectives relevant to issues that are central to the situation of conflict and fragility? Do activities and strategies fit objectives, i.e. is there internal coherence between what the programme is doing and what it is trying to achieve? Has the intervention responded flexibly to changing circumstances over time? Has the conflict analysis been revisited or updated to guide action in changing circumstance? Effectiveness Effectiveness is used to evaluate whether an intervention has met its intended objectives with respect to its immediate peacebuilding environment, or is likely to do so. The key to evaluating effectiveness – and thus the linkage between outputs, outcomes and impacts – is finding out to what degree the envisaged results have been achieved and noting changes that the intervention has initiated or to which it has contributed. Furthermore, as most of the activities undertaken in situations of conflict and fragility are not explicitly geared towards sustainable peace, it is important to draw a distinction between two kinds of results. One is “programme effectiveness”, i.e. to what extent the programme achieved its stated objective. The other is – if the programme met its objectives or goal – the immediate or secondary outcomes as they relate to peacebuilding and conflict dynamics identified in the analysis. A theory of change analysis can be used to assess the criteria of effectiveness. This involves assessing whether an intervention is based on a sound theory and logic and whether these are proving (or have proven) to be true. The understanding of effectiveness is also linked to the conflict analysis. A programme or policy may do good or do well and still not change the underlying dynamics or key driving factors of conflict and fragility identified by the conflict analysis. Also, external factors, unrelated to and beyond the control of the activity in question, may be more significant factors of peace and conflict, in which case effectiveness will be understood relative to these broader dynamics and trends. Box 4.4 describes how Germany evaluated the effectiveness of its civil peace programme using a framework that combined theories of change analysis with empirical research. Box 4.4. Evaluating the effectiveness of the German Civil Peace Service Germany commissioned an independent evaluation of its Civil Peace Service programme, which promotes non-violent ways of dealing with conflict through a variety of education, training and service activities. The evaluators developed a comprehensive evaluation framework that enabled systematic comparison of data across eight country cases. To compensate for insufficient data, a combination of evaluation approaches and methodologies was developed. In addition to reconstructing baselines and describing the Civil Peace Service’s underlying theories of change, the evaluation team assessed the plausibility of outcomes, i.e. the likelihood that the intervention had achieved, or would achieve, the hoped-for outcomes. This was done by checking Civil Peace Service interventions against a list of conditions for effectiveness developed during empirical research. The evaluation team also identified good practice examples from among the interventions evaluated and then identified which factors contributed to their effectiveness. Source: Paffenholz (2011). 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 67 Examples of questions that can help determine effectiveness include: ●Has the intervention achieved its stated (or implicit) purpose, or can it reasonably be expected to do so on the basis of the outputs and outcomes? ●Is the theory of change based on valid/tested assumptions? Are there alternative theories of change? ●To what extent did donors identify and adequately manage context-specific risks? ●Is the intervention achieving progress within a reasonable time frame, or will it do so? Is it possible to accelerate the process? Should the effort be slowed down for any reason? ●What major factors contribute to the achievement or non-achievement of objectives? ●Has the intervention achieved different results for women and men and boys and girls? Depending on the activity being evaluated and the scope of the evaluation, some specific examples of conflict-specific questions on effectiveness include: Does the intervention prompt people to increasingly resist violence and provocation? Do the stakeholders affected have a significant impact on the situation of conflict and fragility? Are the right/key people or many people being addressed? Were gender equality and relevant horizontal inequalities (ethnic, religious, geographical, etc.) that drive conflict taken into consideration and what are the results? Does the intervention result in an increase in people’s safety and in their sense of security? Does it improve non-violent forms of conflict resolution or power management? Does it result in a real improvement in relations among groups in conflict, demonstrated in behaviour changes? Impact The criterion of impact refers to the wider effects produced by an intervention. Such effects may be positive or negative, and may be produced directly or indirectly, intentionally or unintentionally. In fragile and conflict-affected contexts the criterion of impact is used to identify and evaluate the effects of the intervention on the key driving factors and actors of the conflict, as well as on broader development and statebuilding processes, as relevant. Assessment should cover both the desired changes the intervention aimed to achieve and any unintended (or unexpected) positive or negative results. 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 68 Changes in behaviour and attitudes, of the kind many peacebuilding interventions seek, are often difficult to measure and can take a long time to achieve. With this in mind, it may be too soon to reasonably expect significant impacts on conflict drivers (for example, for activities aimed at reforming institutions). In this case the evaluation might focus on outcomes and short term impacts and test the theory and programme logic to predict whether the current strategies are reasonably likely to contribute to peace over the long run. Methods for evaluating impact are covered elsewhere in evaluation literature, for example in the guidance commissioned by the Network of Networks for Impact Evaluation (Leeuw and Vaessen, 2009). The rigorous quantitative methods associated with impact evaluation and randomised control trials are considered not feasible in many situations of conflict and fragility, (although useful experiments are underway at the programme level). Still, it is particularly difficult to apply such methods to high-level questions of peace and conflict across various interventions at country level or to assessments of overall donor engagement in a conflict setting. Where causality cannot be reliably determined using rigorous methods, evaluators may present plausible explanations for their conclusions regarding impact, though limitations should be made explicit. Box 4.5 describes quantitative impact evaluation methods in more detail. When violent conflict is extreme, analysis may have to shift towards monitoring or “real-time” evaluation functions that focus on output indicators. They are easier to track and less prone to attribution failures than impact indicators. Examples of questions that can help determine impact may include: ●What are the primary and secondary, direct and indirect, positive and negative, intended and unintended, immediate and long-term, short-term and lasting effects of the activity or policy in question? Does it exert a significant effect on key factors for conflict or peace? ●Drawing on the conflict analysis, what key drivers of conflict and fragility were affected and how? What changes can be ascertained in attitudes, behaviours, relationships or practices (of how many people and/or classified according to selected criteria such as gender)? Are there any secondary negative effects? ●How has the situation changed over time and what, if any, has been the contribution of the intervention to those changes? ●What impacts have the interventions had on specific indicators of well-being, such as health status or poverty levels, addressed by the intervention? What are the impacts on long-term development trajectories? ●Has the intervention impacted policy? How do these policies relate to the conflict? Sustainability Sustainability is defined as the continuation of benefits on end of assistance. In an environment of conflict and fragility, sustainability includes the probability of continued long-term benefits and resilience to risk over time, as well as lasting benefits in the economy, institutions, human resource management, etc. As in other fields, sustainability also includes “ownership” of the peace and development processes. Experience and peace research demonstrate that peacebuilding processes are long term and call for long-term engagement that can weather setbacks (OECD and CDA, 2007). In conflict-affected regions, Box 4.5. Quantitative methods to evaluate impact in settings of conflict and fragility An impact evaluation measures the net impacts of an intervention by comparing its results with a counterfactual – a measure of what would happen in the absence of the intervention. Although experience with the impact evaluation approach in peacebuilding interventions is limited, it is being used increasingly in the development context and, indeed, in conflict and fragile settings. There are two main methods of conducting a counterfactual comparison. The first is the experimental approach or randomised control trial (RCT) evaluation. RCT evaluations need to be designed and initiated before the intervention begins. The evaluators randomly assign potential beneficiaries to “treatment” and “control” groups. The comparison of impacts between the two groups reveals the net impact of the intervention. This approach can be used to pilot an intervention to inform programme design. The second main method comprises “quasi-experimental” approaches, whereby statistical techniques are used to construct a control group that can serve as a counterfactual. Such techniques include various forms of matching, instrumental variables designs, regression discontinuity designs, and panel methods. These methods estimate the impact of an intervention by ensuring, to the extent that the data allow, that beneficiaries and non-beneficiaries are similar in all ways except for their treatment status. One such technique is propensity score matching, which estimates the net impact of an intervention by matching beneficiaries with non-beneficiaries who are similar to them in all other ways. There are obvious drawbacks to these methods for evaluating peacebuilding interventions. Limitations include the frequent absence of baseline data, the desire to “treat” everyone (the humanitarian imperative), the political dangers of random treatment, and the qualitative nature of many relevant variables which are consequently less compatible with statistical methodologies. Impact evaluations are, nevertheless, often possible at the project or programme level and can provide valuable information on what works. Source: Marie Gaarder (Norad Evaluation Department) and Annette Brown (3iE). 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 69 such engagement requires addressing those who have an interest in sustaining the conflict (sometimes called “spoilers” or the “hard to reach”). Questions regarding sustainability might include: ●Which steps have been taken or are planned to create long-term processes, structures, norms and institutions for peace? To what extent has the building of ownership and participation included both men and women? ●Will new institutions designed to address conflict and fragility survive? Are they being used? By whom? Does the intervention contribute to the momentum for peace by encouraging participants and communities to develop their own initiatives? ●Has a meaningful “hand-over” or exit strategy been developed with local partners or actors to enable them to build or continue their own peacebuilding initiatives? Depending on the activity being evaluated and the scope of the analysis, some specific examples of conflict-related questions on effectiveness include: ●Does the effort result in the creation or reform of political institutions or mechanisms that deal meaningfully with grievances or injustices? 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 70 ●Have those who benefit from and have a vested interest in on-going violence or instability, or who resist movement towards peace (“spoilers”), been addressed adequately? ●Will improvements in inter-group relationships persist in the face of new challenges and risks? ●Will the parties to a negotiated agreement honour and implement it? Efficiency The efficiency criterion is used to assess how economically resources (funds, expertise, time, etc.) are converted to results. In a conflict context, costs associated with prevention work will often be compared with the estimated costs of war or an outbreak of violent hostilities. Averted conflicts, however, are invisible. So there are, unfortunately, not many counterfactuals that compare the efficiency of prevention and conflict – though historical data may be used to reasonably estimate costs. In addition to comparing the costs of supporting peace with those of war, evaluation could look at priorities in relation to key conflict drivers. It could ask this question: “Is this particular way of working against violence the most efficient option?” In a setting of scarce development resources evaluations should shed light on whether a particular activity is “worth it” compared to other actions or no action. Box 4.6 looks at the security costs of operating in a conflict zone, namely Afghanistan. Box 4.6. Security costs A joint evaluation of humanitarian aid and reconstruction assistance delivered in Afghanistan between 2001 and 2005 states that there was an overall security overhead of approximately 20% – i.e. operating costs were 20% higher in Afghanistan than in more stable settings for similar types of projects. The evaluation pointed out that these unexpected (or underestimated) costs made the Afghanistan programmes considerably more expensive than similar ones elsewhere. Further evaluative analysis and other data would be needed, however, to determine whether or not they may still be considered efficient. Source: Chr. Michelsen Institute (2005). When looking at development or humanitarian interventions in a particular conflict-affected area, evaluators should determine efficiency in comparison to other options for supporting development and peacebuilding in this or similar contexts. Questions on efficiency might include: ●Does the intervention deliver its results in an efficient manner compared to the counterfactual? ●How well are resources (human, financial, organisational) used to achieve results? ●Are there better (more efficient) ways of achieving the objectives? ●What was done to ensure the cost efficiency of the intervention? Did the intervention substitute local initiatives or did it come in addition to local initiatives? 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 71 Other criteria: Coherence and co-ordination Much of the peacebuilding, statebuilding and aid effectiveness literature – and, increasingly, government policies themselves – points to the need for more coherent, better co-ordinated approaches. Funding for a particular conflict prevention project or peacebuilding initiative can be overshadowed and contradicted or – conversely – supported and sustained by other interventions that the same government(s) may undertake. However, the evidence base regarding the use and value of such approaches remains weak. An evaluation’s terms of reference may therefore ask evaluators to look at this broader policy context and ask to what extent the activities are coherent with other policies or actions by other parts of government. Evaluations may also look at the degree of co-ordination among or between donors, government and other actors. Evaluations may assess the costs and benefits of investing in co-ordination and coherence and any unintended consequences arising therefrom. It is important that co-ordination with other actors is not automatically presumed to contribute to better results. Potential risks that have been associated with more co-ordinated donor approaches include: heavy management structures which reduce flexibility and increase overhead costs; strong pressure on national stakeholders as a result of external actors’ adopting a block approach (“ganging up” effects); and inappropriate influences on the neutrality, impartiality, and independence of humanitarian and development actors through association with military actors (shrinking humanitarian space). The evaluators must carefully examine the reality of co-ordination mechanisms in relation to the conflict analysis and other criteria (particularly relevance, effectiveness, and efficiency. Bennett et al. (2010) discovered in Southern Sudan that, in certain cases, single donor projects were more effective than co-ordinated pooling mechanisms. Examples of questions that can help assess coherence and co-ordination might include: ●Were coherence and co-ordination factored into inputs and outputs (in other words, were they budgeted for and are they explicitly listed as outputs)? ●What were the roles of and relationships to other actors? ●What aspects of policy were successfully co-ordinated or made more coherent? – Joint analyses or understandings of conflict and fragility in the context? – Common development of priorities for funding or intervention? – Elimination or reduction in duplication of programming? – Useful divisions of labour across sectors, issues or problems? – Joint evaluations of programming at a country or sectoral level? – Useful sharing of experience and generation of lessons? ●Did co-ordination and coherence result in improved effectiveness, efficiency or impacts? ●How much time and what resources were spent on co-ordination? Was it efficient (cost vs. benefits) and appropriate? Did it reduce transaction costs? ●How was co-ordination achieved? What were the main constraints and challenges? Is it replicable in other situations (what contextual factors influenced co-ordination and results of coordination efforts) 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 72 Draw conclusions and make recommendations The ultimate goal of any evaluation is to answer the key questions asked and present the results in a useful way. The evaluation report should clearly and credibly demonstrate the link between findings, conclusions and recommendations. Analysis of the collected data constitutes the basis for conclusions. Readers should be able to follow a clear line of evidence that supports the conclusions. Any recommendations should be well founded, relevant, targeted to the intended users of the evaluations, and actionable (meaning those who receive the information are able to act upon it) and should help improve the evaluated activity and any future activities. Evidence-based findings may be a useful basis for discussion between diverse actors in the conflict prevention and peacebuilding fields. Over time, such conclusions will contribute to better understanding of how peacebuilding and statebuilding processes work, which can help inform strategies for intervention in conflict. Specifically, evaluations can add knowledge by demonstrating the accuracy (or inaccuracy) and use of conflict analyses, or by showing whether or not theories of change work and why. Box 4.7 gives an example of how an evaluation contributed knowledge with findings that yield insight into theories about the links between aid and people’s perceptions of aid. Box 4.7. Can aid win over local communities? Testing the theory in Afghanistan In an evaluation of the impact of development aid on stability in rural communities in North East Afghanistan, the evaluation team examined two theories held by many development actors working in conflict zones. First, the assumption that stability in conflict or post-conflict contexts depends on whether the population perceives the state and its international partners to be legitimate and useful and so co-operates with them, or not. Second, the assumption that aid can influence local people’s perceptions of the government and international actors. The evaluation team collected data on the perceptions of the rural population, and matched it with data on aid delivery to the communities. It was found that aid has little impact on how the population perceives international actors. It does, however, lead to more positive perceptions of the state. Source: Böhnke et al. (2010). Depending on the type of evaluation, conclusions and recommendations may be developed in a participatory format. For example, evaluators could present initial findings to a group of key stakeholders with whom they would then work to draw useful conclusions. On the other hand, if the focus of the evaluation is on accountability, the evaluators are likely to take a more non-participatory approach. Several recommendations from an evaluation of peacebuilding interventions in Southern Sudan are shown in Box 4.8. Evaluators may discover major differences of opinion not only as to what happened, but as to the value of outcomes and impacts – particularly because individual and group understandings are highly determined by conflict and their own roles in it. Evaluation is, ultimately, a values-based exercise. What is viewed as a successful intervention by some groups may be seen as useless or harmful by others. Box 4.8. Making recommendations for donors and the Government of Southern Sudan The evaluation team in charge of assessing donor support to conflict prevention and peacebuilding in Southern Sudan (soon to be the independent state of South Sudan) from 2005 to 2010 (Bennett et al., 2010) drew on the evaluation findings to formulate specific, targeted recommendations to donors and the Government of Southern Sudan. Concluding that donors did not adjust what they were doing on the basis of a sophisticated and nuanced analysis of power relations, causes of vulnerability, drivers of conflict and resilience indicators (conflict analysis), the evaluation team recommended that donors and the wider aid community: ●Ensure that revised and new programmes are always preceded by a conflict analysis that links wider dynamics to those specific to the area of operation. ●Plan, monitor and evaluate interventions according to the critical factors identified. ●Rate interventions on responsiveness to conflict factors. The team analysed which conflict drivers should be prioritised and how this might be done. They suggested the Government of Southern Sudan (and its supporting donors): ●Allocate major resources towards creating and maintaining livelihood programmes for young men who are currently too easily drawn into criminal activity. ●Enable traditional authorities (chiefs) to address root causes of conflict (including disputes over land or bride wealth) at their customary courts by providing capacity-building programmes for these courts. Develop effective oversight mechanisms to monitor security agencies. Source: Bennett et al. (2010). 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 73 Reporting Reporting takes place throughout the evaluation process (which includes the planning and inception phases). There are often three main reporting steps, though specific reporting requirements may be set by the commissioning agent and should be outlined in the terms of reference. Inception report After conducting a conflict analysis and gathering initial information, the evaluators will draft an inception report describing how the team intends to conduct the evaluation and answer the questions set out in the terms of reference. It presents risks and challenges, the methods to be used, data collection tools, indicators if relevant, operationalisation of the main questions, theories of change (implicit or explicit), sample selection tools, case studies (if not selected prior to commissioning the team), the structure of the report, and a work plan for the remaining work. Stakeholders usually comment on the inception reports, often as part of a reference group. (See Chapter 4 for a more detailed discussion of the inception phase.) Draft report A draft of the evaluation report is often circulated widely for comments and is a chance for stakeholders to comment on the evaluation. Sufficient time for comments should be built into the overall time frame. Some agencies may also require an “out-brief” 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 74 before the team departs from a field visit (between the inception and draft reports) to promote accountability, or other types of reports during field visits. Final report Though an evaluation may result in many different outputs, a written report is almost always completed. Reports and presentations will need to be translated into locally relevant language(s) to facilitate sharing with all stakeholders. The final report is sent to stakeholders. Target groups for dissemination should be agreed on at the beginning of the process. Many organisations now use the Internet as an alternative means of publishing the final report, either in part or in its entirety. In all cases, the confidentiality and safety of those who contributed to the evaluation should be carefully protected. Management response and follow-up action The recommendations and conclusions of the evaluation should be systematically responded to. The people or institutions targeted (generally donor agencies, implementing agencies or national governments) by each recommendation will respond and take relevant action. The terms of reference should include the process for reporting and responding to the evaluation. A formal response and follow-up by those in charge of a programme will often be required. To write the management response, to disseminate findings and lessons, and to engage in a learning process is the responsibility of the commissioning agency. The form of the management response and follow-up action required varies by institution and may be different for different types of evaluations. A response will often include an assessment of the quality and limitations of the evaluation. In the case of a joint evaluation, a joint response or multiple individual responses should be used, depending on institutional requirements. In Box 4.8 the Canadian International Development Agency (CIDA) published a formal response to the findings of an evaluation of interventions in what was then Southern Box 4.9. Canada responds to the evaluation of peacebuilding support in Southern Sudan The Multi-donor Evaluation of Support to Conflict Prevention and Peacebuilding Activities in Southern Sudan (Bennett et al., 2010) made recommendations to the Sudanese Government and its international partners, including Canada. Canada has a “whole of government” approach in South Sudan, so relevant departments discussed the report findings. In addition, the Canadian International Development Agency (CIDA) Sudan Programme responded to the evaluation by publishing a formal management response. The response states that CIDA found the evaluation report to be a useful analytical piece. It also pointed out that the evaluation provided a unique learning opportunity because it was the first programme-level multi-donor evaluation in which CIDA had participated. Sudan. Box 4.9. Canada responds to the evaluation of peacebuilding support in Southern Sudan (cont.) The managers described what they had done or planned to do regarding the specific recommendations. For example, on the recommendation to use better conflict analysis, the managers stated that they “[recognised] the importance of conflict analysis throughout the project lifecycle” and described how they undertook initiatives to promote conflict-sensitive programming, including conflict mapping to guide the design and implementation of new projects relating to children and youth and food security. Specifically, the design of projects together with the United Nations Children’s Fund and the Food and Agriculture Organization with the United Nations Industrial Development Organization involved a two-day workshop on a collective mapping exercise to identify potential geographic locations for planned projects and to determine how to address root causes of conflict. Factors related to early recovery needs and opportunities for synergy between food security and youth development programming were taken into consideration. Source: Canadian International Development Agency (2011). 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 75 Disseminate findings Plans for follow-up and dissemination of lessons learned should be implemented, as agreed during the evaluation planning process described in Chapter 3. Appropriate means of communicating the results to specific target group(s), such as managers, staff and decision makers, should be used. Evaluation managers are increasingly using short summaries, local language translations, policy briefs, video clips and other tailored communication tools to reach different audiences with relevant findings. Sharing the outcomes of an evaluation can be difficult when the results are perceived as negative or when they question strategies or approaches to which practitioners feel strongly committed. Stakeholders may resist questioning the effectiveness of their approach. Receptivity can be enhanced by emphasising the learning aspects of evaluation – and engaging stakeholders early on. Feed back into programming and engage in learning A completed evaluation should feed back into the early stages of planning and programme design and help to address challenges by providing more evidence on the validity (or not) of theories of change and data for comparison and reference. Evaluations carried out while a policy or programme is still going on can be used to adjust or redesign it. In addition to the immediate use of the evaluation findings (by those commissioning the evaluation, for example), opportunities may be identified to feed evidence, lessons or broader conclusions of the evaluation into other policy forums and research activities. For instance, an evaluation of a particular programme working with women peace mediators may yield insights into perceptions of women in different conflict-affected communities which could be valuable to working in the conflict context concerned. Alternatively, it may have lessons for the effective implementation of mediation programmes and contextual factors for success which could be relevant to people working on mediation in quite different contexts. 4. CONDUCTING AN EVALUATION IN SITUATIONS OF CONFLICT AND FRAGILITY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 76 Evaluators must, in particular, be transparent about the external validity of evaluation findings – i.e. how applicable they are in other contexts. The consequences of applying findings in the wrong way or in a context where they are not relevant can be very negative. This is one reason why it is important for evaluators to explicitly describe weaknesses and strengths of data and methods used, and how they impact findings, conclusions and recommendations. Having completed the evaluation and learning process, decision makers, managers and staff should be better able to understand and improve strategies, outcomes and impacts, so making more lasting contributions to peace. Evaluating Peacebuilding Activities in Settings of Conflict and Fragility Improving Learning for Results © OECD 2012 77 ANNEX A Conflict analysis and its use in evaluation Introduction This guidance suggests the use of conflict analysis in planning, managing, and evaluating conflict prevention and peacebuilding programmes and policies. Conflict analysis helps to identify what is needed to address the conflict and to understand the context in which an intervention is to be implemented. As such, many practitioners will already be familiar with the use of this tool in designing projects and programmes. This annex seeks to further explain the role of conflict analysis in the context of evaluation. A variety of conflict analysis approaches or frameworks are available and these are often used in combination with each other. The choice of approach will depend on the purpose of evaluation and the actors involved (some development agencies have institutionalised a particular type of conflict analysis, for example). While different in approach and coverage, most frameworks take the user through similar steps to identify the causes and drivers of conflict and fragility: examine the key stakeholders (actors and groups) who are affected by or influence how a conflict develops; understand the multifaceted context in which conflict and peacebuilding takes place – including state society relations and political economy; and assess the dynamics of a conflict, how it might evolve in the future, and what opportunities there are for preventing escalation (International Alert, 2007a). Conducting or reviewing a conflict analysis for an evaluation Evaluation teams are primarily concerned with conflict analysis from three perspectives. First, in assessing relevance, which includes the use of conflict analysis by managers or policy makers in determination of priorities or programme approach. Second, in order to assess the impacts of policies or programmes, the evaluation team needs to understand the conflict that those programmes and policies are attempting to influence or change. An evaluation team thus needs to understand the different approaches to, and tools for, conflict analysis in order to review the analysis performed at the design stage or conduct its own analysis. Finally, evaluators use the analysis to ensure their process is conflict sensitive. Choosing the appropriate kind of conflict analysis If the intervention being evaluated did not use a conflict analysis in the design phase, or if the analysis is implicit, or if it is not clear how the conflict has evolved since the outset/implementation of the programme, the evaluation team will need to obtain or undertake one itself to serve as the basis of the evaluative assessment. The level of effort ANNEX A EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 78 and resources required for this work should be included in the terms of reference and adapted to the scope of the evaluation questions. There are many different models and frameworks for conflict analysis used by development donors and others engaged in working in and on conflict and fragility. Some models are formal, others informal. Informal analysis generally dominates where political sensitivity is high. “Traditional” models of conflict analysis focus on understanding the context and causes and on understanding the conflict’s stakeholders and actors and their interests. Some widely used conflict-analysis methods do not help identify priorities or factors that are important to the conflict and fragility dynamic (OECD and CDA, 2007). Furthermore, some conflict-analysis tools produce a static snapshot – often in the form of lists of factors – without much sense of how they work together. The frameworks and tools that treat conflicts as complex systems and those that explore future scenarios point to ways of addressing this problem. Evaluation managers and evaluation teams might consider a few questions in deciding what tool or combination of tools to use: Purpose ●Does the tool provide sufficient information on causes, actors, dynamics and the context to assess the relevance of the activity to the needs of the peacebuilding process? ●Does the tool provide information on the appropriate issue areas, at the appropriate level and depth, to help evaluate the effectiveness and impacts of the programme or policy? Assumptions ●Do the evaluators share the underlying assumptions about the conflict that form the basis for analysis? Is the tool’s understanding of or assumption about the nature of conflict appropriate to the specific context in which the programme or policy is being implemented? ●Does this perspective correspond to the mandate and values of the organisation being evaluated? Methodology and resource implications ●Does the tool’s proposed methodology match the purpose of the analysis? ●Does the tool’s proposed methodology agree with the ways of working of the evaluation team? ●Does the evaluation team have the capacity (skills, expertise, access, etc.) to use the tool well? ●How long does it take to produce a reliable conflict analysis? ●What are the resource implications of the selected tool (staff time, travel, seminar costs, facilities, data management)? ●Is the evaluation team able to allocate or secure the required resources? Table A.1 outlines a few conflict-analysis tools developed and used by governments, multilateral agencies, research institutions and NGOs. It is not an exhaustive list, but is intended to give a sense of the variety of tools and range of approaches. Further conflict-analysis resources are listed in the bibliography. There is also a useful overview of conflict analysis compiled online at www.conflictsensitivity.org. Table A.1. Summary of selected conflict analysis tools Purpose Potential users Assumptions Methodology and effort Evaluation application Conflict Assessment Framework – USAID Country and programme strategic planning to identify and prioritise causes of conflict based on understanding of impact. Donor desk officers, implementing partners, mission staff, embassy staff, other government officials. Pulls together best research on causes, level and nature of conflict to identify windows of opportunity. – Combination of desk study, in-country visits, workshops and interviews. – Includes significant staff time: about 2 months. – Relevant to both conflict sensitivity, prevention and peacebuilding. – Quality may vary depending on robustness of methodology used to gather data. Conflict-related Development Analysis – UNDP Conflict, related programme planning and review aimed to understand linkages between development and conflict, increasing positive impact of development efforts. Development agency staff and donors working in situations prone to and affected by conflict. – Conflict caused by combination of security, political, economic and social causes and actor interests. – Development can cause violence. – Data collection and analysis followed by workshop or expert study to analyse current responses and suggest ways forward. – Effort depends on method for data collection. – Development-focused and linked to programming. – Useful at country or sector-level, less at micro level. – Quality of analysis depends on rigor of data collection. Manual for Conflict Analysis – SIDA Country/ programme/ project planning to improve effectiveness of development co-operation and humanitarian assistance in areas affected by violent conflict. Development agency staff, implementing partners. Conflicts driven by structural instability, struggle for power and influence, and mutual fear and insecurity. – Desk study, consultations and workshop to consider programme implications – Local ownership of analysis important - 6-12 weeks, pending scope of desk study. – Focus on different levels of programming. – Relevant both for conflict sensitivity and planning at country and sector levels – No methodology. Aid for Peace – Paffenholz and Reychler (2007) Assess peace and conflict relevance, risks and effects of development and humanitarian projects or programmes. Development and foreign ministry officials. – Examines both conflict and peace factors. – Framework for analysis of peacebuilding deficiencies and needs, conflict risks and effects of intervention on conflict. – Desk study/survey of other interventions; field mission with 3-5 day training and workshop. – Potentially time consuming and costly, pending time for baseline study and mapping and number of field visits and workshops. – Addresses both conflict sensitivity and peace and conflict programming. – Provides specific guidance on integrating peace and conflict lens into evaluation. Making Sense of Turbulent Contexts: Analysis tools for Humanitarian Actors – World Vision Aims to improve ability to analyse dynamics of conflicts to impact programme and project planning and advocacy in emergency situations. NGO emergency response, development and advocacy staff. – Focus on chronic political instability, not just violent conflict. – Sees conflict as cyclical with periods of peace followed by conflict. – Collection of tools to analyse actors, symptoms and political economy of conflict, generate future scenarios, and analyse strategic and operational implications. – Effort pending on scope of data collection and workshop. – Focuses on macro level; how conflict will affect programme in future. – Flexible and adaptable to specific contexts. – Can be used for analysis of clusters of countries. Conflict Prognosis: A Conflict and Policy Assessment Framework – Goor and Verstegen (2000), Clingendael Institute Aims to link early warning to policy planning and implementation. Donor and embassy staff involved with foreign policy and development issues. – Focus on indicators of internal conflict and state failure. – Uses Fund for Peace’s measures for sustainable security as goal. – External research and analysis to track indicators and identify problem areas and responses for workshop discussion. – Effort depends on size of workshops, and consultant involvement. – Not programme specific, but focuses on broad policy or programme development. – Facilitates clarity on developments and trends, not causes. ANNEX A EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 79 ANNEX B EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 80 ANNEX B Understanding and evaluating theories of change What are theories of change? Theory of change is a flexible approach meant to encourage critical thinking in the design, implementation and evaluation of development activities. As described by Vogel (2012) “theory of change thinking” is being increasingly used in international development by a wide range of actors. This guidance encourages questioning strategies and activities that impact on peacebuilding and conflict prevention. It offers theories of change as one way to help evaluators assess and programme managers and decision makers think through the hypotheses of change and assumptions that underpin their work. Vogel describes theory of change as a process of analysis and learning that produces insight to support critical thinking throughout the programme cycle. It is also a flexible approach that may be helpful in encouraging innovation in programme strategies to respond and adapt to change in the context. Aid work in relation to conflict and peace is often based on approaches, strategies and tactics that are rooted in theories of change (understandings about why particular inputs or activities are expected to achieve intended results [outputs, outcomes and impacts]) that are unstated or ill-defined. They are embedded in the skills and approaches of individual practitioners and peacebuilding organisations, their capacities and technologies, attachments to favourite methodologies, and the perspectives of different stakeholders about conflict and peace. In the imaginary example of an anti-bias peace programme for journalists in Annex C, one question would be how the planned workshops, consciousness raising, and skills development might actually change conflict reporting. The programme could track the language used in reporting before and after the effort and also survey public attitudes. At the same time, it could see whether the activities were achieving the expected results – or if unexpected obstacles appeared. For instance, it might turn out that individual journalists have very little influence over the use of inflammatory language and that editors determine the use of such language to boost sales. That outcome would suggest that the “theory”, about inducing changes in reporting by training journalists, was flawed. One related task is to identify the sources of theories. Are they a) based on experience (the programme designers’ personal and professional experience or that of the stakeholders and beneficiaries consulted during programme design); or b) research-based? Evaluation can contribute to improving the design and implementation of ongoing programmes. It can also uncover whether success, or lack thereof, is due to programme design and theory or to programme implementation. ANNEX B EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 81 A useful first step in enhancing strategies in conflict prevention and peacebuilding programming and evaluation is to become more explicit about underlying assumptions of how change comes about – that is, theories of how to achieve peace. Conflict prevention and peacebuilding activities are carried out on the basis of specific ideas and goals as to what they hope to achieve. Programme decisions are based on a number of factors – including assumptions about how to bring about peace and theories about how to bring about change. Peace practitioners select methods, approaches and tactics that are rooted in a range of theories of how peace can be achieved in a specific context. It is important to uncover these theories of change, both in order to test the theories against the realities of the conflict and to provide the basis for evaluating progress towards related objectives. Theories of change in peacebuilding include those presented in Table B.1, though a systematic inquiry into ongoing and past conflict prevention and peacebuilding work would likely reveal many other theories underlying peacebuilding programmes. Some theories focus on who needs to change – that is, which individuals and groups in society or which relationships need to change. Other theories concentrate on what needs to change. It may be an institution, a policy, a social norm. Still other theories are tied directly to a particular methodology or approach: how the change could or should happen. Evaluating conflict prevention and peacebuilding theories of change The impacts, effectiveness, relevance, efficiency and sustainability of a conflict prevention and peacebuilding activity rest to a large extent on the accuracy of its underlying theory of change. A false or incomplete theory may be a key explanatory factor for the failure of a programme, project, or policy. In contrast, good theories (based on an up-to-date, thorough conflict analysis) contribute to effective conflict prevention and peacebuilding action and successful interventions. Analysis of the theory of change is therefore a key aspect of any conflict prevention and peacebuilding evaluation. The pertinent theory should be reviewed in the evaluation report and be covered in the evaluation’s findings, conclusions, and lessons learned. Such analysis will help contribute to a more refined understanding of how to bring about change for peace. When conducting an evaluation, the evaluator or evaluation team should ascertain the theories of change of the peacebuilding intervention in question. While they are often variations on the generic theories presented in Table B.1, they should – for the purpose of evaluation – be reframed in relation to the specific context and using the intervention’s particular terms. At times, the theories in operation are obvious, even if unstated, in programme proposals and other documents. More often, the theories need to be uncovered through interviews with implementing staff and other stakeholders – or can be confirmed by those discussions. The evaluation process may also reveal that different staff members are proceeding on different assumptions (theories) about how their efforts will promote change towards peace. The evaluation process itself can thus be useful for helping to clarify this important dimension of intervention strategy. The two real life examples which follow illustrate these points. Example 1: Evaluating grassroots conflict prevention in Liberia In the wake of the 14-year civil war in Liberia, a large international NGO received donor funding to develop Community Peace Councils (CPCs), a community-based mechanism for ANNEX B EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 82 resolving a range of disputes, with an explicitly inter-ethnic approach. The CPCs were designed to promote greater democratic participation through leadership development. The evaluation team first identified underlying theories of change and programme assumptions (derived mainly from discussions with local and international staff members) for the CPCs: Theory #1: Establishing a new community-level mechanism for handling a range of dispute types will contribute to keeping the peace and avoiding incidents that have the potential for escalating into serious violence. Theory #2: Creating inclusive structures for community problem solving can improve communication, respect, and productive interactions among subgroups in the community, and improve the access of disenfranchised groups to decision making. Theory #3: Creating a new leadership group infused with democratic concepts and provided with critical skills can foster more effective and responsive leadership. The evaluation team then discussed whether and how these theories of change were appropriate for the situation in Liberia, and how they were playing out in the programme. To begin, the team conducted an updated conflict analysis, based on interviews and focus groups with a wide range of people in the communities themselves. It then examined whether the programme was having the effects envisioned in the theory of change. For example, the team examined what kinds of conflicts the CPCs handled, and whether those conflicts had the potential for escalating and inciting widespread violence. If they did, then the CPCs would directly contribute to stopping a key factor in violent conflict. If, however, those conflicts were unconnected to the driving factors of the conflict, or the local-level conflict-handling mechanisms were not able to address the types of conflict most likely to escalate, then the CPCs would make little or no contribution to “peace writ large”. The evaluation team found that the CPCs were, for the most part, not handling the most serious and volatile disputes, which concerned land issues. The team then explored whether this was due to a failure in programme implementation or, alternatively, a theory of change that was incomplete or inaccurate. The main conclusion was that, while the CPCs were well set up and trained well, the CPCs became mostly excluded from handling land issues as communities were repopulated and traditional leadership patterns re-established. At the same time, the hope (and theory) regarding alternative leadership models proved unfounded, as traditional leaders gained control over the CPCs or used them to address issues they preferred that someone else deal with. The evaluation recommended that the agency work to expand the mandate and capability of the CPCs for handling land disputes by connecting them to land commissions and other emerging government structures. It should also be said that the CPCs did represent a useful developmental advance, even if they were unable to fulfil, as completely as hoped, a contribution to “peace writ large”. Example 2: The impact of international peacebuilding policies and programming in Kosovo CDA Collaborative Learning Projects performed an extensive study of the reasons for the recurrence of inter-ethnic violence in Kosovo in the spring of 2004 and the relationship of that violence to policies and programmes undertaken by the international community. Among other things, the study identified the theories of change underlying the various approaches to improving ethnic relations. As is often the case, these underlying theories ANNEX B EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 83 were strongly influenced by the policies and (unspoken) assumptions of the international community. The multiple aid and development programmes were directly linked to implementation of internationally-established "Standards for Kosovo" and widely held beliefs regarding refugee returns, inter-ethnic relations, and a future multi-ethnic state as the basis for peacebuilding. The Kosovo example concerns many agencies and multiple programmes. The study identified major programming approaches, and associated theories of change, some of which are listed here, and then examined the effectiveness of each, and their relationship (if any) to preventing violence. A. Inter-ethnic and inter-religious dialogue In Kosovo, the bulk of what agencies and community members identified as peacebuilding was labelled “dialogue.” Dialogue encompassed a wide range of activities: from social contact to structured conversations about identity and the promotion of mutual understanding, to problem-solving related to concrete issues, and to negotiation and mediation of agreements on land use in the Municipal Working Groups on Return. The most frequent theories of change for dialogue efforts in Kosovo were: Theory #1: Involving Kosovar Serbs and Albanians in mutual discussions can help develop the conditions for the safe, successful and peaceful return of internally displaced persons to their homes. This, in turn, will promote reintegration, stabilisation of the environment and will reverse one of the negative consequences of the conflict. Theory #2: Engaging community members in participatory decision making and the implementation of development activities can strengthen community relationships. Theory #3: Promoting co-operation across ethnic lines regarding non-political issues of common interest (HIV/AIDS, drug use, business and entrepreneurship, women’s rights, infrastructure, etc.) can build stronger inter-ethnic ties and understanding. B. Training and peace education Training in conflict resolution, human rights, nonviolent communication and related topics was done in many communities, and, with dialogue, was one of the most popular approaches to peacebuilding programming. Youth camps, peace camps, archaeological camps, art camps and many others were widespread, as were multi-ethnic programmes of technical training in computers, project management, marketing, and other technical or professional topics. To a lesser extent, school-based peace education programmes were developed and included human rights education and tolerance education for children. Theory #1: Providing people with better skills for conflict resolution will increase the ability of communities to settle disputes non-violently and reduce the likelihood of violence. Theory #2: If people talk and play together they will build relationships and break down stereotypes. C. Multi-ethnic projects and institutions Along with dialogue and training, joint (inter-ethnic) projects and institutions comprised a significant proportion of the peacebuilding programming in the communities that were included in the Kosovo study. Some of the projects were the outcome of or ANNEX B EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 84 follow-up to dialogue, aiming to take the communication and relationship building beyond mere talk. Theory #1: Developing activities that provide economic benefits to both ethnic communities (economic interdependence) will give people incentives to resist efforts to incite violence. Theory #2: Providing opportunities for people to work together on practical issues across ethnic lines will help break down mistrust and negative stereotypes, as well as develop habits of co-operation. Theory #3: If people have jobs and economic stability, they will be less hostile to the other ethnic group. D. Democratic governance and capacity-building Many international donors, agencies and NGOs have implemented peacebuilding activities designed to strengthen municipal government institutions to support integration of minorities, better communication and dialogue, and sustainable returns. They work on the following theory: If we can improve administration and service delivery and establish non-discriminatory policies, this will reduce inter-ethnic tensions and demonstrate the viability of a multi-ethnic Kosovo. Many programmes and policies integrated several approaches and theories of change. For example, a programme to facilitate returns of Kosovo Serb minorities included activities and approaches that reflected a combination of different theories: ●dialogue between the host community and returnees was facilitated on the assumption that dialogue would allay fears and re-establish relationships that would allow returnees to return to their homes in peace (Theory A #1); ●multi-ethnic committees to decide community priorities for development aid (Theory A #2); ●provision of equipment and seeds to a multi-ethnic agricultural co-operative (Theories C #1, 2). Once the theories had been identified, they could be assessed in relation to the driving factors of conflict and the factors contributing to the absence of violence in some places in March 2004. The Kosovo study identified patterns of inter-ethnic violence and factors that contributed to the prevention of inter-ethnic violence – through extensive interviews in communities, some of which experienced violence in March 2004 and some which did not. The team then examined the programming approaches and any relationship to the factors that helped communities avoid violence. The study found that the failure of peacebuilding programming to achieve desired impacts was due partly to faulty theories of change and partly to problems in programme design and implementation. Design problems included failures in the participant selection processes, fragmentation of programming, insufficient follow-up and limited resources for “soft” aspects of programming. As for implementation strategy, returnees were not central actors with respect to violence, although they were important victims of the conflict. The channelling of aid to returnees and to communities that accepted them, it turned out, prompted resentment and led to increases in inter-ethnic divisions rather than improved relations between groups. In part, the theory of change on which the programming was based was faulty. With respect to inter-ethnic dialogue between host communities and returnees, the study found Table B.1. Common theories of change Theory of change Examples of methods Individual change: If we transform the consciousness, attitudes, behaviours and skills of many individuals, we will create a critical mass of people who will advocate peace effectively. Individual change through training, personal transformation or consciousness-raising workshops or processes; dialogues and encounter groups; trauma healing. Healthy relationships and connections: Strong relationships are a necessary ingredient for peacebuilding. If we can break down isolation, polarisation, division, prejudice and stereotypes between/among groups, we will enable progress on key issues. Processes of intergroup dialogue; networking; relationship-building processes; joint efforts and practical programmes on substantive problems. Withdrawal of the resources for war: Wars require vast amounts of material (weapons, supplies, transport, etc.) and human capital. If we can interrupt the supply of people and goods to the war-making system, it will collapse and peace will become possible. Campaigns aimed at cutting off funds and national budgets for war; conscientious objection and/or resistance to military service; international arms control; arms (and other) embargoes and boycotts. Reduction of violence: If we reduce the levels of violence perpetrated by combatants and/or their representatives, we will increase the chances of bringing security and peace. Ceasefires; creation of zones of peace; withdrawal or retreat from direct engagement; introduction of peacekeeping forces and interposition; observation missions; accompaniment efforts; promotion of non-violent methods for achieving political, social and economic ends; reform of security sector institutions (military, police, justice system/ courts, prisons). Social justice: If we address the underlying issues of injustice, oppression/exploitation, threats to identity and security, and peoples’ sense of injury/victimisation, it will reduce the drivers of conflict and open up space for peace. Long-term campaigns for social and structural change; truth and reconciliation processes; changes in social institutions, laws, regulations, and economic systems. Good governance: Peace is secured by establishing stable and reliable social institutions that guarantee democracy, equity, justice, and the fair allocation of resources. New constitutional and governance arrangements and entities; power-sharing structures; development of human rights, rule of law, anti-corruption; establishment of democratic, equitable economic structures; economic development; democratisation; elections and election monitoring; increased participation and access to decision making. Political elites: If we change the political calculus and perception of interests of key political (and other) leaders, they will take the necessary steps to bring peace. Raise the costs and reduce the benefits for political elites of continuing war and increase the incentives for peace; engage active and influential constituencies in favour of peace; withdraw international support/ funding for warring parties. Grassroots mobilisation: “When the people lead, the leaders will follow.” If we mobilise enough opposition to war, political leaders will be forced to bring peace. Mobilise grassroots groups to either oppose war or to advocate positive action; use of the media; non-violent direct action campaigns; education and mobilisation effort; organising advocacy groups; dramatic or public events to raise consciousness. Peace agreements/accords: Some form of political settlement is a prerequisite to peace – we must support a negotiation process among key parties to the violence. Official negotiations among representatives; civil society dialogues to support negotiations; track 1½ or 2 dialogue among influential persons. Economic action: People make personal decisions, and decision makers make policy decisions based on a system of rewards and incentives and punishment and sanctions that are essentially economic in nature. If we can change the economies associated with war making, we can bring peace. Use of government or financial institutions to change supply and demand dynamics; control incentive and reward systems; boycotts and embargoes. Public attitudes: War and violence are partly motivated by prejudice, misperceptions, and intolerance of difference. We can promote peace by using the media (television and radio) to change public attitudes and build greater tolerance in society. TV and radio programmes that promote tolerance; modelling tolerant behaviour; symbolic acts of solidarity/unity; dialogue among groups in conflict, with subsequent publicity. Transitional justice: Societies that have experienced deep trauma and social dislocation need a process for handling grievances, identifying what happened, and holding perpetrators accountable. Addressing these issues will enable people to move on to reconstruct a peaceful and prosperous society. Truth and reconciliation commissions; criminal prosecutions and war crimes tribunals; reparations; community reconciliation processes; traditional rites and ceremonies; institutional reforms. Community reintegration: If we enable displaced people (IDPs/refugees) to return to their homes and live in relative harmony with their neighbours, we will contribute to security and economic recovery. Negotiation and problem solving to enable returns; intergroup dialogue; ex-combatant community engagement; processes for handling land claims; trauma healing. Culture of peace: If we transform cultural and societal norms, values and behaviours to reject violence, support dialogue and negotiation, and address the fundamental causes of the conflict, we can develop the long-term conditions for peace. Peace education; poverty eradication; reduction of social inequalities; promotion of human rights; ensuring gender equality; fostering democratic participation; advancing tolerance; enhancing the free flow of information and knowledge; reducing the production of and traffic in arms. ANNEX B EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 85 ANNEX B EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 86 that while dialogue activities opened space for inter-ethnic interaction that might otherwise not have happened, produced some powerful effects on individuals, and led to some co-operative activities across ethnic lines, they neither strengthened community relationships nor led to collective opposition to violence. The assumption that the changes in attitude resulting from dialogue would lead to changes in political attitudes and actions, or trickle out to influence others in the community or trickle up to influence key decision makers, proved to be wrong. In both Kosovo Albanian and Kosovo Serb communities, implicit intra-community pressures, or “rules of the game”, restricted the boundaries of permissible interaction to generally non-visible business interactions and made maintenance and expansion of inter-ethnic linkages difficult. The examples from Liberia and Kosovo illustrate just some of the common theories of change underlying policies and projects working for peace. Others are listed, along with example methods for each, in Table B.1. An initial list of these theories was derived from reviewing the case studies in Reflecting on Peace Practice CDA Collaborative Learning Projects (2004). The table is not intended as a check list or menu of theories. It is designed to be purely illustrative, helping to clarify the concept of theories of change and provide concrete examples. It is by no means exhaustive and the theories it lists are not mutually exclusive – one or more could underlie a single programme. ANNEX C EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 87 ANNEX C Sample terms of reference for a conflict evaluation To consider how an evaluation’s terms of reference (TOR) should be drawn up, this annex takes as an imaginary example the evaluation of a peace journalism programme (schematically outlined in Figure 4.1). It is provided to give readers an idea of the type of information to include in a conflict prevention and peacebuilding TOR. It is indicative and should not be taken as a form model. A real TOR would give greater detail. Further tips on drafting TORs can be found in Quality Standards for Development Evaluation (OECD, 2010c). Terms of Reference: Evaluation of the Agency's "Peace Journalism" programme in conflict area X (2000-03) Define the purpose and use of the evaluation. Is the purpose learning or accountability? Will the evaluation be used to decide on future funding? To inform future support? To provide input to new strategy? The purpose of this evaluation is to determine to what extent the peace journalism programme was implemented according to agency regulations (accountability), what contribution, if any, the peace journalism training course has made to reducing inter-ethnic tensions in country X and how this contribution was made (learning), and if peace journalism makes a significant contribution to long term stability (testing the theory of change). The evaluation will be published and made available to programme managers and country field staff. Management will use the findings to decide on continued funding of the programme. Findings are also of interest to the government of neighbouring country Y, which is considering a similar approach for a nationwide programme. Describe the evaluation object and scope. What are the specific objectives of the evaluation? Is it to document achievements? Assess some or all of the activity's objectives? Will it look at implementation strategies and processes? Will the evaluation have a participatory focus? Will it look at the programme’s underlying assumptions and theory of change? Which DAC evaluation criteria will be used (impact, relevance, sustainability, efficiency, or effectiveness)? The evaluation will examine the results of the peace journalism programme from 2000-03 and its impact on peace and conflict dynamics. Specifically, it will assess whether or not peace journalism is an effective and efficient contribution to peacebuilding. The five DAC evaluation criteria will be used to assess the programme. Describe the rationale of the evaluation: Why this evaluation at this point of time? Describe the longevity, amount of funding, and risks tied to the intervention. Are there any ANNEX C EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 88 specific events that have triggered the evaluation (unveiling of corruption, results that run counter to intentions in the intervention, new research being released)? The conflict situation is worsening in country X and the public in the donor country is demanding to know how our agency has been involved in recent changes. Also, the agency is considering funding similar journalism programmes in other regions and would like to know if this is an effective strategy to pursue. Describe the scope, timeframe, objectives and nature of the activity to be evaluated. Specify issues to be covered, budget and funds spent, the time period to be evaluated, types of activities, geographical coverage, target groups, as well as other elements of the conflict prevention and peace building intervention addressed, such as contextual issues. The peace journalism programme involved the training of 50 journalists from eight municipal districts and four workshops for interior ministry staff. The training took place over the course of two days and were run by agency staff and local organisation partners. The total funds disbursed were EUR 500 000. The programme was meant to contribute to peace by reducing bias in reporting and making journalists more aware of the sources and dynamics of conflict in relation to their work (theory of change). Each training session involved activities led by the agency’s country staff. Workshops were held. Participants included 57% women, while 30% were from the dominant religious group (70% from minority religious groups), 40% from minority ethnic group A and 60% from B. The programme has not been reviewed. Country and programme staff provided twice yearly self-assessments showing the output and achievements of basic outcome objectives, which included the number of journalists trained. Evaluations of workshops and trainings were completed by participants. While staff felt this was a successful programme overall, recent escalations in violence have raised concerns about impact. Many participants have changed their views of the programme in light of the changing situation on the ground. Provide directions for the approaches to be used. What method will be used in the evaluation? How should the evaluation be conducted, via what process and steps, etc.? Will there be an inception phase? What will the level of stakeholder involvement in the evaluation process be? The evaluators will undertake a thorough conflict analysis and then draft an inception report describing how they will answer the key evaluation questions. The evaluation will include a desk review of the programme’s self-evaluations and participants’ evaluations, as well as spending and country reports from the agency and other donors in the region. The evaluation team will visit country X for a participatory workshop with programme staff and embassy staff, as well as to interview programme participants. Logistical and safety concerns: Address ethical behaviour in conflict environments and provide guidance on safety and logistics. Due to safety concerns, the evaluation team will conduct field visits in Regions 1 and 2, but not in Region 3. For Region 3, evaluators will instead meet with proxies in the capital and collect data from a recent OXFAM community study carried out in Regions 3 and 4. The visit should take place during March. Security escorts in Region 2 will be provided by the embassy of country X. Principles: What standards and principles are to be followed. Refer the team to any relevant policy documents or agency agreements. ANNEX C EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 89 The evaluation should follow our agency's “Principles for Engagement in Conflict Situations” and adhere to the DAC Quality Standards. The team is also expected to adhere to our agency's “Guidelines on Gender Sensitive Development Assistance”. Management arrangements, quality control and reporting. Who will be in charge of each task and oversight? To whom will the evaluation team report? Is there a need to establish a steering mechanism for the evaluation? Who will be responsible for ensuring information sharing among team members? Who will be involved in drawing and assessing conclusions? What reports will be generated? Will they be public or confidential? Will they be published or placed on the internet? Will the reports and conclusions be checked? What quality control systems will be used? The team will report directly to the evaluation department country programme manager Ms. X and will also work with a small reference group including X, Y, and Z who will review and comment on the inception report. The team will complete a field report which will be presented in a participatory workshop to country staff before completing the field mission. A and B will review the final report before it is accepted for publication. Team requirements (including team make-up). Who should do the evaluation and what characteristics do they need to have? What is the desired size and composition of the team? What time commitment is involved in terms of person-hours? What types of individuals are needed for this particular evaluation in this particular context? The team should include experts in ethnic conflict and land-based disputes with specific expertise on this conflict. There should be a team leader (40 days), at least four experts (30 days each) and up to two research assistants (30 days total). At least two members should be fluent in language A and language B and all members should be comfortable working under difficult circumstances and should have good communication skills and non-aggressive attitudes. Budget and schedule. How will the evaluation be funded? Have there been arrangements made for security costs or other additional costs associated with working in a conflict environment? Are funds available for conflict analysis? (Bids may also be accepted and then compared to establish the appropriate funding needed.) When will the evaluation be conducted? What criteria will be applied to reports to disburse funding? 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(1999b), “The Influence of Aid in Situations of Violent Conflict: A Synthesis and Commentary on the Lessons Learned from Case Studies on the Limits and Scope for the Use of Development Assistance”, Informal Task Force on Conflict, Peace, and Development Co-operation, Development Assistance Committee, OECD, Paris. Uvin, P. (2002), “The Development/Peacebuilding Nexus: A Typology and History of Changing Paradigms”, Journal of Peacebuilding & Development, Vol. 1, No. 1. Vogel, I. (2012), Review of the use of “Theory of Change” in international development, draft review report for the UK Department for International Development, London. Weiss, C. (1995), “Nothing as Practical as Good Theory: Exploring Theory-based Evaluation for Comprehensive Community Initiatives for Children and Families”, in J. Connell et al. (eds.), New Approaches to Evaluating Community Initiatives: Concepts, Methods, and Contexts, Aspen Institute, Washington, DC. BIBLIOGRAPHY EVALUATING PEACEBUILDING ACTIVITIES IN SETTINGS OF CONFLICT AND FRAGILITY © OECD 2012 97 Wood, B. (2003), Development Dimensions of Conflict Prevention and Peacebuilding, UN Development Programme (UNDP), New York. World Bank (2005a), Towards a Conflict-Sensitive Poverty Reduction Strategy: Lessons from a Retrospective Analysis, World Bank, Washington, DC. World Bank (2005b), “Conducting Quality Impact Evaluations under Budget, Time and Data Constraints”, Independent Evaluation Group, World Bank, Washington, DC. World Bank (2011), Conflict, Security, and Development, World Development Report 2011, World Bank, Washington, DC. ORGANISATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENT The OECD is a unique forum where governments work together to address the economic, social and environmental challenges of globalisation. The OECD is also at the forefront of efforts to understand and to help governments respond to new developments and concerns, such as corporate governance, the information economy and the challenges of an ageing population. The Organisation provides a setting where governments can compare policy experiences, seek answers to common problems, identify good practice and work to co-ordinate domestic and international policies. The OECD member countries are: Australia, Austria, Belgium, Canada, Chile, the Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Israel, Italy, Japan, Korea, Luxembourg, Mexico, the Netherlands, New Zealand, Norway, Poland, Portugal, the Slovak Republic, Slovenia, Spain, Sweden, Switzerland, Turkey, the United Kingdom and the United States. The European Commission takes part in the work of the OECD. OECD Publishing disseminates widely the results of the Organisation’s statistics gathering and research on economic, social and environmental issues, as well as the conventions, guidelines and standards agreed by its members. DEVELOPMENT ASSISTANCE COMMITTEE ( DAC ) In order to achieve its aims, the OECD has set up a number of specialised committees. One of these is the Development Assistance Committee (DAC), whose mandate is to promote development co-operation and other policies so as to contribute to sustainable development – including pro-poor economic growth, poverty reduction and the improvement of living standards in developing countries – and to a future in which no country will depend on aid. To this end, the DAC has grouped the world's main donors, defining and monitoring global standards in key areas of development. The members of the DAC are Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Japan, Korea, Luxembourg, the Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, the United Kingdom, the United States and the European Union. The OECD Development Assistance Committee develops guidelines and reference documents, published in the DAC Guidelines and Reference Series, to inform and assist members in the conduct of their development co-operation programmes. OECD PUBLISHING, 2, rue André-Pascal, 75775 PARIS CEDEX 16 (43 2012 15 1 P) ISBN 978-92-64-10679-6 – No. 60259 2012 Please cite this publication as: OECD (2012), Evaluating Peacebuilding Activities in Settings of Conflict and Fragility: Improving Learning for Results, DAC Guidelines and Reference Series, OECD Publishing. This work is published on the OECD iLibrary, which gathers all OECD books, periodicals and statistical databases. Visit www.oecd-ilibrary.org, and do not hesitate to contact us for more information. DAC Guidelines and Reference Series Evaluating Peacebuilding Activities in Settings of Conflict and Fragility Improving Learning for Results Contents Chapter 1. Conceptual background and the need for improved approaches in situations of conflict and fragility Chapter 2. Addressing challenges of evaluation in situations of conflict and fragility Chapter 3. Preparing an evaluation in situations of conflict and fragility Chapter 4. Conducting an evaluation in situations of conflict and fragility isbn 978-92-64-10679-6 43 2012 15 1 P -:HSTCQE=VU[\^[:
4646	https://artofproblemsolving.com/wiki/index.php/Lifting_the_Exponent_Lemma?srsltid=AfmBOorKwRuYHup6NjMcT2vYT_YEiOGHV8ftNYG9AjLs_f7wuE3BjuRP	Art of Problem Solving Lifting the Exponent Lemma - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Lifting the Exponent Lemma Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Lifting the Exponent Lemma Lifting the exponent allows one to calculate the highest power of an integer that divides various numbers given certain information. It is extremely powerful and can sometimes "blow up" otherwise challenging problems. Let be a prime such that and . LTE comprises of the following identities (where represents the largest power of that divides ): When is odd: , if . , if and is odd. , if and is even. When : , if and is even. if and is odd. Corollaries: if . , if and is even. , if and is odd. External Links This article is a stub. Help us out by expanding it. Retrieved from " Category: Stubs Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
4647	https://en.wikipedia.org/wiki/Sampling_(statistics)	Jump to content Sampling (statistics) العربية Català Čeština Ελληνικά Español Euskara فارسی Français Gaeilge Galego 한국어 Հայերեն हिन्दी Hrvatski Bahasa Indonesia Italiano עברית Latina Lietuvių Magyar मराठी Монгол Nederlands 日本語 Norsk bokmål Oʻzbekcha / ўзбекча Polski Português Русский Simple English Српски / srpski Sunda Suomi Tagalog தமிழ் Türkçe Українська 吴语粵語中文 Betawi Edit links From Wikipedia, the free encyclopedia Selection of data points in statistics For other uses, see Sampling (disambiguation). In this statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a statistical population to estimate characteristics of the whole population. The subset is meant to reflect the whole population, and statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population (in many cases, collecting the whole population is impossible, like getting sizes of all stars in the universe), and thus, it can provide insights in cases where it is infeasible to measure an entire population. Each observation measures one or more properties (such as weight, location, colour or mass) of independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified sampling. Results from probability theory and statistical theory are employed to guide the practice. In business and medical research, sampling is widely used for gathering information about a population. Acceptance sampling is used to determine if a production lot of material meets the governing specifications. History [edit] Random sampling by using lots is an old idea, mentioned several times in the Bible. In 1786, Pierre Simon Laplace estimated the population of France by using a sample, along with ratio estimator. He also computed probabilistic estimates of the error. These were not expressed as modern confidence intervals but as the sample size that would be needed to achieve a particular upper bound on the sampling error with probability 1000/1001. His estimates used Bayes' theorem with a uniform prior probability and assumed that his sample was random. Alexander Ivanovich Chuprov introduced sample surveys to Imperial Russia in the 1870s. In the US, the 1936 Literary Digest prediction of a Republican win in the presidential election went badly awry, due to severe bias . More than two million people responded to the study with their names obtained through magazine subscription lists and telephone directories. It was not appreciated that these lists were heavily biased towards Republicans and the resulting sample, though very large, was deeply flawed. Elections in Singapore have adopted this practice since the 2015 election, also known as the sample counts, whereas according to the Elections Department (ELD), their country's election commission, sample counts help reduce speculation and misinformation, while helping election officials to check against the election result for that electoral division. While the reported sample counts yield a fairly accurate indicative result with a 4% margin of error at a 95% confidence interval, ELD reminded the public that sample counts are separate from official results, and only the returning officer will declare the official results once vote counting is complete. Population definition [edit] Successful statistical practice is based on focused problem definition. In sampling, this includes defining the "population" from which our sample is drawn. A population can be defined as including all people or items with the characteristics one wishes to understand. Because there is very rarely enough time or money to gather information from everyone or everything in a population, the goal becomes finding a representative sample (or subset) of that population. Sometimes what defines a population is obvious. For example, a manufacturer needs to decide whether a batch of material from production is of high enough quality to be released to the customer or should be scrapped or reworked due to poor quality. In this case, the batch is the population. Although the population of interest often consists of physical objects, sometimes it is necessary to sample over time, space, or some combination of these dimensions. For instance, an investigation of supermarket staffing could examine checkout line length at various times, or a study on endangered penguins might aim to understand their usage of various hunting grounds over time. For the time dimension, the focus may be on periods or discrete occasions. In other cases, the examined 'population' may be even less tangible. For example, Joseph Jagger studied the behaviour of roulette wheels at a casino in Monte Carlo, and used this to identify a biased wheel. In this case, the 'population' Jagger wanted to investigate was the overall behaviour of the wheel (i.e. the probability distribution of its results over infinitely many trials), while his 'sample' was formed from observed results from that wheel. Similar considerations arise when taking repeated measurements of properties of materials such as the electrical conductivity of copper. This situation often arises when seeking knowledge about the cause system of which the observed population is an outcome. In such cases, sampling theory may treat the observed population as a sample from a larger 'superpopulation'. For example, a researcher might study the success rate of a new 'quit smoking' program on a test group of 100 patients, in order to predict the effects of the program if it were made available nationwide. Here the superpopulation is "everybody in the country, given access to this treatment" – a group that does not yet exist since the program is not yet available to all. The population from which the sample is drawn may not be the same as the population from which information is desired. Often there is a large but not complete overlap between these two groups due to frame issues etc. (see below). Sometimes they may be entirely separate – for instance, one might study rats in order to get a better understanding of human health, or one might study records from people born in 2008 in order to make predictions about people born in 2009. Time spent in making the sampled population and population of concern precise is often well spent because it raises many issues, ambiguities, and questions that would otherwise have been overlooked at this stage. Sampling frame [edit] Main article: Sampling frame In the most straightforward case, such as the sampling of a batch of material from production (acceptance sampling by lots), it would be most desirable to identify and measure every single item in the population and to include any one of them in our sample. However, in the more general case this is not usually possible or practical. There is no way to identify all rats in the set of all rats. Where voting is not compulsory, there is no way to identify which people will vote at a forthcoming election (in advance of the election). These imprecise populations are not amenable to sampling in any of the ways below and to which we could apply statistical theory. As a remedy, we seek a sampling frame which has the property that we can identify every single element and include any in our sample. The most straightforward type of frame is a list of elements of the population (preferably the entire population) with appropriate contact information. For example, in an opinion poll, possible sampling frames include an electoral register and a telephone directory. A probability sample is a sample in which every unit in the population has a chance (greater than zero) of being selected in the sample, and this probability can be accurately determined. The combination of these traits makes it possible to produce unbiased estimates of population totals, by weighting sampled units according to their probability of selection. Example: We want to estimate the total income of adults living in a given street. We visit each household in that street, identify all adults living there, and randomly select one adult from each household. (For example, we can allocate each person a random number, generated from a uniform distribution between 0 and 1, and select the person with the highest number in each household). We then interview the selected person and find their income. People living on their own are certain to be selected, so we simply add their income to our estimate of the total. But a person living in a household of two adults has only a one-in-two chance of selection. To reflect this, when we come to such a household, we would count the selected person's income twice towards the total. (The person who is selected from that household can be loosely viewed as also representing the person who isn't selected.) In the above example, not everybody has the same probability of selection; what makes it a probability sample is the fact that each person's probability is known. When every element in the population does have the same probability of selection, this is known as an 'equal probability of selection' (EPS) design. Such designs are also referred to as 'self-weighting' because all sampled units are given the same weight. Probability sampling includes: simple random sampling, systematic sampling, stratified sampling, probability-proportional-to-size sampling, and cluster or multistage sampling. These various ways of probability sampling have two things in common: Every element has a known nonzero probability of being sampled and involves random selection at some point. Nonprobability sampling [edit] Main article: Nonprobability sampling Nonprobability sampling is any sampling method where some elements of the population have no chance of selection (these are sometimes referred to as 'out of coverage'/'undercovered'), or where the probability of selection cannot be accurately determined. It involves the selection of elements based on assumptions regarding the population of interest, which forms the criteria for selection. Hence, because the selection of elements is nonrandom, nonprobability sampling does not allow the estimation of sampling errors. These conditions give rise to exclusion bias, placing limits on how much information a sample can provide about the population. Information about the relationship between sample and population is limited, making it difficult to extrapolate from the sample to the population. Example: We visit every household in a given street, and interview the first person to answer the door. In any household with more than one occupant, this is a nonprobability sample, because some people are more likely to answer the door (e.g. an unemployed person who spends most of their time at home is more likely to answer than an employed housemate who might be at work when the interviewer calls) and it's not practical to calculate these probabilities. Nonprobability sampling methods include convenience sampling, quota sampling, and purposive sampling. In addition, nonresponse effects may turn any probability design into a nonprobability design if the characteristics of nonresponse are not well understood, since nonresponse effectively modifies each element's probability of being sampled. Sampling methods [edit] Within any of the types of frames identified above, a variety of sampling methods can be employed individually or in combination. Factors commonly influencing the choice between these designs include: Nature and quality of the frame Availability of auxiliary information about units on the frame Accuracy requirements, and the need to measure accuracy Whether detailed analysis of the sample is expected Cost/operational concerns Simple random sampling [edit] Main article: Simple random sampling In a simple random sample (SRS) of a given size, all subsets of a sampling frame have an equal probability of being selected. Each element of the frame thus has an equal probability of selection: the frame is not subdivided or partitioned. Furthermore, any given pair of elements has the same chance of selection as any other such pair (and similarly for triples, and so on). This minimizes bias and simplifies analysis of results. In particular, the variance between individual results within the sample is a good indicator of variance in the overall population, which makes it relatively easy to estimate the accuracy of results. Simple random sampling can be vulnerable to sampling error because the randomness of the selection may result in a sample that does not reflect the makeup of the population. For instance, a simple random sample of ten people from a given country will on average produce five men and five women, but any given trial is likely to over represent one sex and underrepresent the other. Systematic and stratified techniques attempt to overcome this problem by "using information about the population" to choose a more "representative" sample. Also, simple random sampling can be cumbersome and tedious when sampling from a large target population. In some cases, investigators are interested in research questions specific to subgroups of the population. For example, researchers might be interested in examining whether cognitive ability as a predictor of job performance is equally applicable across racial groups. Simple random sampling cannot accommodate the needs of researchers in this situation, because it does not provide subsamples of the population, and other sampling strategies, such as stratified sampling, can be used instead. Systematic sampling [edit] Main article: Systematic sampling Systematic sampling (also known as interval sampling) relies on arranging the study population according to some ordering scheme, and then selecting elements at regular intervals through that ordered list. Systematic sampling involves a random start and then proceeds with the selection of every kth element from then onwards. In this case, k=(population size/sample size). It is important that the starting point is not automatically the first in the list, but is instead randomly chosen from within the first to the kth element in the list. A simple example would be to select every 10th name from the telephone directory (an 'every 10th' sample, also referred to as 'sampling with a skip of 10'). As long as the starting point is randomized, systematic sampling is a type of probability sampling. It is easy to implement and the stratification induced can make it efficient, if the variable by which the list is ordered is correlated with the variable of interest. 'Every 10th' sampling is especially useful for efficient sampling from databases. For example, suppose we wish to sample people from a long street that starts in a poor area (house No. 1) and ends in an expensive district (house No. 1000). A simple random selection of addresses from this street could easily end up with too many from the high end and too few from the low end (or vice versa), leading to an unrepresentative sample. Selecting (e.g.) every 10th street number along the street ensures that the sample is spread evenly along the length of the street, representing all of these districts. (If we always start at house #1 and end at #991, the sample is slightly biased towards the low end; by randomly selecting the start between #1 and #10, this bias is eliminated.) However, systematic sampling is especially vulnerable to periodicities in the list. If periodicity is present and the period is a multiple or factor of the interval used, the sample is especially likely to be unrepresentative of the overall population, making the scheme less accurate than simple random sampling. For example, consider a street where the odd-numbered houses are all on the north (expensive) side of the road, and the even-numbered houses are all on the south (cheap) side. Under the sampling scheme given above, it is impossible to get a representative sample; either the houses sampled will all be from the odd-numbered, expensive side, or they will all be from the even-numbered, cheap side, unless the researcher has previous knowledge of this bias and avoids it by a using a skip which ensures jumping between the two sides (any odd-numbered skip). Another drawback of systematic sampling is that even in scenarios where it is more accurate than SRS, its theoretical properties make it difficult to quantify that accuracy. (In the two examples of systematic sampling that are given above, much of the potential sampling error is due to variation between neighbouring houses – but because this method never selects two neighbouring houses, the sample will not give us any information on that variation.) As described above, systematic sampling is an EPS method, because all elements have the same probability of selection (in the example given, one in ten). It is not 'simple random sampling' because different subsets of the same size have different selection probabilities – e.g. the set {4,14,24,...,994} has a one-in-ten probability of selection, but the set {4,13,24,34,...} has zero probability of selection. Systematic sampling can also be adapted to a non-EPS approach; for an example, see discussion of PPS samples below. Stratified sampling [edit] Main article: Stratified sampling When the population embraces a number of distinct categories, the frame can be organized by these categories into separate "strata." Each stratum is then sampled as an independent sub-population, out of which individual elements can be randomly selected. The ratio of the size of this random selection (or sample) to the size of the population is called a sampling fraction. There are several potential benefits to stratified sampling. First, dividing the population into distinct, independent strata can enable researchers to draw inferences about specific subgroups that may be lost in a more generalized random sample. Second, utilizing a stratified sampling method can lead to more efficient statistical estimates (provided that strata are selected based upon relevance to the criterion in question, instead of availability of the samples). Even if a stratified sampling approach does not lead to increased statistical efficiency, such a tactic will not result in less efficiency than would simple random sampling, provided that each stratum is proportional to the group's size in the population. Third, it is sometimes the case that data are more readily available for individual, pre-existing strata within a population than for the overall population; in such cases, using a stratified sampling approach may be more convenient than aggregating data across groups (though this may potentially be at odds with the previously noted importance of utilizing criterion-relevant strata). Finally, since each stratum is treated as an independent population, different sampling approaches can be applied to different strata, potentially enabling researchers to use the approach best suited (or most cost-effective) for each identified subgroup within the population. There are, however, some potential drawbacks to using stratified sampling. First, identifying strata and implementing such an approach can increase the cost and complexity of sample selection, as well as leading to increased complexity of population estimates. Second, when examining multiple criteria, stratifying variables may be related to some, but not to others, further complicating the design, and potentially reducing the utility of the strata. Finally, in some cases (such as designs with a large number of strata, or those with a specified minimum sample size per group), stratified sampling can potentially require a larger sample than would other methods (although in most cases, the required sample size would be no larger than would be required for simple random sampling). A stratified sampling approach is most effective when three conditions are met Variability within strata are minimized Variability between strata are maximized The variables upon which the population is stratified are strongly correlated with the desired dependent variable. Advantages over other sampling methods Focuses on important subpopulations and ignores irrelevant ones. Allows use of different sampling techniques for different subpopulations. Improves the accuracy/efficiency of estimation. Permits greater balancing of statistical power of tests of differences between strata by sampling equal numbers from strata varying widely in size. Disadvantages Requires selection of relevant stratification variables which can be difficult. Is not useful when there are no homogeneous subgroups. Can be expensive to implement. Poststratification Stratification is sometimes introduced after the sampling phase in a process called "poststratification". This approach is typically implemented due to a lack of prior knowledge of an appropriate stratifying variable or when the experimenter lacks the necessary information to create a stratifying variable during the sampling phase. Although the method is susceptible to the pitfalls of post hoc approaches, it can provide several benefits in the right situation. Implementation usually follows a simple random sample. In addition to allowing for stratification on an ancillary variable, poststratification can be used to implement weighting, which can improve the precision of a sample's estimates. Oversampling Choice-based sampling or oversampling is one of the stratified sampling strategies. In choice-based sampling, the data are stratified on the target and a sample is taken from each stratum so that rarer target classes will be more represented in the sample. The model is then built on this biased sample. The effects of the input variables on the target are often estimated with more precision with the choice-based sample even when a smaller overall sample size is taken, compared to a random sample. The results usually must be adjusted to correct for the oversampling. Probability-proportional-to-size sampling [edit] Main article: Probability-proportional-to-size sampling In some cases the sample designer has access to an "auxiliary variable" or "size measure", believed to be correlated to the variable of interest, for each element in the population. These data can be used to improve accuracy in sample design. One option is to use the auxiliary variable as a basis for stratification, as discussed above. Another option is probability proportional to size ('PPS') sampling, in which the selection probability for each element is set to be proportional to its size measure, up to a maximum of 1. In a simple PPS design, these selection probabilities can then be used as the basis for Poisson sampling. However, this has the drawback of variable sample size, and different portions of the population may still be over- or under-represented due to chance variation in selections. Systematic sampling theory can be used to create a probability proportionate to size sample. This is done by treating each count within the size variable as a single sampling unit. Samples are then identified by selecting at even intervals among these counts within the size variable. This method is sometimes called PPS-sequential or monetary unit sampling in the case of audits or forensic sampling. Example: Suppose we have six schools with populations of 150, 180, 200, 220, 260, and 490 students respectively (total 1500 students), and we want to use student population as the basis for a PPS sample of size three. To do this, we could allocate the first school numbers 1 to 150, the second school 151 to 330 (= 150 + 180), the third school 331 to 530, and so on to the last school (1011 to 1500). We then generate a random start between 1 and 500 (equal to 1500/3) and count through the school populations by multiples of 500. If our random start was 137, we would select the schools which have been allocated numbers 137, 637, and 1137, i.e. the first, fourth, and sixth schools. The PPS approach can improve accuracy for a given sample size by concentrating sample on large elements that have the greatest impact on population estimates. PPS sampling is commonly used for surveys of businesses, where element size varies greatly and auxiliary information is often available – for instance, a survey attempting to measure the number of guest-nights spent in hotels might use each hotel's number of rooms as an auxiliary variable. In some cases, an older measurement of the variable of interest can be used as an auxiliary variable when attempting to produce more current estimates. Cluster sampling [edit] Main article: Cluster sampling Sometimes it is more cost-effective to select respondents in groups ('clusters'). Sampling is often clustered by geography, or by time periods. (Nearly all samples are in some sense 'clustered' in time – although this is rarely taken into account in the analysis.) For instance, if surveying households within a city, we might choose to select 100 city blocks and then interview every household within the selected blocks. Clustering can reduce travel and administrative costs. In the example above, an interviewer can make a single trip to visit several households in one block, rather than having to drive to a different block for each household. It also means that one does not need a sampling frame listing all elements in the target population. Instead, clusters can be chosen from a cluster-level frame, with an element-level frame created only for the selected clusters. In the example above, the sample only requires a block-level city map for initial selections, and then a household-level map of the 100 selected blocks, rather than a household-level map of the whole city. Cluster sampling (also known as clustered sampling) generally increases the variability of sample estimates above that of simple random sampling, depending on how the clusters differ between one another as compared to the within-cluster variation. For this reason, cluster sampling requires a larger sample than SRS to achieve the same level of accuracy – but cost savings from clustering might still make this a cheaper option. Cluster sampling is commonly implemented as multistage sampling. This is a complex form of cluster sampling in which two or more levels of units are embedded one in the other. The first stage consists of constructing the clusters that will be used to sample from. In the second stage, a sample of primary units is randomly selected from each cluster (rather than using all units contained in all selected clusters). In following stages, in each of those selected clusters, additional samples of units are selected, and so on. All ultimate units (individuals, for instance) selected at the last step of this procedure are then surveyed. This technique, thus, is essentially the process of taking random subsamples of preceding random samples. Multistage sampling can substantially reduce sampling costs, where the complete population list would need to be constructed (before other sampling methods could be applied). By eliminating the work involved in describing clusters that are not selected, multistage sampling can reduce the large costs associated with traditional cluster sampling. However, each sample may not be a full representative of the whole population. Quota sampling [edit] Main article: Quota sampling In quota sampling, the population is first segmented into mutually exclusive sub-groups, just as in stratified sampling. Then judgement is used to select the subjects or units from each segment based on a specified proportion. For example, an interviewer may be told to sample 200 females and 300 males between the age of 45 and 60. It is this second step which makes the technique one of non-probability sampling. In quota sampling the selection of the sample is non-random. For example, interviewers might be tempted to interview those who look most helpful. The problem is that these samples may be biased because not everyone gets a chance of selection. This random element is its greatest weakness and quota versus probability has been a matter of controversy for several years. Minimax sampling [edit] In imbalanced datasets, where the sampling ratio does not follow the population statistics, one can resample the dataset in a conservative manner called minimax sampling. The minimax sampling has its origin in Anderson minimax ratio whose value is proved to be 0.5: in a binary classification, the class-sample sizes should be chosen equally. This ratio can be proved to be minimax ratio only under the assumption of LDA classifier with Gaussian distributions. The notion of minimax sampling is recently developed for a general class of classification rules, called class-wise smart classifiers. In this case, the sampling ratio of classes is selected so that the worst case classifier error over all the possible population statistics for class prior probabilities, would be the best. Accidental sampling [edit] Main article: Accidental sampling Accidental sampling (sometimes known as grab, convenience or opportunity sampling) is a type of nonprobability sampling which involves the sample being drawn from that part of the population which is close to hand. That is, a population is selected because it is readily available and convenient. It may be through meeting the person or including a person in the sample when one meets them or chosen by finding them through technological means such as the internet or through phone. The researcher using such a sample cannot scientifically make generalizations about the total population from this sample because it would not be representative enough. For example, if the interviewer were to conduct such a survey at a shopping center early in the morning on a given day, the people that they could interview would be limited to those given there at that given time, which would not represent the views of other members of society in such an area, if the survey were to be conducted at different times of day and several times per week. This type of sampling is most useful for pilot testing. Several important considerations for researchers using convenience samples include: Are there controls within the research design or experiment which can serve to lessen the impact of a non-random convenience sample, thereby ensuring the results will be more representative of the population? Is there good reason to believe that a particular convenience sample would or should respond or behave differently than a random sample from the same population? Is the question being asked by the research one that can adequately be answered using a convenience sample? In social science research, snowball sampling is a similar technique, where existing study subjects are used to recruit more subjects into the sample. Some variants of snowball sampling, such as respondent driven sampling, allow calculation of selection probabilities and are probability sampling methods under certain conditions. Voluntary sampling [edit] Further information: Self-selection bias The voluntary sampling method is a type of non-probability sampling. Volunteers choose to complete a survey. Volunteers may be invited through advertisements in social media. The target population for advertisements can be selected by characteristics like location, age, sex, income, occupation, education, or interests using tools provided by the social medium. The advertisement may include a message about the research and link to a survey. After following the link and completing the survey, the volunteer submits the data to be included in the sample population. This method can reach a global population but is limited by the campaign budget. Volunteers outside the invited population may also be included in the sample. It is difficult to make generalizations from this sample because it may not represent the total population. Often, volunteers have a strong interest in the main topic of the survey. Line-intercept sampling [edit] Main article: Line-intercept sampling Line-intercept sampling is a method of sampling elements in a region whereby an element is sampled if a chosen line segment, called a "transect", intersects the element. Panel sampling [edit] Panel sampling is the method of first selecting a group of participants through a random sampling method and then asking that group for (potentially the same) information several times over a period of time. Therefore, each participant is interviewed at two or more time points; each period of data collection is called a "wave". The method was developed by sociologist Paul Lazarsfeld in 1938 as a means of studying political campaigns. This longitudinal sampling-method allows estimates of changes in the population, for example with regard to chronic illness to job stress to weekly food expenditures. Panel sampling can also be used to inform researchers about within-person health changes due to age or to help explain changes in continuous dependent variables such as spousal interaction. There have been several proposed methods of analyzing panel data, including MANOVA, growth curves, and structural equation modeling with lagged effects. Snowball sampling [edit] Main article: Snowball sampling Snowball sampling involves finding a small group of initial respondents and using them to recruit more respondents. It is particularly useful in cases where the population is hidden or difficult to enumerate. Theoretical sampling [edit] \| \| \| --- \| \| \| This section needs expansion. You can help by adding to it. (July 2015) \| Main article: Theoretical sampling Theoretical sampling occurs when samples are selected on the basis of the results of the data collected so far with a goal of developing a deeper understanding of the area or develop theories. An initial, general sample is first collected with the goal of investigating general trends, where further sampling may consist of extreme or very specific cases might be selected in order to maximize the likelihood a phenomenon will actually be observable. Active sampling [edit] In active sampling, the samples which are used for training a machine learning algorithm are actively selected, also compare active learning (machine learning). Judgmental selection [edit] Main article: Judgment sample Judgement sampling, also known as expert or purposive sampling, is a type of non-random sampling where samples are selected based on the opinion of an expert, who can select participants based on how valuable the information they provide is. Haphazard sampling [edit] \| \| \| --- \| \| \| This section needs expansion. You can help by adding to it. (July 2024) \| Haphazard sampling refers to the idea of using human judgement to simulate randomness. Despite samples being hand-picked, the goal is to ensure that no conscious bias exists within the choice of samples, but often fails due to selection bias. Haphazard sampling is generally opted for due to its convenience, when the tools or capacity to perform other sampling methods may not exist. The major weakness of such samples is that they often do not represent the characteristics of the entire population, but just a segment of the population. Because of this unbalanced representation, results from haphazard sampling are often biased. Replacement of selected units [edit] See also: urn model Sampling schemes may be without replacement ('WOR' – no element can be selected more than once in the same sample) or with replacement ('WR' – an element may appear multiple times in the one sample). For example, if we catch fish, measure them, and immediately return them to the water before continuing with the sample, this is a WR design, because we might end up catching and measuring the same fish more than once. However, if we do not return the fish to the water or tag and release each fish after catching it, this becomes a WOR design. Sample size determination [edit] Main article: Sample size determination See also: Sample complexity Formulas, tables, and power function charts are well known approaches to determine sample size. Steps for using sample size tables: Postulate the effect size of interest, α, and β. Check sample size table Select the table corresponding to the selected α Locate the row corresponding to the desired power Locate the column corresponding to the estimated effect size. The intersection of the column and row is the minimum sample size required. Sampling and data collection [edit] Good data collection involves: Following the defined sampling process Keeping the data in time order Noting comments and other contextual events Recording non-responses Applications of sampling [edit] Sampling enables the selection of right data points from within the larger data set to estimate the characteristics of the whole population. For example, there are about 600 million tweets produced every day. It is not necessary to look at all of them to determine the topics that are discussed during the day, nor is it necessary to look at all the tweets to determine the sentiment on each of the topics. A theoretical formulation for sampling Twitter data has been developed. In manufacturing different types of sensory data such as acoustics, vibration, pressure, current, voltage, and controller data are available at short time intervals. To predict down-time it may not be necessary to look at all the data but a sample may be sufficient. Errors in sample surveys [edit] Main article: Sampling error Survey results are typically subject to some error. Total errors can be classified into sampling errors and non-sampling errors. The term "error" here includes systematic biases as well as random errors. Sampling errors and biases [edit] Sampling errors and biases are induced by the sample design. They include: Selection bias: When the true selection probabilities differ from those assumed in calculating the results. Random sampling error: Random variation in the results due to the elements in the sample being selected at random. Non-sampling error [edit] Main article: Non-sampling error Non-sampling errors are other errors which can impact final survey estimates, caused by problems in data collection, processing, or sample design. Such errors may include: Over-coverage: inclusion of data from outside of the population Under-coverage: sampling frame does not include elements in the population. Measurement error: e.g. when respondents misunderstand a question, or find it difficult to answer Processing error: mistakes in data coding Non-response or Participation bias: failure to obtain complete data from all selected individuals After sampling, a review is held of the exact process followed in sampling, rather than that intended, in order to study any effects that any divergences might have on subsequent analysis. A particular problem involves non-response. Two major types of non-response exist: unit nonresponse (lack of completion of any part of the survey) item non-response (submission or participation in survey but failing to complete one or more components/questions of the survey) In survey sampling, many of the individuals identified as part of the sample may be unwilling to participate, not have the time to participate (opportunity cost), or survey administrators may not have been able to contact them. In this case, there is a risk of differences between respondents and nonrespondents, leading to biased estimates of population parameters. This is often addressed by improving survey design, offering incentives, and conducting follow-up studies which make a repeated attempt to contact the unresponsive and to characterize their similarities and differences with the rest of the frame. The effects can also be mitigated by weighting the data (when population benchmarks are available) or by imputing data based on answers to other questions. Nonresponse is particularly a problem in internet sampling. Reasons for this problem may include improperly designed surveys, over-surveying (or survey fatigue),[need quotation to verify] and the fact that potential participants may have multiple e-mail addresses, which they do not use anymore or do not check regularly. Survey weights [edit] In many situations, the sample fraction may be varied by stratum and data will have to be weighted to correctly represent the population. Thus for example, a simple random sample of individuals in the United Kingdom might not include some in remote Scottish islands who would be inordinately expensive to sample. A cheaper method would be to use a stratified sample with urban and rural strata. The rural sample could be under-represented in the sample, but weighted up appropriately in the analysis to compensate. More generally, data should usually be weighted if the sample design does not give each individual an equal chance of being selected. For instance, when households have equal selection probabilities but one person is interviewed from within each household, this gives people from large households a smaller chance of being interviewed. This can be accounted for using survey weights. Similarly, households with more than one telephone line have a greater chance of being selected in a random digit dialing sample, and weights can adjust for this. Weights can also serve other purposes, such as helping to correct for non-response. Methods of producing random samples [edit] Random number table Mathematical algorithms for pseudo-random number generators Physical randomization devices such as coins, playing cards or sophisticated devices such as ERNIE See also [edit] Mathematics portal Wikimedia Commons has media related to Sampling (statistics). Data collection Design effect Estimation theory Gy's sampling theory German tank problem Horvitz–Thompson estimator Latin hypercube sampling Official statistics Ratio estimator Replication (statistics) Random-sampling mechanism Resampling (statistics) Pseudo-random number sampling Sample size determination Sampling (case studies) Sampling bias Sampling distribution Sampling error Sortition Survey sampling Notes [edit] The textbook by Groves et alia provides an overview of survey methodology, including recent literature on questionnaire development (informed by cognitive psychology) : Robert Groves, et alia. Survey methodology (2010 2nd ed. ) ISBN 0-471-48348-6. The other books focus on the statistical theory of survey sampling and require some knowledge of basic statistics, as discussed in the following textbooks: David S. Moore and George P. McCabe (February 2005). "Introduction to the practice of statistics" (5th edition). W.H. Freeman & Company. ISBN 0-7167-6282-X. Freedman, David; Pisani, Robert; Purves, Roger (2007). Statistics (4th ed.). New York: Norton. ISBN 978-0-393-92972-0. The elementary book by Scheaffer et alia uses quadratic equations from high-school algebra: Scheaffer, Richard L., William Mendenhal and R. Lyman Ott. Elementary survey sampling, Fifth Edition. Belmont: Duxbury Press, 1996. More mathematical statistics is required for Lohr, for Särndal et alia, and for Cochran: Cochran, William G. (1977). Sampling techniques (Third ed.). Wiley. ISBN 978-0-471-16240-7. Lohr, Sharon L. (1999). Sampling: Design and analysis. Duxbury. ISBN 978-0-534-35361-2. Särndal, Carl-Erik; Swensson, Bengt; Wretman, Jan (1992). Model assisted survey sampling. Springer-Verlag. ISBN 978-0-387-40620-6. The historically important books by Deming and Kish remain valuable for insights for social scientists (particularly about the U.S. census and the Institute for Social Research at the University of Michigan): Deming, W. Edwards (1966). Some Theory of Sampling. Dover Publications. ISBN 978-0-486-64684-8. OCLC 166526. Kish, Leslie (1995) Survey Sampling, Wiley, ISBN 0-471-10949-5 References [edit] ^ Lance, P.; Hattori, A. (2016). Sampling and Evaluation. Web: MEASURE Evaluation. pp. 6–8, 62–64. ^ Salant, Priscilla, I. Dillman, and A. Don. How to conduct your own survey. No. 300.723 S3. 1994. ^ Seneta, E. (1985). "A Sketch of the History of Survey Sampling in Russia". Journal of the Royal Statistical Society. Series A (General). 148 (2): 118–125. doi:10.2307/2981944. JSTOR 2981944. ^ David S. Moore and George P. McCabe. "Introduction to the Practice of Statistics". ^ Freedman, David; Pisani, Robert; Purves, Roger. Statistics. ^ "SAMPLE COUNT - Elections Department Singapore" (PDF). Retrieved 3 September 2023. ^ Ho, Timothy (1 September 2023). "Presidential Election 2023: How Accurate Will The Sample Count Be Tonight?". DollarsAndSense.sg. Retrieved 3 September 2023. ^ Jump up to: a b c d Robert M. Groves; et al. (2009). Survey methodology. John Wiley & Sons. ISBN 978-0470465462. ^ Lohr, Sharon L. Sampling: Design and analysis. ^ Särndal, Carl-Erik; Swensson, Bengt; Wretman, Jan. Model Assisted Survey Sampling. ^ Scheaffer, Richard L.; William Mendenhal; R. Lyman Ott. (2006). Elementary survey sampling. ^ Jump up to: a b c Shahrokh Esfahani, Mohammad; Dougherty, Edward (2014). "Effect of separate sampling on classification accuracy". Bioinformatics. 30 (2): 242–250. doi:10.1093/bioinformatics/btt662. PMID 24257187. ^ Scott, A.J.; Wild, C.J. (1986). "Fitting logistic models under case-control or choice-based sampling". Journal of the Royal Statistical Society, Series B. 48 (2): 170–182. doi:10.1111/j.2517-6161.1986.tb01400.x. JSTOR 2345712. ^ Jump up to: a b Lohr, Sharon L. Sampling: Design and Analysis. Särndal, Carl-Erik; Swensson, Bengt; Wretman, Jan. Model Assisted Survey Sampling. ^ Ariyaratne, Buddhika (30 July 2017). "Voluntary Sampling Method combined with Social Media advertising". heal-info.blogspot.com. Health Informatics. Retrieved 18 December 2018.[unreliable source?] ^ Lazarsfeld, P., & Fiske, M. (1938). The" panel" as a new tool for measuring opinion. The Public Opinion Quarterly, 2(4), 596–612. ^ Jump up to: a b Groves, et alia. Survey Methodology ^ "Examples of sampling methods" (PDF). ^ "Haphazard sampling definition". AccountingTools. 7 January 2024. ^ IRS Statistical Sampling Handbook. USA: Department of the Treasury, Internal Revenue Service. 1988. p. 8. ^ Cohen, 1988 ^ Deepan Palguna; Vikas Joshi; Venkatesan Chakaravarthy; Ravi Kothari; L. V. Subramaniam (2015). Analysis of Sampling Algorithms for Twitter. International Joint Conference on Artificial Intelligence. ^ Berinsky, A. J. (2008). "Survey non-response". In: W. Donsbach & M. W. Traugott (Eds.), The Sage handbook of public opinion research (pp. 309–321). Thousand Oaks, CA: Sage Publications. ^ Jump up to: a b Dillman, D. A., Eltinge, J. L., Groves, R. M., & Little, R. J. A. (2002). "Survey nonresponse in design, data collection, and analysis". In: R. M. Groves, D. A. Dillman, J. L. Eltinge, & R. J. A. Little (Eds.), Survey nonresponse (pp. 3–26). New York: John Wiley & Sons. ^ Dillman, D.A., Smyth, J.D., & Christian, L. M. (2009). Internet, mail, and mixed-mode surveys: The tailored design method. San Francisco: Jossey-Bass. ^ Vehovar, V., Batagelj, Z., Manfreda, K.L., & Zaletel, M. (2002). "Nonresponse in web surveys". In: R. M. Groves, D. A. Dillman, J. L. Eltinge, & R. J. A. Little (Eds.), Survey nonresponse (pp. 229–242). New York: John Wiley & Sons. ^ Porter; Whitcomb; Weitzer (2004). "Multiple surveys of students and survey fatigue". In Porter, Stephen R (ed.). Overcoming survey research problems. New directions for institutional research. San Francisco: Jossey-Bass. pp. 63–74. ISBN 9780787974770. Retrieved 15 July 2019. ^ Cochran, William G. (1977-01-01). Sampling Techniques, 3rd Edition (3rd ed.). New York, NY: John Wiley & Sons. ISBN 978-0-471-16240-7. Further reading [edit] Singh, G N, Jaiswal, A. K., and Pandey A. K. (2021), Improved Imputation Methods for Missing Data in Two-Occasion Successive Sampling, Communications in Statistics: Theory and Methods. DOI:10.1080/03610926.2021.1944211 Chambers, R L, and Skinner, C J (editors) (2003), Analysis of Survey Data, Wiley, ISBN 0-471-89987-9 Deming, W. Edwards (1975) On probability as a basis for action, The American Statistician, 29(4), pp. 146–152. Gy, P (2012) Sampling of Heterogeneous and Dynamic Material Systems: Theories of Heterogeneity, Sampling and Homogenizing, Elsevier Science, ISBN 978-0444556066 Korn, E.L., and Graubard, B.I. (1999) Analysis of Health Surveys, Wiley, ISBN 0-471-13773-1 Lucas, Samuel R. (2012). doi:10.1007/s11135-012-9775-3 "Beyond the Existence Proof: Ontological Conditions, Epistemological Implications, and In-Depth Interview Research."], Quality & Quantity, doi:10.1007/s11135-012-9775-3. Stuart, Alan (1962) Basic Ideas of Scientific Sampling, Hafner Publishing Company, New York [ISBN missing] Smith, T. M. F. (1984). "Present Position and Potential Developments: Some Personal Views: Sample surveys". Journal of the Royal Statistical Society, Series A. 147 (The 150th Anniversary of the Royal Statistical Society, number 2): 208–221. doi:10.2307/2981677. JSTOR 2981677. Smith, T. M. F. (1993). "Populations and Selection: Limitations of Statistics (Presidential address)". Journal of the Royal Statistical Society, Series A. 156 (2): 144–166. doi:10.2307/2982726. JSTOR 2982726. (Portrait of T. M. F. Smith on page 144) Smith, T. M. F. (2001). "Centenary: Sample surveys". Biometrika. 88 (1): 167–243. doi:10.1093/biomet/88.1.167. Smith, T. M. F. (2001). "Biometrika centenary: Sample surveys". In D. M. Titterington and D. R. Cox (ed.). Biometrika: One Hundred Years. Oxford University Press. pp. 165–194. ISBN 978-0-19-850993-6. Whittle, P. (May 1954). "Optimum preventative sampling". Journal of the Operations Research Society of America. 2 (2): 197–203. doi:10.1287/opre.2.2.197. JSTOR 166605. Standards [edit] ISO [edit] ISO 2859 series ISO 3951 series ASTM [edit] ASTM E105 Standard Practice for Probability Sampling Of Materials ASTM E122 Standard Practice for Calculating Sample Size to Estimate, With a Specified Tolerable Error, the Average for Characteristic of a Lot or Process ASTM E141 Standard Practice for Acceptance of Evidence Based on the Results of Probability Sampling ASTM E1402 Standard Terminology Relating to Sampling ASTM E1994 Standard Practice for Use of Process Oriented AOQL and LTPD Sampling Plans ASTM E2234 Standard Practice for Sampling a Stream of Product by Attributes Indexed by AQL ANSI, ASQ [edit] ANSI/ASQ Z1.4 U.S. federal and military standards [edit] MIL-STD-105 MIL-STD-1916 External links [edit] Wikiversity has learning resources about Sampling (statistics) Media related to Sampling (statistics) at Wikimedia Commons \| v t e Statistics \| \| --- \| \| Outline Index \| \| \| \| Descriptive statistics \| \| --- \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- \| \| Continuous data \| \| \| \| --- \| \| Center \| Mean + Arithmetic + Arithmetic-Geometric + Contraharmonic + Cubic + Generalized/power + Geometric + Harmonic + Heronian + Heinz + Lehmer Median Mode \| \| Dispersion \| Average absolute deviation Coefficient of variation Interquartile range Percentile Range Standard deviation Variance \| \| Shape \| Central limit theorem Moments + Kurtosis + L-moments + Skewness \| \| \| Count data \| Index of dispersion \| \| Summary tables \| Contingency table Frequency distribution Grouped data \| \| Dependence \| Partial correlation Pearson product-moment correlation Rank correlation + Kendall's τ + Spearman's ρ Scatter plot \| \| Graphics \| Bar chart Biplot Box plot Control chart Correlogram Fan chart Forest plot Histogram Pie chart Q–Q plot Radar chart Run chart Scatter plot Stem-and-leaf display Violin plot \| \| \| \| \| \| \| Data collection \| \| --- \| \| \| \| \| --- \| \| Study design \| Effect size Missing data Optimal design Population Replication Sample size determination Statistic Statistical power \| \| Survey methodology \| Sampling + Cluster + Stratified Opinion poll Questionnaire Standard error \| \| Controlled experiments \| Blocking Factorial experiment Interaction Random assignment Randomized controlled trial Randomized experiment Scientific control \| \| Adaptive designs \| Adaptive clinical trial Stochastic approximation Up-and-down designs \| \| Observational studies \| Cohort study Cross-sectional study Natural experiment Quasi-experiment \| \| \| \| \| \| \| Statistical inference \| \| --- \| \| \| \| \| --- \| \| Statistical theory \| Population Statistic Probability distribution Sampling distribution + Order statistic Empirical distribution + Density estimation Statistical model + Model specification + Lp space Parameter + location + scale + shape Parametric family + Likelihood (monotone) + Location–scale family + Exponential family Completeness Sufficiency Statistical functional + Bootstrap + U + V Optimal decision + loss function Efficiency Statistical distance + divergence Asymptotics Robustness \| \| Frequentist inference \| \| \| \| --- \| \| Point estimation \| Estimating equations + Maximum likelihood + Method of moments + M-estimator + Minimum distance Unbiased estimators + Mean-unbiased minimum-variance - Rao–Blackwellization - Lehmann–Scheffé theorem + Median unbiased Plug-in \| \| Interval estimation \| Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling + Bootstrap + Jackknife \| \| Testing hypotheses \| 1- & 2-tails Power + Uniformly most powerful test Permutation test + Randomization test Multiple comparisons \| \| Parametric tests \| Likelihood-ratio Score/Lagrange multiplier Wald \| \| \| Specific tests \| \| \| \| --- \| \| Z-test (normal) Student's t-test F-test \| \| \| Goodness of fit \| Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality (Shapiro–Wilk) Likelihood-ratio test Model selection + Cross validation + AIC + BIC \| \| Rank statistics \| Sign + Sample median Signed rank (Wilcoxon) + Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova + 1-way (Kruskal–Wallis) + 2-way (Friedman) + Ordered alternative (Jonckheere–Terpstra) Van der Waerden test \| \| \| Bayesian inference \| Bayesian probability + prior + posterior Credible interval Bayes factor Bayesian estimator + Maximum posterior estimator \| \| \| \| \| \| \| Correlation Regression analysis \| \| --- \| \| \| \| \| --- \| \| Correlation \| Pearson product-moment Partial correlation Confounding variable Coefficient of determination \| \| Regression analysis \| Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS) \| \| Linear regression \| Simple linear regression Ordinary least squares General linear model Bayesian regression \| \| Non-standard predictors \| Nonlinear regression Nonparametric Semiparametric Isotonic Robust Homoscedasticity and Heteroscedasticity \| \| Generalized linear model \| Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions \| \| Partition of variance \| Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom \| \| \| \| \| \| \| Categorical / multivariate / time-series / survival analysis \| \| --- \| \| \| \| \| --- \| \| Categorical \| Cohen's kappa Contingency table Graphical model Log-linear model McNemar's test Cochran–Mantel–Haenszel statistics \| \| Multivariate \| Regression Manova Principal components Canonical correlation Discriminant analysis Cluster analysis Classification Structural equation model + Factor analysis Multivariate distributions + Elliptical distributions - Normal \| \| Time-series \| \| \| \| --- \| \| General \| Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality \| \| Specific tests \| Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey \| \| Time domain \| Autocorrelation (ACF) + partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR) \| \| Frequency domain \| Spectral density estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood \| \| \| Survival \| \| \| \| --- \| \| Survival function \| Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time \| \| Hazard function \| Nelson–Aalen estimator \| \| Test \| Log-rank test \| \| \| \| \| \| \| \| \| \| --- \| \| \| \| \| --- \| \| Biostatistics \| Bioinformatics Clinical trials / studies Epidemiology Medical statistics \| \| Engineering statistics \| Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification \| \| Social statistics \| Actuarial science Census Crime statistics Demography Econometrics Jurimetrics National accounts Official statistics Population statistics Psychometrics \| \| Spatial statistics \| Cartography Environmental statistics Geographic information system Geostatistics Kriging \| \| \| \| \| \| Category Mathematics portal Commons WikiProject \| \| \| v t e Social survey research \| \| --- \| \| Data collection \| Collection methods Questionnaire Interview + Structured + Semi-structured + Unstructured + Couple \| \| Methodology \| Census Sampling frame Statistical sample Sampling for surveys + Random sampling + Simple random sampling + Quota sampling + Stratified sampling + Nonprobability sampling + Sample size determination Research design + Panel study + Cohort study + Cross-sectional study + Cross-sequential study \| \| Survey errors \| Sampling error Standard error Sampling bias Systematic errors Non-sampling error + Specification error + Frame error + Measurement error + Response errors + Non-response bias + Coverage error + Pseudo-opinion + Processing errors \| \| Data analysis \| Categorical data Contingency table Level of measurement Descriptive statistics Exploratory data analysis Multivariate statistics Psychometrics Statistical inference Statistical models + Graphical + Log-linear + Structural \| \| Applications \| Audience measurement Demography Market research Opinion poll Public opinion \| \| Major surveys \| List of comparative social surveys Afrobarometer American National Election Studies Asian Barometer Survey Comparative Study of Electoral Systems Emerson College Polling Eurobarometer European Social Survey Gallup Poll General Social Survey Household, Income and Labour Dynamics in Australia Survey International Social Survey Latinobarómetro List of household surveys in the United States National Health and Nutrition Examination Survey New Zealand Attitudes and Values Study Suffolk University Political Research Center The Phillips Academy Poll Quinnipiac University Polling Institute World Values Survey \| \| Associations \| American Association for Public Opinion Research European Society for Opinion and Marketing Research International Statistical Institute Pew Research Center World Association for Public Opinion Research \| \| Category Projects + Business + Politics + Psychology + Sociology + Statistics \| \| \| Authority control databases \| \| --- \| \| National \| Germany United States Japan Czech Republic + 2 Israel \| \| Other \| NARA Yale LUX \| Retrieved from " Categories: Sampling (statistics) Survey methodology Scientific method Hidden categories: All articles lacking reliable references Articles lacking reliable references from December 2018 Articles with short description Short description matches Wikidata Articles to be expanded from July 2015 All articles to be expanded Articles to be expanded from July 2024 Wikipedia articles needing factual verification from July 2019 Commons category link is on Wikidata Pages with missing ISBNs Commons category link from Wikidata
4648	https://www.youtube.com/watch?v=U--1YD5iP7I	ANGLE BISECTOR (Part 2). Vector Method. 2D & 3D Problems. Cornerstones of Math 1810 subscribers 8 likes Description 962 views Posted: 26 Sep 2023 CHECK PART 1 HERE (The Non-Vector Method): 0:00 Video Starts 0:10 Method 2 (Vector Method). 2D Problem, With a Fair Amount of Theoretical Background on Vectors. 10:53 Method 3 (Vector Method). 3D Problem, Which is Honestly Simpler Than 2D Problem. Continuing the task of finding the angle bisector between two lines, now using VECTORS. For a two-dimensional (2D) problem this looks rather lengthy and tedious compared to the non-vector method in my last video. But this method is actually very powerful in three-dimensional (3D) cases, because in 3D coordinate geometry, using vectors is the default. You have to use vectors to indicate directions in 3D space. So I can confidently say that this method definitely worth the attention. Geometry #EuclideanVectors #AngleBisector #AnalyticGeometry #3DGeometry #vectors #lines #VectorEquations CORNERSTONES OF MATH features quality math problems to strengthen your math fundamentals and problem-solving ability. Problems are generally on high school level (with some deviations), spanning over topics such as algebra, discrete mathematics, calculus, geometry, statistics, trigonometry, etc. I hope that this channel provides some intellectual pleasure and make you appreciate the beauty of math itself. Please consider giving a Like to this video and Subscribing to my channel, it really means a lot for the creator like me, and you will be introduced to many more interesting math videos! 3 comments Transcript: Method 2 (Vector Method). 2D Problem, With a Fair Amount of Theoretical Background on Vectors. now let's solve the same problem with different method the vector method in this method I will denote arbitrary point on the angle bisector as point x Now using small X and small y because that does not create much confusion in this method and I will also call this intersection point x 0. in order to understand this Vector method first we need to think about what do we need in order to uniquely Define a line well you might have learned in your basic coordinate geometry course that in order to uniquely Define a line you have to know at least one point that the line passes and the slope of the line if you know these two you can find the equation of the line now in Vector equivalent finding the equation of the line means finding the expression for this point x and point x can be written in a vector form as the position Vector o x which is this Vector right here where this o is the origin which works as a reference point for our position vector so we need to know the coordinates of at least one point and a slope in terms of vectors knowing the coordinates of at least one point means knowing the position Vector of at least one point on the line in our example we know the coordinates of this intersection point x 0 which lies on the angle bisector this means that we know this position vector vector ox0 next we need to know the slope of the line here another method of expressing the slope is by using the direction Vector which is a vector that is aligned in the direction of the line like this blue Vector here let's call this direction Vector D1 I used subscript 1 here because there exists another angle bisector with different direction Vector which can be called Vector D2 now let's see what kind of equation we can create with these vectors by vector addition this position Vector o x equals position vector ox0 plus this Vector x 0 x here this vector x0x is parallel to the direction Vector D1 this means that Vector x 0 x can be expressed as a scholar multiple or real number multiple of the direction Vector D1 here this T is a varying real number so we have Vector o x equals Vector ox0 plus T times Vector D1 where T is a real number or if you write Vector o x as simply Vector X and Vector ox0 as Vector x 0 we can express everything in terms of single error vectors which is Vector x equals Vector x 0 plus T times Vector D1 this is called the vector equation of a line because now everything is expressed in terms of vectors and here comes the next step this position Vector x0 is given as 5 comma 3 however we don't know this direction Vector D1 yet so how can we find this direction Vector D1 or the other direction Vector D2 well we can find these Direction vectors using these two vectors that is using the direction vectors of two given lines and the angle bisector condition these Direction vectors denoted as vectors A and B can be determined from the slopes of two lines then we obtain Direction Vector a as 3 comma four because from the equation you can easily find out that if X increases by 3 units then y increases by 4 units corresponding to the slope of four over three similarly we can obtain Direction Vector b as 12 comma 5. let us draw these vectors right here with the same starting point because we are about to do some vector addition but we must add up carefully though because in order to obtain a vector aligned with this angle bisector we must add up two vectors in directions of vectors A and B that have the same length therefore let us think about the vectors in these directions with length 1. that is the unit vectors in directions of A and B let us denote these vectors as a hat and B hat then Vector a hat can be obtained by Vector a divided by the magnitude of vector a or Norm of vector a and the same thing goes for Vector B hat the norm of these two vectors are square root of 3 squared plus 4 squared so 5 and square root of 12 squared plus 5 squared which is 13 respectively therefore a hat is one-fifth of vector a which is 3 comma four so we obtain 3 over 5 comma 4 over 5 and Vector B hat is 1 13 of vector 12 comma 5 so we have 12 over 13 comma 5 over 13. therefore if you calculate the sum of these two unit vectors a hat and B hat which is this plus this you will obtain 99 over 65 comma 77 over 65. now this Vector can be expressed as 11 over 65 times 9 comma 7 and since D1 can be any vectors in this Direction let us choose one that's as simple as possible such as 9 comma 7. so we obtained Vector D1 but what about this Vector D2 the direction Vector for the outer angle bisector well this is rather simple you just draw Vector minus B hat the unit Vector having opposite direction to B hat and if you add a vector a hat and Vector minus B hat you will obtain a vector in the direction of vector D2 the actual calculation gives Vector minus 21 over 65 comma 27 over 65 which can be written as 3 over 65 times -7 comma 9. so let's choose -7 comma 9 as Vector D2 here I have used Vector a hat minus B hat but you can also use B hat minus a hat which is the vector in this direction so now with these two direction vectors we can obtain the equation for each angle bisector first the angle bisector with Direction Vector D1 let us recall our vector equation position Vector o x for arbitrary points equals Vector o x 0 plus this Vector which is T times the direction Vector D1 where T is the real number here Vector o x is Vector X comma y so we have this equals Vector ox0 is 5 comma three so this plus T times the direction Vector 9 comma seven so we have X comma y equals nine t plus 5 comma 70 plus 3 where T can have real values this is called the parametric equation of the line because now it is expressed using this real number t as a parameter in order to obtain a single equation for the line we simply eliminate this parameter T since we have x equals nine t plus 5 and Y equals 70 plus 3 from the first equation we have t equals x minus 5 over 9. substituting this into the second equation eliminates the parameter T and we obtain y equals 7 times x minus 5 over 9 plus 3 which is 7x minus 8 over 9 which rearranges to 7x minus 9y minus 8 equals zero also for the angle bisector with Direction Vector D2 the position Vector o x equals o x 0 plus now this Vector which is T times the direction Vector D2 so we have X comma y equals 5 comma 3 plus T times now Vector minus seven comma 9 so X comma y equals now minus seven t plus five comma nine t plus three so x equals minus seven t plus five and Y equals nine t plus three from the first equation we have t equals minus X plus five over seven so using this to eliminate T in the second equation we obtain y equals nine minus X plus five over seven plus three which is minus nine X Plus 66 over seven so we have nine X plus seven y minus 66 equals zero so this is the vector method and although this method looks much more complicated than the first method it is the only practical method when it comes to 3D cases Method 3 (Vector Method). 3D Problem, Which is Honestly Simpler Than 2D Problem. so let us take a look at 3D case right now the most notable feature in 3D coordinate geometry is in order to indicate directions in 3D space vectors are absolutely necessary for example let us take a look at these two equations of lines in 3D space now I don't want to go too deep into 3D coordinate geometry so for now let me just tell you that in this expression the numerator tells us the one point that this line passes through which is 2 comma minus 1 comma 3 and the denominator contains the information about the direction Vector of the line in this case this line has the direction Vector minus 2 comma 3 comma six which is this Vector a for the second equation the numerator tells us that the line passes through 0.2 comma minus 1 comma 3 and the denominator tells us that the direction Vector of this line is 2 comma 2 comma 1 right which is this Vector B so these two lines definitely meet at this point 2 comma minus 1 comma three from here in order to find the direction vectors D1 and D2 of angle bisectors we do the following first we calculate the unit Direction vectors a hat and B hat then we calculate the vector sum a hat plus b hat and a hat minus B hat so let's do that so the unit Vector a hat is Vector a divided by the norm of vector a so Vector minus 2 comma 3 comma 6 divided by square root of minus 2 squared plus 3 squared plus 6 squared and since it is 7 we have minus two over seven three over seven and six over seven and similarly B hat is Vector B divided by the norm of vector B so 2 comma 2 comma 1 divided by square root of 2 squared plus 2 squared plus 1 squared and since this is three we have 2 over 3 comma 2 over 3 comma 1 over 3. therefore there are some a hat plus b hat is 8 over 21 comma 23 over 21 comma 25 over 21 and their difference a hat minus B hat is minus 20 over 21 comma minus 5 over 21 comma 11 over 21. here for direction vectors D1 and D2 let us take the simplest ones that is vectors with least possible integer components and that would be Vector D1 as 8 comma 23 comma 25 and Vector D2 as 20 comma 5 comma minus 11. using this we can totally find the equations of these two angle bisectors for this one here is the line that passes through this intersection Point 2 comma minus 1 comma 3 and the direction Vector is 8 comma 23 comma 25 therefore if you recall what I have just explained about the equation of the line in 3D space the equation should be x minus 2 over 8 equals y plus 1 over 23 equals z minus 3 over 25. and for this one is the line that passes through this point 2 comma minus one comma three and the direction Vector is 20 comma 5 comma minus 11 so the equation is x minus 2 over 20 equals y plus 1 over 5 equals z minus 3 over minus 11. and with that we also had a brief Taste of the fascinating word that is three-dimensional chord in a geometry and that was all for today's video if you want to know how you can find a point that is symmetric with respect to a line or a plane in 2D or 3D please go check out these videos as well and I will see you in another video
4649	https://www.mayocliniclabs.com/test-catalog/Overview/606273	GBAW - Overview: Beta-Glucosidase, Leukocytes Environment is Prod MayoACCESS MayoLINK Register My Dashboard \| Web: \| mayocliniclabs.com \| \| Email: \| mcl@mayo.edu \| \| Telephone: \| 800-533-1710 \| \| International: \| +1 855-379-3115 \| \| Values are valid only on day of printing \| Test Catalog AlgorithmsDownload Test Catalog & Interpretive HandbookNew TestsNY State Informed Consent TestsPerforming LocationsReferred TestsSpecialty TestingTest Updates Ordering & Results Critical Values & ResultsCustom Gene OrderingFormsLIS ResourcesMayoACCESS ResourcesMayoLINK ResourcesOrder Your First TestPatient Information and Signature FormsRequest Additional TestingSend Additional Specimen Specimen Handling Collection & PreparationDangerous Goods TrainingShipping & Courier GuidesShipping RegulationsSupplies Customer Service BillingContact UsFAQIT UpdatesOperational UpdatesQuality & Compliance Education & Insights Educational ResourcesNewsOutreachPodcastsStories My Dashboard Register Contact Log Out Home Test Catalog Overview Share Print Test Catalog A B C D E F G H I J K L M N O P Q R S T U V W X Y Z # Test Id : GBAW Order This Test Beta-Glucosidase, Leukocytes Download Overview Overview Specimen Clinical & Interpretive Performance Fees & Codes Setup & Updates Test Catalog A B C D E F G H I J K L M N O P Q R S T U V W X Y Z # Useful For Suggests clinical disorders or settings where the test may be helpful Diagnosis of Gaucher disease This test is not intended for carrier detection. Genetics Test Information Provides information that may help with selection of the correct genetic test or proper submission of the test request This test provides diagnostic testing for patients with clinical signs and symptoms suspicious for Gaucher disease. Enzyme testing is included in the diagnostic workup for infants following a positive newborn screen result for Gaucher disease. Testing Algorithm Delineates situations when tests are added to the initial order. This includes reflex and additional tests. For additional information see Newborn Screen Follow-up for Gaucher Disease If the patient has abnormal newborn screening results for Gaucher disease, refer to the appropriate American College of Medical Genetics and Genomics Newborn Screening ACT Sheet.(1) Method Name A short description of the method used to perform the test Flow Injection Analysis-Tandem Mass Spectrometry NY State Available Indicates the status of NY State approval and if the test is orderable for NY State clients. Yes Reporting Name Lists a shorter or abbreviated version of the Published Name for a test Beta-Glucosidase, Leukocytes Aliases Lists additional common names for a test, as an aid in searching Acid Beta-Glucosidase Beta-Glucosidase Gaucher's Disease GBA Deficiency GD 1 Glucocerebrosidase Glucocerebrosidase Deficiency Glucosidase, Beta Testing Algorithm Delineates situations when tests are added to the initial order. This includes reflex and additional tests. For additional information see Newborn Screen Follow-up for Gaucher Disease If the patient has abnormal newborn screening results for Gaucher disease, refer to the appropriate American College of Medical Genetics and Genomics Newborn Screening ACT Sheet.(1) Specimen Type Describes the specimen type validated for testing Whole Blood ACD Ordering Guidance This test is preferred for diagnostic testing but does not reliably detect carriers. For carrier detection, order GBAZ / Gaucher Disease, Full Gene Analysis, Varies or CGPH / Custom Gene Panel, Hereditary, Next-Generation Sequencing, Varies (specify GBA Gene List ID IEMCP-M4F13T). Call 800-533-1710 to discuss testing options. Shipping Instructions For optimal isolation of leukocytes, it is recommended the specimen arrive refrigerated within 6 days of collection to be stabilized. Collect specimen Monday through Thursday only and not the day before a holiday. Specimen should be collected and packaged as close to shipping time as possible. Specimen Required Defines the optimal specimen required to perform the test and the preferred volume to complete testing Container/Tube: Preferred: Yellow top (ACD solution B) Acceptable: Yellow top (ACD solution A) or lavender top (EDTA) Specimen Volume: 6 mL Collection Instructions: Send specimen in whole blood original tube. Do not aliquot. Special Instructions Library of PDFs including pertinent information and forms related to the test Informed Consent for Genetic Testing Biochemical Genetics Patient Information Newborn Screen Follow-up for Gaucher Disease Informed Consent for Genetic Testing (Spanish) Forms New York Clients-Informed consent is required. Document on the request form or electronic order that a copy is on file. The following documents are available: -Informed Consent for Genetic Testing (T576) -Informed Consent for Genetic Testing-Spanish (T826) 2.Biochemical Genetics Patient Information (T602) If not ordering electronically, complete, print, and send a Biochemical Genetics Test Request (T798) with the specimen. Specimen Minimum Volume Defines the amount of sample necessary to provide a clinically relevant result as determined by the testing laboratory. The minimum volume is sufficient for one attempt at testing. 2 mL Reject Due To Identifies specimen types and conditions that may cause the specimen to be rejected Gross hemolysis Reject Specimen Stability Information Provides a description of the temperatures required to transport a specimen to the performing laboratory, alternate acceptable temperatures are also included \| Specimen Type \| Temperature \| Time \| Special Container \| --- --- \| \| Whole Blood ACD \| Refrigerated (preferred) \| 6 days \| \| \| \| Ambient \| 6 days \| \| Useful For Suggests clinical disorders or settings where the test may be helpful Diagnosis of Gaucher disease This test is not intended for carrier detection. Genetics Test Information Provides information that may help with selection of the correct genetic test or proper submission of the test request This test provides diagnostic testing for patients with clinical signs and symptoms suspicious for Gaucher disease. Enzyme testing is included in the diagnostic workup for infants following a positive newborn screen result for Gaucher disease. Testing Algorithm Delineates situations when tests are added to the initial order. This includes reflex and additional tests. For additional information see Newborn Screen Follow-up for Gaucher Disease If the patient has abnormal newborn screening results for Gaucher disease, refer to the appropriate American College of Medical Genetics and Genomics Newborn Screening ACT Sheet.(1) Clinical Information Discusses physiology, pathophysiology, and general clinical aspects, as they relate to a laboratory test Gaucher disease (GD) is an autosomal recessive lysosomal storage disorder caused by reduced or absent acid beta-glucosidase (glucocerebrosidase) enzyme activity resulting in accumulation of glucosylceramide (glucocerebroside) and glucopsychosine (glucosylsphingosine) in the lysosomes. This interferes with the normal functioning of cells and leads to clinical abnormalities characteristic of the disease. While clinical features and severity of symptoms are widely variable within Gaucher disease, common features include abnormal blood parameters such as decreased red blood cells (anemia) and/or platelets (thrombocytopenia), bone disease, and hepatosplenomegaly. Three clinical subtypes have been identified based on the presence and progression of central nervous system (CNS) involvement. Type 1 is the most common type, representing 95% of all cases, and is generally characterized by bone disease, hepatosplenomegaly, anemia and thrombocytopenia, coagulation abnormalities, lung disease, and no CNS involvement. Type 2 or acute neuronopathic (GD2), typically has a very severe progression with onset in the first 2 years of life including neurologic disease, hepatosplenomegaly, and lung disease, with death usually between 2 and 4 years due to lung failure. Individuals with type 3 or chronic neuronopathic (GD3) may have onset prior to 2 years of age, but the progression is not as severe, and they may survive into the third and fourth decade. Finally, within the spectrum, there is a perinatal lethal form associated with skin abnormalities and nonimmune hydrops fetalis and a cardiovascular form presenting with calcification of the aortic and mitral valves, mild splenomegaly, and corneal opacities. Treatment is available in the form of enzyme replacement therapy (ERT), substrate reduction therapy, and chaperone therapy for types 1 and 3. Individuals with type 3 may benefit from bone marrow transplantation. Currently, only supportive therapy is available for type 2. Emerging therapies currently listed at Clinicaltrials.gov include gene therapy and in utero ERT. The incidence of type 1 ranges from 1 in 20,000 to 200,000 in the general population, but it is much more frequent among Ashkenazi Jewish population with an incidence between 1 in 400 and 900. Types 2 and 3 both have an incidence of approximately 1 in 100,000 in the general population. A diagnostic workup for Gaucher disease may demonstrate the characteristic finding of "Gaucher cells" on bone marrow examination. Significantly reduced or absent enzyme activity of acid beta-glucosidase along with elevation of the biomarker, glucopsychosine (GPSY / Glucopsychosine, Blood Spot; GPSYP / Glucopsychosine, Plasma; GPSYW / Glucopsychosine, Blood) is diagnostic. Molecular analysis of the GBA gene allows for detection of disease-causing variants in affected patients (GBAZ / Gaucher Disease, Full Gene Analysis, Varies or CGPH / Custom Gene Panel, Hereditary, Next-Generation Sequencing, Varies [specify GBA Gene List ID IEMCP-M4F13T]). Reference Values Describes reference intervals and additional information for interpretation of test results. May include intervals based on age and sex when appropriate. Intervals are Mayo-derived, unless otherwise designated. If an interpretive report is provided, the reference value field will state this. or =3.53 nmol/hour/mg protein An interpretative report will be provided. Note:Results from this assay do not reflect carrier status because of individual variation of beta-glucosidase enzyme levels. Interpretation Provides information to assist in interpretation of the test results Individuals affected with Gaucher disease will have enzyme levels less than 3.53 nmol/h/mg protein. In our experience some carriers will also have less than 3.53 nmol/h/mg protein activity. Cautions Discusses conditions that may cause diagnostic confusion, including improper specimen collection and handling, inappropriate test selection, and interfering substances Enzyme levels may be normal in individuals receiving enzyme replacement therapy. Clinical Reference Recommendations for in-depth reading of a clinical nature Newborn Screening ACT Sheet [Decreased beta-glucocerebrosidase] Gaucher Disease. American College of Medical Genetics and Genomics; 2022. Revised March 2022. Accessed June 10, 2024. Available at www.acmg.net/PDFLibrary/Gaucher.pdf Martins AM, Valadares ER, Porta G, et al: Recommendations on diagnosis, treatment, and monitoring for Gaucher disease. J Pediatr. 2009 Oct;155(4 Suppl):S10-S18 Daykin EC, Ryan E, Sidransky E: Diagnosing neuronopathic Gaucher disease: New considerations and challenges in assigning Gaucher phenotypes. Mol Genet Metab. 2021 Feb;132(2):49-58. doi: 10.1016/j.ymgme.2021.01.002 Pastores GM, Hughes DA: Gaucher disease. In: Adam MP, Ardinger HH, Pagon RA, et al, eds. GeneReviews [Internet]. University of Washington, Seattle; 2000. Updated June 21, 2018. Accessed March 1, 2022. Available at www.ncbi.nlm.nih.gov/books/NBK1269/ Weinreb NJ, Andersson HC, Banikazemi M, et al: Prevalence of type 1 Gaucher disease in the United States. Arch Intern Med. 2008 Feb;168:326-328 Elliott S, Buroker N, Cournoyer JJ, et al: Pilot study of newborn screening for six lysosomal storage diseases using tandem mass spectrometry. Mol Genet Metab. 2016 Aug;118(4):304-309 Method Description Describes how the test is performed and provides a method-specific reference The specimens are incubated with a mix of substrate and internal standard for acid sphingomyelinase, beta-glucocerebrosidase, acid alpha-glucosidase, alpha-galactosidase, galactocerebrosidase and alpha-L-iduronidase. The sample is then purified by liquid-liquid extraction. The extract is evaporated and reconstituted before analysis by tandem mass spectrometry.(Unpublished Mayo method) PDF Report Indicates whether the report includes an additional document with charts, images or other enriched information No Day(s) Performed Outlines the days the test is performed. This field reflects the day that the sample must be in the testing laboratory to begin the testing process and includes any specimen preparation and processing time before the test is performed. Some tests are listed as continuously performed, which means that assays are performed multiple times during the day. Preanalytical processing: Monday through Saturday Testing performed: Monday, Wednesday Report Available The interval of time (receipt of sample at Mayo Clinic Laboratories to results available) taking into account standard setup days and weekends. The first day is the time that it typically takes for a result to be available. The last day is the time it might take, accounting for any necessary repeated testing. 5 to 9 days Specimen Retention Time Outlines the length of time after testing that a specimen is kept in the laboratory before it is discarded WBC homogenate: 1 month Performing Laboratory Location Indicates the location of the laboratory that performs the test Mayo Clinic Laboratories - Rochester Main Campus CLIA Number: 24D0404292 Fees : Several factors determine the fee charged to perform a test. Contact your U.S. or International Regional Manager for information about establishing a fee schedule or to learn more about resources to optimize test selection. Authorized users can sign in to Test Prices for detailed fee information. Clients without access to Test Prices can contact Customer Service 24 hours a day, seven days a week. Prospective clients should contact their account representative. For assistance, contact Customer Service. Test Classification Provides information regarding the medical device classification for laboratory test kits and reagents. Tests may be classified as cleared or approved by the US Food and Drug Administration (FDA) and used per manufacturer instructions, or as products that do not undergo full FDA review and approval, and are then labeled as an Analyte Specific Reagent (ASR) product. This test was developed and its performance characteristics determined by Mayo Clinic in a manner consistent with CLIA requirements. It has not been cleared or approved by the US Food and Drug Administration. CPT Code Information Provides guidance in determining the appropriate Current Procedural Terminology (CPT) code(s) information for each test or profile. The listed CPT codes reflect Mayo Clinic Laboratories interpretation of CPT coding requirements. It is the responsibility of each laboratory to determine correct CPT codes to use for billing. CPT codes are provided by the performing laboratory. 82963 LOINC® Information Provides guidance in determining the Logical Observation Identifiers Names and Codes (LOINC) values for the order and results codes of this test. LOINC values are provided by the performing laboratory. \| Test Id \| Test Order Name \| Order LOINC Value \| --- \| GBAW \| Beta-Glucosidase, Leukocytes \| 32540-7 \| \| Result Id \| Test Result Name \| Result LOINC Value Applies only to results expressed in units of measure originally reported by the performing laboratory. These values do not apply to results that are converted to other units of measure. \| --- \| 606273 \| Beta-Glucosidase, Leukocytes \| 32540-7 \| \| 606274 \| Interpretation \| 59462-2 \| \| 606275 \| Reviewed By \| 18771-6 \| Test Setup Resources Setup Files Test setup information contains test file definition details to support order and result interfacing between Mayo Clinic Laboratories and your Laboratory Information System. 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4650	https://math.stackexchange.com/questions/1763978/can-you-use-both-sides-of-an-equation-to-prove-equality	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Can you use both sides of an equation to prove equality? Ask Question Asked Modified 4 years, 3 months ago Viewed 22k times $\begingroup$ For example: $\color{red}{\text{Show that}}$$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$ In high school my maths teacher told me To prove equality of an equation; you start on one side and manipulate it algebraically until it is equal to the other side. So starting from the LHS: $$\frac{4\cos(2x)}{1+\cos(2x)}=\frac{4(2\cos^2(x)-1)}{2\cos^2(x)}=\frac{2(2\cos^2(x)-1)}{\cos^2(x)}=\frac{4\cos^2(x)-2}{\cos^2(x)}=4-2\sec^2(x)$$ $\large\fbox{}$ At University, my Maths Analysis teacher tells me To prove a statement is true, you must not use what you are trying to prove. So using the same example as before: LHS = $$\frac{4\cos(2x)}{1+\cos(2x)}=\frac{4(2\cos^2(x)-1)}{2\cos^2(x)}=\frac{2(2\cos^2(x)-1)}{\cos^2(x)}=\frac{2\Big(2\cos^2(x)-\left[\sin^2(x)+\cos^2(x)\right]\Big)}{\cos^2(x)}=\frac{2(\cos^2(x)-\sin^2(x))}{\cos^2(x)}=\bbox[yellow]{2-2\tan^2(x)}$$ RHS =$$4-2\sec^2(x)=4-2(1+\tan^2(x))=\bbox[yellow]{2-2\tan^2(x)}$$ So I have shown that the two sides of the equality in $\color{red}{\rm{red}}$ are equal to the same highlighted expression. But is this a sufficient proof? Since I used both sides of the equality (which is effectively; using what I was trying to prove) to show that $$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$ One of the reasons why I am asking this question is because I have a bounty question which is suffering from the exact same issue that this post is about. EDIT: Comments and answers below seem to indicate that you can use both sides to prove equality. So does this mean that my high school maths teacher was wrong? $$\bbox[#AFF]{\text{Suppose we have an identity instead of an equality:}}$$ $$\bbox[#AFF]{\text{Is it possible to manipulate both sides of an identity to prove that the identity holds?}}$$ Thank you. soft-question proof-writing Share edited Jun 11, 2021 at 20:36 PNT 4,30511 gold badge1414 silver badges3939 bronze badges asked Apr 29, 2016 at 10:55 BLAZEBLAZE 1 $\endgroup$ 13 31 $\begingroup$ You did not use both side of the equality. You showed that each is equal to the same thing. That's ok. Using what you want to prove would be using the fact that both sides are equal, but you don't. $\endgroup$ Captain Lama – Captain Lama 2016-04-29 10:59:17 +00:00 Commented Apr 29, 2016 at 10:59 5 $\begingroup$ You can simplify both sides separately to get get to a common point $\endgroup$ Archis Welankar – Archis Welankar 2016-04-29 10:59:19 +00:00 Commented Apr 29, 2016 at 10:59 17 $\begingroup$ Just another comment: proving that $a=b$ is the same as proving that $a-b=0$. So there is no meaningful difference between "manipulating one side" and "manipulating both sides". $\endgroup$ Nefertiti – Nefertiti 2016-04-29 11:10:48 +00:00 Commented Apr 29, 2016 at 11:10 4 $\begingroup$ This has already been answered but I'd add that you are really using the fact that $=$ sign is an equivalence relation and hence transitive. This might be an issue with more complicated proof whereby the relation is not transitive. $\endgroup$ Karl – Karl 2016-04-29 12:14:16 +00:00 Commented Apr 29, 2016 at 12:14 3 $\begingroup$ You did not "use" either side. What does "use" mean here? It means making an assertion, stating a sentence. The sides of the Eq'n are not sentences. If you take either side's formula and show that it is equal to some other formula, without any unsupported or unwarranted assumptions, then you are logical. E.g.: The RHS is always equal to $4-4/(2 \cos^2 x)=4-4/(1+\cos 2 x)=4\cos 2 x/(1+\cos 2 x)$ regardless of what is written on the LHS. $\endgroup$ DanielWainfleet – DanielWainfleet 2016-04-29 14:07:17 +00:00 Commented Apr 29, 2016 at 14:07 \| Show 8 more comments 7 Answers 7 Reset to default 46 $\begingroup$ There's no conflict between your high school teacher's advice To prove equality of an equation; you start on one side and manipulate it algebraically until it is equal to the other side. and your professor's To prove a statement is true, you must not use what you are trying to prove. As in Siddarth Venu's answer, if you prove $a = c$ and $b = c$ ("working from both sides"), then $a = c = b$ by transitivity of equality. This conforms to both your teacher's and professor's advice. Both your high school teacher and university professor are steering you away from "two-column proofs" of the type: \begin{align} -1 &= 1 &&\text{To be shown;} \ (-1)^{2} &= (1)^{2} && \text{Square both sides;} \ 1 &= 1 && \text{True statement. Therefore $-1 = 1$.} \end{align} Here, you assume what you want to prove, deduce a true statement, and assert that the original assumption was true. This is bad logic for at least two glaring reasons: If you assume $-1 = 1$, there's no need to prove $-1 = 1$. Logically, if $P$ denotes the statement "$-1 = 1$" and $Q$ denotes "$1 = 1$", the preceding argument shows "$P$ implies $Q$ and $Q$ is true", which does not eliminate the possibility "$P$ is false". What you can do logically is start ("provisionally", on scratch paper) with the statement $P$ you're trying to prove and perform logically reversible operations on both sides until you reach a true statement $Q$. A proof can then be constructed by starting from $Q$ and working backward until you reach $P$. Often times, the backward argument can be formulated as a sequence of equalities, conforming to your teacher's advice. (Note that in the initial phase of seeking a proof, you aren't bound by anything: You can make inspired guesses, additional assumptions, and the like. Only when you write up a final proof must you be careful to assume no more than is given, and to make logically-valid deductions.) Share answered Apr 29, 2016 at 13:01 Andrew D. HwangAndrew D. Hwang 82.3k66 gold badges111111 silver badges208208 bronze badges $\endgroup$ 5 2 $\begingroup$ Suppose you start with $1=1$, take the square root of both sides except use $-1$ on the left and $1$ on the right, and come up with $-1=1$. It seems like there is more going on here than assuming what you want to prove. $\endgroup$ Frank Hubeny – Frank Hubeny 2016-04-29 13:16:37 +00:00 Commented Apr 29, 2016 at 13:16 11 $\begingroup$ @FrankHubeny To the best of my knowledge, you've just equated the two branches of a multifunction restricted to the real axis - which isn't valid. $\endgroup$ QuantumFool – QuantumFool 2016-04-29 15:43:52 +00:00 Commented Apr 29, 2016 at 15:43 1 $\begingroup$ @QuantumFool That's right. The problem is not with assuming what one has to prove but with invalid steps along the way. $\endgroup$ Frank Hubeny – Frank Hubeny 2016-04-29 18:44:57 +00:00 Commented Apr 29, 2016 at 18:44 2 $\begingroup$ @Frank: You're perfectly correct that assuming the conclusion is not the only fatal error in this argument; as noted, proving the converse is another. The (logically valid!) argument for "$-1 = 1$ implies $1 = 1$" does not consist entirely of reversible steps, so (as you note) does not prove that $-1 = 1$. I mentioned this example to illustrate what the OP has been cautioned against, and because a non-negligible fraction of American university students instinctively attempt to prove algebraic identities by starting with the desired conclusion and manipulating until they obtain a tautology. $\endgroup$ Andrew D. Hwang – Andrew D. Hwang 2016-04-29 20:55:29 +00:00 Commented Apr 29, 2016 at 20:55 1 $\begingroup$ @BLAZE: An "identity" (such as $\cos^{2} x + \sin^{2} x = 1$) is just an equation that holds for all values of one or more variables (possibly with a "small number of exceptions or restrictions"), so "yes", any proof technique that works for numerical equations also works for identities. $\endgroup$ Andrew D. Hwang – Andrew D. Hwang 2016-05-01 00:04:01 +00:00 Commented May 1, 2016 at 0:04 Add a comment \| 12 $\begingroup$ It is enough.. Consider this example: To prove: $a=b$ Proof: $$a=c$$ $$b=c$$ Since $a$ and $b$ are equal to the same thing, $a=b$. That is the exact technique you are using and it sure can be used. Share answered Apr 29, 2016 at 11:06 Siddharth VenuSiddharth Venu 34711 gold badge44 silver badges1717 bronze badges $\endgroup$ 1 4 $\begingroup$ I think we also have to consider the domain of each side and make this explicit up front. For example, let $a=(x-1)/(x-1)$ and $b=1$. Then we algebraically manipulate this and get $a = 1$. If we do not consider the constraint that $x$ is not equal to $1$ in the domain of $a$, we get $a = b = 1$ for all $x$. That is incorrect. $\endgroup$ Frank Hubeny – Frank Hubeny 2016-04-29 15:12:12 +00:00 Commented Apr 29, 2016 at 15:12 Add a comment \| 5 $\begingroup$ To prove a statement is true, you must not use what you are trying to prove. The main problem is that you've misunderstood what the second teacher said (and possibly the meaning of the equals sign). What that second teacher is saying is do not take as prior facts what you're trying to prove. As one textbook puts it (Setek and Gallo, Fundamentals of Mathematics, 10th Edition, Sec. 3.8), "An argument, or proof, consists basically of two parts: the given statements, which are called premises, and the conclusion". Wikipedia says this (sourced from Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic Press, 2001. Page 3.): "In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions." Now consider a simple equation/identity like $6 = 2 \times 3$. The separate sides are expressions; if you just said "6" in English that's a sentence fragment, not an assertion of any fact. It can't be evaluated as either "true" or "false", because it has no assertive content. It cannot be used as a premise because it's not a proposition. What makes something a fully-formed statement in mathematical language is a relation, most commonly equals (but alternatively "is lesser than", "is greater than", etc., effectively the verbs of the language). Translating the equation $6 = 2 \times 3$ to English we get "6 is the same as 2 times 3", which is indeed a full sentence. This can be checked as being true or false; it makes an assertion. It can be used as a premise because it is a proposition of a particular fact. In conclusion, both your teachers are correct, and both of your proofs are correct (although most of us would prefer the more concise one). When one says "don't use what you're trying to prove" they're not talking about the appearance of any particular expression in an algebraic transformation; expressions are neither premises nor things that can be proven; they are sentence fragments. They're talking about an assertion of fact, which in math has to be a statement including a relational symbol (most commonly an equation). The fact that you didn't start by assuming that equality means that in both cases you've complied with your second teacher's warning. Share answered Apr 29, 2016 at 15:45 Daniel R. CollinsDaniel R. Collins 9,34111 gold badge2727 silver badges4747 bronze badges $\endgroup$ Add a comment \| 4 $\begingroup$ short answer: equality is symmetric, implication is not (both are however transitive) longer answer: You are right: if you prove $A = C$ and $B = C$ for some terms/expressions/objects $A,B,C$ then you are allowed to conclude that $A = B$ (because "=" is transitive and symmetric) Your teacher is right: if you prove that something true follows from $A = B$, i.e. $A = B \implies \text{true}$, you are not allowed to conclude the converse i.e. that $A = B$ since "true is true" (because implication is not symmetric) Share edited Jun 11, 2021 at 12:21 PNT 4,30511 gold badge1414 silver badges3939 bronze badges answered Apr 29, 2016 at 15:04 D.F.FD.F.F 67033 silver badges88 bronze badges $\endgroup$ Add a comment \| 3 $\begingroup$ Unfortunately your high-school teacher (and some of the other answers) is wrong. It is false that proving something of the form "$A = B$" is always possible by starting from one side and algebraically manipulating it to get a sequence of equal expressions ending with the other side, not to say necessary. Not necessary Let us first deal with the false misconception that you can only go in one direction. Suppose you have proven the following, where $A,B,C$ are any expressions: $A = C$. $B = C$. Then you can use the second sentence to substitute $C$ for $B$ in the first to obtain: $A = B$. This is logically valid because "$=$" means "is exactly the same as". Not always possible Let us now consider an example where it is simply impossible to manipulate from one side to the other to prove an equality! $\def\zz{\mathbb{Z}}$ $\def\qq{\mathbb{Q}}$ $\def\rr{\mathbb{R}}$ Theorem: Take any $x \in \rr$ such that $x^2 \le 0$. Then $x = 0$. Proof: $x = 0$ or $x > 0$ or $x < 0$. [by trichotomy] If $x > 0$: $x^2 = x \times x > 0$. [by positive multiplication] This contradicts $x^2 \le 0$. If $x < 0$: $0 < -x$. [by subtraction] $x^2 = (-x) \times (-x) > 0$. [by positive multiplication] This contradicts $x^2 \le 0$. Therefore $x = 0$. Here "positive multiplication" denotes "multiplying by a positive real", which preserves the inequality sign. Similarly subtraction preserves an inequality. The above theorem cannot be proven directly by algebraic manipulation from one side "$x$" to the other "$0$", simply because the only given condition is an inequality. Similarly the construction of $\sqrt{2}$ in elementary real analysis, as shown below, does not permit a proof by algebraic manipulation. Theorem: Let $S = { r : r \in \qq_{\ge 0} \land r^2 \le 2 }$. Then $S$ is non-empty and has an upper bound in $\rr$. Let $x = \sup_\rr(S)$. Then $x^2 = 2$. Proof: [Exercise! Or see a good textbook like Spivak's.] General identities To address the new sub-question in blue, it suffices to generalize the above theorem to the cube-root of arbitrary real numbers, namely: $\sup( { r : r \in \qq \land r^3 \le x } )^3 = x$ for any $x \in \rr$. This is an identity but cannot be proven by direct manipulation. You may not be satisfied with this counter-example, but in higher mathematics this kind of identity is in fact the usual kind that we are interested in, rather than identities that can be proven by algebraic manipulation (which are usually considered trivial). Here are some more examples: $\lceil \frac{m}{n} \rceil = \lfloor \frac{m-1}{n}+1 \rfloor$ for any $m \in \zz$ and $n \in \zz_{>0}$. $\sum_{k=0}^{n-1} 2^k = 2^n-1$ for any $n \in \zz_{>0}$. Nevertheless, if you desire an arithmetic identity, the question becomes much more interesting and depends heavily on what you mean by "arithmetic" and what axioms you are allowed to use. Tarski's problem asked whether an arithmetic identity concerning positive integers can be proven using only the basic high-school identities about them, and Wilkie gave an explicit and simple identity that cannot be so proven, basically because any proof needs to use subtraction and hence negative integers. Share edited Jun 12, 2020 at 10:38 CommunityBot 1 answered Apr 30, 2016 at 5:59 user21820user21820 61.1k99 gold badges109109 silver badges282282 bronze badges $\endgroup$ 7 1 $\begingroup$ It seems clear to me that the OP's teacher's advice, "To prove equality of an equation; you start on one side and manipulate it algebraically until it is equal to the other side.", was given in a particular context (avoiding "two-column proofs"), not asserted as the only way of establishing an equality. After re-reading every answer here, I can't find anyone claiming (explicitly or implicitly) that "proving something of the form "$A=B$" is always possible by starting from one side and algebraically manipulating it to get a sequence of equal expressions ending with the other side [...]." $\endgroup$ Andrew D. Hwang – Andrew D. Hwang 2016-04-30 13:31:09 +00:00 Commented Apr 30, 2016 at 13:31 3 $\begingroup$ @AndrewD.Hwang: I know what you're saying, but let me explain why I make that comment. Imagine a student who is not good at mathematics whose teacher tells him/her exactly the words cited in the question. It's easy to see that the student will go away with the wrong impression as I specified. Indeed, standard English only allows it to be interpreted that way, because "To do X, you do Y." only means "In order to do X, you { have to / ought to / should / better } do Y." Furthermore, I have seen so many students with exactly that wrong conception. Any good answer ought to correct that. $\endgroup$ user21820 – user21820 2016-04-30 14:29:02 +00:00 Commented Apr 30, 2016 at 14:29 $\begingroup$ @BLAZE: I didn't want to criticize the other answers too much, but actually Andrew's argument about avoiding two column proofs is in fact not so good because that is how we can justify a formal proof. If you learn natural deduction (especially Fitch-style, which can be extended to quantifiers like at math.stackexchange.com/a/1684204), you will understand what I mean. The issue in the 'proof' Andrew shows is not its two-column nature but because no axiom allows writing the first statement, and in some sense it doesn't address the actual issue of your high-school teacher's teaching. $\endgroup$ user21820 – user21820 2016-05-01 04:17:58 +00:00 Commented May 1, 2016 at 4:17 1 $\begingroup$ @BLAZE: To be precise, it is in my opinion pointless to steer a student away from an incorrect proof form by saying things that aren't correct, and furthermore without even explaining what form is incorrect! Your university teacher at least said the right thing, though it is not well explained enough to dismantle the prior misconceptions of many students that they gain from high-school! $\endgroup$ user21820 – user21820 2016-05-01 04:20:42 +00:00 Commented May 1, 2016 at 4:20 1 $\begingroup$ This seems to be the only answer that (correctly) points out that the high-school teacher's prescription is wrong. $\endgroup$ ryang – ryang 2022-02-10 18:53:34 +00:00 Commented Feb 10, 2022 at 18:53 \| Show 2 more comments 2 $\begingroup$ I agree with @Siddharth Venu. If we have to prove a=b, At times, on proceeding from LHS you may end up in a stage(intermediate step) which you cannot solve any further and you continue to solve RHS to arrive at the same stage i.e, a=c and b=c so you can conclude that a=b these kind of equality problems often comes in trigonometry Many chapters like matrices and our normal algaebra always have a final step where you can directly establish a relation (a=b). Share answered Apr 29, 2016 at 11:19 SudharsanSudharsan 6166 bronze badges $\endgroup$ Add a comment \| 2 $\begingroup$ There are two common errors that these methods are preventing: \begin{align} 1 &< 2 &&\text{True.} \ 0 \cdot 1 &< 0 \cdot 2 &&\text{Um...} \ 0 &< 0 &&\text{False.} \end{align} \begin{align} 1 &= 2 &&\text{False.} \ 0 \cdot 1 &= 0 \cdot 2 &&\text{Um...} \ 0 &= 0 &&\text{True.} \end{align} Much confusion ensues when the two "$0$"s in the middle steps are obscured by being large, complicated expressions. For instance, is it evident that you are multiplying by zero when you multiply by $\sin(2x) - 2\sin(x)\cos(x)$? (This uses the double angle formula for sine to get an expression that is always zero.) The correct form of inference when you multiply or divide by something complicated is "$A = B$" becomes "$A/C = B/C$ or $C = 0$". That is, you got what you expected or you have inadvertently done something crazy. It seems to take a lot of practice to remember that second clause. The example with $A$, $B$, and $C$ can be altered somewhat without changing the need for the second clause. You can also multiply both sides by $C$. The relation need not be an equality. Note however, that the sense of an inequality may change if $C$ is ever negative. Share answered Apr 29, 2016 at 23:37 Eric TowersEric Towers 71.5k33 gold badges5555 silver badges124124 bronze badges $\endgroup$ Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions soft-question proof-writing See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Linked Predicate logic: How do you self-check the logical structure of your own arguments? 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4651	https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/resources/mit18_01sc_pset5prb-1/	18.01 EXERCISES Unit 5. Integration techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A-1 Evaluate a) tan −1 √3 b) sin −1(√3/2) c) If θ = tan −1 5, then evaluate sin θ, cos θ, cot θ, csc θ, and sec θ. d) sin −1 cos( π/ 6) e) tan −1 tan( π/ 3) f) tan −1 tan(2 π/ 3) g) lim tan −1 x. x→−∞ � 2 � 2b � 1 dx dx dx 5A-2 Calculate a) b) c) . x2 + 1 x2 + b2 1b−1 √1 − x2 5A-3 Calculate the derivative with respect to x of the following � � a) sin −1 x − 1 x + 1 b) tanh x c) ln( x + √x2 + 1) d) y such that cos y = x, 0 ≤ x ≤ 1 and 0 ≤ y ≤ π/ 2. e) sin −1(x/a ) f) sin −1(a/x ) g) tan −1(x/ √1 − x2) h) sin −1 √1 − x 5A-4 a) If the tangent line to y = cosh x at x = a goes through the origin, what equation must a satisfy? b) Solve for a using Newton’s method. 5A-5 a) Sketch the graph of y = sinh x, by finding its critical points, points of inflection, symmetries, and limits as x → ∞ and −∞ . COPYRIGHT DAVID JERISON AND MIT 1996, 2003 1 � � � � � � � � � � � � � � E. 18.01 Exercises 5. Integration techniques b) Give a suitable definition for sinh −1 x, and sketch its graph, indicating the domain of definition. (The inverse hyperbolic sine.) c) Find d sinh −1 x. dx dx d) Use your work to evaluate √a2 + x2 5A-6 a) Find the average value of y with respect to arclength on the semicircle x2 + y2 = 1, y > 0, using polar coordinates. b) A weighted average of a function is � b � � b f (x)w(x)dx w(x)dx aa Do part (a) over again expressing arclength as ds = w(x)dx . The change of variables needed to evaluate the numerator and denominator will bring back part (a). c) Find the average height of √1 − x2 on −1 < x < 1 with respect to dx . Notice that this differs from part (b) in both numerator and denominator. 5B. Integration by direct substitution Evaluate the following integrals � ln xdx 5B-1. x x2 − 1dx 5B-2. e8xdx 5B-3. x 5B-4. cos xdx 5B-5. sin 2 x cos xdx 5B-6. sin 7xdx 2 + 3 sin x 5B-7. 6xdx 5B-8. tan 4xdx 5B-9. ex(1 + ex)−1/3dx √x2 + 4 5B-10. sec 9xdx 5B-11. sec 2 9xdx 5B-12. xe −x2 dx x2dx 5B-13. . Hint: Try u = x3 . 1 + x6 Evaluate the following integrals by substitution and changing the limits of integration. 2� � � � � � � � � � � � � � � � � � � � Integration techniques E. 18.01 Exercises � π/ 3 � e (ln x)3/2dx � 1 tan −1 xdx 5B-14. sin 3 x cos xdx 5B-15. 5B-16. 01 x −1 1 + x2 5C. Trigonometric integrals Evaluate the following 5C-1. sin 2 xdx 5C-2. sin 3(x/ 2) dx 5C-3. sin 4 xdx 5C-4. cos 3(3 x)dx 5C-5. sin 3 x cos 2 xdx 5C-6. sec 4 xdx 5C-7. sin 2(4 x) cos 2(4 x)dx 5C-8. tan 2(ax ) cos( ax )dx 5C-9. sin 3 x sec 2 xdx 5C-10. (tan x + cot x)2dx 5C-11. sin x cos(2 x)dx (Use double angle formula.) � π 5C-12. sin x cos(2 x)dx (See 27.) 0 5C-13. Find the length of the curve y = ln sin x for π/ 4 ≤ x ≤ π/ 2. 5C-14. Find the volume of one hump of y = sin ax revolved around the x-axis. 5D. Integration by inverse substitution Evaluate the following integrals dx x3dx (x + 1) dx 5D-1. 5D-2. 5D-3. (a2 − x2)3/2 √a2 − x2 4 + x2 � √a2 − x2dx � 5D-4. a2 + x2dx 5D-5. 5D-6. x2 a2 + x2dx x2 (For 5D-4,6 use x = a sinh y, and cosh 2 y = (cosh(2 y) + 1) /2, sinh 2 y = 2 sinh y cosh y.) √x2 − a2dx 5D-7. 5D-8. x x2 − 9dx x2 3� � � � � � � � � � � � E. 18.01 Exercises 5. Integration techniques 5D-9. Find the arclength of y = ln x for 1 ≤ x ≤ b. Completing the square Calculate the following integrals � dx � � � � 5D-10. (x2 + 4 x + 13) 3/2 5D-11. x −8 + 6 x − x2dx 5D-12. −8 + 6 x − x2dx dx xdx √4x2 − 4x + 17 dx 5D-13. 5D-14. 5D-15. √2x − x2 √x2 + 4 x + 13 2x − 1 5E. Integration by partial fractions dx xdx xdx 5E-1. dx 5E-2. dx 5E-3. dx (x − 2)( x + 3) (x − 2)( x + 3) (x2 − 4)( x + 3) 3x2 + 4 x − 11 3x + 2 2x − 9 5E-4. dx 5E-5. dx 5E-6. dx (x2 − 1)( x − 2) x(x + 1) 2 (x2 + 9)( x + 2) 5E-7 The equality (1) of Notes F is valid for x = 1 �, −2. Therefore, the equality (4) is also valid only when x = 1 � , −2, since it arises from (1) by multiplication. Why then is it legitimate to substitute x = 1 into (4)? 5E-8 Express the following as a sum of a polynomial and a proper rational function 232 x x x a) b) c) x2 − 1 x2 − 1 3x − 1 x + 2 x8 d) e) (just give the form of the solution) 3x − 1 (x + 2) 2(x − 2) 2 5E-9 Integrate the functions in Problem 5E-8 . 5E-10 Evaluate the following integrals dx (x + 1) dx (x2 + x + 1) dx a) b) c) x3 − x (x − 2)( x − 3) x2 + 8 x 4� � � � � � � � � Integration techniques E. 18.01 Exercises (x2 + x + 1) dx dx (x2 + 1) dx d) e) f) x2 + 8 x x3 + x2 x3 + 2 x2 + x x3dx (x2 + 1) dx g) h) (x + 1) 2(x − 1) x2 + 2 x + 2 5E-11 Solve the differential equation dy/dx = y(1 − y). 5E-12 This problem shows how to integrate any rational function of sin θ and cos θ using the substitution z = tan( θ/ 2). The integrand is transformed into a rational function of z, which can be integrated using the method of partial fractions. a) Show that 1 − z2 2z 2dz cos θ = , sin θ = , dθ = . 1 + z2 1 + z2 1 + z2 Calculate the following integrals using the substitution z = tan( θ/ 2) of part (a). � π � π � π dθ dθ b) c) d) sin θdθ (Not the easiest 1 + sin θ (1 + sin θ)2 000 way!) 5E-13 a) Use the polar coordinate formula for area to compute the area of the region 0 < r < 1/(1 + cos θ), 0 ≤ θ ≤ π/ 2. Hint: Problem 12 shows how the substi tution z = tan( θ/ 2) allows you to integrate any rational function of a trigonometric function. b) Compute this same area using rectangular coordinates and compare your answers. 5F. Integration by parts. Reduction formulas Evaluate the following integrals 5F-1 a) xa ln xdx (a =� −1) b) Evaluate the case a = −1 by substitu tion. 5F-2 a) xe xdx b) x2 exdx c) x3 exdx 5� � � � � � � E. 18.01 Exercises 5. Integration techniques n d) Derive the reduction formula expressing x e ax dx in terms of xn−1 eax dx . 5F-3 Evaluate sin −1(4 x)dx 5F-4 Evaluate ex cos xdx . (Integrate by parts twice.) 5F-5 Evaluate cos(ln x)dx . (Integrate by parts twice.) 5F-6 Show the substitution t = ex transforms the integral xn exdx, into (ln t)ndt . Use a reduction procedure to evaluate this integral. 6MIT OpenCourseWare 18.01SC Single Variable Calculus Fall 20 10 For information about citing these materials or our Terms of Use, visit: .
4652	https://misclaseslocas.com/product/el-imperfecto-spanish-imperfect-tense-bundle/?srsltid=AfmBOooMhz64PpNbk0TlE8aqGSs2-xHo4VwwnYZndK4Il7NalMkSMy4L	Imperfect Tense Spanish Lesson Bundle - Mis Clases Locas Skip to content Home About Blog PD Shop My account Cart Checkout FAQs & Support Home About Blog PD Shop My account Cart Checkout FAQs & Support Shop TPT Search Imperfect Tense Spanish Lesson Bundle ~~$37.88~~Original price was: $37.88.$27.99 Current price is: $27.99. Imperfect Tense Spanish Lesson Bundle quantity Add to cart Description Reviews (0) Description Struggle to teach the imperfect past tense in Spanish? This Spanish past tense el imperfecto bundle teaches the Super 7 high frequency verbs(era, tenía, le gustaba, había, estaba, iba, quería). This bundle includes all irregular imperfect verbs. Teach the irregular verbs of the Imperfect tense in Spanish with a mix of in-person and digital options to cover any situation you might be teaching with. Teaching the simple past tense of of the Spanish language? Teach the imperfect Spanish tense focusing on high frequency Spanish verbs. Help students understand the ongoing action using the Spanish Imperfect tense. This Bundle includes these Imperfect Spanish units and activities High Frequency Verb Unit – Imperfect Tense Novice Spanish(with in person handouts and resources, all in Google) Distance Learning High Frequency Verb Unit – Imperfect tense(all online Google Slides for distance or hybrid learning when needed) Find Someone Who – high frequency verbs – imperfect(PDF for in class speaking activity) IMPERFECT High Frequency Verbs Google Slides(50 task slides to review the unit) NEWImperfect Tense Spanish Conversation Cards (PDF Conversation cards for practice of Imperfect tense Speaking assessment) The original IMPERFECT tense unit for Spanish class includes: AllEditable Google documentsin a Google Folder Possibleunit planlasting up to 2 weeks 40 pages Editable Google slideshow 19 persona especial questions which show specific uses of the imperfect tense with example sentences 7 brain break songs(one for each verb with examples of the imperfect tense) Open-ended free-writequiz Student help sheet Google Docwith each question in Spanish, English, 1st person response, and 3rd person description OptionalGoogle Doc guided notessheet (great for accommodations) story asking script& extension activity idea Poster of personalized question BONUS 10 Quizlet sets are now included as well!! The Digital Imperfect Spanish Super 7 unit includes: Multiple possible unit plans w/ up to 7 weeks w/ 15-30 mins of distance learning per day, with both synchronous and asynchronous digital learning options, as well as sample block schedule. 63 Editable Google Slides – share with students on Google Classroom. 7 slides of vocabulary, each with a separate Quizlet 7 slides of personalized questions using the high-frequency verbs about past habitual actions, past events &past habits 7 slides of comprehensible story videos with optional extensions 7 authentic songs with extension activity for listening practice (mix of songs included in original & 4 new ones). 7 comprehensible song intros using the vocabulary for reading practice 7 authentic jokes for reading practice 7 guess the classic story (1 for each verb) for authentic reading practice 7 open-ended free writes (with vocab support) A comprehensible story, story script, mad lib version & extension activity idea 3 extra Quizlets to review the whole unit with all vocab, questions/answers practice & questions/translation practice. Overall assessment ideas The El Imperfecto Speaking Activity Includes The PDF Printable Imperfect Spanish Find Someone Who contains multiple differentiated versions to fit a variety of class needs to REVIEW el imperfecto Page 1 – questions with yo form of the answer provided – most basic Page 2 – write about friends with the support of s/he forms included – most basic Page 3 – questions (without yo form) Page 4 – write about friends & self with the support of a word bank Page 5 – write about friends & self – most advanced The Imperfect Tense Spanish Google Slides Digital Review The Google Slides contains 50 task slides to complete with el imperfecto 10 each of Matching questions & answers, Question Translations, Fill in the Blank, Correct Errors & Open-Ended Personalized questions. Each set of 10 is in order of the verbs in the unit. Everything is editable, so you could modify it to fit your students or only pick certain parts to assign. Plus there are 3 optional bonus slides at the end for students to practice writing & speaking. These would be a great extension or review for end-of-unit assessments. NEW Imperfect Tense Conversation Cards PDF Differentiated Conversation cards for practice of Imperfect tense Great for supportive speaking assessments about Spanish childhood The imperfect tense can be the easiest tense to learn in Spanish. The sets of endings are mostly regular. Teach students about ongoing past actions, habitual past actions, and continuous actions using the imperfect tense. These general rules will help students learn the appropriate ending. What another Spanish teacher is saying about this Imperfect Bundle for Spanish Class This was a great activity to review the imperfect and give the students some speaking practice! (9th-12th grades) ACTFL World-Readiness Standards for Learning Languages Interpersonal Communication– Learners interact and negotiate meaning in spoken conversations to share information, reactions, feelings, and opinions. Presentational Communication– Learners present information, concepts, and ideas to inform, explain, persuade, and narrate on a variety of topics. Interpretive Communication– Learners understand, interpret, and analyze what is heard, read, or viewed on a variety of topics. Bundle up for the best deals possible! Get thisIMPERFECT Bundlefor the best deal on all of my imperfect tense Super 7 resources all together including the originalImperfect Tenseunit plusDigital IMPERFECT unitgreat for distance or paperless learning, and these reviews activitiesImperfect Google Slides Review Activity,Imperfect Tense Spanish Conversation Cards&Find Someone Who – Imperfect Super 7.Use these to prepare for assessments, or spiral review these important verbs all year long. Or if you teach multiple levels of Spanish, the best deal is theHigh-Frequency Verbs MEGA BUNDLEwhich includes every single Super 7 and Sweet 16 Spanish resource in every tense including, these original unitsSuper 7 Present, Sweet 16 – present,Imperfect Tense,Preterite Tense& Future Tense, plus all digital version and accompanying review activities. You might also like these other Super 7 Spanish High-Frequency Spanish Verb Bundles Super 7 Present Tense Spanish BUNDLE Spanish PRETERITE tense BUNDLEto compare the main past tenses & different past tenses Spanish IMPERFECT Tense Bundle Spanish Future Tense BUNDLE Have difficulty with a file? This is a mix of PDF & Google Resources. This is a Google Resource & will be found linked in the instructions page of the PDF. The Find Someone Whos are PDF. You will need an updated PDF reader to open them Visit theFAQs section for more support Let’s Connect! For more information check out my blog postHow to Use High Frequency Verb Units,Preterite & Imperfect Spanish Class&Introducing past tense in Spanish class Sign up for mymailing listto get resources & inspiration delivered to your inbox Copyright © Mis Clases Locas Permission to copy for single classroom use only. Please purchase additional licenses if you intend to share this product. Reviews There are no reviews yet. Be the first to review “Imperfect Tense Spanish Lesson Bundle” Cancel reply You must be logged in to post a review. You might also like... Related products Sale! Spanish Future Tense BUNDLE --------------------------- ~~$14.98~~Original price was: $14.98.$12.99 Current price is: $12.99.Add to cart Spanish Preterite Worksheet Find Someone Who -------------------------------------------- $3.99Add to cart Sale! Essential Middle School Spanish 1 Curriculum -------------------------------------------- ~~$92.91~~Original price was: $92.91.$74.99 Current price is: $74.99.Add to cart Spanish Preterite Tense Super 7 Unit ------------------------------------ $10.99Add to cart Imperfect Tense Spanish Lesson Bundle ~~$37.88~~Original price was: $37.88.$27.99 Current price is: $27.99. GET FREE SPANISH BELL RINGERS! Get Editable Google Slide Bell Ringers & signed up for Mis Clases Locas newsletter when you enter your email. First Z name First na p me F G irst name First n b ame Email k address Email a k ddress Em d ail address Email g address Subscribe Thank you for subscribing! Find what you need Home About Blog PD Shop Account Contact Home About Blog PD Shop Account Contact Search Search Shopping-cartFacebookInstagramYoutubeX-twitterPinterest © Mis Clases Locas • Website by KristenDoyle.co Privacy Affiliates & Disclaimer FAQs & Support Privacy Affiliates & Disclaimer FAQs & Support Spanish Freebie! 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4653	https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/flight-equations-with-drag/	Glenn Research Center Home > Beginners Guide to Aeronautics Flight Equations with Drag A ball in flight has no engine to produce thrust, so the resulting flight is similar to the flight of shell from a cannon, or a bullet from a gun. This type of flight is called ballistic flight and assumes that weight is the only force acting on the ball. In reality, a baseball or a soccer ball in flight generates a moderate amount of aerodynamic drag and is not strictly ballistic. On this page we develop the equations which describe the motion of a flying ball including the effects of drag. Vertical Location At launch the ball is inclined at some angle to the vertical, so we resolve the initial velocity into a vertical and horizontal component. Unlike the ballistic flight equations, the horizontal equation includes the action of aerodynamic drag on the ball. We will first consider the vertical component and then develop the equations for the horizontal component. In the vertical plane, the only forces acting on the ball are the forces of weight and drag. There is a characteristic velocity which appears in many of the equations that is called the terminal velocity because it is the constant velocity that the object sustains during a coasting descent. Terminal velocity is noted by the symbol Vt. Vertical Descent During the vertical descent, for a light object, the weight and drag of an object are equal and opposite. There is no net force acting on the ball and the vertical acceleration is zero. (\LARGE a=0) (\LARGE W=D) where a is the acceleration, W is the weight, and D is the drag. The weight of any object is given by the weight equation: (\LARGE W=mg) where m is the mass of the object and g is the gravitational acceleration equal to 32.2 ft/sec2 or 9.8 m/sec2 on the surface of the Earth. (The gravitational acceleration has different values on the Moon and on Mars.) The drag is given by the drag equation: (\LARGE D=\frac{\mathit{Cd}\cdot\rho AV_{t}^{2}}{2}) where rho ((\bf\rho)) is the gas density, Cd is the drag coefficient which characterizes the effects of shape of the ball, A is the cross-sectional area of the ball, and Vt is the terminal velocity. Velocity The gas density has different surface values on the Earth and on Mars and varies with altitude. On the Moon the gas density is zero. Combining the last three equations, we can determine the terminal velocity: (\LARGE mg=\frac{\mathit{Cd}\cdot\rho AV_{t}^2}{2}) (\LARGE V_{t}=\sqrt{\frac{2mg}{\mathit{Cd}\cdot\rho A}}) Now, turning to the ascent trajectory, the ball is traveling at an initial vertical velocity V0. With the positive vertical coordinate denoted by y, the net vertical force Fnet acting on the ball is given by: (\LARGE F_{net}=-W-D) Because the weight of the object is a constant, we can use the simple form of Newton’s second law to solve for the vertical acceleration: (\LARGE F_{net}=ma=-W-D) (\LARGE ma=-mg-\frac{\mathit{Cd}\cdot\rho Av^{2}}{2}) (\LARGE a=-g-\frac{\mathit{Cd}\cdot\rho Av^{2}}{2m}) Notice that the acceleration changes with time. Multiply the last term by (\bf \frac{g}{g})and use the definition of the terminal velocity to obtain: (\LARGE a=-g\cdot (1+\frac{v^{2}}{V_{t}^2})) Acceleration The acceleration is the time rate of change of velocity: (\LARGE a=\frac{\text{d}v}{\text{d}t}=-g\cdot (1+\frac{v^{2}}{V_{t}^{2}})) Integrating this differential equation: (\LARGE \frac{{\text{d}v}}{1+\frac{v^{2}}{V_{t}^{2}}}=-g{\text{d}t}) (\LARGE V_{t}\tan^{-1}(\frac{v}{V_{t}})=-gt) where tan-1 is the inverse tangent function, and t is time. The limits of integration for velocity v are from V0 to V and the limits for time t is from 0 to t: (\LARGE \tan^{-1}(\frac{V}{V_{t}})-\tan^{-1}(\frac{V_{0}}{V_{t}})=\frac{-gt}{V_{t}}) (\LARGE \tan^{-1}(\frac{V}{V_{t}})=\tan^{-1}(\frac{V_{0}}{V_{t}})-\frac{gt}{V_{t}}) Now take the tangent function of both sides of the equation using the trigonometric identity: (\LARGE \tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}) on the right-hand side to obtain: (\LARGE \frac{V}{V_{t}}=\frac{\frac{V_{0}}{V_{t}}-\tan(\frac{gt}{V_{t}})}{1+\frac{V_{0}}{V_{t}}\tan(\frac{gt}{V_{t}})}) (\LARGE \frac{V}{V_{t}}=\frac{V_{0}-V_{t}-\tan(\frac{gt}{V_{t}})}{V_{t}+V_{0}\tan(\frac{gt}{V_{t}})}) Vertical Ascent This is the equation for the velocity at any time during the vertical ascent. At the top of the trajectory, the velocity is zero. We can solve the velocity equation to determine the time when this occurs: (\LARGE \frac{V_{0}}{V_{t}}=\tan(\frac{gt(v=0)}{V_t})) (\LARGE t(v=0)=\frac{V_{t}}{g}\tan^{-1}(\frac{V_{0}}{V_{t}})) To determine the vertical location during the ascent, we have to use another identity from differential calculus: (\LARGE \frac{\text{d}v}{\text{d}t}=\frac{\text{d}v}{\text{d}y}\cdot\frac{\text{d}y}{\text{d}t}) (\LARGE \frac{\text{d}v}{\text{d}t}=v\frac{\text{d}v}{\text{d}y}) We previously determined that: (\LARGE \frac{\text{d}v}{\text{d}t}=-g\cdot (1+\frac{v^{2}}{V_t^{2}})) (\LARGE v\frac{\text{d}v}{\text{d}y}=-g\cdot (1+\frac{v^{2}}{V_{t}^{2}})) (\LARGE \frac{v}{1+\frac{v^{2}}{V_{t}^2}}\text{d}v=-g\text{d}y) Integrating both sides: (\LARGE \frac{V_{t}^2}{2}\ln(v^{2}+V_{t}^{2})=-gy) where ln is the natural logarithmic function. The limits of integration for velocity v is from V0 to V and the limits for direction y is from 0 to y: (\LARGE \frac{V_{t}^2}{2}\ln(V^{2}+V_{t}^2)-\ln(V_{0}^2+V_{t}^2)=-gy) (\LARGE y=\frac{V_{t}^{2}}{2g}\ln(\frac{V_{0}^2+V_{t}^{2}}{V^{2}+V_{t}^{2}})) Notice that the location equation is pretty messy! For a given time t, we would have to find the local velocity V, and then plug that value into the location equation to get the location y. At the maximum height ymax, the velocity is equal to zero: (\LARGE y_{max}=\frac{V_{t}^{2}}{2g}\ln(\frac{V_{0}^{2}+V_{t}^{2}}{V_{t}^{2}})) Horizontal Location The horizontal equations are a little easier, since the only net force acting on the ball is the drag: (\LARGE F_{net}=ma=-D) (\LARGE ma=-\frac{\mathit{Cd}\cdot\rho Au^{2}}{2}) (\LARGE a=-\frac{\mathit{Cd}\cdot\rho Au^{2}}{2m}) where u is the horizontal velocity. We can use the terminal velocity to simplify this equation: (\LARGE a=\frac{\text{d}u}{\text{d}t}=\frac{-gu^{2}}{V_{t}^2}) (\LARGE \frac{1}{u^{2}}{\text{d}u}=-\frac{g}{V_{t}^{2}}{\text{d}t}) Integrating the equations, with the limits on the velocity from the initial velocity U0 to U, we obtain: (\LARGE u=\frac{\text{d}x}{\text{d}t}=\frac{V_{t}^{2}U_{0}}{V_{t}^2+gU_{0}t}) The horizontal velocity is inversely dependent on the time. We can similarly solve for the location x at any time by integrating the velocity equation: (\LARGE x=\frac{V_{t}^{2}}{g}\ln(\frac{V_{t}^{2}+gU_{0}t}{V_{t}^2})) Provide feedback ×
4654	https://www.biostars.org/p/484596/	DESeq2 and NA adj.pvalue LatestOpenJobsTutorialsTagsAboutFAQ CommunityPlanetNew Post Log In New PostLatestOpenNewsJobsTutorialsForumTagsPlanetUsers Log InSign UpAbout DESeq2 and NA adj.pvalue 2 0 Entering edit mode 4.7 years ago Francois Piumi ▴ 70 Hi, Could you explain me how it's possible to have large FCs and adj.pvalues equal to "NA" ? The counts table didn't show any missing or aberrant values. Thanks ! DESeq2 • 5.1k views ADD COMMENT • link updated 4.7 years ago by i.sudbery 22k • written 4.7 years ago by Francois Piumi ▴ 70 2 Entering edit mode 4.7 years ago rpolicastro 13k DESeq2 gives a few reasons in their documentation. Note on p-values set to NA: some values in the results table can be set to NA for one of the following reasons: If within a row, all samples have zero counts, the baseMean column will be zero, and the log2 fold change estimates, p value and adjusted p value will all be set to NA. If a row contains a sample with an extreme count outlier then the p value and adjusted p value will be set to NA. These outlier counts are detected by Cook’s distance. Customization of this outlier filtering and description of functionality for replacement of outlier counts and refitting is described below If a row is filtered by automatic independent filtering, for having a low mean normalized count, then only the adjusted p value will be set to NA. Description and customization of independent filtering is described below ADD COMMENT • link 4.7 years ago by rpolicastro 13k 0 Entering edit mode Thanks, but in my case not all samples have zero counts, only those from one condition : log2FC adj.pValue Cond1 Cond2 Cond3 Mock1 Mock2 Mock3 6.88709872531625 NA 5.31277170703943 27.6763981516444 31.2711272029594 0 0 0 So I was expecting to get a low adj.pValue since there's a difference between the Mock - Cond counts. Maybe the problem comes from "Cond1" which have a lower count vs "Cond2" et "Cond3" ADD REPLY • link 4.7 years ago by Francois Piumi ▴ 70 0 Entering edit mode That only addresses point 1 in their documentation. See @i.sudbery's answer on how points 2 and 3 factor in. ADD REPLY • link 4.7 years ago by rpolicastro 13k 2 Entering edit mode 4.7 years ago i.sudbery 22k There are several stages at which an NA might be included in the adj.pvalues column in DESeq2 output. The two most likely are low expression filtering and outlier exclusion. Independent hypothesis filtering: DESeq2 automatically finds the base expression level at which to filter in order maximise the number of genes that pass the specified level of alpha (i.e. the FDR). Any genes whose expression level is below this have their adj.pvalue's set to NA, but should still have pvalues (I think). This can be disabled by setting bash independentFiltering = TRUE in the call to bash results() . For more details see this part of the DESeq2 user guide. Outlier exclusion: DESeq2 tests each sample in each gene for outliers using the Cook's distance. Genes to have outliers will also have their padj.pvalues set to zero (I think also their pvalues as well, in contrast to low count filtering). This can be disabled with bash cooksCutoff=FALSE in the call to bash results() . For more details see this part of the DESeq2 user guide. ADD COMMENT • link 4.7 years ago by i.sudbery 22k 0 Entering edit mode Using independentFiltering=FALSE, some genes get a significant padj (i.e. < 0.05). So do you think I should absolutely try to recover those genes which have strong log2FCs, very low pvalues but padj set to NA? ADD REPLY • link 4.7 years ago by Francois Piumi ▴ 70 1 Entering edit mode You don't have to use this option, but using it maximises the number of significant genes you will get. This does not mean that none of the genes it filters out will significant, but rather than filtering out those genes means that a larger number of other genes become significant. The other thing to be aware of is that the default setting for significance in DESeq2 is padj<0.1, and this is the level that bash independentFiltering uses as its benchmark. If you are using bash 0.05 , then you should set bash alpha=0.05 in your call to results. ADD REPLY • link 4.7 years ago by i.sudbery 22k Login before adding your answer. Similar Posts Same results in Limma Toptable as in DESEQ2 results • updated 3.0 years ago by jared.andrews07 ★ 19k • written 3.0 years ago by Nona ▴ 90 Dear all, I am trying to generate exactly the same results in Limma for my result table as I did in DESEQ2 (Differential Expression Anal… Weird looking volcano plot • updated 12 months ago by Juan ▴ 30 • written 12 months ago by Harini ▴ 10 Hello! 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4655	https://fiveable.me/key-terms/honors-statistics/rise-run	Rise Over Run - (Honors Statistics) - Vocab, Definition, Explanations \| Fiveable \| Fiveable new!Printable guides for educators Printable guides for educators. Bring Fiveable to your classroom ap study content toolsprintablespricing my subjectsupgrade All Key Terms Honors Statistics Rise Over Run 📊honors statistics review key term - Rise Over Run Citation: MLA Definition Rise over run, also known as the slope of a line, is a way to quantify the steepness or inclination of a straight line. It represents the ratio of the change in the vertical (y) direction to the change in the horizontal (x) direction between two points on the line. 5 Must Know Facts For Your Next Test The rise over run formula is calculated as $\frac{\Delta y}{\Delta x}$, where $\Delta y$ represents the change in the y-coordinate and $\Delta x$ represents the change in the x-coordinate between two points on the line. The slope of a line can be positive, negative, zero, or undefined, depending on the relationship between the rise and the run. A positive slope indicates that the line is sloping upward from left to right, while a negative slope indicates that the line is sloping downward from left to right. A slope of zero means the line is horizontal, and a slope of undefined means the line is vertical. The slope of a line can be used to determine the angle of inclination of the line relative to the horizontal axis. Review Questions How can you calculate the rise over run of a line given two points on the line? To calculate the rise over run of a line, you need to identify two points on the line, (x1, y1) and (x2, y2). The rise over run is then calculated as $\frac{y_2 - y_1}{x_2 - x_1}$. This gives you the slope of the line, which represents the steepness or inclination of the line in the coordinate plane. Explain how the rise over run is related to the slope of a linear equation and its graphical representation. The rise over run is directly equivalent to the slope of a linear equation. In the general linear equation $y = mx + b$, the slope 'm' represents the rise over run, which is the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on the line. This slope determines the steepness and direction of the line when graphed on the coordinate plane. A positive slope indicates an upward-sloping line, a negative slope indicates a downward-sloping line, a slope of zero indicates a horizontal line, and a slope of undefined indicates a vertical line. Discuss how the rise over run can be used to analyze and compare the characteristics of different linear equations. The rise over run, or slope, of a linear equation is a critical characteristic that can be used to analyze and compare the properties of different linear equations. By calculating the rise over run, you can determine the steepness and direction of the line, which provides insights into the relationship between the variables. For example, a steeper line with a higher absolute value of the slope will have a greater rate of change between the x and y variables compared to a line with a lower slope. Additionally, the sign of the slope can indicate whether the line is increasing or decreasing. This information can be used to make comparisons between linear equations and understand their underlying relationships. Related terms Slope: The slope of a line is a number that describes both the direction and the steepness of the line. It is commonly represented by the letter 'm' and is calculated as the rise over run. Linear Equation: A linear equation is an equation that represents a straight line. The general form of a linear equation is $y = mx + b$, where 'm' is the slope (rise over run) and 'b' is the y-intercept. Coordinate Plane: The coordinate plane is a two-dimensional grid used to represent and plot points, lines, and other shapes. The horizontal axis is the x-axis, and the vertical axis is the y-axis, forming the rise and run components of the rise over run calculation. 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4656	https://math.stackexchange.com/questions/2859543/understanding-fractional-exponents	exponential function - Understanding fractional exponents - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Understanding fractional exponents Ask Question Asked 7 years, 2 months ago Modified7 years, 2 months ago Viewed 1k times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. How do I understand fractional exponents like 1/2, 3/2, 1/5, etc? Like y y raised to a power 2 is y.y y.y and when the power is increased or decreased by one, it means we multiply the number by another y y or divide it by y y respectively. But how do I think about fractional exponential powers by this same logic? Thank you. exponential-function Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications asked Jul 22, 2018 at 16:23 Ram KeswaniRam Keswani 1,013 3 3 gold badges 13 13 silver badges 24 24 bronze badges Add a comment\| 4 Answers 4 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. "how do I think about fractional exponential powers by this same logic?" Carefully. You have to expand the definition. We notice that b n∗b m=b∗...∗bn b∗...∗bm=b∗...∗b=b n+m b n∗b m=b∗...∗b⏟n b∗...∗b⏟m=b∗...∗b⏟=b n+m for positive integers n n and m m and we take b n+m=b n b m b n+m=b n b m as a rule. Then we think. Well what that means b 0∗b m=b 0+m=b m b 0∗b m=b 0+m=b m. That would mean b 0=1 b 0=1, doesn't it? Well, in one sense it does not make sense that we can't mulitple b b times itself zero times. That's meaningless. But that is no longer what the definition of b n b n means. It means that for n∈Z n∈Z and n≥2 n≥2 but... for n∈Z n∈Z and n≤1 n≤1 we are going to have it mean.... whatever it has to be in order for the rule b n+m=b n∗b m b n+m=b n∗b m to work. That means: b 1=b b 1=b and b 0=0 b 0=0 and for a negative integer −n<0−n<0 then b−n=1 b n b−n=1 b n (becuase we must have b−n∗b n=b−n+n=b 0=1 b−n∗b n=b−n+n=b 0=1. Okay, but we also have the rule (b n)m=b n∗...∗b nm=b n+....+nm=b n m(b n)m=b n∗...∗b n⏟m=b n+....+n⏟m=b n m. So what does that make b 1 m b 1 m have to be? Well, (b 1 m)m=b 1 m∗m=b 1=b(b 1 m)m=b 1 m∗m=b 1=b. So that means b 1 m b 1 m has to be b√m b m, right? And that's that. b m n=(b√n)m b m n=(b n)m. That's it. .... Except we have to be careful. x 2=4 x 2=4 has two possible answer −2,2−2,2 so which will it be. ANd x 2=−4 x 2=−4 has none. So we add a caveat rule: we only feel safe talking about b m n b m n if b≥0 b≥0 if n n is even and in that case the value is positive. We can talk about (−4)m n(−4)m n if n n is odd (in which case the value is negative) but for most of the time, we should avoid it. Many mathematicians will simply say b q;q∈Q b q;q∈Q is only defined if b≥0 b≥0. And for x∈R x∈R, we say b x b x is only defined fro b≥0 b≥0. Unless we are doing complex analysis where we accept the square roots of negative numbers. In that case, things get very different. But it's still a matter of defining things by what the must be in order for the rules to work. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Jul 22, 2018 at 16:50 answered Jul 22, 2018 at 16:43 fleabloodfleablood 132k 5 5 gold badges 52 52 silver badges 142 142 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. Notice that, for example, if x 1 2=a x 1 2=a, then by squaring both sides we get x=(x 1 2)2=a 2 x=(x 1 2)2=a 2. In other words x−−√=a x=a. Following the same logic you can get that for every n∈N n∈N, x 1 n=x−−√n x 1 n=x n After you accepted that, by exponentiation rules, you know that for every p,q∈N p,q∈N: x p q=x p⋅1 q=(x p)1 q=x p−−√q x p q=x p⋅1 q=(x p)1 q=x p q So in general, raising to a rational number p q p q means to take the p p-th power and the q q-th root. And, of course, it also generalize to negative rational numbers. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Jul 22, 2018 at 16:33 GuySaGuySa 513 3 3 silver badges 18 18 bronze badges 1 This implies that if p p and q q are coprime integers, with q q positive, then all nonzero complex numbers have exactly q q (p/q p/q)th powers.Michael Ejercito –Michael Ejercito 2024-02-11 02:53:18 +00:00 Commented Feb 11, 2024 at 2:53 Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. The exponents follow the rule (a b)c=a b c(a b)c=a b c meaning that the power of a power is the power to the product of the exponents. Then assume a rational power p/q p/q. We write a p/q=b a p/q=b and raise both members to the q t h q t h: a(p/q)q=a p=b q.a(p/q)q=a p=b q. So b b is the number such that when raised to the q t h q t h power yields a p a p. In other words, it is the q t h q t h root of a p a p. If you didn't learn the roots yet, understand that this is the "inverse" operation of exponentiation. For instance, 3 5=243⟺243 1/5=243−−−√5=3.3 5=243⟺243 1/5=243 5=3. Very often, roots are irrational numbers, 1.24573093⋯5=3⟺3 1/5=3–√5=1.24573093⋯1.24573093⋯5=3⟺3 1/5=3 5=1.24573093⋯ Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Jul 22, 2018 at 17:08 answered Jul 22, 2018 at 16:34 user65203 user65203 Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. Let us start with 1/2 1/2 If I want to find 16 1/2 16 1/2, I expect to find a number whose square is 16.16. That is if x=16 1/2 x=16 1/2 we like to have x 2=16 x 2=16 Well we have two such real numbers, namely 4 4 and −4−4 We denote them by 16−−√=4 16=4 and −16−−√=−4−16=−4 Sometimes you do not get a real solution at all, for example (−16)1/2(−16)1/2 is not a real number, because when you square a real number the answer is always non-negative. Sometimes you get only one solution for example 27 1/3=3 27 1/3=3 because 3^3=27 and no other real number satisfies x 3=27.x 3=27. Once you find a 1/n a 1/n then a m/n a m/n is found by (a 1/n)m(a 1/n)m For example 27 2/3=(27 1/3)2=3 2=9 27 2/3=(27 1/3)2=3 2=9 You may practice with some fractional powers and check your answers with a calculator to build confidence. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Jul 22, 2018 at 17:09 Mohammad Riazi-KermaniMohammad Riazi-Kermani 70.1k 4 4 gold badges 44 44 silver badges 93 93 bronze badges 1 One corollary is is that x−−√a/b x a/b is also well-defined. For example, 4–√1/2=16 4 1/2=16 Michael Ejercito –Michael Ejercito 2024-02-11 02:51:03 +00:00 Commented Feb 11, 2024 at 2:51 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions exponential-function See similar questions with these tags. 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4657	https://geekymedics.com/respiratory-examination-2/	Save 15% on our Everything Bundle with discount code EVERY15 🎓 1,300+ OSCE Stations ✨ Dr Lewis Potter· Respiratory examination ·· Last updated: Respiratory Examination – OSCE Guide Access all our resources with a subscription ⚡️ Supercharge your learning with access to OSCE stations, virtual patients, question banks, flashcards and more. Learn more ✨ Respiratory examination frequently appears in OSCEs and you’ll be expected to pick up the relevant clinical signs using your examination skills. This respiratory examination OSCE guide provides a clear step-by-step approach to examining the respiratory system, with an included video demonstration. Download the respiratory examination PDF OSCE checklist, or use our interactive OSCE checklist. You may also be interested in our paediatric respiratory examination guide. Introduction Wash your hands and don PPE if appropriate. Introduce yourself to the patient including your name and role. Confirm the patient’s name and date of birth. Briefly explain what the examination will involve using patient-friendly language. Gain consent to proceed with the examination. Adjust the head of the bed to a 45° angle. Adequately expose the patient’s chest for the examination (offer a blanket to allow exposure only when required and if appropriate, inform patients they do not need to remove their bra). Exposure of the patient’s lower legs is also helpful to assess for peripheral oedema. Ask the patient if they have any pain before proceeding with the clinical examination. Explore our premium collection of 1,300+ OSCE stations, including clinical examination stations ✨ General inspection Clinical signs Inspect the patient from the end of the bed whilst at rest, looking for clinical signs suggestive of underlying pathology: Age: the patient’s approximate age is helpful when considering the most likely underlying pathology, with younger patients more likely to have diagnoses such as asthma or cystic fibrosis (CF) and older patients more likely to have chronic obstructive pulmonary disease (COPD), interstitial lung disease or malignancy. Cyanosis: bluish discolouration of the skin due to poor circulation (e.g. peripheral vasoconstriction secondary to hypovolaemia) or inadequate oxygenation of the blood (e.g. right-to-left cardiac shunting). Shortness of breath: signs may include nasal flaring, pursed lips, use of accessory muscles, intercostal muscle recession and the tripod position (sitting or standing leaning forward and supporting the upper body with hands on knees or other surfaces). Shortness of breath is a common feature of most respiratory pathology, however possible underlying diagnoses in an OSCE could include asthma, pulmonary oedema, pulmonary fibrosis, lung cancer and COPD. The inability to speak in full sentences is an indicator of significant shortness of breath. Cough: a productive cough can be associated with several respiratory pathologies including pneumonia, bronchiectasis, COPD and CF. A dry cough may suggest a diagnosis of asthma or interstitial lung disease. Wheeze:a continuous, coarse, whistling sound produced in the respiratory airways during breathing. Wheeze is often associated with asthma, COPD and bronchiectasis. Stridor:a high-pitched extra-thoracic breath sound resulting from turbulent airflow through narrowed upper airways. Stridor has a wide range of causes, including foreign body inhalation (acute) and subglottic stenosis (chronic). Pallor: a pale colour of the skin that can suggest underlying anaemia (e.g. haemorrhage/chronic disease) or poor perfusion (e.g. congestive cardiac failure). It should be noted that healthy individuals may have a pale complexion that mimics pallor. Oedema: typically presents with swelling of the limbs (e.g. pedal oedema) or abdomen (i.e. ascites) and is often associated with right ventricular failure. Pulmonary oedema often occurs secondary to left ventricular failure. Cachexia:ongoing muscle loss that is not entirely reversed with nutritional supplementation. Cachexia is commonly associated with underlying malignancy (e.g. lung cancer) and other end-stage respiratory diseases (e.g. COPD). Objects and equipment Look for objects or equipment on or around the patient that may provide useful insights into their medical history and current clinical status: Oxygen delivery devices:note the type of oxygen device (e.g. Venturi mask, non-rebreathing mask, nasal cannulae) and the current flow rate of oxygen (e.g. 2L, 4L, 10L, 15L). Look for other forms of respiratory support such as CPAP or BiPAP. Sputum pot: note the volume and colour of the contents (e.g. COPD/bronchiectasis). Other medical equipment: ECG leads, medications (e.g. inhalers/nebulisers in asthma/COPD), catheters (note volume/colour of urine) and intravenous access. Cigarettes or vaping equipment: smoking is a significant risk factor for lung cancer and chronic lung disease (e.g. COPD). Mobility aids: items such as wheelchairs and walking aids give an indication of the patient’s current mobility status. Vital signs: charts on which vital signs are recorded will give an indication of the patient’s current clinical status and how their physiological parameters have changed over time. Fluid balance: fluid balance charts will give an indication of the patient’s current fluid status which may be relevant if a patient appears fluid overloaded or dehydrated. Prescriptions: prescribing charts or personal prescriptions can provide useful information about the patient’s recent medications. Inspect the patient from the end of the bed Hands The hands can provide lots of clinically relevant information and therefore a focused, structured assessment is essential. Inspection General observations Observe the hands and note your findings: Colour: cyanosis of the hands may suggest underlying hypoxaemia. Tar staining: caused by smoking, a significant risk factor for respiratory disease (e.g. COPD, lung cancer). Skin changes:bruising and thinning of the skin can be associated with long-term steroid use (e.g. asthma, COPD, interstitial lung disease). Joint swelling or deformity: may be associated with rheumatoid arthritis which has several extra-articular manifestations that affect the respiratory system (e.g. pleural effusions/pulmonary fibrosis). Finger clubbing Finger clubbing involves uniform soft tissue swelling of the terminal phalanx of a digit with subsequent loss of the normal angle between the nail and the nail bed. Finger clubbing is associated with several underlying disease processes, but those most likely to appear in a respiratory OSCE station include lung cancer, interstitial lung disease, cystic fibrosis and bronchiectasis. To assess for finger clubbing: Ask the patient to place the nails of their index fingers back to back. In a healthy individual, you should be able to observe a small diamond-shaped window (known as Schamroth’s window). When finger clubbing develops, this window is lost Fine tremor Assess for the presence of a fine tremor: Ask the patient to hold out their hands in an outstretched position and observe for a fine tremor which is typically associated with beta-2-agonist use (e.g. salbutamol). Asterixis (flapping tremor) Asterixis (also known as ‘flapping tremor’) is a type of negative myoclonus characterised by irregular lapses of posture causing a flapping motion of the hands. In the context of a respiratory examination, the most likely underlying cause is CO2 retention in conditions that result in type 2 respiratory failure (e.g. COPD). Other causes of asterixis include uraemia and hepatic encephalopathy. Whilst the patient still has their hands stretched outwards, ask them to cock their hands backwards at the wrist joint and hold the position for 30 seconds. Observe for evidence of asterixis during this time period. Palpation Temperature Place the dorsal aspect of your hand onto the patient’s to assess temperature: In healthy individuals, the hands should be symmetrically warm, suggesting adequate perfusion. Cool hands may suggest poor peripheral perfusion. Excessively warm and sweaty hands can be associated with CO2 retention. Heart rate Assessing heart rate: Palpate the patient’s radial pulse, located at the radial side of the wrist, with the tips of your index and middle fingers aligned longitudinally over the course of the artery. Once you have located the radial pulse, assess the rate and rhythm. Calculating heart rate: You can calculate the heart rate in a number of ways, including measuring for 60 seconds, measuring for 30 seconds and multiplying by 2 or measuring for 15 seconds and multiplying by 4. The shorter the interval used, the higher the risk of obtaining an inaccurate result, so wherever possible, you should palpate for a full 60 seconds. For irregular rhythms, you should measure the pulse for a full 60 seconds to improve accuracy. Pulse abnormalities Bounding pulse: can be associated with underlying CO2 retention (e.g. type 2 respiratory failure). Pulsus paradoxus: pulse wave volume decreases significantly during the inspiratory phase. This is a late sign of cardiac tamponade, severe acute asthma and severe exacerbations of COPD (therefore it is unlikely to be relevant to most OSCE scenarios). Respiratory rate Assessing respiratory rate: Whilst still palpating the radial pulse (but no longer counting it), assess the patient’s respiratory rate (palpation of the radial pulse at this stage purely to avoid making the patient aware you are directly observing their breathing, as this can itself alter the respiratory rate). Note any asymmetries in the expiratory and inspiratory phases of respiration (e.g. the expiratory phase is often prolonged in asthma exacerbations and in patients with COPD). Calculating respiratory rate: Assess the patient’s respiratory rate for 60 seconds to calculate the number of breaths per minute. Respiratory rate abnormalities In healthy adults, the respiratory rate should be between 12-20 breaths per minute. A respiratory rate of fewer than 12 breaths per minute is referred to as bradypnoea (e.g. opiate overdose). A respiratory rate of more than 20 breaths per minute is referred to as tachypnoea (e.g. acute asthma). Inspect the hands Inspect for finger clubbing Inspect for fine tremor Inspect for flapping tremor (asterixis) Assess and compare temperature Palpate the radial pulse Assess the respiratory rate Peripheral cyanosis 2 Pallor due to anaemia 3 Tar staining on the hands 4 Finger clubbing 5 Jugular venous pressure (JVP) Jugular venous pressure (JVP) provides an indirect measure of central venous pressure. This is possible because the internal jugular vein (IJV) connects to the right atrium without any intervening valves, resulting in a continuous column of blood. The presence of this continuous column of blood means that changes in right atrial pressure are reflected in the IJV (e.g. raised right atrial pressure results in distension of the IJV). The IJV runs between the medial end of the clavicle and the ear lobe, under the medial aspect of the sternocleidomastoid, making it difficult to visualise (its double waveform pulsation is, however, sometimes visible due to transmission through the sternocleidomastoid muscle). Because of the inability to easily visualise the IJV, it’s tempting to use the external jugular vein (EJV) as a proxy for assessment of central venous pressure during clinical assessment. However, because the EJV typically branches at a right angle from the subclavian vein (unlike the IJV which sits in a straight line above the right atrium) it is a less reliable indicator of central venous pressure. See our guide to jugular venous pressure (JVP) for more details. Measure the JVP 1. Position the patient in a semi-recumbent position (at 45°). 2. Ask the patient to turn their head slightly to the left. 3. Inspect for evidence of the IJV, running between the medial end of the clavicle and the ear lobe, under the medial aspect of the sternocleidomastoid (it may be visible between just above the clavicle between the sternal and clavicular heads of the sternocleidomastoid. The IJV has a double waveform pulsation, which helps to differentiate it from the pulsation of the external carotid artery. 4. Measure the JVP by assessing the vertical distance between the sternal angle and the top of the pulsation point of the IJV (in healthy individuals, this should be no greater than 3 cm). Respiratory causes of a raised JVP A raised JVP indicates the presence of venous hypertension. Respiratory causes of a raised JVP include: Pulmonary hypertension: causes right-sided heart failure, often occurring due to COPD or interstitial lung disease. There are several other causes of a raised JVP that relate to the cardiovascular system (e.g. congestive heart failure, tricuspid regurgitation and constrictive pericarditis). Hepatojugular reflux test The hepatojugular reflux test involves the application of pressure to the liver whilst observing for a sustained rise in JVP. See our cardiovascular examination guide for details on how to elicit hepatojugular reflux. Assess the jugular venous pressure (JVP) Assess for hepatojugular reflex Face General Inspect the face for any signs relevant to the respiratory system: Plethoric complexion: a congested red-faced appearance associated with polycythaemia (e.g. COPD) and CO2 retention (e.g. type 2 respiratory failure). Eyes Inspect the eyes for signs relevant to the respiratory system: Conjunctival pallor: suggestive of underlying anaemia. Ask the patient to gently pull down their lower eyelid to allow you to inspect the conjunctiva. Ptosis, miosis and enophthalmos: all features of Horner’s syndrome (anhydrosis is another important sign associated with the syndrome). Horner’s syndrome occurs when the sympathetic trunk is damaged by pathology such as lung cancer affecting the apex of the lung (e.g. Pancoast tumour). Mouth Inspect the mouth for signs relevant to the respiratory system: Central cyanosis: bluish discolouration of the lips and/or the tongue associated with hypoxaemia. Oral candidiasis: a fungal infection commonly associated with steroid inhaler use (due to local immunosuppression). It is characterised by pseudomembranous white slough which can be easily wiped away to reveal underlying erythematous mucosa. Inspect the eyes Inspect the mouth Conjunctival pallor Horner's syndrome (ptosis and miosis) 6 Central cyanosis 7 Oral candidiasis 8 Central cyanosis 13 Inspection of the chest Scars Closely inspect the chest wall for scars and other abnormalities: Median sternotomy scar: located in the midline of the thorax. This surgical approach is used for cardiac valve replacement and coronary artery bypass grafts (CABG). Axillary thoracotomy scar: located between the posterior border of the pectoralis major and anterior border of latissimus dorsi muscles, through the 4th or 5th intercostal space. This surgical approach is used for the insertion of chest drains. Posterolateral thoracotomy scar: located between the scapula and mid-spinal line, extending laterally to the anterior axillary line. This surgical approach is used for lobectomy, pneumonectomy and oesophageal surgery. Infraclavicular scar: located in the infraclavicular region (on either side). This surgical approach is used forpacemaker insertion. Radiotherapy-associated skin changes: may be present in patients who have been treated for lung cancer. Clinical features can include xerosis (dry skin), scale, hyperkeratosis (thickened skin), depigmentation and telangiectasia. Chest wall deformities Inspect for evidence of chest wall deformities: Asymmetry: typically associated with pneumonectomy (e.g. lung cancer) and thoracoplasty (e.g. tuberculosis). Pectus excavatum: a caved-in or sunken appearance of the chest. Pectus carinatum: protrusion of the sternum and ribs. Hyperexpansion (a.k.a. ‘barrel chest’):chest wall appears wider and taller than normal. Associated with chronic lung diseases such as asthma and COPD. Inspect the anterior chest Inspect the axilla Posterolateral thoracotomy scar Anterolateral thoracotomy scar Pectus excavatum 9 Pectus carinatum 10 Barrel chest Trachea and cricosternal distance Assess tracheal position Gently assess the position of the trachea, which should be central in healthy individuals (this can be uncomfortable, so warn the patient in advance): 1. Ensure patient’s neck musculature is relaxed by asking them to position their chin slightly downwards. 2. Dip your index finger into the thorax beside the trachea. 3. Gently apply side pressure to locate the border of the trachea. 4. Compare this space to the other side of the trachea using the same process. 5. A difference in the amount of space between the sides suggests the presence of tracheal deviation. Causes of tracheal deviation The trachea deviates away from tension pneumothorax and large pleural effusions. The trachea deviates towards lobar collapse and pneumonectomy. Palpation of the trachea can be uncomfortable, so warn the patient and apply a gentle technique Assess cricosternal distance Cricosternal distance is the distance between the inferior border of the cricoid cartilage and the suprasternal notch: 1. Measure the distance between the suprasternal notch and cricoid cartilage using your fingers. 2. In healthy individuals, the distance should be 3-4 fingers. Cricosternal distance is actually based on the size of the patient’s fingers so if their fingers are significantly different in size from your own, it may be worth using their fingers for the assessment. Causes of abnormal cricosternal distance A distance of fewer than 3 fingers suggests underlying lung hyperinflation (e.g. asthma, COPD). Assess tracheal position Assess cricosternal distance Palpation of the chest Palpate the apex beat 1. Palpate the apex beat with your fingers placed horizontally across the chest. 2. In healthy individuals, it is typically located in the 5th intercostal space in the midclavicular line. Causes of a displaced apex beat Respiratory causes of a displaced apex beat: Right ventricular hypertrophy (e.g. pulmonary hypertension, COPD, interstitial lung disease) Large pleural effusion Tension pneumothorax Assess chest expansion 1. Place your hands on the patient’s chest, inferior to the nipples. 2. Wrap your fingers around either side of the chest. 3. Bring your thumbs together in the midline so that they touch but are not fixed to the chest wall. 4. Ask the patient to take a deep breath in. 5. Observe the movement of your thumbs (in healthy individuals they should move symmetrically upwards/outwards during inspiration and symmetrically downwards/inwards during expiration). 6. Reduced movement of one of your thumbs indicates reduced chest expansion on that side. Chest expansion can also be assessed posteriorly using the same technique and may be more comfortable for patients with breasts. Respiratory causes of reduced chest expansion Symmetrical: pulmonary fibrosis reduces lung elasticity, restricting overall chest expansion. Asymmetrical: pneumothorax, pneumonia and pleural effusion would all cause ipsilateral reduced chest expansion. Palpate the apex beat Assess anterior chest expansion Percussion of the chest Percussion of the chest involves listening to the volume and pitch of percussion notes across the chest to identify underlying pathology. Correct technique is essential to generating effective percussion notes. Percussion technique 1. Place your non-dominant hand on the patient’s chest wall. 2. Position your middle finger over the area you want to percuss, firmly pressed against the chest wall. 3. With your dominant hand’s middle finger, strike the middle phalanx of your non-dominant hand’s middle finger using a swinging movement of the wrist. 4. The striking finger should be removed quickly, otherwise, you may muffle the resulting percussion note. Areas to percuss Percuss the following areas of the chest, comparing side to side as you progress (see image example below): Supraclavicular region: lung apices Infraclavicular region Chest wall: percuss over 3-4 locations bilaterally Axilla Types of percussion note Resonant: a normal finding (listen to the example in the video demonstration). Dullness: suggests increased tissue density (e.g. cardiac dullness, consolidation, tumour, lobar collapse). Stony dullness: typically caused by an underlying pleural effusion. Hyper-resonance: the opposite of dullness, suggestive of decreased tissue density (e.g. pneumothorax). Tactile vocal fremitus Assessing tactile vocal fremitus involves palpating over different areas of the chest wall whilst the patient repeats a word or number consistently (e.g. “ninety-nine”). The presence of increased tissue density or fluid affects the strength at which the patient’s speech is transmitted as vibrations through the chest wall to the examiner’s hands. Technique 1. Ask the patient to say “99” repeatedly at the same volume and in the same tone. 2. Palpate the chest wall on both sides, using the ulnar border of your hand. 3. Cover all major regions of the chest wall, comparing each side at each location. Abnormal tactile vocal fremitus Increased vibration over an area suggests increased tissue density (e.g. consolidation, tumour, lobar collapse). Decreased vibration over an area suggests the presence of fluid or air outside of the lung (e.g. pleural effusion, pneumothorax). An alternative method of assessment Vocal resonance (see below) is an alternative method of assessing the conduction of sound through lung tissue and involves auscultating over different areas of the chest wall whilst the patient repeats a word or number consistently. The presence of increased tissue density or fluid affects the volume at which the patient’s speech is transmitted to the diaphragm of the stethoscope. Given both tests assess the same thing, there is no reason to perform both vocal resonance and tactile vocal fremitus in the same examination. Percuss the anterior chest wall Chest percussion locations Auscultation of the chest When auscultating the chest, it is important that you have a systematic approach that allows you to compare each area on both the left and the right as you progress. Auscultate the chest Technique 1. Ask the patient to relax and breathe deeply in and out through their mouth (prolonged deep breathing should, however, be avoided). 2. Position the diaphragm of the stethoscope over each of the relevant locations on the chest wall to ensure all lung regions have been assessed and listen to the breathing sounds during inspiration and expiration. Assess the quality and volume of breath sounds and note any added sounds. 3. Auscultate each side of the chest at each location to allow for direct comparison and increased sensitivity at detecting local abnormalities. Quality of breath sounds Vesicular: the normal quality of breath sounds in healthy individuals. Bronchial: harsh-sounding (similar to auscultating over the trachea), inspiration and expiration are equal and there is a pause between. This type of breath sound is associated with consolidation. Volume of breath sounds Quiet breath sounds: suggest reduced air entry into that region of the lung (e.g pleural effusion, pneumothorax). When presenting your findings, state ‘reduced breath sounds’, rather than ‘reduced air entry’. Added sounds Wheeze:a continuous, coarse, whistling sound produced in the respiratory airways during breathing. Wheeze is often associated with asthma, COPD and bronchiectasis. Stridor:a high-pitched extra-thoracic breath sound resulting from turbulent airflow through narrowed upper airways. Stridor has a wide range of causes, including foreign body inhalation (acute) and subglottic stenosis (chronic). Coarse crackles:discontinuous, brief, popping lung sounds typically associated with pneumonia, bronchiectasis and pulmonary oedema. Fine end-inspiratory crackles: often described as sounding similar to the noise generated when separating velcro. Fine end-inspiratory crackles are associated with pulmonary fibrosis. Assess vocal resonance Assessing vocal resonance involves auscultating over different areas of the chest wall whilst the patient repeats a word or number consistently. The presence of increased tissue density or fluid affects the volume at which the patient’s speech is transmitted to the diaphragm of the stethoscope. Technique 1. Ask the patient to say “99” repeatedly at the same volume and in the same tone. 2. Auscultate all major regions of the anterior chest wall, comparing each side at each location. Abnormal vocal resonance Increased volume over an area suggests increased tissue density (e.g. consolidation, tumour, lobar collapse). Decreased volume over an area suggests the presence of fluid or air outside of the lung (e.g. pleural effusion, pneumothorax). An alternative method of assessment Tactile vocal fremitus is an alternative method of assessing the conduction of sound through lung tissue and involves feeling for sound vibrations on the chest wall with your hands as the patient speaks. Given both tests assess the same thing, there is no reason to perform both vocal resonance and tactile vocal fremitus in the same examination. Auscultate the anterior chest Auscultation locations on the anterior chest Assess vocal resonance Lymph nodes Palpate the patient’s lymph nodes 1.Position the patient sitting upright and examine from behind if possible. Ask the patient to tilt their chin slightly downwards to relax the muscles of the neck and aid palpation of lymph nodes. You should also ask them to relax their hands in their lap. 2.Inspect for any evidence of lymphadenopathy or irregularity of the neck. 3.Stand behind the patient and use both hands to start palpating the neck. 4.Use the pads of the second, third and fourth fingers to press and roll the lymph nodes over the surrounding tissue to assess the various characteristics of the lymph nodes. By using both hands (one for each side) you can note any asymmetry in size, consistency and mobility of lymph nodes. 5.Start in the submental area and progress through the various lymph node chains. Any order of examination can be used, but a systematic approach will ensure no areas are missed: Submental Submandibular Pre-auricular Post-auricular Superficial cervical Deep cervical Posterior cervical Supraclavicular – left supraclavicular region is where Virchow’s node may be noted (associated with upper gastrointestinal malignancy) Take caution when examining the anterior cervical chain that you do not compromise cerebral blood flow (due to carotid artery compression). It may be best to examine one side at a time here. A common mistake is a “piano-playing” or “spider’s legs” technique with the fingertips over the skin rather than correctly using the pads of the second, third and fourth fingers to press and roll the lymph nodes over the surrounding tissue. Example of logical systematic examination of the lymph nodes 1.Start under the chin (submental lymph nodes), then move posteriorly palpating beneath the mandible (submandibular), turn upwards at the angle of the mandible and feel anterior (preauricular lymph nodes) and posterior to the ears (posterior auricular lymph nodes). 2.Follow the anterior border of the sternocleidomastoid muscle (anterior cervical chain) down to the clavicle, then palpate up behind the posterior border of the sternocleidomastoid (posterior cervical chain) to the mastoid process. 3. Ask the patient to tilt their head (bring their ear towards their shoulder) each side in turn, and palpate behind the posterior border of the clavicle in the supraclavicular fossa (supraclavicular and infraclavicular lymph nodes). Respiratory causes of lymphadenopathy Lung cancer with metastases Tuberculosis Sarcoidosis Palpate the submental lymph nodes Palpate the submandibular lymph nodes Palpate the auricular lymph nodes Palpate the cervical lymph nodes Palpate the supraclavicular lymph nodes Lymph nodes of the head and neck Cervical lymphadenopathy 11 Posterior chest assessment With the patient still sitting forwards, ask them to fold their arms across their chest so that their hands are touching the opposite shoulder. This results in rotation of the scapulae to better expose the underlying chest wall for assessment. Assess the posterior chest including inspection, chest expansion, percussion, tactile vocal fremitus (or vocal resonance) and auscultation. Allocate adequate time to assessing the posterior aspect of the chest as this is where you are most likely to identify clinical signs. Inspect the posterior chest Assess posterior chest expansion Percuss the posterior chest wall Chest percussion locations Auscultate the posterior chest Posterior chest auscultation locations Assess vocal resonance Final steps Assess for evidence of pitting sacral and pedal oedema (e.g. congestive heart failure). Assess the calves for signs of deep vein thrombosis (e.g. swelling, increased temperature, erythema, visible superficial veins) as the patient may have shortness of breath secondary to pulmonary embolism. Inspect for evidence of erythema nodosum, which can be associated with sarcoidosis. Assess for sacral oedema Assess for pedal oedema Assess the calves Pitting pedal oedema 12 To complete the examination… Explain to the patient that the examination is now finished. Thank the patient for their time. Dispose of PPE appropriately and wash your hands. Summarise your findings. Example summary “Today I examined Mrs Smith, a 64-year-old female. On general inspection, the patient appeared comfortable at rest, with no evidence of shortness of breath. There were no objects or medical equipment around the bed of relevance.” “The hands had no peripheral stigmata of respiratory disease and were symmetrically warm. There was no evidence of a fine tremor or asterixis.” “The pulse was regular at 70 beats per minute and the respiratory rate was 16 breaths per minute.” “On inspection of the face, there were no stigmata of respiratory disease.” “Assessment of the JVP did not reveal any abnormalities. The trachea was centrally located and the cricosternal distance was within the normal range.” “Closer inspection of the chest did not reveal any scars or chest wall deformities. The apex beat was palpable in the 5th intercostal space, in the mid-clavicular line and chest expansion was equal.” “Percussion of the chest revealed normal resonance throughout all lung fields.” “Auscultation of the chest revealed normal vesicular breath sounds, with no added sounds. Vocal resonance was also normal.” “There was no lymphadenopathy on assessment.” “There was no evidence of peripheral oedema and the calves were soft and non-tender.” “In summary, these findings are consistent with a normal respiratory examination.” “For completeness, I would like to perform the following further assessments and investigations.” Further assessments and investigations Suggest further assessments and investigations to the examiner: Check oxygen saturation (SpO2) and provide supplemental oxygen if indicated. Check other vital signs including temperature and blood pressure. Take a sputum sample. Perform peak flow assessment if relevant (e.g. asthma) Request a chest X-ray (if abnormalities were noted on examination) Take an arterial blood gas if indicated (also see ABG interpretation) Perform a full cardiovascular examination if indicated (e.g. cor pulmonale) Reviewer Dr Gareth Hynes Respiratory Registrar References Respiratory sounds. Provided by EasyAuscultation and Andy Howes. Adapted by Geeky Medics. James Heilman, MD. Cyanosis. Licence: CC BY-SA. Adapted by Geeky Medics. James Heilman, MD. Peripheral pallor. Licence: CC BY-SA. Adapted by Geeky Medics. James Heilman, MD. Tar staining. Licence: CC BY-SA. Adapted by Geeky Medics. Desherinka. Finger clubbing. Licence: CC BY-SA. Adapted by Geeky Medics. Waster. Horner’s syndrome. Licence: CC BY. Adapted by Geeky Medics. Ankit Jain, MBBS, corresponding author Anuradha Patel, MD, FRCA and Ian C. Hoppe, MD. Central cyanosis. Licence: CC BY-SA. Adapted by Geeky Medics. James Heilman, MD. Oral candidiasis. Licence: CC BY-SA. Adapted by Geeky Medics. Aurora Bakalli, Tefik Bekteshi, Merita Basha, Afrim Gashi, Afërdita Bakalli and Petrit Ademaj. Pectus excavatum. Licence: CC BY-SA. Adapted by Geeky Medics. Jprealini. Pectus carinatum. Licence: CC BY-SA. Adapted by Geeky Medics. Coronation Dental Specialty Group. Lymphadenopathy. Licence: CC BY-SA. Adapted by Geeky Medics. James Heilman, MD. Pedal oedema. Licence: CC BY-SA. Adapted by Geeky Medics. Singhai A et al. Journal of Dr. NTR University of Health Sciences. Methemoglobinaemia. Licence: CC BY-SA. Suggest an improvement Introduction General inspection Hands Jugular venous pressure Face Inspection of the chest Trachea and cricosternal distance Palpation of the chest Percussion of the chest Auscultation of the chest Lymph nodes Posterior chest assessment Final steps To complete the examination... 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4658	https://www.damtp.cam.ac.uk/user/dbs26/PQM/chap5.pdf	Principles of Quantum Mechanics University of Cambridge Part II Mathematical Tripos David Skinner Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA United Kingdom d.b.skinner@damtp.cam.ac.uk Abstract: These are the lecture notes for the Principles of Quantum Mechanics course given to students taking Part II Maths in Cambridge during Michaelmas Term of 2019. The main aim is to discuss quantum mechanics in the framework of Hilbert space, following Dirac. Along the way, we talk about transformations and symmetries, angular momentum, composite systems, dynamical symmetries, perturbation theory (both time–independent and time–dependent, degenerate and non–degenerate). We’ll ﬁnish with a discussion of various interpretations of quantum mechanics, entanglement and decoherence. Contents 1 Introduction 1 2 Hilbert Space 4 2.1 Deﬁnition of Hilbert Space 4 2.1.1 Examples 5 2.1.2 Dual Spaces 7 2.1.3 Dirac Notation and Continuum States 8 2.2 Operators 10 2.3 Composite Systems 14 2.3.1 Tensor Product of Hilbert Spaces 14 2.4 Postulates of Quantum Mechanics 18 2.4.1 The Generalized Uncertainty Principle 21 3 The Harmonic Oscillator 22 3.1 Raising and Lowering Operators 22 3.2 Dynamics of Oscillators 25 3.2.1 Coherent States 27 3.2.2 Anharmonic Oscillations 27 3.3 The Three-Dimensional Isotropic Harmonic Oscillator 29 4 Transformations and Symmetries 31 4.1 Transformations of States and Operators 31 4.1.1 Continuous Transformations 33 4.2 Translations 34 4.3 Rotations 38 4.3.1 Translations Around a Circle 42 4.3.2 Spin 43 4.4 Time Translations 45 4.4.1 The Heisenberg Picture 46 4.5 Dynamics 47 4.5.1 Symmetries and Conservation Laws 50 4.6 Parity 51 5 Angular Momentum 56 5.1 Angular Momentum Eigenstates (A Little Representation Theory) 57 5.2 Rotations and Orientation 59 5.2.1 Rotation of Diatomic Molecules 60 5.3 Spin 62 5.3.1 Large Rotations 62 5.3.2 The Stern–Gerlach Experiment 63 – i – 5.3.3 Spinors and Projective Representations 65 5.3.4 Spin Matrices 68 5.3.5 Paramagnetic Resonance and MRI Scanners 71 5.4 Orbital Angular Momentum 74 5.4.1 Spherical Harmonics 74 6 Addition of Angular Momentum 78 6.1 Combining the Angular Momenta of Two States 79 6.1.1 j ⊗0 = j 83 6.1.2 1 2 ⊗1 2 = 1 ⊕0 83 6.1.3 1 ⊗1 2 = 3 2 ⊕1 2 84 6.1.4 The Classical Limit 85 6.2 Angular Momentum of Operators 86 6.2.1 The Wigner–Eckart Theorem 86 6.2.2 Dipole Moment Transitions 86 6.2.3 The Runge–Lenz Vector in Hydrogen 86 6.2.4 Enhanced Symmetry of the 3d Isotropic Harmonic Oscillator 89 6.3 Angular Momentum of the Isotropic Oscillator 91 6.4 The Radial Hamiltonian for the Coulomb Potential 93 7 Identical Particles 97 7.1 Bosons and Fermions 97 7.1.1 Pauli’s Exclusion Principle 99 7.1.2 The Periodic Table 100 7.1.3 White Dwarfs, Neutron Stars and Supernovae 104 7.2 Exchange and Parity in the Centre of Momentum Frame 104 7.2.1 Identical Particles and Inelastic Collisions 104 8 Perturbation Theory I: Time Independent Case 107 8.1 An Analytic Expansion 107 8.1.1 Fine Structure of Hydrogen 110 8.1.2 Hyperﬁne Structure of Hydrogen 114 8.1.3 The Ground State of Helium 115 8.1.4 The Quadratic Stark Eﬀect 118 8.2 Degenerate Perturbation Theory 120 8.2.1 The Linear Stark Eﬀect 123 8.3 Does Perturbation Theory Converge? 124 9 Perturbation Theory II: Time Dependent Case 127 9.1 The Interaction Picture 127 9.2 Fermi’s Golden Rule 130 9.2.1 Ionization by Monochromatic Light 132 9.2.2 Absorption and Stimulated Emission 135 9.2.3 Spontaneous Emission 137 – ii – 9.2.4 Selection Rules 140 10 Interpreting Quantum Mechanics 142 10.1 The Density Operator 142 10.1.1 Von Neumann Entropy 144 10.1.2 Reduced density operators 146 10.1.3 Decoherence 147 10.2 Quantum Mechanics or Hidden Variables? 150 10.2.1 The EPR Gedankenexperiment 150 10.2.2 Bell’s Inequality 151 – iii – Acknowledgments Nothing in these lecture notes is original. In particular, my treatment is heavily inﬂuenced by the textbooks listed below, especially Weinberg’s Lectures on Quantum Mechanics, Hall’s Quantum Theory for Mathematicians and Binney & Skinner’s The Physics of Quan-tum Mechanics (though I feel more justiﬁed in plagiarising this one). I’ve also borrowed from previous versions of the lecture notes for this course, by Profs. R. Horgan and A. Davis, which are available online. I am supported by the European Union under an FP7 Marie Curie Career Integration Grant. – iv – Preliminaries These are the notes for the Part II course on Principles of Quantum Mechanics oﬀered in Part II of the Maths Tripos, so I’ll feel free to assume you took the IB courses on Quantum Mechanics, Methods and Linear Algebra last year. For Part II, I’d expect the material presented here to complement the courses on Classical Dynamics, Statistical Physics and perhaps Integrable Systems very well on the Applied side, in addition to being a prerequisite for Applications of Quantum Mechanics next term. On the pure side, the courses Linear Analysis or Functional Analysis and Representation Theory are probably the most closely related to this one. Books & Other Resources There are many textbooks and reference books available on Quantum Mechanics. Diﬀerent ones emphasise diﬀerent aspects of the theory, or diﬀerent applications in physics, or give prominence to diﬀerent mathematical structures. QM is such a huge subject nowadays that it is probably impossible for a single textbook to give an encyclopaedic treatment (and absolutely impossible for me to do so in a course of 24 lectures). Here are some of the ones I’ve found useful while preparing these notes; you might prefer diﬀerent ones to me. • Weinberg, S. Lectures on Quantum Mechanics, CUP (2013). This is perhaps the single most appropriate book for the course. Written by one of the great physicists of our times, this book contains a wealth of information. Highly recommended. • Binney, J.J. and Skinner, D. The Physics of Quantum Mechanics, OUP (2014). The number 1 international bestseller... Contains a much fuller version of these notes (with better diagrams!). Put it on your Christmas list. • Dirac, P. Principles of Quantum Mechanics, OUP (1967). The notes from an earlier version of this course! Written with exceptional clarity and insight, this is a genuine classic of theoretical physics by one of the founders of Quantum Mechanics. • Messiah, A. Quantum Mechanics, Vols 1 & 2, Dover (2014). Another classic textbook, originally from 1958. It provides a comprehensive treat-ment of QM, though the order of the presentation is slightly diﬀerent to the one we’ll follow in this course. • Shankar, R. Principles of Quantum Mechanics, Springer (1994). A much-loved textbook, with particular emphasis on the physical applications of quantum mechanics. • Hall, B. Quantum Theory for Mathematicians, Springer (2013). If you’re more interested in the functional analysis & representation theory aspects of QM rather than the physical applications, this could be the book for you. Much of it is at a more advanced level than we’ll need. – v – • Sakurai, J.J. Modern Quantum Mechanics, Addison–Wesley (1994). By now, another classic book on QM, presented from a perspective very close to this course. • Townsend, J. A Modern to Approach to Quantum Mechanics, 2nd edition, Univer-sity Science Books (2012). Very accessible, but also fairly comprehensive textbook on QM. Starts by considering spin-1 2 particle as a simple example of a two–state system. Textbooks are expensive, so check what’s available in your college library before shelling out – any reasonably modern textbook on QM will be adequate for this course. There are also lots of excellent resources available freely online, including these: • Here are the lecture notes from another recent version of this course. • Here are the lecture notes from the Part II Further Quantum Mechanics course in the Cavendish. • These are the lecture notes from Prof. W. Taylor’s graduate course on QM at MIT. The ﬁrst half of his course covers material that is relevant here, while scattering theory will be covered in next term’s Applications of Quantum Mechanics Part II course. • This is the lecture course on Quantum Mechanics from Prof. R. Fitzpatrick at the University of Texas, Austin. – vi – 1 Introduction In classical mechanics, a particle’s motion is governed by Newton’s Laws. These are second order o.d.e.s, so to determine the fate of our particle we must specify two initial conditions. We could take these to be the particle’s initial position x(t0) and velocity v(t0), or it’s initial position x(t0) and momentum p(t0). Once these are speciﬁed the motion is determined (provided of course we understand how to describe the forces that are acting). This means that we can solve Newton’s Second Law to ﬁnd the values (x(t), p(t)) for t > t01. So as time passes, our classical particle traces out a trajectory in the space M of possible positions and momenta, sketched in ﬁgure 1. The space M is known as phase space and in our case, for motion in three dimensions, M is just R6. In general M comes with a rich geometry known as a Poisson structure; you’ll study this structure in detail if you’re taking the Part II courses on Classical Dynamics or Integrable Systems, and we’ll touch on it later in this course, too. Classical observables are represented by functions f : M →R f : (x, p) 7→f(x, p) . For example, we may be interested in the kinetic energy T = p2/2m, potential energy V (x), angular momentum L = x × p, or a host of other possible quantities. A priori, these functions are deﬁned everywhere over M, but if we want to know about the energy or angular momentum of our speciﬁc particle then we should evaluate them not at some random point (x, p) ∈M, but along the particle’s phase space trajectory. For example, if at time t the particle has position x(t) and momentum p(t), then its angular momentum is x(t) × p(t). Thus the values of the particle’s energy, angular momentum etc. may depend on time, though of course our deﬁnition of these quantities does not2. In this way, 1We can certainly do this numerically, at least for t in a suﬃciently small open subset around t0. 2Technically, we take the pullback of f by the embedding map ı : [t0, ∞) →M. z x y − − + + (x0, p0) (x(t), p(t)) R2d R f Figure 1: A particle’s trajectory in phase space. Observables are represented by functions f : R2d →R, evaluated along a given particle’s trajectory. – 1 – everything we could possibly want to know about a single, pointlike particle is encoded in its phase space trajectory. But that is not our World. To the best of our current experimental knowledge, our World is a quantum, not classical, one. Initially, these experiments were based on careful studies of atomic spectroscopy and blackbody radiation, but nowadays I’d prefer to say that the best evidence of quantum mechanics is simply that we use it constantly in our everyday lives. Each time you listen to music on your stereo, post a photo on Instagram or make a call on your phone you’re relying on technology that’s only become possible due to our understanding of the quantum structure of matter. Whenever you plug some-thing into the mains, you’re using electricity that’s in part generated by nuclear reactions in which quantum mechanics is essential, while much of modern medicine relies on new drugs designed with the beneﬁt of the improved understanding of chemistry that quantum mechanics provides. In such a quantum world, instead of a phase space trajectory, everything we could want to know about a particle is encoded in a vector ψ in Hilbert space H. As you met in IB Quantum Mechanics, this state vector evolves in time according to Schr¨ odinger’s equation. In the quantum world, observables are represented by certain operators O. The operators you saw in IB QM had the same sort of form as observables classical mechanics, such as the kinetic energy operator ˆ T = ˆ p2/2m or angular momentum operator ˆ L = ˆ x × ˆ p. However, rather than being functions, these operators are (roughly) linear maps O : H →H . Again, in the ﬁrst instance these operators are deﬁned throughout H, but if we’re interested in knowing about the energy or angular momentum of our particular quantum particle, then we should ﬁnd out what happens when they act on the speciﬁc ψ ∈H that describes the state of our particle at time t. (See ﬁgure 2.) z x y − − − + + + (x0, p0) (x(t), p R2d f H 0 = O O Figure 2: In Quantum Mechanics, complete knowledge of a particle’s states is determined by a vector in Hilbert space. Observables are represented by Hermitian linear operators O : H →H. – 2 – In the following chapters we’ll study what Hilbert space is and what it’s operators do in a more general framework than you saw last year, building your insight into the mathematical structure of quantum mechanics. Much of this is just linear algebra, but the Hilbert spaces we’ll care about in QM are often inﬁnite-dimensional, so we also make contact with Functional Analysis. Furthermore, although last year you ‘guessed’ the form of quantum operators by analogy with their classical counterparts, we’ll see that at a deeper level many of them can be understood to have their origins in symmetries of space and time; the operators just reﬂect the way these symmetry transformations act on Hilbert space, rather than on (non-relativistic) space-time. In this way, Quantum Mechanics makes contact with Representation Theory. So, mathematically, much of Quantum Mechanics boils down to a mix of Functional Analysis and Representation Theory. It’s even true that it provides a particularly interest-ing example of these subjects. But this is not the reason we study it. We study Quantum Mechanics in an eﬀort to understand and appreciate our World, not some abstract mathe-matical one. You’re all intimately familiar with vector spaces, and you’re (hopefully!) also very good at solving Sturm–Liouville type eigenfunction / eigenvalue problems. But the real skill is in understanding how this formalism relates to the world we see around us. It’s not obvious. Newton’s laws are (at least generically) non-linear diﬀerential equa-tions and we can’t usually superpose solutions. General Relativity teaches us that space-time is not ﬂat. So it’s not at all clear that our particle should in fact be described by a point in a vector space, any more than it was obvious to Aristotle that bodies actually stay in uniform motion unless acted on by a force, or clear to the Ancients that the arrival of solar eclipses, changes of the weather, or any other natural phenomenon are actually gov-erned by calculable Laws, rather than the whims of various gods. For this reason, instead of emphasising how weird and diﬀerent Quantum Mechanics is, I’d prefer to make you appreciate how it actually underpins the physics you’re already familiar with. No matter how good you are at solving eigenvalue problems, if you don’t see how these relate to your everyday physical intuition, knowledge you’ve built up since ﬁrst opening your eyes and learning to crawl, then you haven’t really understood the subject. Let’s begin. – 3 – 2 Hilbert Space The realm of Quantum Mechanics is Hilbert space3, so we’ll begin by exploring the prop-erties of these. This chapter will necessarily be almost entirely mathematical; the physics comes later. 2.1 Deﬁnition of Hilbert Space Hilbert space is a vector space H over C that is equipped with a complete inner product. Let’s take a moment to understand what this means; much of it will be familiar from IB Linear Algebra or IB Methods. Saying that H is a vector space means that it is a set on which we have an operation + of addition, obeying commutativity ψ + φ = φ + ψ associativity ψ + (φ + χ) = (ψ + φ) + χ identity ∃! o ∈H s.t. ψ + o = ψ (2.1) for all ψ, φ, χ ∈H. Furthermore, since it is a vector space over C, we can multiply our vectors by numbers a, b, c, . . . ∈C, called scalars. This multiplication is distributive over H c(ψ + φ) = cψ + cφ distributive in C (a + b)ψ = aψ + bψ . (2.2) In addition, H comes equipped with an inner product. This is a map ( , ) : H × H →C that obeys conjugate symmetry (φ, ψ) = (ψ, φ) linearity (φ, aψ) = a (φ, ψ) additivity (φ, ψ + χ) = (φ, ψ) + (φ, χ) positive–deﬁniteness (ψ, ψ) ≥0 ∀ψ ∈H, with equality iﬀψ = o . (2.3) Note that the ﬁrst two of these imply (aφ, ψ) = a (φ, ψ) so that ( , ) is antilinear in its ﬁrst (leftmost) argument4. Note also that (ψ, ψ) = (ψ, ψ) so that this is necessarily real. Whenever we have inner product, we can deﬁne5 the norm of a state to be ∥ψ∥= p (ψ, ψ) . (2.4) The properties (2.3) also ensure that the Cauchy–Schwarz inequality \|(φ, ψ)\|2 ≤(φ, φ) (ψ, ψ) holds. 3In this course, we’ll focus on Dirac’s formulation of QM, which is based on Hilbert space. This is by far the most commonly used approach. However, there are some approaches to QM (notably deformation quantization and the theory of C∗-algebras) in which Hilbert spaces do not play a prominent role. We won’t discuss any of them. 4In the maths literature, the inner product is often taken to be linear in the left entry and antilinear in the right. We’ll follow the QM literature, which always uses the opposite convention to the maths literature. 5A norm is a more fundamental notion than an inner product: there are normed vector spaces that do not have inner products, while every inner product space has a norm, deﬁned as in (2.4). – 4 – As always, a set of vectors {φ1, φ2, . . . , φn} are linearly independent iﬀthe only solution to c1φ1 + c2φ2 + · · · + cnφn = o (2.5) for ci ∈C is c1 = c2 = · · · = cn = 0. The dimension of the vector space is the largest possi-ble number of linearly independent vectors we can ﬁnd. If there is no such largest number, then we say the vector space has inﬁnite dimension. A set of vectors {φ1, φ2, . . . , φn} is orthonormal if (φi, φj) = ( 0 when i ̸= j, 1 when i = j. (2.6) An orthonormal set {φ1, φ2, . . . , φn} forms a basis of an n-dimensional Hilbert space if every ψ ∈H can be uniquely expressed as a sum ψ = Pn a=1 caφa, with some coeﬃcients ca ∈C. We can determine these coeﬃcients by taking the inner product with φb, since (φb, ψ) = φb, X a caφa ! = X a ca(φb, φa) = cb (2.7) using the linearity of the inner product and orthonormality of the φa. Quantum mechanics makes use of both ﬁnite– and inﬁnite–dimensional Hilbert spaces, as we’ll see. In the inﬁnite dimensional case, we have to decide what we mean by an ‘inﬁnite linear combination’ of (e.g. basis) vectors; not every such inﬁnite sum makes sense as inﬁnite sums such as P∞ a=1 caφa might not converge. To prevent this, one requires that H is complete in the norm (2.4), meaning that every Cauchy sequence {S1, S2, . . .} converges in H. We say that the sum P∞ a=1 caψa converges to a vector Ψ if lim N→∞∥SN −Ψ∥= 0 (2.8) where SN = PN a=1 caψa is the sum of just the ﬁrst N terms. The interplay between the vector space structure and this requirement of completeness can be very subtle in inﬁnite dimensions — as you’ll see if you take the Part II course on Linear Analysis or (next term) Functional Analysis. In this course, we’ll largely ignore such subtleties, not because they’re not interesting, but because they’re a distraction from all the interesting physics we need to learn! 2.1.1 Examples It turns out that all ﬁnite dimensional (complex) Hilbert spaces are isomorphic to Cn for some n, with norm ∥u∥= v u u t n X i=1 \|ui\|2 (2.9) and inner product (v, u) = n X i=1 viui (2.10) – 5 – where {ui} are complex numbers – the components of the vector u in the canonical basis. Being able to diﬀerentiate vectors in Cn requires that we can take limits, and to do this we need to know when the limits exist. Given an inﬁnite sequence {u(1), u(2), . . .} of vectors, their sum P∞ a=1 u(a) is absolutely convergent to a vector in Cn iﬀthe sum of the lengths P∞ a=1 ∥u(a)∥converges as a series of real numbers. Just as with real numbers, if a sequence of vectors converges absolutely, then there is some U ∈Cn to which it converges, in the sense that U − N X a=1 u(a) →0 as N →∞. (2.11) The expresses the completeness of the space Cn – heuristically, there are no points ‘missing’. A simple inﬁnite dimensional generalisation of this is the space of inﬁnite sequences of complex numbers u = (u1, u2, u3, . . .) such that ∞ X i=1 \|ui\|2 < ∞. (2.12) This space is known as ℓ2 and, heuristically, you can think of it as ‘Cn with n = ∞’. The inner product between two such sequences u and v is deﬁned as (v, u) = ∞ X i=1 viui (2.13) as an obvious generalisation of the ﬁnite dimensional case (2.10). The Cauchy–Schwartz inequality gives \|(v, u)\| ≤∥v∥∥u∥< ∞, so this inner product converges provided the norms do. Again, the notion of completeness of ℓ2 with respect to this norm enables us to meaningfully take limits and, ultimately, diﬀerentiate vectors u ∈ℓ2. We will often also meet inﬁnite dimensional vector spaces that are spaces of functions, such as the wavefunctions you dealt with throughout 1B QM. Given two functions ψ : Rn →C and φ : Rd →C, we can deﬁne linear combinations aψ + bφ for any a, b ∈C in the obvious way (aψ + bφ)(x) = aψ(x) + bφ(x) (2.14) where the multipication and addition on the rhs are just those in C, so spaces of functions are naturally inﬁnite dimensional vector spaces (as you saw in IB Methods). To turn this space of functions into a Hilbert space, we ﬁrst give it the norm ∥ψ∥= sZ Rd \|ψ(x)\|2 ddx (2.15) and require that ∥ψ∥< ∞– i.e. that the integral converges. In physics, functions for which ∥ψ∥< ∞holds are called normalisable. Note that just asking for (2.15) to converge is not a very strong restriction, and so our normed function space contains a very wide class of functions, including all piecewise continuous functions that decay suﬃciently rapidly as – 6 – \|x\| →∞, and even some functions that are singular for some discrete values of x, provided these singularities are not strong enough to cause the integral (2.15) to diverge6. Finally, we take the inner product between two functions ψ, φ whose norms are ﬁnite to be (φ, ψ) = Z Rd ¯ φ(x)ψ(x) ddx (2.16) and again this converges by Cauchy–Schwarz. In physics, we often call ψ(x) the wavefunc-tion of the particle, and sometimes call (φ, ψ) the overlap integral between two wavefunc-tions. I’d like to promote Tensor Products to sit here (or hereabouts). I don’t think it’s pos-sible to properly discuss spin, parity etc without ﬁrst admitting that the H corresponding to a physical system may be more involved than the basic examples considered above. I doubt this will be too hard for them to swallow at this stage, as they’ve seen tensor prod-ucts in IB Linear Algebra. (Also in Methods, and especially in L2(R3, d3x) ∼ = ˆ N3L2(R, dx) though of course they won’t recognize this as a tensor product yet. This will require some reorganising, especially of the introduction to rotations, spin & parity. I don’t think this should be too diﬃcult to achieve. 2.1.2 Dual Spaces As with any vector space, the dual H∗of a Hilbert space H is the space of linear maps H →C. That is, an element ϕ ∈H∗deﬁnes a map ϕ : ψ 7→ϕ(ψ) ∈C for every ψ ∈H, such that ϕ : aψ1 + bψ2 7→aϕ(ψ1) + bϕ(ψ2) (2.17) for all ψ1, ψ2 ∈H and a, b ∈C. One way to construct such a map is to use the inner product: given some state φ ∈H we can deﬁne an element (φ, ) ∈H∗which acts by (φ, ) : ψ 7→(φ, ψ) (2.18) i.e. we take the inner product of ψ ∈H with our chosen element φ. The linearity properties of the inner product transfer to ensure that (φ, ) is indeed a linear map — note that since the inner product is antilinear in its ﬁrst entry, it’s important that our chosen element φ sits in this ﬁrst entry. In 1B Linear Algebra you proved that, in ﬁnite dimensions, every element of H∗arises this way. That is, any linear map ϕ : H →C can be written as (φ, ) 6The requirement that the space be complete in the norm (??) is rather subtle. If ∥ψ −φ∥= 0, then we must identify ψ and φ as the same object in our space. This does not necessarily mean that they’re identical as functions, because e.g. they could take diﬀerent at some discrete points xi ⊂R, as the non-zero value of ψ −φ at these discrete points would not contribute to (??). In particular, any function that is non-zero only at a discrete set of points should be identiﬁed with the zero function. The resulting space is known as L2(Rd, ddx) or sometimes just L2 for short. (The L stands for Lebesgue, and is an example of a more general type of normed function space.) L2(Rd, ddx) consists of equivalence classes of Cauchy sequences of functions that are convergent in the norm (2.15). In this course we’ll mostly gloss over such technicalities, and they’re certainly non-examinable. For a deeper discussion of Hilbert space, see the Part II Linear Analysis & Functional Analysis courses. – 7 – for some ﬁxed choice of φ ∈H. This means in particular that the inner product ( , ) provides a vector space isomorphism H∗( , ) ∼ = H . (2.19) Comfortingly, the same result also holds in inﬁnite dimensions7, but it’s non–trivial to prove and is known as the Riesz Representation Theorem. The isomorphism H∗∼ = H is what is special about Hilbert spaces among various other inﬁnite dimensional vector spaces, and makes them especially easy to handle. 2.1.3 Dirac Notation and Continuum States From now on, in this course we’ll use a notation for Hilbert spaces that was introduced by Dirac and is standard throughout the theoretical physics literature. Dirac denotes an element of H as \|ψ⟩, where the symbol “\| ⟩” is known as a ket. An element of the dual space is written ⟨φ\| and the symbol “⟨ \|” is called a bra. The relation between the ket \|φ⟩∈H and the bra ⟨φ\| ∈H∗is what we would previously have written as φ vs (φ, ). The inner product between two states \|ψ⟩, \|φ⟩∈H is then written ⟨φ\|ψ⟩forming a bra-ket or bracket. Note that this implicitly uses the isomorphism H∗∼ = H provided by the inner product, building it into the notation. Recall also that (at least in all physics courses!) ⟨φ\|ψ⟩is antilinear in \|φ⟩. Given an orthonormal basis {\|ea⟩} of H, at least for the cases H ∼ = Cn or H ∼ = ℓ2, in Dirac notation we can expand a general ket \|ψ⟩as \|ψ⟩= X a ψa\|ea⟩ (2.20) in terms of this basis. Then ⟨χ\|ψ⟩= P a,b χbψa⟨eb\|ea⟩= P a χaψa as usual. It’s very useful to be able to extend this idea also to function spaces. In this case, we introduce a ‘continuum basis’ with elements \|a⟩labelled by a continuous variable a, normalised so that ⟨a′\|a⟩= δ(a′ −a) (2.21) using the Dirac δ-function. Then we write \|ψ⟩= Z ψ(a)\|a⟩da (2.22) to expand a general \|ψ⟩in terms of the \|a⟩’s. The point of the normalization (2.22) is that ⟨χ\|ψ⟩= Z χ(b)ψ(a) ⟨b\|a⟩db da = Z χ(b)ψ(a) δ(b −a) db da = Z χ(a)ψ(a) da (2.23) 7We should really be more careful here, though we won’t be concerned with the following subtleties in this course. In inﬁnite dimensions we distinguish the algebraic dual space – the space of all linear functionals ϕ : H →C from the continuous dual, where the number ϕ(ψ) is required to be vary continuously as ψ varies in H. The Riesz Representation applies to the continuous dual. For example, the algebraic dual also contains distributions such as the Dirac δ, acting as δ : ψ 7→δ[ψ] ≡ψ(0) “ = R R δ(x) ψ(x) dx”, with ψ ∈L2(R, dx). Certainly δ(x) is not itself square-integrable, so is not in the Hilbert space. However, since L2(R, dx) contains discontinuous functions, nor does δ[ψ] vary smoothly with ψ. Functional analysis is almost always interested in the continuous dual, and this if often called just the dual space. – 8 – which is just the inner product (and also norm) we gave L2(R, da) before. Indeed, a key example of a ‘continuum basis’ is the position basis {\|x⟩}, where x ∈R. Expanding a general state \|ψ⟩as an integral \|ψ⟩= Z R ψ(x′)\|x′⟩dx′ , (2.24) we see that the complex coeﬃcients are ⟨x\|ψ⟩= Z R ψ(x′) ⟨x\|x′⟩dx′ = ψ(x). (2.25) In other words, the position space wavefunctions we’re familiar with are nothing but the coeﬃcients of a state \|ψ⟩∈H in a particular (position) continuum basis. As always, we could equally choose to expand this same vector in any number of diﬀerent bases. For example, our state \|ψ⟩= R R ψ(x)\|x⟩dx from above can equally be expanded in the momentum basis as \|ψ⟩= R ˜ R ˜ ψ(p)\|p⟩dp, where the new coeﬃcients ˜ ψ(p) = ⟨p\|ψ⟩are the momentum space wavefunction. Later, we’ll show that ⟨x\|p⟩= eixp/ℏ/ √ 2πℏ, so these two sets of coeﬃcients are related by ⟨x\|ψ⟩= Z ˜ R ˜ ψ(p) ⟨x\|p⟩dp = 1 √ 2πℏ Z ˜ R eixp/ℏ˜ ψ(p) dp ⟨p\|ψ⟩= Z R ψ(x) ⟨p\|x⟩dx = 1 √ 2πℏ Z R e−ixp/ℏψ(x) dx . (2.26) (In the ﬁrst line here, we expanded \|ψ⟩in the momentum basis, then took the inner product with \|x⟩, while in the second we expanded the same \|ψ⟩in the postion basis and took the inner product with \|p⟩.) Equation (2.26) is just the statement that the position and momentum space wavefunctions are eachother’s Fourier transforms, again familiar from IB QM. The real point I wish to make is that the fundamental object is the abstract vector \|ψ⟩∈H. All the physical information about a quantum system is encoded in its state vector \|ψ⟩; the wavefunctions ψ(x) or ˜ ψ(p) are merely the expansion coeﬃcients in some basis. Like any other choice of basis, this expansion may be useful for some purposes and unhelpful for others. Finally, a technical8 point. Although using ‘continuum bases’ such as {\|x⟩} or {\|p⟩} is convenient, because it emphasizes the similarities between the inﬁnite and ﬁnite–dimensional cases, it blurs the analysis. In particular, if ⟨x′\|x⟩= δ(x′ −x) then the norm ∥\|x⟩∥2 = δ(x −x) = δ(0) ??? (2.27) so, whatever these objects \|x⟩are, they certainly do not lie in our Hilbert space. It is possible to make good mathematical sense of these by appropriately enlarging our spaces to include spaces of distributions, but for the most part (and certainly in this course) physicists are content to say that continuum states such as \|x⟩are allowable as basis elements, but call them non–normalizable states: actual physical particles are never represented by a non-normalizable state. If you’re interested in a more detailed discussion, I recommend the book by Hall listed above. 8And deﬁnitely non-examinable, at least in this course. – 9 – 2.2 Operators A linear operator A is a map A : H →H that is compatible with the vector space structure in the sense that A(c1\|φ1⟩+ c2\|φ2⟩) = c1A\|φ1⟩+ c2A\|φ2⟩, (2.28) in Dirac notation. All the operators we meet in Quantum Mechanics will be linear9, so henceforth we’ll just call them ‘operators’. Operators form an algebra: Given two such linear operators A, B, we deﬁne their sum αA + βB as (αA + βB) : \|φ⟩7→αA\|φ⟩+ βB\|φ⟩ (2.29) for all α, β ∈C and all \|φ⟩∈H, and take their product AB to be the composition AB : φ 7→A ◦B\|φ⟩= A(B\|φ⟩) (2.30) for all \|φ⟩∈H. One can check that both the sum and product of two linear operators is again a linear operator in the sense of (2.28)10. The operator algebra is associative, so A(BC) = (AB)C, but not commutative and in general AB ̸= BA so that the order in which the operators act is important. The diﬀerence between these two actions is known as the commutator [A, B] = AB −BA . (2.31) This commutator obeys the following properties: antisymmetry [A, B] = −[B, A] linearity [α1A1 + α2A2, B] = α1[A1, B] + α2[A2, B] ∀α1, α2 ∈C Leibniz identity [A, BC] = [A, B]C + B[A, C] Jacobi identity [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 . (2.32) Each of these may easily be veriﬁed from the deﬁnitions (2.31) & (2.28). Commutators of operators arise often in quantum mechanics, as I’m sure you remember from 1B QM. A state ψ ∈H is said to be an eigenstate of an operator A if A\|ψ⟩= aψ\|ψ⟩ (2.33) 9Non-linear operators do play an important role in Quantum Field Theory, where interactions can change the number of particles. 10In functional analysis, a linear operator A : H →H is bounded if ∥A\|ψ⟩∥≤M ∥\|ψ⟩∥for some ﬁxed M > 0, and the space of such bounded linear operators on H is denoted as B(H). Bounded linear operators thus map normalisable states to normalisable states, and so act on the whole Hilbert space H. This usually makes them the nicest ones to deal with. One of the reasons we’ll be largely ignoring the analysis aspects of Hilbert space in this course is that the operators we commonly deal with in quantum mechanics, such as the position and momentum operators X and P, are unbounded. (For example, even when its wavefunction is correctly normalised, a particle can be located at arbitrarily large x ∈R3, or have arbitrarily large momentum.) It’s of course possible to handle such unbounded operators rigorously, but doing so involves an extra layer of technicality that would take us too far aﬁeld here. For further discussion, see that Part II course on Analysis of Functions, or the recommended book by Hall. – 10 – where the number aψ ∈C is known as the eigenvalue. Thus, if \|ψ⟩is an eigenstate of A, acting on \|ψ⟩with A returns exactly the same vector, except that it is rescaled by a number that may depend both on the state and the particular operator. The set of all eigenvalues of an operator A is sometimes called the spectrum of A, while the number of linearly independent eigenstates having the same eigenvalue is called the degeneracy of this eigenvalue. One place where Dirac notation is particularly convenient is that it allows us to label states by their eigenvalues. We let \|q⟩denote an eigenstate of some operator Q with eigenvalue q, so that Q\|q⟩= q\|q⟩. For example, a state that is an eigenstate of the position operator X with position x ∈R3 (representing a particle that is deﬁnitely located at x) is written \|x⟩, while the state \|p⟩is an eigenstate of the momentum operator P with eigenvalue p. There’s a potential confusion here: \|x⟩does not refer to the function x, but rather a state in H whose only support is at x. We saw above that the wavefunction corresponding to \|x⟩was really a δ-function. In this course, I’ll mostly use the (unconventional!) convention that operators are written using capital letters, with their eigenvalues being labelled by the same letter in lowercase. However, I won’t stick to this religiously; a notable and deeply ingrained excep-tion is to use E for the eigenvalues of the Hamiltonian operator H. The extra structure of the inner product on H allows us to also deﬁne the adjoint A† of an operator A by ⟨φ\|A†\|ψ⟩= ⟨ψ\|A\|φ⟩ for all \|φ⟩, \|ψ⟩∈H . (2.34) One can easily11 check that (A+B)† = A†+B† , (AB)† = B†A† , (αA)† = αA† and (A†)† = A . (2.35) Note that it follows that [A, B]† = −[A†, B†]. Note also that the adjoint of the eigenvalue equation A\|a⟩= a\|a⟩is ⟨a\|A† = ⟨a\|a . (2.36) An operator Q is called Hermitian12 if Q† = Q, so that ⟨φ\|Q\|ψ⟩= ⟨ψ\|Q\|φ⟩for all \|φ⟩, \|ψ⟩∈H. Hermitian operators are very special and have a number of important prop-erties. Firstly, suppose \|q⟩is an eigenstate of a Hermitian operator Q, with eigenvalue q. 11At least in the ﬁnite dimensional case. We’ll assume without being careful that it holds in the inﬁnite dimensional case too. 12The terminology ‘Hermitian’ is used in the physics literature, coming from the fact that if H is ﬁnite dimensional, we can represent linear operators obeying Q† = Q by Hermitian matrices. In functional analysis, we’d have to be a little more careful about exactly which states we allow our operator to act on. For example, diﬀerentiating a non-smooth but square–integrable function will typically sharpen its singularities and may render it non-normalisable. Thus the momentum operator −iℏd/dx may take a state in H ∼ = L2(R, dx) out of L2(R, dx), and to keep analytic control we should ﬁrst limit the states on which we allow the momentum operator to act. Being more careful, we’d say that operators obeying ⟨ψ\|Q\|φ⟩= ⟨φ\|Q\|ψ⟩for all \|φ⟩, \|ψ⟩∈H are symmetric, while symmetric operators for which the domain Dom(Q) ⊆H of the operator is the same as the domain Dom(Q†) of the adjoint are called self-adjoint. The distinction between operators that are self-adjoint and those that are merely symmetric is a little technical, but has important consequences for their spectrum and the existence of eigenstates. See e.g. this Wikipedia page for a basic discussion with examples. We’ll largely ignore such subtleties in our course. – 11 – Then q⟨q\|q⟩= ⟨q\|Q\|q⟩= ⟨q\|Q\|q⟩= q⟨q\|q⟩, (2.37) so q = q and the eigenvalues of a Hermitian operator are real. Second, suppose \|q1⟩and \|q2⟩are both eigenstates of Q with distinct eigenvalues q1 and q2. Then (q1 −q2)⟨q1\|q2⟩= ⟨q1\|Q†\|q2⟩−⟨q1\|Q\|q2⟩= 0 , (2.38) where the ﬁrst equality uses the reality of q1 and the second uses the fact that Q is Hermitian. Since q1 ̸= q2 by assumption, we must have ⟨q1\|q2⟩= 0, so eigenstates of Hermitian operators with distinct eigenvalues are orthogonal and, perhaps after rescaling, we can choose them to be orthonormal. In the ﬁnite dimensional case, you proved in 1B Linear Algebra that the set of eigen-vectors of a given Hermitian operator form a basis of H; that is, they form a complete, orthonormal set. This property allows us to express Hermitian operators in a form that is often useful. If {\|n⟩} is an orthonormal basis of eigenstates of a Hermitian operator Q, with eigenvalues {qn}, then we can write Q = X n qn \|n⟩⟨n\| (2.39) where we think of this as acting on some \|ψ⟩∈H by Q\|ψ⟩= X n qn \|n⟩⟨n\|ψ⟩. (2.40) Note that indeed Q\|ψ⟩∈H, since the ⟨n\|ψ⟩are just complex numbers, whereas each term in the sum also involves \|n⟩∈H. In particular, expressing \|ψ⟩in this basis as \|ψ⟩= P m cm \|m⟩gives Q\|ψ⟩= X n,m qncm \|n⟩⟨n\|m⟩= X n qncn \|n⟩ (2.41) assuming the basis is orthonormal. One reason this representation of Q is useful is that it allows us to deﬁne functions of operators. We set f(Q) = X n f(qn) \|n⟩⟨n\| (2.42) which makes sense provided f(qn) is deﬁned for all eigenvalues qn of the original operator. For example, the inverse of an operator Q is the operator Q−1 = X n 1 qn \|n⟩⟨n\| (2.43) which makes sense provided qn ̸= 0 for all n, while ln Q = X n ln(qn) \|n⟩⟨n\| (2.44) is meaningful whenever Q has eigenvalues that are all positive–deﬁnite. – 12 – A particularly important example of this is that we can represent the identity operator 1H on H as 1H = X n \|n⟩⟨n\| or 1H = Z \|q⟩⟨q\| dq (2.45) where {\|n⟩} or {\|q⟩} are any bases of H, respectively for discrete or continuous labels. This is often convenient for moving between diﬀerent bases. For example, if {\|p⟩} denotes a complete set of eigenstates of the momentum operator P, with P\|p⟩= p\|p⟩, then to represent this operator acting on a state \|ψ⟩in the position basis we write ⟨x\|P\|ψ⟩= Z ⟨x\|P\|p⟩⟨p\|ψ⟩dp = Z ⟨x\|p⟩p ˜ ψ(p) dp = 1 √ 2πℏ Z eixp/ℏp ˜ ψ(p) dp = −iℏ∂ ∂x 1 √ 2πℏ Z eixp/ℏ˜ ψ(p) dp . (2.46) In the ﬁrst equality here we’ve squeezed the identity operator 1H = R \|p⟩⟨p\| dp in between the momentum operator P and the general state \|ψ⟩, then used the fact that \|p⟩is an eigenstate of P, and ﬁnally used our previous expression for ⟨x\|p⟩and standard properties of Fourier transforms. We recognise the integral in the ﬁnal expression as the position space wavefunction of \|ψ⟩, so altogether we have ⟨x\|P\|ψ⟩= −iℏ∂ ∂x⟨x\|ψ⟩, (2.47) or ˆ Pψ(x) = −iℏ∂ψ/∂x in 1B notation. This and similar manipulations will be used re-peatedly throughout the course, so it’s important to make sure you’re comfortable with them. If H is ﬁnite dimensional, we can represent any linear operator on H by a matrix. We pick an orthonormal basis {\|n⟩} and deﬁne Anm := ⟨n\|A\|m⟩ (2.48) as usual. In particular, inserting the identity operator in the form I = P n \|n⟩⟨n\| shows that the matrix elements of the product operator AB are (AB)km = ⟨k\|AB\|m⟩= ⟨k\|A X n \|n⟩⟨n\| ! B\|m⟩ = X n ⟨k\|A\|n⟩⟨n\|B\|m⟩= X n AknBnm , (2.49) which just corresponds to usual matrix multiplication. Hermitian operators are represented by a Hermitian matrices, i.e. ones whose elements obey (A†)nm = Amn, hence the name13. 13Though see the previous footnote for subtleties in the inﬁnite dimensional case. – 13 – 2.3 Composite Systems We often have to deal with systems with more than a single degree of freedom. This could be just a single particle moving in three dimensions rather than one, or a composite system such as a Hydrogen atom consisting of both an electron and a proton, or a diamond which consists of a very large number of carbon atoms. In the case of a macroscopic object, our system could be made up of a huge number of constituent parts, each able to behave diﬀerently, at least in principle. It should be intuitively clear that the more complicated our system is, meaning the more independent degrees of freedom it possesses, the larger a Hilbert space we’ll need if we wish to encode all its possible states. We’ll now understand how quantum mechanics handles systems with more than one degree of freedom by taking the tensor product of the Hilbert spaces of its constituent parts. 2.3.1 Tensor Product of Hilbert Spaces Hilbert spaces are vector spaces over C, so we can try to deﬁne their tensor product in the same way as we would for any pair of vector spaces. Recall from IB Linear Algebra that if H1 and H2 are two, ﬁnite dimensional Hilbert spaces, with {\|ea⟩}m a=1 a basis of H1 and {\|fα⟩}n α=1 a basis of H2, then the tensor product H1 ⊗H2 is again a vector space over C spanned by all pairs of elements \|ea⟩⊗\|fα⟩chosen from the two bases14. It follows that dim(H1 ⊗H2) = dim(H1) dim(H2) (2.50) provided these are ﬁnite. This should be familiar from IB Linear Algebra. It’s important to stress that, as with any vector space, a general element of H1 ⊗H2 is a linear combination of these basis elements. In particular, although general elements of H1 and H2 can be written \|ψ1⟩= m X a=1 ca\|ea⟩ and \|ψ2⟩= m X α=1 dα\|fα⟩, (2.51) 14If you’re a purist (and the PQM Tripos examiners are not), then there’s something slightly unsatisfactory about this deﬁnition of the tensor product vector space, which is that it apparently depends on a choice of basis on both H1 and H2. We can do better, just using the abstract vector space structure, as follows. For any pair of vector spaces V and W, we ﬁrst deﬁne the free vector space F(V × W) to be the set of all possible pairs of elements chosen from V and W. This is a vector space with the sum of a pair (v, w) and a pair (v′, w′) denoted simply by (v, w) + (v′, w′) (Do not confuse the pair (v, w) ∈F(V × W) with an inner product! Here v and w live in diﬀerent spaces.) By itself, F(V × W) does not care that V and W are already vector spaces. For example, if {ei} is a basis for V and v = c1e1 + c2e2 for some coeﬃcients c1,2, the elements (v, w) and c1(e1, w) + c2(e2, w) are nonetheless distinct in F(V × W). We’d like to teach the free vector space about the structure inherited from V and W individually. We thus deﬁne the tensor product V ⊗W to be the quotient F(V × W)/ ∼under the equivalence relations (v, w) + (v′, w) ∼(v + v′, w) , (v, w) + (v, w′) ∼(v, w + w′) and c(v, w) ∼(cv, w) ∼(v, cw) for all v ∈V , w ∈W and all c ∈C. We denote the equivalence class of the pair (v, w) by v ⊗w. – 14 – it is not true that a general element of H1 ⊗H2 necessarily takes the form \|ψ1⟩⊗\|ψ2⟩. Rather, a general element of H1 ⊗H2 may be written as \|Ψ⟩= X a,α raα \|ea⟩⊗\|fα⟩ (2.52) In particular, elements of the form \|ψ1⟩⊗\|ψ2⟩∈H1 ⊗H2 are speciﬁed by only dim H1 + dim H2 complex coeﬃcients, vastly fewer than is required to specify a generic element (2.52). Elements of the form \|ψ⟩⊗\|φ⟩are sometimes called simple, while in physics generic ele-ments of the form (2.52) are said to be entangled. In section ?? we’ll explore some of the vast resources this entanglement opens up to quantum mechanics; you’ll see much more of it next term if you take the course on Quantum Information & Computation. In order to make H1⊗H2 a Hilbert space, rather than just a vector space, we must give it an inner product. We do this in the obvious way: if ⟨\| ⟩1 and ⟨\| ⟩2 denote inner products on H1 and H2, we ﬁrst deﬁne the inner product between each pair of basis elements of H1 ⊗H2 by (⟨ea\| ⊗⟨fα\|) (\|eb⟩⊗\|fβ⟩) = ⟨ea\|eb⟩1 ⟨fα\|fβ⟩2 , (2.53) and then extend it to arbitrary states \|Ψ⟩by linearity. Note in particular that if {\|ea⟩} and {\|fα⟩} are orthonormal bases of their respective Hilbert spaces, then {\|ea⟩⊗\|fα⟩} will also be an orthonormal basis of H1 ⊗H2. The most common occurrence of this is simply a single particle moving in more than one dimension. For example, suppose a quantum particle moves in R2, described by Cartesian coordinates (x, y). If {\|x⟩}x∈R is a complete set of position eigenstates for the x-direction, and {\|y⟩}y∈R likewise a complete set for the y-direction, then the state of the particle can be expanded as \|ψ⟩= Z R×R ψ(x, y) \|x⟩⊗\|y⟩dx dy (2.54) where {\|x⟩⊗\|y⟩}x,y ∈R form a continuum basis of the tensor product space. Note that in general, our wavefunction is not a simple product ψ(x, y) ̸= ψ1(x)ψ2(y) of wavefunctions in the two directions separately, though it may be possible to write it as the sum of such products, as you met repeatedly when studying separation of variables in IB Methods. In this continuum case, the inner product between any two states is ⟨χ\|ψ⟩= Z R2×R2 χ(x′, y′) ψ(x, y) ⟨x′\|x⟩⟨y′\|y⟩dx′ dy′ dx dy = Z R2×R2 χ(x′, y′) ψ(x, y) δ(x −x′) δ(y −y′) dx′ dy′ dx dy = Z R2 χ(x, y) ψ(x, y) dx dy . (2.55) This is exactly the structure of L2(R2 d2x), and we identify15 L2(R2, d2x) ∼ = L2(R, dx) ⊗ L2(R, dy). 15There’s actually one more – again non-examinable! – step in the inﬁnite dimensional case: we must take the completion of L2(R, dx) ⊗L2(R, dy) in the norm associated to this inner product. There is no problem – 15 – More commonly, physics considers quantum systems that live in R3, described by a Hilbert space H ∼ = L2(R3, d3x) obtained from the tensor product of three copies of the Hilbert space for a one-dimensional system. We can expand a general state in this Hilbert space as \|ψ⟩= Z R3 ψ(x, y, z) \|x⟩⊗\|y⟩⊗\|z⟩dx dy dz = Z R3 ψ(x) \|x⟩d3x (2.57) where the last expression is just a convenient shorthand. Going further, to describe a composite system such as a Hydrogen atom that consists of two particles (an electron and proton), we need to take the tensor product of the individual Hilbert spaces of each particle. Thus (neglecting the spin of the electron and proton) the atom is described by a state \|Ψ⟩= Z R6 Ψ(xe, xp) \|xe⟩⊗\|xp⟩d3xe d3xp (2.58) where Ψ(xe, xp) = (⟨xe\| ⊗⟨xp\|)\|Ψ⟩is the amplitude for the atom’s electron to be found at xe while its proton is at xp. Once we know the Hilbert spaces appropriate to describe the fundamental constituents of our system, we can build up the Hilbert space for the combined system by taking tensor products. We should then ask ‘What is the correct Hilbert space to use to describe the fundamental particles in our system?’. Ultimately, this question can only be determined by carrying out an experiment. For example, experiments performed by Stern & Gerlach (see section 5.3.2) showed that a single electron is in fact described by a Hilbert space He = L2(R3, d3x) ⊗C2, formed from the tensor product of the electron’s position space wavefunction with the two-dimensional Hilbert space C2 describing the electron’s ‘internal’ degrees of freedom. Thus, letting {\|↑⟩, \|↓⟩} be an orthonormal basis of C2, a generic state of the electron (whether or not it’s part of a Hydrogen atom) is \|Ψ⟩= \|φ⟩⊗\|↑⟩+ \|χ⟩⊗\|↓⟩, (2.59) if our general state ψ(x, y) can be written as a ﬁnite sum of products of square-integrable wavefunctions, say φa(x) and ρb(y) in each of the two directions separately, for then Z R2 \|ψ(x, y)\|2 dx dy = Z R2 X a,a′,b,b′ φa′(x)ρb′(y) φa(x)ρb(y) dx dy =  X a,a′ Z R φa′(x) φa(x) dx    X b,b′ Z R ρb′(y) ρb(y) dy  < ∞ with convergence guaranteed by the properties of the inner product on each copy of L2(R, dx). However, there are functions in L2(R2, d2x) that cannot be so expressed as a ﬁnite sum. A simple example is the function f : R2 →C given by f : (x, y) 7→ ( 1 when p x2 + y2 ≤a 0 else . (2.56) It is clear that this function is square-integrable over R2, but it cannot be written as a ﬁnite sum of products of functions of x and y separately. It thus lies in the completion ofL2(R, dx) ⊗L2(R, dy), but not in this space itself. The Hilbert space completion of H1 ⊗H2 is often denoted H1 ˆ ⊗H2, so more correctly we have L2(R2, d2x) ∼ = L2(R, dx)ˆ ⊗L2(R, dy). We’ll ignore such subtleties in this course, saving a proper treatment for the Functional Analysis course. – 16 – given in the position representation by ⟨xe\|Ψ⟩= φ(xe)\|↑⟩+ χ(xe)\|↓⟩ (2.60) involving a pair of wavefunctions φ, χ ∈L2(R3, d3x). We’ll investigate this in detail in section 5.3. Let’s now understand how operators act on our composite system. Given linear oper-ators A : H1 →H1 and B : H2 →H2, we deﬁne the linear operator A ⊗B : H1 ⊗H2 → H1 ⊗H2 by (A ⊗B) : \|ea⟩⊗\|fα⟩7→(A\|ea⟩) ⊗(B\|fα⟩) (2.61) and we extend this deﬁnition to arbitrary states in H1 ⊗H2 by linearity. In particular, the operator A acting purely on H1 corresponds to the operator A ⊗1H2 when acting on the tensor product, and similarly 1H1 ⊗B acts non–trivially just on the second factor in the tensor product. Note that since they act on separate Hilbert spaces, [ A ⊗1H2 , 1H1 ⊗B ] = 0 (2.62) for any operators A, B, even if these operators happen not to commute when acting on the same Hilbert space. As an example, let’s again consider the case of the Hydrogen atom, where the Hamil-tonian takes the form H = P2 e 2me ⊗1p + 1e ⊗P2 p 2mp + V (Xe, Xp) . (2.63) The kinetic terms for the electron and proton each act non–trivially only on one factor in the tensor product, while the Coulomb interaction V (Xe, Xp) = − e2 \|Xe −Xp\| (2.64) depends non–trivially on the location of each particle. In this case, since V depends only on the relative separation of the two, it’s actually more convenient to view the tensor product diﬀerently, writing HHyd = He ⊗Hp = Hcom ⊗Hrel (2.65) to split it in terms of the states describing the behaviour of the centre of mass and those describing the relative states. Deﬁning the centre of mass and relative operators Xcom = meXe + mpXp me + mp Pcom = Pe + Pp X = Xe −Xp P = mpPe −mePp me + mp , (2.66) the Hamiltonian becomes H = P2 com 2M ⊗1rel + 1com ⊗ P2 2µ −e2 \|X\| (2.67) – 17 – where M = me + mp is the total mass and µ = memp/M the reduced mass. Such calcula-tions should be familiar from studying planetary orbits in IA Dynamics & Relativity. Henceforth we’ll often omit the ⊗symbol — both in states and in operators — when the meaning is clear. 2.4 Postulates of Quantum Mechanics So far, we’ve just been doing linear algebra, talking about Hilbert spaces in a fairly abstract way. Let’s now begin to connect this to some physics. The ﬁrst postulate of quantum mechanics says that our system is described (up to a redundancy discussed below) by some state \|ψ⟩∈H, and that any complete set of orthogo-nal states {\|φ1⟩, \|φ2⟩, . . .} is in one-to-one correspondence with all the possible outcomes of the measurement of some quantity. Further, if a system is prepared to be in some general state \|ψ⟩= X a ca\|φa⟩ (2.68) then the probability that the measurement will yield an outcome corresponding to the state \|φb⟩is Prob(\|ψ⟩→\|φb⟩) = \|⟨φb\|ψ⟩\|2 ⟨ψ\|ψ⟩⟨φb\|φb⟩ (2.69) or equivalently Prob(\|ψ⟩→\|φb⟩) = \|cb\|2 ⟨φb\|φb⟩/ ⟨ψ\|ψ⟩. The denominators in (2.69) allow for the possibility that the states \|ψ⟩and \|φb⟩may have arbitrary norm. Notice that Prob(\|ψ⟩→\|φb⟩) ≥0 and also, since ⟨ψ\|ψ⟩= P a,b cacb⟨φb\|φa⟩= P b \|cb\|2⟨φb\|φb⟩by the orthogonality of the \|φb⟩s, that X b Prob(\|ψ⟩→\|φb⟩) = X b \|cb\|2⟨φb\|φb⟩ ⟨ψ\|ψ⟩ = ⟨ψ\|ψ⟩ ⟨ψ\|ψ⟩= 1 . (2.70) Thus the probabilities sum to one. Let me make some remarks. Firstly, note that we’ve not speciﬁed exactly which Hilbert space H we should use; is H to be Cn, or ℓ2, or L2(R3, d3x), or something else? In fact, the appropriate choice depends on the system we’re trying to describe. For example, a single point particle with no internal structure could be described using16 H ∼ = L2(R3, d3x) whereas to specify the state of a more complicated system with several degrees of freedom we’ll need a larger Hilbert space, encoding for example the location of its centre of mass, but also details of the system’s orientation. We’ll consider how to do this in chapter 2.3. Second, the fact that quantum mechanics only predicts the probabilities of obtaning a given experimental outcome – even in principle – is one of its most puzzling features. Orig-inally, this probabilistic interpretation of wavefunctions was due to Max Born, and (2.69) is sometimes known as the Born rule. He found that, solving the Schr¨ odinger equation for scattering a particle oﬀa generic obstacle in R3, the particle’s wavefunction would typically 16Note that we only require H to be isomorphic to the space L2(R3, d3x) of normalizable position-space wavefunctions. We could equally well describe our structureless particle using a momentum space wave-function, and indeed L2(e R3, d3p) ∼ = L2(R3, d3x), with the isomorphism provided by the Fourier transform. – 18 – become very spread out so as to have appreciable magnitude over a wide region. This is in contrast to our experience of particles bouncing oﬀtargets, each in a certain speciﬁc way. For example, recall that when performing his gold foil experiment to probe the structure of the atom, Rutherford usually observed α-particles to plough straight through, but oc-casionally saw them rebound back revealing the presence of the dense nucleus. Nucleus or not, each α-particle was observed at a speciﬁc angle from the target, and did not itself ‘spread out’. Reconciling this observation with the behaviour of the wavefunction is what led Born to suggest that quantum mechanics should be inherently probabilistic. His rea-soning was perhaps not totally sound, and we’ll explore the interpretative meaning of QM further and from a more modern perspective in chapter 10. As a ﬁnal remark on this postulate, to clean up the basic probability rule (2.69), it’s usually convenient to assume our states are normalised, meaning that ⟨ψ\|ψ⟩= 1 , (2.71) and to expand them in an orthonormal basis {\|φb⟩}. In this case, the above probability is simply Prob(\|ψ⟩→\|φb⟩) = \|⟨φb\|ψ⟩\|2 , (2.72) while the quantity ⟨φb\|ψ⟩∈C itself is often called the probability amplitude, or just am-plitude for short. In what follows, we’ll almost always use this simpler expression, but it’s important to recall that it only holds in the case of a properly normalised state expanded in an orthonormal basis. Even when all states are properly normalised, we’re still free to redeﬁne \|ψ⟩→eiα\|φ⟩for some constant phase α. This phase drops out of both the normalisation condition and the Born rule (2.72) for the probability. Taking into account both the normalisation condition and the phase freedom, in quantum mechanics physical states do not correspond to elements \|ψ⟩∈H but rather to rays through the origin. That is, provided \|ψ⟩̸= 0, the entire family of states \|ψλ⟩= λ\|ψ⟩ for λ ∈C∗ (2.73) deﬁne the same physical system — by tuning \|λ\| = 1/⟨ψ\|ψ⟩we can always ﬁnd a member this family that’s correctly normalised, and the (constant) phase of λ cannot aﬀect the probability of any outcome of a physical experiment. (This is the redundancy referred to above.) Note in particular that the zero vector 0 – the unique element of H with vanishing norm – never represents a physical state. Geometrically, the equivalence \|ψ⟩∼λ\|ψ⟩for all \|ψ⟩∈H/{0} and all λ ∈C∗means that physical states actually correspond to elements of the projective Hilbert space PH. As with any projective space, it’s often most convenient to work simply with normalised vectors \|ψ⟩∈H themselves, and recall that the overall phase is irrelevant. The second postulate of quantum mechanics states that observable quantities are rep-resented by Hermitian linear operators. In particular, upon measurement of a quantity corresponding to a Hermitian operator Q, a state is certain to return the deﬁnite value q iﬀit is an eigenstate of Q with eigenvalue q. Let \|n⟩be a complete, orthonormal set of – 19 – eigenstates of some Hermitian operator Q, with Q\|n⟩= qn\|n⟩. Then we can expand an arbitrary state \|ψ⟩in this basis as \|ψ⟩= P n cn\|n⟩. The expectation value of Q in this state is ⟨Q⟩ψ = ⟨ψ\|Q\|ψ⟩= X m,n cm cn⟨m\|Q\|n⟩= X n qn\|cn\|2 , (2.74) using the orthonormality of the basis. This is just the sum of values of Q possessed by the states \|n⟩, weighted by the probability that \|ψ⟩agrees with \|n⟩. Since operators representing observables are Hermitian, for any such operator we have ⟨ψ\|Q2\|ψ⟩= ⟨ψ\|Q†Q\|ψ⟩= ∥Q\|ψ⟩∥2 ≥0 (2.75) and hence 0 ≤⟨ψ\|(Q −⟨Q⟩ψ)2\|ψ⟩= ⟨Q2⟩ψ −⟨Q⟩2 ψ . (2.76) This shows that ⟨Q2⟩ψ ≥⟨Q⟩2 ψ, with equality iﬀ\|ψ⟩is an eigenstate of Q. We deﬁne the rms deviation ∆ψQ of Q from its mean ⟨Q⟩ψ in a state \|ψ⟩by ∆ψQ = q ⟨ψ\|(Q −⟨Q⟩ψ)2\|ψ⟩. (2.77) This is just the usual deﬁnition familiar from probability. As always, it gives us a measure of how ‘spread’ a state is around the eigenstate of Q. This implies that we can be sure of the value we’ll obtain when measuring an observable quantity only if we somehow know that our state is in an eigenstate of the corresponding operator before carrying out the measurement. Let me emphasize that these two postulates do not say anything about how the physical process of actually carrying out a measurement is described in the formalism of QM. (Nor do they even tell us what constitutes ‘makes a measurement’.) In particular, we do not say that measuring the observable corresponding to some Hermitian operator Q has anything to do with the mathematical operation of acting on our state \|ψ⟩with Q. According to the Copenhagen interpretation, if we measure the observable corresponding Q and ﬁnd the result q, then immediately after this measurement, our system must be the state \|q⟩, because we’ve just learned that it does indeed have this value. The wavefunction is assumed to have collapsed from whatever it was before we measured it to \|q⟩. The Copenhagen interpretation is the most widely accepted version of quantum mechanics, but it’s not uncontroversial. We’ll try to understand measurement from a deeper perspective in section 10.2. The ﬁnal postulate of quantum mechanics is that the state \|ψ⟩of our system evolves in time according to the Schr¨ odinger equation iℏ∂ ∂t\|ψ⟩= H\|ψ⟩ (2.78) where H is some distinguished operator known as the Hamiltonian. – 20 – 2.4.1 The Generalized Uncertainty Principle Suppose we have two Hermitian operators A and B. Let’s deﬁne \|ψA⟩to be the state A\|ψ⟩−⟨A⟩ψ\|ψ⟩so that (∆ψA)2 = ∥\|ψA⟩∥2, and similarly \|ψB⟩= B\|ψ⟩−⟨B⟩ψ\|ψ⟩. Then the Cauchy–Schwarz inequality says (∆ψA)2(∆ψB)2 = ∥\|ψA⟩∥2 ∥\|ψB⟩∥2 ≥\|⟨ψA\|ψB⟩\|2 . (2.79) Expanding the rhs, we have ⟨ψA\|ψB⟩= ⟨ψ\|(A −⟨A⟩ψ)(B −⟨B⟩ψ)\|ψ⟩= ⟨ψ\|AB −⟨A⟩ψ⟨B⟩ψ\|ψ⟩ (2.80) Since we’re considering Hermitian opertaors A, B, we have ⟨ψ\|AB\|ψ⟩= ⟨ψ\|BA\|ψ⟩, so Im ⟨ψA\|ψB⟩= 1 2i⟨ψ\|[A, B]\|ψ⟩. (2.81) Combining this with the Cauchy–Schwarz inequality gives the generalised uncertainty re-lation (∆ψA)2(∆ψB)2 ≥\|⟨ψA\|ψB⟩\|2 ≥1 4\|⟨[A, B]⟩ψ\|2 . (2.82) In particular, if [A, B] ̸= 0 we cannot in general ﬁnd states that simultaneously have deﬁnite values for both the quantities represented by A or B. As a particular case, recall from IB QM that the position and momentum operators obey the commutation relation [X, P] = iℏ. Using this in (2.82) gives ∆ψX ∆ψP ≥ℏ 2 (2.83) so that no quantum state can have a deﬁnite value of both position and momentum17. This is the original uncertainty principle of Heisenberg. 17The inequality is in fact saturated by states whose position space wavefunctions are Gaussians, so despite the fact that our derivation above neglected several positive semi–deﬁnite terms, we cannot place a higher bound on the minimum uncertainty. – 21 – 3 The Harmonic Oscillator I now want to use Dirac’s formalism to study a simple system – the one-dimensional harmonic oscillator – with which you should already be familiar. Our aim here is not to learn new things about harmonic oscillators; indeed, we’ll mostly just recover results you’ve known about since you ﬁrst heard of simple harmonic motion. Rather, our aim is to get accustomed to this more abstract approach to QM, seeing how it’s a very powerful way to think about the subject. The steps we follow in our treatment of the harmonic oscillator will form a good prototype for the way we’ll approach many other problems in QM. We’ll see these same steps repeated in various contexts throughout thef course. 3.1 Raising and Lowering Operators The Hamiltonian of a harmonic oscillator of mass m and classical frequency ω is H = P 2 2m + 1 2mω2X2 . (3.1) where X and P are the position and momentum operators, respectively. To analyse this, we begin by deﬁning the dimensionless combination A = 1 √ 2mℏω (mωX + iP) . (3.2) A is not a Hermitian operator, but since X and P are both Hermitian, we see that the adjoint of A is A† = 1 √ 2mℏω (mωX −iP) . (3.3) Roughly, the motivation for introducing these operators is that they allow us to ‘factorize’ the Hamiltonian. More precisely, we have A†A = 1 2mℏω (mωX −iP) (mωX + iP) = 1 2mℏω P 2 + m2ω2X2 + imω[X, P] = H ℏω −1 2 (3.4) so we can write our Hamiltonian as H = ℏω A†A + 1 2 = ℏω N + 1 2 , (3.5) where N = A†A . (3.6) A and A† are often called lowering and raising operators, respectively, whilst N is often called the number operator. (The reason for these names will become apparent soon.) No-tice that computing the spectrum of H is obviously equivalent to computing the spectrum of N. – 22 – Whenever we’re presented with some new operators, the ﬁrst thing to do is to work out their commutation relations. In this case, the fundamental commutation relations [X, P] = iℏshow that h A, A†i = 1 2mℏω m2ω2[X, X] −imω[X, P] + imω[P, X] + [P, P] = −imω 2mℏω ([X, P] −[P, X]) = 1 , (3.7) whilst [A, A] = 0 = [A†, A†] trivially. It will also be useful to compute commutators involving N. We have [N, A†] = [A†A, A†] = A†[A, A†] + [A†, A†]A = A† , (3.8) where the ﬁnal step uses (3.7). We learn that, similarly, [N, A] = −A by taking the adjoint of (3.8). Let’s see what these commutators teach us about the possible energy levels. Suppose that \|n⟩is a correctly normalised eigenstate of N, so that N\|n⟩= n\|n⟩and ⟨n\|n⟩= 1. We have NA†\|n⟩= (A†N + [N, A†])\|n⟩= A†(N + 1)\|n⟩= (n + 1)A†\|n⟩ (3.9) so A†\|n⟩is an eigenstate of N with eigenvalue n + 1. Similarly, NA\|n⟩= (AN + [N, A])\|n⟩= (n −1)A\|n⟩ (3.10) so A\|n⟩is an eigenstate of N with eigenvalue n−1. Acting with A† thus raises the eigenvalue of N by one unit, whilst acting with A lowers it by one, giving these operators their names. The above algebra shows that if we can ﬁnd just one energy eigenstate \|n⟩, then we can construct a whole family of them by repeatedly applying either A† or A. The energies of these new states will be (n+k + 1 2)ℏω for some k ∈Z. However, we can’t yet conclude that the energy levels are quantized, because we don’t yet know that there isn’t a continuum of possible starting-points \|n⟩(or indeed any!). This brings us to the second key step: we must now investigate the norm of our states. Since N is Hermitian all its eigenvalues must certainly be real, but in fact they’re also non-negative, because n = n⟨n\|n⟩= ⟨n\|N\|n⟩= ⟨n\|A†A\|n⟩= ∥A\|n⟩∥2 ≥0 (3.11) with n = 0 iﬀA\|n⟩= 0, by properties of the norm. If A\|n⟩̸= 0 then it is an eigenstate of N with eigenvalue n −1, but we’ve just shown that there are no states in H with negative eigenvalue for N, so this lowering process must terminate. That will be the case iﬀn is a non-negative integer. Putting all this together, we have a ground state \|0⟩of energy 1 2ℏω and an inﬁnite tower of excited states \|n⟩of energy En = n + 1 2 ℏω , where n ∈N0 . (3.12) – 23 – These energy levels should be familiar from IB QM. Acting on an energy eigenstate with the raising operator does not necessarily give us an excited state that is correctly normalised. In d = 1, energy eigenstates of the harmonic oscillator are non-degenerate18, so we must have A†\|n⟩= cn\|n + 1⟩for some constant cn (which may depend on n). Taking the norm of both sides shows that \|cn\|2 = ∥A†\|n⟩∥2 = ⟨n\|AA†\|n⟩= ⟨n\|N + 1\|n⟩= n + 1 . (3.13) Therefore, the correctly normalised state is \|n + 1⟩= 1 √n + 1A†\|n⟩= 1 p (n + 1)! (A†)n+1\|0⟩. (3.14) Likewise, you can check that \|n −1⟩= 1 √nA\|n⟩for n ≥1, whilst A\|0⟩= 0. We’ve seen that everything about the energy levels of the harmonic oscillator follows from i) the algebra (i.e. the commutation relations) of raising and lowering operators, together with ii) considering the norm. Again, we’ll follow these same two steps in analysing the eigenvalues and eigenstates of various operators throughout this course. This also parallels what you did in IB QM: there, by looking for series solutions of Schr¨ odinger’s equation, you could ﬁnd an energy eigenstate for all E ∈R. However, these eigenstates were only normalizable if the series you found terminated. It was requiring this termination (i.e. normalizability) that lead to quantization of the energies. I hope I’ve persuaded you that the algebraic approach above is somewhat cleaner and more eﬃcient than looking for series solutions to Schr¨ odinger’s equation. Nothing has been lost, and we can easily use Dirac’s formalism to recover the explicit wavefunctions of our eigenstates. For example, in the position representation, the deﬁning equation A\|0⟩= 0 of the ground state becomes 0 = √ 2mℏω ⟨x\|A\|0⟩= ⟨x\|mωX + iP\|0⟩= mωx + ℏd dx ψ0(x) , (3.15) where ψ0(x) = ⟨x\|0⟩is the position space wavefunction of our ground state. This is a ﬁrst order o.d.e. whose solution is ψ0(x) = mω πℏ 1/4 exp −mω 2ℏx2 , (3.16) 18Here is a sketch of the proof. Suppose ψ1 and ψ2 are two normalizable solutions of the d = 1 Schr¨ odinger equation −ℏ2ψ′′/2m + V (x)ψ = Eψ with the same energy. Then the Wronskian W(ψ1, ψ2) = ψ1ψ′ 2 −ψ2ψ′ 1 must be constant ∀x, as follows by diﬀerentiating and using the TISE. Since ψ1 and ψ2 are both normaliz-able, they must decay to zero as \|x\| →∞. Provided their derivatives remain ﬁnite, limx→∞W(ψ1, ψ2) = 0 and hence it will vanish everywhere. This implies that ψ2 ∝ψ1, so our two normalizable solutions would be linearly dependent and agree (up to a constant phase) once normalized. Must the derivatives remain ﬁnite as x →∞? A heuristic argument says that for any energy eigenstate E = ⟨T⟩+ ⟨V ⟩and ⟨T⟩= ∥P\|ψ⟩∥2/2m, so if E is ﬁnite and the potential is bounded from below, we must have ∥P\|ψ⟩∥2 = ℏ2 R R \|ψ′(x)\|2 dx < ∞also. This should not be convincing, and a more careful argument is given in pp. 96–105 of Messiah, A. Quantum Mechanics, vol. 1. The careful proof shows that in d = 1 energy eigenstates with energy E are indeed non-degenerate provided ∃x0 ∈R s.t. V (x) > E ∀x > x0. In other words, if the particle does not have enough energy to exist classically anywhere beyond x = x0. – 24 – where the overall constant is ﬁxed (up to phase) by the normalization requirement ⟨0\|0⟩= R R \|ψ0(x)\|2 dx = 1. This is the same Gaussian wavefunction for the ground state of the harmonic oscillator familiar from IB QM. If needed, the wavefunctions of the excited states may be found by acting with A† = (mωX −iP)/ √ 2mℏω, which in the position represen-tation is the diﬀerential operator 1 √ 2mℏω −ℏd dx + mωx . You can check (or just Google) that these operators generate the usual Hermite polynomials. 3.2 Dynamics of Oscillators At this stage, we’ve learned everything about the set {\|n⟩} of energy eigenstates of the quantum harmonic oscillator and their corresponding energies En = (n + 1 2)ℏω. However, we still haven’t said anything about the physics of how quantum oscillators actually behave. Classically, we know that a harmonic oscillator would undergo periodic motion with a period T = 2π/ω. Furthermore, the energy of the classical oscillator is independent of the period, but is proportional to the square of the amplitude of oscillation. To what extent is the same true of our quantum oscillator? To say anything about the motion of a quantum system, we need to examine the TDSE. To get started, ﬁrst suppose our system is prepared at time t = 0 to be in some energy eigenstate \|n⟩. Then by the TDSE, at a later time t it will have evolved to \|n, t⟩= e−i(n+1/2)ωt\|n⟩. (3.17) Consequently, no eigenstate has a time dependence which oscillates at the classical fre-quency ω. Even worse, no matter which energy eigenstate our oscillator is in, the expected position of the oscillator at any given time t is ⟨n, t\|X\|n, t⟩= e+i(n+1/2)ωt ⟨n\|X\|n⟩e−i(n+1/2)ωt = ⟨n\|X\|n⟩, (3.18) where we’ve used the linearity / antilinearity properties of the inner product. The rhs is independent of time, so none of these states move – our oscillator does not appear to be oscillating! To ﬁnd some interesting dynamics, we must consider not a single energy eigenstate, but rather a superposition. This is a much more realistic assumption: there’s no practical way we could prepare a macroscopic system to be in just one energy eigenstate. Let’s now suppose our oscillator is prepared at t = 0 to be in some generic state \|ψ, 0⟩= P n cn\|n⟩. The cn should be chosen so that ⟨ψ\|ψ⟩= 1, but are otherwise arbitrary. Then at time t this state will have evolved to \|ψ, t⟩= ∞ X n=0 cn e−iEnt/ℏ\|n⟩ (3.19) by the TDSE. Now let’s examine where we expect to ﬁnd such a generic state. We have ⟨ψ, t\|X\|ψ, t⟩= X m cm eiEmt/ℏ⟨m\| ! X X n cn e−iEnt/ℏ\|n⟩ ! = X n,m cm cn ei(m−n)ωt ⟨m\|X\|n⟩. (3.20) – 25 – To evaluate the inner product ⟨m\|X\|n⟩, we write19 X = r ℏ 2mω A + A† (3.21) in terms of the raising and lowering operators. Recalling that A†\|n⟩= √n + 1 \|n + 1⟩and A\|n⟩= √n \|n −1⟩, we have ⟨m\|X\|n⟩= r ℏ 2mω √n ⟨m\|n −1⟩+ √ n + 1 ⟨m\|n + 1⟩ , (3.22) showing that ⟨m\|X\|n⟩is non-zero only when m = n ± 1. The double sum in (3.20) thus reduces to ⟨ψ, t\|X\|ψ, t⟩= r ℏ 2mω ∞ X n=1 √n cn−1cn e−iωt + √n cncn−1 e+iωt ! = X n xn cos(ωt + φn) , (3.23) where the xn ∈R>0 and phases φn ∈R are deﬁned by r 2nℏ mω cncn−1 = xn eiφn . (3.24) Equation (3.23) shows that ⟨X⟩oscillates sinusoidally at exactly the classical frequency ω whenever our oscillator is prepared in any generic20 superposition of energy eigenstates. In particular, as for the classical oscillator, the frequency of oscillation is independent of the amplitude. In the calculation above, this occurs because the separation between every adjacent pair of energy levels is always ℏω. For a macroscopic oscillator, the only non-negligible amplitudes will be those where n ≈ncl for some ncl ≫1. Consequently, a measurement of the energy is certain to yield some value close to Encl = (ncl + 1 2)ℏω ≈nclℏω. Similarly, in any energy eigenstate we have ⟨n\|X2\|n⟩= ℏ 2mω ⟨n\|AA† + A†A\|n⟩= En mω2 . (3.25) Thus for our macroscopic oscillator, ⟨ψ\|X2\|ψ⟩will be very close to Encl/(mω2). Classically, the time average of x2 is proportional to the average potential energy, which for a harmonic oscillator is just half the total energy. Thus, classically we have x2 = E/(mω2) in agreement with the quantum result. The correspondence principle requires that the quantum and classical results agree for large values of E. That this result agrees even for individual energy eigenstates (even low-lying ones) is a coincidence due to special symmetries of the harmonic oscillator. 19To do this calculation using the techniques of IB QM, you’d have worked in the position representation and said ⟨m\|X\|n⟩= Z ∞ −∞ Hm(x) e−x2/2α∗ x Hn(x) e−x2/2α dx where Hn(x) are Hermite polynomials of degree n and α = ℏ/mω. This is correct, but the integral looks rather unpleasant to evaluate. Fortunately, our operator formalism means we never even have to try! 20Here, ‘generic’ simply means we must have non-zero amplitudes cn = ⟨n\|ψ⟩for at least one pair of adjacent energy levels. – 26 – 3.2.1 Coherent States To be written: 2019 3.2.2 Anharmonic Oscillations Suppose that, instead of the pure harmonic oscillator, we have a potential that has a mini-mum at x = 0 around which it grows approximately quadratically while \|x\| ≪a, but then asymptotes to a constant value at large \|x\|. Particles of low energy will have position space wavefunctions that are supported near the minimum of V (x), so we’d expect the low-lying energy levels to be roughly equal to those of a corresponding harmonic oscillator. Particles of higher energy would start to see the fact that the potential is not purely harmonic. In fact, for any such asymptotically constant potential with a single extremum, the separa-tion between energy levels gets smaller and smaller as we approach the asymptotic value limx→∞V (x) of the potential (beyond which we have a continuum of non-bound states)21. Let’s prepare a state to be in two adjacent energy levels, say \|ψ⟩= cn\|n⟩+ cn+1\|n + 1⟩, where \|n⟩is the nth energy eigenstate of our anharmonic potential, whatever it may be. As long as the potential is symmetric around x = 0, its eigenstates will have deﬁnite parity and so ⟨n\|X\|n⟩= 0 for all \|n⟩. Therefore, at time t, ⟨ψ, t\|X\|ψ, t⟩= cncn+1 ei(En−En+1)t/ℏ⟨n\|X\|n + 1⟩+ c.c. . (3.26) This is again a sinusoidally oscillating function of time, but we no longer expect the energy levels to be equally spaced, so now the period T = 2πℏ/(En+1 −En) will depend on n. In fact, since the energy levels get closer together as n increases, if we give our particle a larger amplitude of oscillation – and hence more energy – it will take longer to execute a complete an oscillation. This is just what we’d expect classically. As an example, consider the P¨ oschl-Teller potential V (X) = −V0 sech2(κX) (3.27) for some constant V0 > 0 and length scale a = 1/κ (see ﬁgure 3). This has bound state energy levels22 En = −ℏ2κ2 2m (ν −n)2 for 0 ≤n < ν , (3.28) where ν is the positive root of ν(ν + 1) = 2mV0/ℏ2κ2. The separation between adjacent bound state energy levels is thus En+1 −En = (2(ν −n) −1)ℏ2κ2/2m (3.29) 21For a proof, see e.g. chapter 6 of S. Gustafson & I.M. Sigal, Mathematical Concepts of Quantum Mechanics, Springer (2010). (Both this result and its proof are non-examinable in this course.) 22Exercise: Try to prove this. As for the harmonic oscillator, the spectrum can be found algebraically, though doing so will require a certain amount of ingenuity. The required method is similar to one of the questions on Problem Sheet 1. – 27 – Figure 3: The P¨ oschl-Teller potential V (x) = −V0 sech2(κX) and its energy levels. Figure by Nicoguaro, taken from this Wikipedia page. and deceases as n increases towards ν. (These formulæ break down when n ≥ν, where we enter the continuum of non-normalizable states.) If V0 ≫ℏ2κ2/2m then the potential is much deeper than it is wide, and contains many bound states. In this regime, we have ν ≈ p 2mV0/ℏ2κ2 ≫1 and so from (3.29) we ﬁnd that a superposition of low-lying states will oscillate with a frequency ω ≈ℏκ2 m ν ≈κ r 2V0 m , (3.30) as adjacent low-lying energy levels are always separated by roughly ℏκ2ν/m. Now, when x ≪1/κ, we may approximate −V0 sech2(κx) ≈−V0 + V0κ2x2. We see that this frequency is indeed just what we’d expect for the corresponding harmonic oscillator. On the other hand, a superposition of states clustered around n = ν/2 will oscillate at around half this frequency as neighbouring energy levels in this region are only separated by about half as much. If we include a wider range of states in our initial superposition, then we’ll instead ﬁnd a sequence of terms23 ⟨ψ, t\|X\|ψ, t⟩= · · · + cn−1cn ei(En−1−En)t/ℏ⟨n −1\|X\|n⟩+ cn+1cn ei(En+1−En)t/ℏ⟨n + 1\|X\|n⟩ + cn+3cn ei(En+3−En)t/ℏ⟨n + 3\|X\|n⟩+ · · · . (3.31) Since this is no longer a harmonic oscillator, we do not generically expect ⟨n + 3\|X\|n⟩= 0. Let’s deﬁne a frequency Ω= (Encl+1 −Encl)/ℏwhere ncl ≫1. Provided the cn’s are clustered around this large central value n = ncl suﬃciently tightly that the diﬀerence 23Note that for a symmetric potential, we again have ⟨n + 2k\|X\|n⟩= 0 by parity. – 28 – between adjacent energy levels is roughly constant over the range of n for which the cn are appreciable, then, to reasonable accuracy, all the terms that contribute to (3.31) oscillate with frequencies that are integer multiples of Ωncl. Thus the motion will be periodic, but anharmonic, just as we expect classically. If we release the anharmonic oscillator from some large extension x0, then initially the wavefunction will be a superposition of many energy levels, with coeﬃcients that ensure ⟨x\|ψ⟩= P n cn⟨x\|n⟩is sharply peaked around x0. At time t, this state will evolve to \|ψ, t⟩= e−iEnclt/ℏX n cn ei(Encl−En)t/ℏ\|n⟩. Since the separation between energy levels varies, the frequencies appearing in this sum are not all integer multiples of any Ω. Consequently, after a time of order 2π/Ω, most terms in the sum will have not quite returned to their original values, so the wavefunction at t = 2π/Ωwill be less sharply peaked around x0. With each subsequent passing of time 2π/Ω, the wavefunction will become more and more diﬀusely spread. Classically, if we release an oscillator with a rather uncertain value for it’s energy, then for a pure harmonic oscillator we always know where to ﬁnd the oscillator at later times, since it’s period is independent of the energy. However, for an anharmonic oscillator, the period depends on the energy, so after a long time our oscillator is equally likely to be located anywhere. 3.3 The Three-Dimensional Isotropic Harmonic Oscillator Let’s now consider an oscillator in R3 that is isotropic in the sense that the potential depends only on the radial distance from the origin. If the oscillator is harmonic, its Hamiltonian is H = P2 2m + 1 2mω2X2 = 1 2m P 2 x + P 2 y + P 2 z + 1 2mω2 X2 + Y 2 + Z2 . (3.32) This is just the sum of the Hamiltonians for three one-dimensional harmonic oscillators, each with the same frequency ω. We introduce raising and lowering operators A† = 1 √ 2mℏω (mω X −iP) and A = 1 √ 2mℏω (mω X + iP) , (3.33) one for each Cartesian direction in R3. Their commutation relations are h Ai, A† j i = δij , [Ai, Aj] = 0 , [A† i, A† j] = 0 (3.34) as an obvious generalization of (3.7), and again the Hamiltonian may be written as H = ℏω 3 X i=1 A† iAi + 1 2 = ℏω A† · A + 3 2 (3.35) in terms of these raising and lowering operators. As above, for any energy eigenstate \|En⟩ we have En ℏω −3 2 = En H ℏω −3 2 En = ⟨En\| A† · A \|En⟩ = 3 X i=1 ∥Ai\|En⟩∥2 ≥0 , (3.36) – 29 – so the ground state has energy 3ℏω/2 – three times the ground state energy of a one-dimensional quantum harmonic oscillator with the same frequency. We’ll call this ground state \|0⟩. Every component of the lowering operator A annihi-lates \|0⟩, so its position space wavefunction obeys ⟨x\|Ax\|0⟩= ⟨x\|Ay\|0⟩= ⟨x\|Az\|0⟩= 0 (3.37) and hence is given by ⟨x\|0⟩= mω πℏ 3/4 exp −mωx2 2ℏ exp −mωy2 2ℏ exp −mωz2 2ℏ = mω πℏ 3/4 exp −mωr2 2ℏ , (3.38) where again the overall constant is ﬁxed by the normalization requirement in R3. Note that this ground state is spherically symmetric. Just as in 1d, we can obtain excited states by acting on \|0⟩with any component of the raising operators A†. Acting on any energy eigenstate by any of the components of A† raises the state’s energy by one unit of ℏω. Since the ground state has energy 3/2ℏω, all of the states \|n⟩= \|nx⟩⊗\|ny⟩⊗\|nz⟩= (A† x)nx (A† y)ny (A† z)nz\|0⟩ p (nx + 1)! (ny + 1)! (nz + 1)! (3.39) involving a ﬁxed total number N = nx + ny + nz of raising operators are degenerate, each with energy EN = N + 3 2 ℏω . (3.40) The total degeneracy of the Nth energy level is (N + 1)(N + 2)/2, the number of ways to partition N into three non-negative integers (nx, ny, nx) (think ‘stars & bars’). This large degeneracy relies in part on the oscillator being isotropic, so that the potential is spherically symmetric. Even so, the degeneracy is rather greater than we’d expect for a generic central potential. We’ll understand the origin of this large degeneracy more fully in section ??. – 30 – 4 Transformations and Symmetries In this chapter, we’ll return to examine the sinews of how the formalism of QM works. While classical physics is ﬁrmly rooted in space-time, we’ve seen that quantum mechanics takes place in the more abstract Hilbert space. Here, we’ll understand how transformations such as spatial translations or rotations of R3 aﬀect states in Hilbert space, linking together the world of quantum mechanics with our familiar experience in ‘normal’ space. In so doing, we’ll obtain a deeper appreciation of the origin of many of the most common commutation relations, including all the ones I expect you’re familiar with. For example, we’ll understand why [X, P] = iℏ, a fact which probably seemed rather mysterious in IB QM. 4.1 Transformations of States and Operators Suppose we move our physical system about in some way, perhaps translating it through R3 or rotating it around some chosen origin. In QM, we’d expect our system to be described a diﬀerent state before the transformation compared to after the transformation. For example, suppose particle is prepared to be in a state \|ψ⟩whose position space wavefunction is strongly peaked around the origin 0 ∈R3. If we translate our system through a vector a, we should expect to ﬁnd our particle in a new state \|ψ′⟩whose wavefunction is strongly peaked around x = a. This suggests that the eﬀect of any given spatial transformation should be represented on Hilbert space by a linear operator U : H →H U : \|ψ⟩7→\|ψ′⟩= U\|ψ⟩. (4.1) where the operator in question depends on the speciﬁc type of transformation. Whatever state we start with, after applying the transformation, we still expect to ﬁnd it somewhere in space. Thus, properly normalised states should remain so after applying U, or in other words 1 = ⟨ψ\|ψ⟩= ⟨ψ′\|ψ′⟩= ⟨ψ\|U †U\|ψ⟩ for all \|ψ⟩∈H. (4.2) We’ll now show that the requirement that U preserves the norm of every state ﬁxes it to be a unitary operator24. The required trick is a form of the polarization identity. We ﬁrst write \|ψ⟩as \|ψ⟩= \|φ⟩+ λ\|χ⟩for some states \|φ⟩, \|χ⟩∈H and some λ ∈C. Equation (4.2) becomes ⟨φ\|φ⟩+¯ λ⟨χ\|φ⟩+λ⟨φ\|χ⟩+\|λ\|2⟨χ\|χ⟩= ⟨φ\|U †U\|φ⟩+¯ λ⟨χ\|U †U\|φ⟩+λ⟨φ\|U †U\|χ⟩+\|λ\|2⟨χ\|U †U\|χ⟩ (4.3) where we recall that the inner product is antilinear in its left entry. By assumption U preserves the norm of every state, so ⟨φ\|U †U\|φ⟩= ⟨φ\|φ⟩and ⟨χ\|U †U\|χ⟩= ⟨χ\|χ⟩. Thus our 24Because physical states are represented by rays in H, rather than actual vectors, there is one other possibility, transformations may be represented by operators that are both anti-linear, in the sense that U(c1\|ψ1⟩+ c2\|ψ2⟩) = c1U\|ψ1⟩+ c2U\|ψ2⟩), and anti-unitary, meaning that ⟨χ\|U †U\|φ⟩= ⟨φ\|χ⟩= ⟨χ\|φ⟩for all \|φ⟩, \|χ⟩∈H. Such antiunitary operators turn out to be related to reversing the direction of time; we won’t discuss them further in this course. Wigner showed that all transformations are represented by either linear, unitary operators or anti-linear, anti-unitary ones. – 31 – identity simpliﬁes to λ ⟨φ\|χ⟩−⟨φ\|U †U\|χ⟩ = λ ⟨χ\|U †U\|φ⟩−⟨χ\|φ⟩ . (4.4) In order for this to hold for every \|ψ⟩∈H, it must continue to hold as we vary λ. In particular, as the phase of λ varies, the only way for (4.4) to be satisﬁed is if ⟨φ\|χ⟩= ⟨φ\|U †U\|χ⟩. (4.5) Finally, for this hold for arbitrary states \|φ⟩and \|χ⟩we must have U †U = 1. Multiplying through on the right by U −1 shows that U † = U −1 (4.6) which says that the adjoint of U is equal to its inverse. Operators with this property are said to be unitary25. There’s one further condition on our operators U. Transformations of space such as translations and rotations form a group. For example, the composition of two rotations is again a rotation, the trivial rotation forms the identity, and the inverse of a rotation is a rotation of the same amount around the same axis, but in the opposite sense (e.g. clockwise instead of anticlockwise). Let’s suppose that our spatial transformations form a group G. To reﬂect this group structure, our operators should26 provide a homomorphism from G to the group of unitary operators in the sense that, for all g1, g2 ∈G, U(g2) ◦U(g1) = U(g2 · g1) and U(1) = 1H (4.7) where · denotes the group multiplication in G and ◦denotes the composition of linear operators acting on H. (We’ll typically suppress these composition symbols from now on.) Note that if U1 and U2 are each unitary operators, then (U2U1)−1 = U −1 1 U −1 2 = U † 1U † 2 = (U2U1)† and so the composite operator U2U1 is also unitary. Note also that the identity operator 1H is trivially unitary. Mathematically, the search for homomorphisms from G to a group of linear operators on a vector space is the subject of Representation Theory. In our case, the vector space in question is our chosen Hilbert space H. Beyond just being a vector space, this already comes with the structure of an inner product and we’re asking that our representation of G on H preserves this inner product. Thus, mathematically, the structure of QM involves classifying unitary representations of various groups of transformations of R3. We can also understand how transformations act on operators. Suppose A is some operator whose matrix elements we’re interested in. If our transformation maps \|ψ⟩→\|ψ′⟩, then the expectation value of A will be mapped as ⟨ψ\|A\|ψ⟩7→⟨ψ′\|A\|ψ′⟩= ⟨ψ\| U †(θ)AU(θ) \|ψ⟩. (4.8) 25If (and only if!) you’re doing some functional analysis , note that unitary operators are certainly bounded, and in fact have unit norm in the operator topology. They are thus somewhat better behaved than generic Hermitian (or symmetric) operators. 26This statement is not quite accurate – there’s an important reﬁnement that we’ll return to later in the course. – 32 – Consequently, we can ﬁnd the expectation value A will take after the transformation by working with the original states, but instead transforming the operator as A 7→A′ = U †(θ)AU(θ) = U −1(θ)AU(θ) , (4.9) using the fact that U is unitary. This is known as a similarity transform. Note that [A′, B′] = [U −1AU, U−1BU] = U −1[A, B]U , (4.10) so the commutator of two similarity transforms is the similarity transform of the commu-tator. Also, similarity transforms do not change the spectrum of any operator: if \|ψ⟩is an eigenstate of A with eigenvalue a, then U −1\|ψ⟩is an eigenstate of A′ with the same eigenvalue. 4.1.1 Continuous Transformations A particularly important class of transformations are those that depend smoothly on some parameter θ. For example, we can smoothly vary both the axis about which and the angle through which we rotate, or the magnitude and direction of a translation vector. If θ = 0 is trivial transformation, represented on H by the identity operator, then for inﬁnitesimal transformation we have U(δθ) = 1 −i δθ T + O(δθ2) (4.11) where T is some operator that is independent of θ. (The factor of −i in this equation is just a convention for later convenience.) T is called the generator of the transformation. For such inﬁnitesimal transformations, the condition (4.6) that U is unitary becomes 1 + i δθ T † + O(δθ2) = 1 + iδθ T + O(δθ2) , (4.12) which to ﬁrst order in δθ gives T = T † . (4.13) Thus the generator T is Hermitian and hence a good candidate for an observable quantity. A ﬁnite transformation can be generated by repeatedly performing an inﬁnitesimal one. Speciﬁcally, if we set δθ = θ/N and transform N times with U(δθ), then in the limit N →∞we have U(θ) = lim N→∞ 1 −i θ N T N = e−iθT , (4.14) where the exponential of an operator may be deﬁned by (2.42), or equivalently by its power series expansion. This form of the unitary operator U(θ) is especially useful when states are expressed in terms of the basis of eigenstates of the Hermitian generator T. We obtain an important equation by using (4.11) to evaluate \|ψ′⟩= U(δθ)\|ψ⟩. Sub-tracting \|ψ⟩from both sides and dividing through by δθ, in the limit δθ →0 we obtain i∂\|ψ⟩ ∂θ = T\|ψ⟩ (4.15) There is no assumption here that the wavefunction ψ(x) = ⟨x\|ψ⟩should be diﬀerentiable as a function of space. We merely say that as our transformation varies smoothly away – 33 – from the identity, so too does the state \|ψ⟩vary smoothly inside H. The rate at which \|ψ⟩ varies is governed by the generator. For an inﬁnitesimal transformation, our transformation law (4.9) for operators becomes U †(δθ) A U(δθ) = A + i δθ [T, A] + O(δθ2) , (4.16) so δA = i δθ [T, A] . (4.17) Thus, while the rate of change of states themselves is given by the action of the generator, the rate of change of operators under a transformation is determined by the commutator of the generator with the operator. We’ll see that most of the commutation relations we meet in quantum mechanics (and certainly all the familiar ones) arise this way. Furthermore, because Hermitian operators represent observable quantities with which we are familiar, in practice it’s often easier to understand how a transformation should act on an operator, rather than on the more abstract notion of a state in Hilbert space. The above discussion has been rather abstract. Let’s ground it by looking at a few of the most important examples of transformations and their corresponding generators. 4.2 Translations When we move an object around, we expect to ﬁnd it in a new place. Speciﬁcally, suppose ⟨ψ\|X\|ψ⟩= x0 for some normalised state \|ψ⟩. Since x0 ∈R3 just labels a spatial point, it must behave under translations and rotations like any vector. Thus, after the translation we expect our object to be described by a new state \|ψ′⟩with ⟨ψ′\|X\|ψ′⟩= x0 + a. The general theory of the previous section asserts that this translation is represented on Hilbert space by a unitary operator27 U(a) so that \|ψ′⟩= U(a)\|ψ⟩. Therefore we can write the expected location of the translated state as ⟨ψ′\|X\|ψ′⟩= ⟨ψ\|X\|ψ⟩+ a = ⟨ψ\|(X + 1Ha)\|ψ⟩ = ⟨ψ\|U −1(a)XU(a)\|ψ⟩, (4.18) where 1H represents the identity operator on H. (We’ll often drop the identity operator where it’s unambiguous.) Since this is true for any state \|ψ⟩, the same argument as above shows that U −1(a) X U(a) = X + a1H . (4.19) Let’s be clear about what this equation means. In this course, we’re interested in Quantum Mechanics on R3, so the (boldface) position operator X : H →H is really a collection of three operators corresponding to the three coordinates of R3. In more detail, taking the standard Cartesian basis of R3, (4.19) says    U −1(a)XU(a) U −1(a)Y U(a) U −1(a)ZU(a)   =    X + ax1H Y + ay1H Z + az1H   . (4.20) 27Strictly, we should label this as Utrans(a) to emphasise that it is the unitary operator generating translations, as opposed to anything else we might associate to a. We typically drop such additional labels. Exactly which unitary operator we’re talking about should hopefully be clear from the context. – 34 – In other words, conjugating the X (component!) operator by U(a) gives a new operator U −1(a)XU(a) : H →H that we identify with X + ax1H, and similarly for Y and Z. If U(a) acted on the position operator in any other way, it could not represent a translation, so equation (4.19) may be taken as the deﬁning property of the translation operator, distinguishing it from other unitary transformations. For an inﬁnitesimal translation δa, equation (4.11) states that the translation operator can be written as28 U(δa) = 1 −i ℏδa · P + O(δa2) (4.21) for some Hermitian operator P. I’ve named this operator P, so I’m not going to be able to fool any of you that it will turn out to be anything other than the momentum operator. But that’s not how I want you to think of it yet. By deﬁnition, P/ℏis the generator of inﬁnitesimal translations. The translation generator P/ℏmust have units 1/(length) so that it makes sense to add the second term of (4.21) to the ﬁrst. Because we’ve included a factor of ℏ, our P must have dimensions of (mass×velocity). In fact, this is just a convention. We could choose to measure masses in units of 1/(length×velocity), using ℏas a conversion factor for our units. In relativity, the speed of light c provides a further natural conversion factor between lengths and times, so in units of ℏ/c, masses are equivalent to inverse lengths. In natural units, we work with length and time scales adapted to Nature rather than our civilisation, so we set ℏ= 1 and c = 1. In these units, the translation generator is precisely P. Incidentally, you may be worried about setting ℏ= 1 when it is such a ‘small’ number; ℏ≈1.0547 × 10−34 Js/rad. But this is quite wrong. Each atom in the Universe knows that ℏis not at all small or insigniﬁcant. It only appears small when measured in units such as Joules and seconds that are relevant for steam engines and pocketwatches. The real, deep question is not why ℏappears small, but why humans are so big. Plugging (4.21) into the deﬁning equation (4.19) for the translation operator we ﬁnd i ℏ[δa · P , X ] = δa (4.22) and since this holds for any inﬁnitesimal translation, [Xi , Pj] = iℏδij . (4.23) These, as I’m sure you recognise, are the fundamental commutation relations of quantum mechanics. I’m prepared to bet that the ﬁrst time you met them (and perhaps even up until now) you thought they were a weird and mysterious feature of quantum mechanics. Here they’re revealed as little more than common sense: asking where you are and then translating is not the same as ﬁrst translating and then asking where you are. What is weird and quantum about this equation is not that X and P don’t commute, but the fact that they are operators acting on a Hilbert space. 28Here, δa · P = P3 i=1 δai Pi involves the standard dot product in R3. Note that the result is still an operator on H, a linear combination of the three operators P. – 35 – By repeatedly performing the same inﬁnitesimal translation, we can write U(a) = exp −i ℏa · P (4.24) for a translation through the ﬁnite vector a. Since U(a) provides a homomorphism of this group of translations to the group of linear operators on H, we have U(b) U(a) = U(b + a) = U(a + b) = U(a) U(b) , (4.25) where in the second equality we used the fact that a + b = b + a, stating that the order in which we perform two translations doesn’t matter29. Using (4.21) to expand the far left & far right of this equation to lowest non–trivial order in aibj shows that the generators P obey [Pi, Pj] = 0 for all i, j (4.26) and so commute with eachother. Again, the fundamental meaning of this equation is simply that translations form an Abelian group — the order of translations doesn’t matter. It only says anything about our ability to simultaneously know all three components of a particle’s momentum once we identify P as corresponding to momentum, which we haven’t yet done. Now let’s think about what the translation operator does to states. Let \|x⟩repre-sent an eigenstate of the position operator with eigenvalue x, so that X\|x⟩= x\|x⟩, with normalisation condition ⟨x′\|x⟩= δ3(x′ −x) (4.27) as usual for continuum states30. This state represents a particle which is deﬁnitely located at x ∈R3. Applying the translation operator, X(U(a)\|x⟩) = ([X, U(a)] + U(a)X) \|x⟩. (4.28) We can evaluate the commutator by multiplying (4.19) through by U(a) on the left, ﬁnd-ing [X, U(a)] = U(a) a. The constant a and the eigenvalue x each commute with the translation operator, so X(U(a)\|x⟩) = (x + a) (U(a)\|x⟩) . (4.29) As anticipated, our new state U(a)\|x⟩is certainly located at x + a. Consequently, U(a)\|x⟩ must be proportional to the state \|x + a⟩, the normalised state that is deﬁnitely located 29Here this is of course the statement that the group of spatial translations is (R3, +), where the group operation ‘+’ is commutative. Groups for which the group operation is commutative are often called Abelian. 30Technically, \|x⟩is a non-normalizable state, which really means it doesn’t live in the Hilbert space L2(R3, d3x). This is related to the fact that the position operator X is unbounded – there is no constant M such that ∥X\|ψ⟩∥≤M∥\|ψ⟩∥for all \|ψ⟩∈H. Na¨ ıvely, this means that ‘the eigenvalues of X can be arbitrarily large’, which is physically reasonable since our particle may be located very far away. However, in general, unbounded operators do not have any eigenstates in H, so do not really have eigenvalues. The distinction between bounded and unbounded operators is tremendously important in functional analysis, but will not play a role in this course. – 36 – at x + a. Setting U(a)\|x⟩= c\|x + a⟩for some c ∈C and taking the inner product with another position eigenstate \|x′⟩we have31 c δ3(x′ −x −a) = c ⟨x′\|x + a⟩= ⟨x′\|U(a)\|x⟩= U −1(a)\|x′⟩ † \|x⟩ = 1 c \|x′ −a⟩ † \|x⟩= 1 c δ3(x′ −a −x) (4.30) where we used the fact that U(a) is unitary and the fact that if U(a)\|x⟩= c\|x + a⟩then U −1(a)\|x + a⟩= c−1\|x⟩. Comparing both sides shows that \|c\|2 = 1, so c is a pure phase and without loss of generality we can set c = 1. Now suppose we consider translating an arbitrary state \|ψ⟩. The position space wave-function of this state is simply its coeﬃcient ψ(x) = ⟨x\|ψ⟩ (4.31) in the position basis. Thus the wavefunction ψtrans(x) of the translated state \|ψtrans⟩= U(a)\|ψ⟩is ψtrans(x) = ⟨x\|ψtrans⟩= ⟨x\|U(a)\|ψ⟩= U −1(a)\|x⟩ † \|ψ⟩ = ⟨x −a\|ψ⟩= ψ(x −a) . (4.32) Eq. (4.32) says that the wavefunction of the translated state takes the same value at x as the original wavefunction took at x−a. This is completely natural as we’ve translated our state through a! In particular, for an inﬁnitesimal translation δa, on the one hand we can Taylor expand the translated wavefunction to ﬁnd32 ψtrans(x) −ψ(x) = −δa · ∇ψ(x) , (4.33) while on the other we expand the operator U(δa) to ﬁnd ψtrans(x) −ψ(x) = x 1 −i ℏδa · P ψ −⟨x\|ψ⟩= −i ℏδa · ⟨x\|P\|ψ⟩ (4.34) using (4.21) to lowest order in δa. Consequently, for any state \|ψ⟩the momentum operator P acts in the position representation as ⟨x\|P\|ψ⟩= −iℏ∇ψ(x) . (4.35) This is just what you would have said in IB QM last year, here derived from the point of view of the eﬀect a translation of R3 has on states in Hilbert space. As a special case, let’s apply this argument to the state \|p⟩that obeys33 P\|p⟩= p\|p⟩, representing a particle whose momentum is certainly p. In this case the above argument 31Note the slight awkwardness of the Dirac notation here. Because operators act to the right, we have to write ⟨χ\|U\|φ⟩= (U −1\|χ⟩)†\|φ⟩for any unitary operator U, as we used in the third equality. In mathematical notation we would simply write (χ, Uφ) = (U −1χ, φ). 32At least if the wavefunction is suﬃciently smooth. 33As with the position operator, the momentum operator P is unbounded, so the ‘momentum eigenstates’ \|p⟩are necessarily non-normalizable, as we’ll soon see explicitly. – 37 – Figure 4: An inﬁnitesimal rotation transforms v 7→v + δα × v. becomes ψp(x −a) = ⟨x −a\|p⟩= ⟨x\|U(a)\|p⟩= ⟨x\|e−ia·P/ℏ\|p⟩ = e−ia·p/ℏ⟨x\|p⟩= e−ia·p/ℏψp(x) , (4.36) where in going to the second line we used the fact that \|p⟩is an eigenstate of P. Comparing the ﬁrst and last expression, we deduce that the position space wavefunction of a momentum eigenstate must take the form ψp(x) = C eip·x/ℏ (4.37) for some x-independent factor C. It’s convenient to choose C to be (2πℏ)−3/2 since in this case we have ⟨p′\|p⟩= Z d3y d3x ⟨p′\|y⟩⟨y\|x⟩⟨x\|p⟩= Z d3x d3y (2πℏ)3 e−ip′·y/ℏδ3(y −x) eip·x/ℏ = Z d3x (2πℏ)3 e−i(p′−p)·x/ℏ= δ3(p′ −p) (4.38) so that the momentum eigenstates are normalised in the same way as the position eigen-states (and in the same way as for any continuum states). Again, the form ψp(x) = 1 (2πℏ)3/2 eip·x/ℏ (4.39) of the wavefunction for momentum eigenstates is familiar, but here we’ve derived it from ﬁrst principles rather than using the position space representation (4.35) of the momentum operator. (We assumed this result earlier when showing that the position and momen-tum space wavefunctions are eachother’s Fourier transforms.) Note also that \|ψp(x)\|2 = 1/(2πℏ)3, so R R3 \|ψp(x)\|2 d3x diverges. Like position eigenstates, momentum eigenstates are also non-normalizable. 4.3 Rotations We now consider the eﬀect rotating R3 has on a quantum state \|ψ⟩∈H. For a vector v ∈R3 a rotation anticlockwise around the axis ˆ α by an amount \|α\| is a linear transformation R(α) : R3 →R3 R(α) : v 7→v′ = R(α)v (4.40) – 38 – that obeys v′ · v′ = v · v and det(R(α)) = +1 . (4.41) The ﬁrst of these conditions says that rotations preserve lengths, and implies that R(α) must be an orthogonal transformation. The second ensure that R(α) preserves the orien-tation. For inﬁnitesimal rotations, ﬁgure 4 shows that v′ = v + δα × v + O(δα2) . (4.42) (Note that this obeys (4.41).) Given two orthogonal transformations R(α) and R(β) the composite R(β) R(α) is again an orthogonal transformation with unit determinant. Spatial rotations form a group, known as SO(3), with the identity being the trivial rotation and the inverse of R(α) being a rotation of the same amount around the same axis α, but now clockwise. However, in general R(β) R(α) ̸= R(α) R(β), so the order in which we apply rotations is important and the rotation group is non–Abelian. As for any group of transformations, in quantum mechanics the group of rotations is represented on H by unitary operators. We denote the unitary operator corresponding to R(α) as U(α). U(α) acts on the position operator as U −1(α) X U(α) = R(α)X (4.43) where the lhs involves composition of operators in H, while the rhs here is the usual action of a rotation on the three components of X. For example, if α = αˆ z so that the rotation is around the z-axis in R3, then in more detail (4.43) says    U −1(α)XU(α) U −1(α)Y U(α) U −1(α)ZU(α)   =    cos α −sin α 0 sin α cos α 0 0 0 1       X Y Z   . (4.44) Thus, conjugating the operator X by this rotation operator changes it to a new operator U −1(α)XU(α) on Hilbert space, which we identity with the usual linear combination X cos α −Y sin α. Again, U(α) must act on X this way if it is indeed to correspond to a rotation. For an inﬁnitesimal transformation, following (4.11) we can write U(δα) = 1 −i ℏδα · J + O(δα2) (4.45) for some Hermitian generators J/ℏ. Since angles are dimensionless, J, like ℏ, must have dimensions of (length × momentum). Later we will see that J corresponds to the angular momentum operator, though by deﬁnition we have that J/ℏis the generator of rotations. Using this and (4.42) in (4.43) shows that we have i ℏ[ δα · J , X ] = δα × X , (4.46) and since this is true for any axis of rotation, we have the commutation relations [Ji, Xj] = iℏ X k ϵijkXk . (4.47) – 39 – These relations are nothing more than the statement that the operator X transforms as a vector under rotations. They are the inﬁnitesimal version of (4.43) – they must hold if J is indeed to generate rotations. Just as for translations, the mutual commutation relations among the components of J themselves follow from the fact that the U(α)s provide a homomorphism from the group34 SO(3) of rotations to the space of unitary operators on H. However, the non–Abelian nature of the rotation group means we should expect these commutators to be non–trivial in general. Performing two inﬁnitesimal rotations through angles δα and δβ, for any vector v we have R(δβ) R(δα) v = R(δβ) (v + δα × v) = (v + δα × v) + δβ × (v + δα × v) = v + δα × v + δβ × v + δβ × (δα × v) (4.48) to lowest non–trivial order in δα and δβ. Consequently, the diﬀerence between these rotations and the two rotations performed in the opposite order is [R(δβ) R(δα) −R(δα) R(δβ)] v = δβ × (δα × v) −δα × (δβ × v) = (δβ × δα) × v = R(δβ × δα) v −v (4.49) using standard properties of the vector triple product. The rhs again involves a rotation, through the angle δβ × δα. Applying the homomorphism U we obtain [U(δβ), U(δα)] = U(δβ × δα) −I (4.50) for our operators in Hilbert space. Using (4.45), this is −1 ℏ2 [δβ · J , δα · J] = −i ℏ(δβ × δα) · J (4.51) and since this must hold for arbitrary successive inﬁnitesimal rotations, we have ﬁnally [Ji, Jj] = iℏ X k ϵijkJk (4.52) as the commutation relations among the rotation generators. Again, there’s nothing ‘weird’ or ‘quantum’ about (4.52), beyond the fact that they involve operators on Hilbert space. In particular, their form just reﬂects the fact that the order of rotations around diﬀerent axes matters, and that the diﬀerence between the two orderings is itself a rotation around the axis perpendicular to the original two. Compared to the relations [Pi, Pj] = 0, the non–triviality of the commutation relations (4.52) arises purely because SO(3) is a non–Abelian group, unlike the group of translations. These non– trivial commutation relations do not prevent us from exponentiating (4.45) to write U(α) = e−iα·J/ℏfor a ﬁnite rotation around a ﬁxed axis α, because the exponentiation always involved the same component ˆ α · J of the rotation generator, which certainly commutes with itself. 34Again, later we’ll see that this statement needs to be slightly reﬁned. – 40 – The combined group of translations and rotations of Euclidean R3 is sometimes known as E(3), or else ISO(3). For translations through a1 and a2, and rotations R1 and R2, E(3) has the group composition law (a2, R2) · (a1, R1) = (a2 + R2a1, R2R1) , (4.53) where the we note that the second rotation also acts on the ﬁrst translation. Clearly, both R3 and SO(3) are subgroups of E(3), but the fact that rotations act non-trivially on previous translations means that E(3) is the semi-direct product E(3) ∼ = R3 ⋊SO(3). This group law shows that rotations and translations do not commute: translating any vector v through a and then rotating around ˆ α gives us R(α)(v + a), whereas ﬁrst rotating then translating gives a + R(α)v instead. In particular, for inﬁnitesimal translations and rotations we have R(δα)(v + δa) −(R(δα)v + δa) = δα × δa (4.54) independent of v. Applying the homomorphism U, in Hilbert space we obtain commutation relations35 [UR(δα), UT (δa)] = UT (δα × δa) −I (4.55) or equivalently [Ji, Pj] = iℏ X k ϵijkPk (4.56) in terms of the rotation and translation generators. More generally, any 3-component linear operator V : H →H is said to transform as a vector under rotations if it obeys U −1(α)VU(α) = R(α)V (4.57) for any α. (Depending on its behaviour under parity transformations, discussed in sec-tion 4.6, such a V could either be a vector or pseudovector operator.) Just as above, for inﬁnitesimal rotations this implies the commutation relations [Ji, Vj] = iℏ X k ϵijkVk (4.58) among the components of V and J. We see that X, P and J itself36 each transform as vectors under rotations. This of course is just what we’d expect for position, momentum and angular momentum in classical mechanics. On the other hand, if an operator S obeys U −1(α) S U(α) = S (4.59) 35Here, for the sake of clarity, we have labelled the translations operator by UT and rotation operator by UR. In particular, the operator on the rhs of (4.55) is a translation. 36This is always true of X and P, but turns out to be a three-dimensional coincidence for J: the rotation group SO(d) of Rd has dimension d(d −1)/2 and the generators are generically represented by antisymmetric matrices Jij. When d = 3, a 3 × 3 antisymmetric matrix has 3 independent components, so we can equivalently package the Jij as vectors through Ji = ϵijkJjk. – 41 – for any α, so that it is unchanged by the rotation operator, we say S is a scalar operator37. The corresponding inﬁnitesimal version is [J, S] = 0 . (4.60) Just as we can form a scalar by taking the Euclidean inner product v·w of the two vectors v, w ∈R3, so we can form scalar operators from the Euclidean inner product of two vector operators. Indeed, if U −1(α)VU(α) = R(α)V and similarly for W, then U −1(α) (V · W) U(α) = (U −1(α)VU(α)) · (U −1(α)WU(α)) = (R(α)V) · (R(α)W) = V · W (4.61) by the standard rotational invariance of the dot product. As always, we can write this in terms of commutators by considering inﬁnitesimal rotations:  Ji , X j VjWj  = X j [Ji, Vj]Wj + X j Vj[Ji, Wj] = iℏ X jk ϵijkVkWj + iℏ X jk Vj ϵijkWk = iℏ X jk ϵijk(−VjWk + VjWk) = 0 , (4.62) where in going to the ﬁnal line we relabelled dummy indices j ↔k in the ﬁrst term and used the antisymmetry of ϵijk. Note that our calculations always preserved the order of V and W — our result holds irrespective of whether V and W commute. An important special case of (4.60) is to take S = J2 = J·J. The fact that the rotation generators obey the commutation relations [Ji, Jj] = iℏ X k ϵijkJk and [Ji, J2] = 0 (4.63) means that we cannot ﬁnd a complete set of simultaneous eigenstates of all of the Jis, but we can ﬁnd a complete set of simultaneous eigenstates of J2 and any one component of J. Looking ahead to our interpretation of J as the angular momentum operator, Born’s 2nd postulate of QM tells us that we cannot say a particle has a deﬁnite angular momentum vector, but we can know the magnitude of the angular momentum and the amount aligned along any given axis. 4.3.1 Translations Around a Circle In IB QM, you deﬁned the orbital angular momentum operator L = X × P which also has the commutation relations (4.58) with itself and with X and P. We’ll understand the 37More precisely, we say that S transforms as a scalar under rotations — depending on its behaviour under parity transformations it could be either a scalar or pseudoscalar operator. – 42 – relation between J and L below, but it’s important to realise that in general J ̸= L. We can understand L from the present perspective as follows. When a system is displaced through the vector a, its state is transformed by the unitary translation operator U(a) = e−ia·P/ℏ. We now imagine successively performing N translations, each along one edge of a regular N-sided polygon that lies in the plane n · x and is centred on the origin. In the limit N →∞that our regular polygon has an inﬁnite number of sides, this corresponds to translating our system around a circular path. Thus, we can move our system around a circle by applying a succession of small translations, each in a slightly diﬀerent direction. Speciﬁcally, if the system initially lies at some location x, making an angle α with some axis in the plane n · x = 0 of our polygon, then when move it to lie at an angle α + δα we translate it through δa = δα n × x. Thus the associated unitary translation operator obeys U −1(δa) X U(δa) = X + δα (n × X) (4.64) and so can be written as U(δa) = 1 −i ℏδα (n × X) · P + O(δα2) = 1 −i ℏδα n · L + O(δα2) , (4.65) where, interchanging dot and cross, L is the Hermitian operator L = X × P . (4.66) We now see that L/ℏis the generator of circular transformations. We recall from IB QM that the components of L obey the commutation relations38 [Li, Xj] = iℏ X k ϵijkXk , [Li, Pj] = iℏ X k ϵijkPk and [Li, Lj] = iℏ X k ϵijkLk , all of which follow from the more primitive commutation relations [Xi, Pj] = iℏδij and [Xi, Xj] = 0 = [Pi, Pj]. Equation (4.65) contains only one operator, n · L, so it inevitably commutes with itself and we can exponentiate to ﬁnd U(acirc) = e−iα n·L/ℏfor ﬁnite trans-lations around our circle. 4.3.2 Spin If our system is completely described by the Hilbert space L2(R3, d3x), so that we com-pletely specify the state of the system by giving (say) its position space wavefunction ψ(x), then L and J are indistinguishable. However, if our system has any internal structure its Hilbert space will be more complicated.Translating it around a circular arc is then not the same thing as rotating it through the same angle: the diﬀerence is best explained by a picture, which you can ﬁnd in ﬁgure 5. Thus, when H ≇L2(R3, d3x) the action of J and L may not coincide. We deﬁne the spin operator S to be the diﬀerence S = J −L (4.67) 38Check that you’re able to derive these! – 43 – Figure 5: The rotation operator U(α) swings a system around the origin and also rotates its orientation in R3, while a circular translation merely moves the system around a circular path, without aﬀecting its orientation. The diﬀerence is a rotation of the body around its own centre of mass, reorientating it without changing its location. so that J = L + S. From ﬁgure 5 we see that diﬀerence between rotating an object around some ﬁxed origin and a translating it’s centre of mass along a circular path is a rotation around the object’s centre of mass. generates a rotation of the body around its own centre of mass. We thus expect (and will conﬁrm below) that S generates rotations of a body around its own centre of mass, reorienting it in space. This is why S is called the spin operator. In the case of a macroscopic body, made up of many constituent particles, it’s reason-able to suppose that the spin operator just account for the diﬀerence between translating the body as a whole and translating each individual particle around their own arcs, with slightly diﬀerent radii according to where in the body the particle is located. That is, S ? = X a Xa × Pa ! −X × P = X a xa × pa (4.68) where Xa and Pa are position and translation operators for each individual particle and xa and pa are their positions and momenta relative to the centre of mass39. However, for an object consisting of many particles such a description is clearly going to be very cumbersome. More fundamentally, we do not know what the ‘fundamental’ constituents of our object really are. (For example, if I sit on a merry-go-round, is the ‘right’ quantum description of my motion given in terms of my cells, or my atoms, or protons and neutrons, or quarks and gluons, or bits of string, or . . .?) It’s thus crucial that we can consider objects as a whole in quantum mechanics, just as we can classically. For rotations, understanding S will allow us to do this. In addition, as we’ll see in section 5.3.2, one of the surprises of quantum mechanics is that even fundamental particles such as electrons and photons may have an ‘intrinsic’ spin that (as far as we know) is not related to any composite structure. 39We’ll understand how to properly describe the quantum mechanics of a system with many degrees of freedom (e.g. many particles) in chapter 2.3. – 44 – Fortunately, commutation relations involving the spin operator S are easy to obtain from the ones we already have for J and L. Firstly, the fundamental relations [Ji, Xj] = iℏ X k ϵijkXk and [Ji, Pj] = iℏ X k ϵijkPk (4.69) show that [Ji, Lj] = iℏ X k ϵijkLk , (4.70) so that the angular momentum operator L also transforms as a vector under rotations. Therefore we have [Si, Sj] = [Ji −Li, Jj −Lj] = [Ji, Jj] −[Ji, Lj] −[Li, Jj] + [Li, Lj] = iℏ X k ϵijk(Jk −Lk) = iℏ X k ϵijkSk , (4.71) where we used the fact that [Li, Lj] = iℏP k ϵijkLk. This algebra is the same as that obeyed by the components of J and L. It conﬁrms that the spin operators S generate some form of rotation. In particular, it immediately follows from (4.71) that [Si, S2] = 0 , (4.72) just as for J and L. On the other hand, since [Li, Xj] = iℏP k ϵijkXk and [Li, Pj] = iℏP k ϵijkPk, we have [Si, Xj] = [Ji, Xj] −[Li, Xj] = 0 [Si, Pj] = [Ji, Pj] −[Li, Pj] = 0 (4.73) If we view X and P as referring to the system’s centre of mass, then these commutation relations conﬁrm that the spin operator has nothing to do with our object’s location in or motion through space, but is purely to do with rotating its intrinsic orientation. Finally, since S commutes with both X and P, it also commutes with L: [Si, Lj] = 0 ∀i, j . (4.74) This allows us to factorize the operator U(α) describing ﬁnite rotations as U(α) = e−iα·J/ℏ= e−iα·(L+S)/ℏ= e−iα·L/ℏe−iα·S/ℏ= e−iα·S/ℏe−iα·L/ℏ. (4.75) As in ﬁgure 5, these equations conﬁrm that we can think of a quantum rotation as consisting of a translation of a body’s centre of mass along an arc centred on the origin together with a simultaneous rotation of the body around it’s own centre of mass by the same amount. The order in which we perform these two operations makes no diﬀerence. 4.4 Time Translations It’s not only transformations of space that are represented on H by unitary operators. Consider translations in time. These again form an Abelian group, since sending t0 to t0 + t and then to t0 + t + t′ gives the same end result as does t0 →t0 + t′ →t0 + t′ + t. – 45 – If we want the total probability of our particle being found somewhere in R3 at all times, then time translation must also be represented by a unitary operator U(t) : H →H. Since we can translate through an arbitrarily small time, the time translation operator takes the form U(t) = exp −i ℏHt (4.76) where H/ℏmust be Hermitian and is, by deﬁnition, the generator of time translations. This generator must have dimensions 1/(time), so the conventional factor of ℏmeans that H itself has dimensions of energy. Of course, H will turn out to be the Hamiltonian, but for now I’d like you to think of it more abstractly just as the generator of time translations. The fact that U(t) represents the eﬀect of a time translation on H means that if a particle is in some state \|ψ(0)⟩at time t0 = 0, translating forward to time t we will ﬁnd it in the state \|ψ(t)⟩= U(t)\|ψ(0)⟩= e−iHt/ℏ\|ψ(0)⟩. (4.77) In particular, the diﬀerence between \|ψ(t)⟩and the state we ﬁnd a short time later is \|ψ(t + δt)⟩−\|ψ(t)⟩= −i ℏδt H\|ψ(t)⟩+ O(δt2) (4.78) or, taking the limit δt →0, iℏ∂ ∂t\|ψ(t)⟩= H\|ψ(t)⟩. (4.79) This, famously, is the Time Dependent Schr¨ odinger Equation, which I’ll often abbreviate to TDSE in what follows. It’s just the inﬁnitesimal version of the statement that all states in H evolve in time according to the action of a unitary operator U(t). 4.4.1 The Heisenberg Picture In 1B QM, we were used to the idea that states evolve in time according to the TDSE. However, just as we did for spatial transformations, we can instead work in a picture where the states are time independent and instead the operators evolve in time. To see how this works, suppose OS is some operator that contains no explicit time dependence. Then the amplitude for the state OS\|ψ(t)⟩to agree at time t with the state \|χ(t)⟩is ⟨χ(t)\|OS\|ψ(t)⟩= ⟨χ(0)\|U −1(t) OS U(t)\|ψ(0)⟩ (4.80) The right hand side here only makes explicit reference to the initial values of the states \|ψ⟩ and \|χ⟩. Since it holds true for any pair of states, just as above we can obtain the same results by always working with the initial states and instead evolving the operators as OH(t) = U −1(t) OS U(t) . (4.81) The version of quantum mechanics where we use the TDSE to evolve the states in time, leaving operators unaltered, is known as the Schr¨ odinger picture, whereas the version where the states are ﬁxed at their initial values and instead the operators evolve in time is called the Heisenberg picture. – 46 – In fact, all this has a precise analogue in classical mechanics. Classically, there are also two ways of thinking about time evolution. On the one hand, we can think of a particle moving in some way through phase space M. If we know it’s location (x(t), p(t)) ∈M for every time t we can compute any quantity we wish, represented by some function f : M → R, by evaluating f at the location of our particle, obtaining the value f(x(t), p(t)). (This is the perspective we took in the Introduction.) However, Newton’s Laws are deterministic, so, given a force, the entire trajectory is determined by the initial conditions (x0, p0). This suggests a perspective in which the ‘state’ of our particle is simply a choice of initial conditions. These initial conditions do not themselves evolve, rather, it is the quantities we measure that vary in time. Thus, instead of thinking of a physical quantity f as a map from phase space, we treat it just as a map from time, so f : [t0, ∞) →R. You’ll examine these classical pictures further if you’re taking the Classical Dynamics course, in the context of Hamiltonian mechanics. (It’s really for this reformulation of classical mechanics, not the particular H, that Hamilton is famous.) Diﬀerentiating (4.81) wrt t shows that d dtOH(t) = d dt U −1(t) OS U(t) = i ℏ U −1(t) HOS U(t) −U −1 OSH U(t) = i ℏ[H, OH(t)] , (4.82) where in the last step we used the fact that [U(t), H] = 0, since U(t) depends only on H. This is Heisenberg’s equation of motion. It’s completely equivalent to Schr¨ odinger’s equation (4.79), and is also just a (very important!) special case of our general argument of how operators transform under the action of unitary groups of transformations. More generally, if the original operator OS had some explicit time dependence of its own — independent of any particular particle’s motion — then we would obtain a further term in this equation, modifying it to d dtOH(t) = U −1(t)∂OS ∂t U(t) + i ℏ[H, OH(t)] . (4.83) For example, we may wish to understand the behaviour of a charged particle in the presence of an applied electric ﬁeld. If the electric ﬁeld is itself changing in time, then the potential in which the particle moves will be time–dependent even in the Schr¨ odinger picture. 4.5 Dynamics So far, we’ve simply deﬁned some unitary operators U(a), U(α) and U(t) that are re-sponsible for translating, rotating or evolving our system through space and time. The commutation relations of the generators P and J, and their commutation relations with X were determined purely by the properties of the corresponding group of transformations of R3. However, while we’ve seen that commutators such as [H, X], [H, P] and [H, J] are important in the Heisenberg picture, telling us how these operators change in time, we haven’t yet given any way to actually calculate what such commutators should be. Simi-larly, although we’ve said that in the Schr¨ odinger picture, states evolve in time according to – 47 – \|ψ(t)⟩= U(t)\|ψ(0)⟩, we haven’t given any way to work out what the action of the generator H on a state should actually be. To do so, we must specify the dynamics: we must provide a relation H = H(X, P) giving H in terms of the other operators whose commutators we already understand. The simplest (non-trivial) such relation is40 H = 1 2mP2 (4.84) where m is a constant with dimensions of mass. Equation (4.84) relates how a state evolves in time (via H) to how its location is translated through space (via P); in other words, it’s telling us something about how particles move! In particular, with this form of Hamiltonian the diﬀerence between the expected location of a state \|ψ⟩at an initial time and a short time δt later is ⟨ψ\|U −1(δt) XU(δt)\|ψ⟩−⟨ψ\|X\|ψ⟩= i ℏδt ⟨ψ\|[H, X]\|ψ⟩+ O(δt2) = δt m ⟨ψ\|P\|ψ⟩+ O(δt2) . (4.85) In the limit δt →0 we see that the expected velocity of the particle is ⟨ψ\|P\|ψ⟩/m. Equiv-alently, since this is true for all \|ψ⟩∈H, we can write d dtX(t) = P(t) m (4.86) in the Heisenberg picture. Furthermore, since [H, P] = 0 for this Hamiltonian, P(t) = P(0) and all states travel in uniform motion. In the real world, we observe that particles do not always travel with constant velocities: they may slow down, speed up, or change direction as they encounter various obstacles. These obstacles are typically located at various diﬀerent points in space. To allow for this, we generalise our dynamical relation (4.84) to H = 1 2mP2 + V (X) . (4.87) The ﬁrst term on the right is the kinetic term and is the contribution to the Hamiltonian due to the state’s travelling through space. The second, potential term is the contribution due to the state’s location. This familiar form is still a very special case of the general 40Note that this is the ﬁrst time we’ve pinned ourselves down to a non–relativistic theory. Up to this point, everything we’ve said holds good in relativistic quantum mechanics and, suitably interpreted, also in quantum ﬁeld theory. In the lectures we studied translations and rotations, but in the ﬁrst problem set you’ll extend this to also consider Galilean boosts and will show that the non–relativistic Hamiltonian is compatible with the Galilean algebra. If one instead wishes to study a relativistic version of quantum theory, one starts by ﬁnding unitary operators representing the action of the Poincar´ e group R3,1 ⋊SO(3, 1) of special relativity. The relation (4.84) between H and P does not respect the commutation relations appropriate for Poincar´ e symmetry, but of course the dynamical relation H2 = c2P2 + m2c4 does. The diﬃculty with relativistic quantum mechanics lies not with symmetries, but with interactions. The proper treatment of these requires Quantum Field Theory, a far deeper subject. – 48 – statement H = H(X, P) and later in the course we’ll meet examples of Hamiltonians that don’t ﬁt this form. Repeating the calculation of (4.86) we still ﬁnd that dX(t)/dt = P(t)/m, but now [H, P] ̸= 0 so the rate of motion through space will not always be the same. Rather, using (4.82), the Heisenberg picture operators obey d dtP(t) = i ℏ[H, P(t)] = −∇V (t) , (4.88) where ∇V (t) = U −1(t) (∂V /∂X) U(t) = −∇V (X(t)). Thus the motion slows down or speeds up according to the gradient of V (X). Most of our common intuition about momentum comes from equation (4.88). We ‘feel’ that a tennis ball travelling with a certain speed has less momentum than a cannonball travelling with the same speed, and less than the same tennis ball travelling faster, because we’ve known from an early age that it will cause us less damage to stand in the way of the ﬁrst tennis ball than either of the other two. This is really a statement about what our unfortunate bodies will have to do in order to bring these projectiles to rest (or how much eﬀort we will have to exert to launch them in the ﬁrst place). In other words, our intuitive notion of momentum is built on a feeling for the energy our body will gain as we slow the projectile down during the impact. This energy then excites the atoms that ultimately make up our nerves, muscles and bones, becoming dissipated through our bodies. Ultimately, it is the dynamical relation between the Hamiltonian and other operators that justiﬁes us identifying the operator P/ℏ— by deﬁnition, the generator of translations — with our pre-existing notion of momentum (in units of ℏ). Only after we specify our dynamics do commutation relations such as [Xi, Pj] = iℏδij tell us, via Born’s 2nd postulate of QM, that a particle cannot simultaneously be in a state of well-deﬁned position and of well-deﬁned momentum. We’ll often ﬁnd it useful to write the kinetic term in a slightly diﬀerent form. Keeping careful track of the operator ordering, you’ll show in the problem sets that L2 = (X × P) · (X × P) = X2 P2 −P 2 r , (4.89) where for ˆ X = X/\|X\|, Pr = 1 2 ˆ X · P + P · ˆ X (4.90) is the radial momentum operator. Consequently, we can write the Hamiltonian (4.87) in terms of the generator of circular translations and the radial momentum operator Pr as H = P 2 r 2m + 1 2mX2 L2 + V (X) (4.91) where we note that [L, X2] = 0, so the order of the operators in the second term is irrelevant provided we keep X2 together as a composite operator. (In particular, in the position representation, X2 acts as multiplication by x2 = r2, which is certainly unaﬀected by translation in a circle.) Most of our intuitive feeling about angular momentum — mine comes from happy childhood hours spent on merry-go-rounds — is encapsulated in this – 49 – relation between the Hamiltonian and L2. Again, it’s the form of the Hamiltonian which is the basis of our identiﬁcation of the generator L/ℏof circular transformations with angular momentum (in units of ℏ). 4.5.1 Symmetries and Conservation Laws One of the immediate uses of the Heisenberg picture is to deduce a relation between sym-metries of the Hamiltonian and conserved quantities41. If it happens that an operator Q commutes with the Hamiltonian, then it also commutes with e−iHt/ℏ, so the Heisenberg picture operator Q(t) = U −1(t) Q U(t) = U −1(t) U(t) Q = Q , (4.92) coinciding with the Schr¨ odinger operator for all t. Inﬁnitesimally, provided Q has no explicit time dependence, this is d dtQ(t) = i ℏ[H, Q(t)] = i ℏU −1(t) [H, Q] U(t) = 0 . (4.93) Operators that are time independent even in the Heisenberg picture are said to be con-served. We’ve shown that conserved operators are just those that commute with the Hamil-tonian. Suppose we prepare a particle to be in an eigenstate of some conserved operator at time t = 0, so Q\|ψ(0)⟩= q\|ψ(0)⟩. Then at any later time we have Q\|ψ(t)⟩= Q U(t)\|ψ(0)⟩= U(t) Q\|ψ(0)⟩= q U(t)\|ψ(0)⟩= q\|ψ(t)⟩ (4.94) so our particle remains in an eigenstate of Q at all subsequent times (provided the state evolves according to the TDSE). For this reason, it’s usually sensible to expand our states in a basis of eigenstates of a maximal set of conserved operators, rather than a maximal commuting set of any old operators. We label the states in this basis by their corresponding eigenvalues, because the same labelling will remain valid at subsequent times. For example, provided H has no explicit time dependence42 [H, H] = 0 trivially, so the Hamiltonian itself is always conserved, and it is often useful to work in a basis of energy eigenstates. By far the most important source of conserved quantities is symmetries of the Hamil-tonian. These are transformations that leave the Hamiltonian invariant. We’ve seen that a generic unitary operator U(θ) = e−iθT representing some transformation of space on H acts on operators as in equation (4.9). In particular, its action on the Hamiltonian is H 7→U −1(θ)HU(θ) . (4.95) If the Hamiltonian is invariant under this transformation – i.e., if the transformation is a symmetry – then U −1(θ)HU(θ) = H , (4.96) 41Those of you taking the Classical Dynamics course will see a corresponding relation in Hamilton’s approach to classical physics. 42Even if the Hamiltonian does contain explicit time dependence, for example because it describes the dynamics of a charged particle in a varying electric ﬁeld, [H(t), H(t)] = 0 trivially. However, [H(t′), H(t)] may not vanish as the form of the Hamiltonian itself changes. – 50 – or equivalently [T, H] = 0 (4.97) for an inﬁnitesimal symmetry transformation, where T is the Hermitian generator. This is exactly the same condition (4.93) we had for the generator T to be conserved, so symmetries of the Hamiltonian correspond to conserved quantities43. For example, if the Hamiltonian is translationally invariant then [P, H] = 0, so for such Hamiltonians, momentum will be conserved. Note that it’s not necessary for the Hamiltonian to be independent of each particle’s position xa if H is to be translationally invariant, so long as any potential term V (xa −xb) depends only on the relative positions. Similarly, if the Hamiltonian is rotationally invariant then [J, H] = 0 so angular momentum will be conserved. In this case, Jz, J2 and H form a set of mutually commuting operators, so we can expand a general state in a basis {\|n, ℓ, m, . . . ⟩} where the labels (n, ℓ, m) refer to the eigenvalues of H, J2 and Jz. We’ve left open the possibility that our states have further conserved properties, indexed by further labels. These will often contain information about the ‘internal’ state of our system. 4.6 Parity Not all transformations are continuous. In physics, the most prominent example of a discrete transformation is the parity transformation P, acting on R3 as P : x 7→−x. Since det(P) = −1, this is diﬀerent from a rotation, so may have consequences that cannot be deduced by considering only rotations. On Hilbert space, parity transformations will still be represented by a unitary operator Π : H →H, but this operator does not involve any parameter that may be taken inﬁnitesimally small. Consequently there’s no generator associated with parity transformations. Parity transformations form the group G = Z2, since carrying out a non-trivial parity transformation twice just brings us back to where we were. Thus P2 = 1R3 and applying our homomorphism shows that Π2 = 1H (4.98) for the parity operator on Hilbert space, too. Thus, the spectrum of Π is just {+1, −1}. Note that since Π is unitary its quantum numbers η behave multiplicatively, whilst those of a Hermitian generator of a continuous symmetry (such as P) behave additively. Since Π is supposed to represent a reﬂection in R3, it must act on the position operator as Π−1 X Π = PX = −X , (4.99) or equivalently {Π , X} = Π X + X Π = 0 . (4.100) where the bracket {A, B} is sometimes called the anticommutator of A and B. Translating through a and then applying the parity operator P : R3 →R3 is the same as ﬁrst applying 43If you’re taking the Classical Dynamics or Integrable Systems courses, you’ll meet the corresponding statement in classical mechanics in the guise of N¨ other’s theorem. – 51 – P, then translating through −a, so Π−1U(a)Π = U(−a) , (4.101) showing that Π−1PΠ = −P , (4.102) so the translation generators P also anticommute with the parity operator. More generally, if V is any operator that transforms as a vector under rotations, in the sense of (4.57), we say V is a vector operator if also Π−1VΠ = −V , (4.103) so that {Π, V } = 0. If instead Π−1VΠ = +V (4.104) or equivalently [Π, V] = 0, then V is a pseudovector operator (provided it transforms appropriately under rotations). The most prominent example of a pseudovector operator is the rotation generator J: since the parity tranformation P acts on R3 as −I3×3, rotations obey R(α)P = PR(α), or P−1R(α)P = R(α). This gives Π−1U(α)Π = U(α) (4.105) for the parity and rotation operators in Hilbert space, or Π−1JΠ = J for the rotation generator. Likewise, the orbital angular momentum operator obeys Π−1LΠ = Π−1(X × P)Π = (Π−1XΠ) × (Π−1PΠ) = (−X) × (−P) = +L (4.106) so is a pseudovector as in classical mechanics. It follows that the spin operator S = J −L is also a pseudovector. Similarly, operators S that are invariant under rotations are scalar operators if in addi-tion Π−1SΠ = S so that they are unchanged under parity, and are pseudoscalar operators if instead Π−1SΠ = −S. (Notice that the signs for scalars and pseudoscalars are the opposite way round to those for vectors and pseudovectors.) As usual, the parity of a system will be conserved if [H, Π] = 0. As a simple example, if our system is governed by the dynamical relation H = P2/2m + V (X) where the potential is an even function, then Π−1HΠ = 1 2m(Π−1PΠ) · (Π−1PΠ) + Π−1V (X)Π = P2 2m + V (Π−1XΠ) = P2 2m + V (X) = H , (4.107) so parity is conserved for such Hamiltonians. Now let’s understand how Π acts on a generic state \|ψ⟩∈H, starting with the case where H ∼ = L2(R3, d3x) so that all the information about our system is contained in e.g. – 52 – its position space wavefunction ⟨x\|ψ⟩. Note that if \|x⟩is an eigenstate of the position operator, then from (4.103) the parity-reversed state \|x′⟩= Π\|x⟩obeys X\|x′⟩= XΠ\|x⟩= −ΠX\|x⟩= −x Π\|x⟩= −x\|x′⟩ (4.108) and hence Π\|x⟩= c\| −x⟩for some constant c. Since Π2 = id, at most c can be a sign44. None of the properties of the Π operator above ﬁx this sign45, and we ﬁx the ambiguity by declaring Π to be the operator such that Π\|x⟩= +\|−x⟩. (4.109) More generally, given any state \|ψ⟩∈L2(R3, d3x), the wavefunction of the parity trans-formed state Π\|ψ⟩is ⟨x\|Π\|ψ⟩= ⟨−x\|ψ⟩= ψ(−x) , (4.110) since Π† = Π−1 = Π. Thus, the wavefunction of the new state takes the same value at x as the original one did at −x. For example, if our particle is completely described by the a state \|n, ℓ, m⟩with position space wavefunction ⟨x\|n, ℓ, m⟩= Rn(\|x\|) Ym ℓ(x) (where the radial part Rn(r) of the wavefunction determines the energy level n), applying the parity operator gives ⟨x\|Π\|n, ℓ, m⟩= ⟨−x\|n, ℓ, m⟩= Rn(\| −x\|) Ym ℓ(−x) = (−1)ℓ⟨x\|n, ℓ, m⟩, (4.111) where we recall from IB QM (or Methods) that the spherical harmonics Ym ℓ(x) obey Ym ℓ(−x) = (−1)ℓYm ℓ(x). When we have a complicated object, such as a macroscopic system with some internal structure, the Hilbert space is not simply L2(R3, d3x) and parity transformations can act in a slightly more complicated way. This is clear classically: the parity transformation of a book initially located at x is not simply a book located at −x, but rather a mirror image (simultaneously left–right, up–down and front–back) of the book. As with the distinction between J and L, it’s tempting to try to account for this by declaring that X just refers to the location of the macroscopic object’s centre of mass, and deﬁne a set of parity operators Πa, one for each constituent particle, each acting as Π−1 a XaΠa = −Xa so as to reﬂect that one particle. Just as with spin and rotations, the problem with this (aside from ineﬃciency) is that it’s dangerous to assume we know what the correct choice of Hilbert space actually is for a subatomic particle – does the electron have an internal structure? Whatever the choice of H on which the full Π acts, its eigenstates must obey Π\|ψ⟩= η\|ψ⟩with either η = +1 or η = −1. These signs can certainly arise from the behaviour of the position space wavefunction as an even/odd function, but there can also be ‘intrinsic’ parities that tell us something about the internal structure of the system. In the subatomic world, such intrinsic parities depend just on the ‘species’ of particle our state describes (i.e. whether it’s an electron, a proton, a photon or something else). 44Note that it is important here that Π\|x⟩= c\| −x⟩for all values of x. See e.g. Weinberg, The Quantum Theory of Fields vol. 1, sec. 2.6 for a fuller discussion. 45The operator Π′ = −Π obeys (Π′)2 = +id and {Π′, X} = 0, so is just as good a choice for the parity operator as Π itself. – 53 – Figure 6: Most atomic transitions are due to the absorption or emission of a photon from a single electron in the atom. The photon’s angular frequency ω = ∆E/ℏ. The intrinsic parity of a speciﬁc type of particle needs to be determined experimentally. For example, consider transitions between diﬀerent energy levels of an atom. These are usually mediated by electromagnetic interactions: during such a transition, the atom’s electrons either emit or absorb a photon whose energy ℏω is equal to the diﬀerence in energy ∆E between the two levels involved in the transition. As we’ll study in more detail later, when the corresponding wavelength is much larger than the typical size of the atom, the transition rate Γ(i →f) is given by Γ(i →f) = 4(∆E)3 c3ℏ4 \|⟨f\|D\|i⟩\|2 , (4.112) where \|i⟩and \|f⟩are the initial and ﬁnal states of the atom, and D = P eaXa is the operator corresponding to the atom’s electric dipole moment. (The sum runs over all the electrons in the atom.) Reversing the parity of space aﬀects all the electrons equally, so the parity operator obeys Π−1XaΠ = −Xa for each Xa. Thus Π−1DΠ = −D. Now suppose that the initial and ﬁnal states of the atom are eigenstates of Π, with eigenvalues ηi and ηf, respectively. Then −⟨f\|D\|i⟩= ⟨f\|Π−1DΠ\|i⟩= ηi ηf⟨f\|D\|i⟩ (4.113) so the amplitude for the transition will vanish unless the initial and ﬁnal atomic states have opposite parity, ηi ηf = −1. If our initial and ﬁnal states kept track of the photon as well as the atom, we’d be able to say overall parity is conserved in such transitions if we assign an intrinsic parity −1 to the photon that is emitted or absorbed in the transition. The intrinsic parity of the photon is related to the fact that the electromagnetic vector potential A, whose quantum version provides the description of the photon, indeed transforms as a vector, not a pseudovector. (Note that we cannot use this experiment to determine the intrinsic parity of the electron, since we have the same number of electrons in the initial and ﬁnal state.) In the most common case, just a single electron is involved in the transition and in this case (4.111) shows that ηi ηf = η2 e(−1)ℓi+ℓf , where ℓi,f are the total orbital angular – 54 – momentum quantum numbers of the initial and ﬁnal states of this electron, and ηe is the intrinsic parity of the electron. Since necessarily η2 e = 1, in any radiative atomic transition involving just a single electron, the orbital angular momentum quantum number ℓmust change by an odd number. We’ll explore intrinsic parity further in section 7.2.1 and in the second problem set. In fact, [H, Π] = 0 holds in general for particles travelling in the presence of any type of electromagnetic interaction. Having [H, Π] be zero means that if at time t = 0 you set up a system to be a mirror image of another system, then their subsequent evolution will be identical in the sense that if you observe them at a later time, they will still be mirror images of one another. Hence, when [H, Π] = 0 it is impossible to tell whether a system is being observed directly or through a mirror. One of the major surprises of twentieth–century physics was an experiment by Wu et al. in 1957, which showed that in fact the Hamiltonian of our Universe has [H, Π] ̸= 0 — you can see things in a mirror that are impossible in our World! – 55 – 5 Angular Momentum In the previous chapter we obtained the fundamental commutation relations among the position, momentum and angular momentum operators, together with an understanding of how a dynamical relation H = H(X, P) allows us to understand how such quantities evolve in time. However, with the exception of the parity operator, we’ve not yet said anything about the spectrum — the set of possible eigenvalues — of these operators, nor have we been speciﬁc about the precise Hilbert space on which they act. Answering such questions is an important part of the subject of representation theory in mathematics; the study of how group actions can be realised on a given vector space. In physics, understanding this will provide essential information about the nature of our quantum system. To a large extent, both questions are determined by the operator algebra itself. For example, taking the trace of the commutation relations [Xi, Pj] = iℏδij IH in any ﬁnite-dimensional Hilbert space H gives dim(H) δij = −i ℏtrH(XiPj −PjXi) = 0 (5.1) for all i, j by the cyclic property of the trace. So there is no realisation of the position and translation operators on any non-trivial ﬁnite dimensional Hilbert space (and if dim(H) = 0 then X and P necessarily act trivially). The above argument fails in an inﬁnite dimensional Hilbert space, where neither trH(IH) nor trH(XiPj) is deﬁned. Thus, if we wish to discuss the position and momentum of a quantum system, we’re necessarily in the world of function spaces46. On the other hand, taking the trace of the commutation relations [Ji, Jj] = iℏ X k ϵijkJk (5.2) for the rotation generators just gives iℏϵijk trH(Jk) = trH(JiJj −JjJi) = 0, so it is possible to represent each component of J on a ﬁnite dimensional Hilbert space in terms of a traceless matrix47. This is often a useful thing to do, particularly when discussing the internal structure of a system. In this chapter, we’ll consider ﬁnite–dimensional Hilbert spaces Hj whose elements transform among themselves under rotations. In mathematics, the Hj are known as ir-reducible representations of the rotation group, whilst in physics they’re called multiplets of deﬁnite total angular momentum. This chapter will also substantiate the link between the rotation generators J and the physics of angular momentum, by examining a simple Hamiltonian for rotating diatomic molecule. We’ll also ﬂesh out the details of exactly how the orientation of a system is encoded in the amplitudes for it to be found in diﬀerent eigenstates of appropriate angular momentum operators. 46Even here we should really be more careful. As mentioned in a previous footnote, X and P are not deﬁned as linear operators on the whole of a Hilbert space such as L2(R3, d3x), because a state that is initially square-integrable may not remain so after the application of the unbounded operator X or P. 47This traceless condition in H is closely related to – but distinct from – the fact that the generators of SO(3) can also be represented by antisymmetric (hence traceless) matrices acting on R3. – 56 – 5.1 Angular Momentum Eigenstates (A Little Representation Theory) Let’s begin by seeing how the algebra [Ji, Jj] = iℏP k ϵijkJk determines the spectrum of the angular momentum operators. Since no two components of J commute, we cannot ﬁnd a complete set of simultaneous eigenstates of two components of J. However, since [J, J2] = 0 we can ﬁnd a complete set of simultaneous eigenstates of (any) one component of J and J2. Without loss of generality, we orient our coordinate system so that the chosen component of J is Jz. We let \|β, m⟩denote a simultaneous eigenstate of Jz and J2, where J2\|β, m⟩= βℏ2\|β, m⟩ and Jz\|β, m⟩= mℏ\|β, m⟩ (5.3) and the factors of ℏare for later convenience. We also choose all the states {\|β, m⟩} to be properly normalised. Since they are eigenstates of Hermitian operators, they must then be orthonormal: ⟨β′, m′\|β, m⟩= δββ′ δmm′ . (5.4) Finally, since we’re looking for the simplest possible way to realize our commutation rela-tions, we’ll assume that the states {\|β, m⟩} are non-degenerate; that is, we want a Hilbert space where \|β, m⟩is the unique eigenstate of J2 and Jz with the given eigenvalues. I’ll comment more on this assumption in the next chapter. We now deﬁne J± = Jx ± iJy (5.5) which obey J† ± = J∓and so are eachother’s adjoint. As they are built from linear combina-tions of the components of J, clearly J± each commute with J2, while their commutation relations with Jz are [Jz, J±] = [Jz, Jx] ± i[Jz, Jy] = iℏ(Jy ∓iJx) = ±ℏJ± . (5.6) We learn that J2(J±\|β, m⟩) = J±J2\|β, m⟩= ℏ2β J±\|β, m⟩ (5.7) and Jz(J±\|β, m⟩) = ([Jz, J±] + J±Jz) \|β, m⟩= (m ± 1)ℏ(J±\|β, m⟩) . (5.8) This shows that the new states J±\|β, m⟩, if not zero, are still eigenstates of both J2 and Jz, with the same eigenvalue βℏ2 for J2. However, their Jz eigenvalue is shifted up or down (respectively for J+\|β, m⟩and J−\|β, m⟩) by one unit of ℏ. We see that the role of J± in angular momentum is similar to the role of A and A† for energy of the harmonic oscillator. Again, J+ and J−are often called raising and lowering operators. Their role is to rotate our system, aligning more or less of its total angular momentum along the z-axis without changing the total angular momentum available. (See ﬁgure 7.) Just as for the harmonic oscillator, examining the algebra of our raising and lower-ing operators has told us the separation between angular momentum eigenstates, given a starting point \|β, m⟩. To ﬁx our initial states, we must examine the norm. By assumption, \|β, m⟩itself is correctly normalized and we compute ∥J+\|β, m⟩∥2 = ⟨β, m\|J−J+\|β, m⟩= ⟨β, m\|(Jx −iJy)(Jx + iJy)\|β, m⟩ = ⟨β, m\|(J2 −J2 z −ℏJz)\|β, m⟩= ℏ2(β −m(m + 1)) (5.9) – 57 – Figure 7: The angular momentum raising and lowering operators J± realign the system’s angular momentum to place more or less of it along the z-axis. where we’ve used the fact that J† + = J−. Similarly, we ﬁnd ∥J−\|β, m⟩∥2 = ℏ2(β−m(m−1)). But since this is a norm, whatever state \|beta, m⟩we started with we must have ∥J+\|β, m⟩∥2 = β −m(m + 1) ≥0 (5.10) with equality iﬀJ+\|β, m⟩= 0 is the trivial state. This shows that it cannot be possible to always keep applying J+, repeatedly raising the Jz eigenvalue m whilst leaving β un-changed. There must be some maximum value of m – let’s call it j – for which J+\|β, j⟩= 0. By (5.10) this can only be the case if β obeys β = j(j + 1) (5.11) and so is ﬁxed in terms of the maximum allowed value of m. Similarly, for a generic m value we ﬁnd ∥J−\|β, m⟩∥2 = β −m(m −1) (5.12) so it also cannot be possible to keep lowering the Jz eigenvalue whilst remaining in the Hilbert space. There must be some minimum value of m, say j′, for which J−\|β, j′⟩= 0 and by (5.12) this can only be the case if β = j′(j′ −1) . (5.13) Applying J+ or J−doesn’t change the J2 eigenvalue β, so these two values of β must agree. Comparing them, we obtain a quadratic equation for j′ that we solve in terms of j, ﬁnding j′ = j + 1 or j′ = j. Since the minimum value of m can’t be greater than the maximum value, we must choose the root j′ = −j. The J2 eigenvalue βℏ2 = j(j + 1)ℏ2 is determined by j, so we henceforth label our states as \|j, m⟩; this labelling is simply less cluttered than \|j(j + 1), m⟩would be. Finally, we note that since applying J−repeatedly will take us from the highest state \|β, j⟩through \|β, j −1⟩, . . . down to \|β, −j⟩, it must be that 2j is a non-negative integer. In other words, the possible values for j are j ∈{0, 1/2, 1, 3/2, . . .} . (5.14) – 58 – Once the total angular momentum quantum number j is ﬁxed, we have m ∈{−j, −j + 1, . . . , j −1, j} . (5.15) Thus there’s a total of 2j + 1 states in any given j multiplet, and Hj is the Hilbert space of dimension 2j + 1 spanned by the {\|j, m⟩} of ﬁxed j. We can move between states in Hj using the raising and lowering operators J+ and J−which obey J+\|j, m⟩= ℏ p j(j + 1) −m(m + 1) \|j, m + 1⟩ J−\|j, m⟩= ℏ p j(j + 1) −m(m −1) \|j, m −1⟩ (5.16) These operators only change the Jz eigenvalue, not the J2 one. They just realign a given system, placing more (J+) or less (J−) of its angular momentum along the z-axis. Although we know we can choose multiplets with diﬀerent values of j, we’ve not yet discovered a mathematical operator that alters this J2 eigenvalue; this will be done in section 6.2.1. 5.2 Rotations and Orientation This section should talk about how eigenfunctions of angular momentum are related to an object’s orientation. In particular, a classical object has a well-deﬁned orientation because it is in a superposition of very many nearby angular momentum eigenstates. I think this is likely to mean lots of values of j as well as m, but not sure of the best example to use to illustrate this. Note that, since Jx and Jy can be written in terms of the raising and lowering operators J±, rotations around an arbitrary axis transform states in a given Hj into other states in the same Hj. Since Jx = (J+ + J−)/2, we see that when j > 0, the state \|j, m⟩is never an eigenstate of Jx. Hence, for j > 0, we can never be certain of the outcome of measurements of both Jx and Jz. This argument applies equally to Jy, so it’s impossible to be certain of the outcome of measurements of more than one component of J. If j = 0, every component of J yields zero, but a null vector in R3 has no direction. Consequently, there are no states in which the vector J has a well-deﬁned direction. This situation contrasts with the momentum vector P which can have a well–deﬁned direction since all its components commute with one another. In the mathematical literature, states with m = j are known as highest weight states. They play a key role in the representation theory of any group G, because once we know the highest weight state the rest of the multiplet can be constructed by applying lowering operators such as J−. Physically, a highest weight state is one in which the body’s angular momentum is most nearly aligned along the z-axis. In this state, the ratio of the squared angular momentum that lies in the xy-plane to that parallel to the z-axis is ⟨j, j\|J2 x + J2 y\|j, j⟩ ⟨j, j\|J2 z \|j, j⟩ = ⟨j, j\|J2 −J2 z \|j, j⟩ ℏ2j2 = 1 j . (5.17) As Jx = (J+ + J−)/2 we have ⟨j, j\|Jx\|j, j⟩= 0 and similarly ⟨j, j\|Jy\|j, j⟩= 0. Thus we have no information about the direction in the xy-plane any angular momentum points. – 59 – Macroscopic bodies, for which j ≫1, have only a tiny fraction of their total angular momentum in the xy-plane when they’re in state \|j, j⟩, so the uncertain direction of this component of angular momentum does not lead to signiﬁcant uncertainty in the total direction of the angular momentum vector. By contrast, when j = 1/2, even in state \|1/2, 1/2⟩there’s twice as much angular momentum associated with the xy-plane as with the z-axis and the uncertain direction of this planar component makes it impossible to say anything more speciﬁc than that the angular momentum vector lies somewhere in the northern, rather than southern, hemisphere. Even when j = 1 and m = 1, the state \|1, 1⟩ has as much angular momentum along the xy-plane as it has parallel to the z-axis, so the direction of the angular momentum vector is still very uncertain. This is no surprise: since dim(Hj) = 2j + 1, when j = 1 2 there are only two independent states the orientation of our body can take. With such a limited Hilbert space it’s no wonder we only have a fuzzy notion of where the body’s angular momentum lies. 5.2.1 Rotation of Diatomic Molecules Our deduction of the possible eigenvalues and eigenstates of J2 and Jz came from con-sidering rotations: the states \|j, m⟩simply enable us to describe what happens when an object is rotated around some axis, as we’ll understand in more detail and with many examples later in the chapter. In particular, it’s important to understand that a priori these states have nothing to do with the energy levels of any given particle. As always, there’s no way to tell how rotating an object may or may not change its energy until we specify a form for the Hamiltonian. In this section, we’ll choose a simple form of dynamical relation H = H(J) that will enable us to understand an important part of the dynamics of a diatomic molecule. For some purposes, a diatomic molecule such as CO molecule can be considered to consist of two point masses, the nuclei of the oxygen and carbon atoms, joined by a ‘light rod’ provided by the electrons. Following the analogous formula in classical mechanics, we model the dynamics of this molecule by the Hamiltonian H = 1 2 J2 x Ix + J2 y Iy + J2 z Iz ! . (5.18) where Ix is the moment of inertia along the x-axis, and similarly for Iy and Iz. Though motivated by classical mechanics, the best justiﬁcation for this form of Hamiltonian is ultimately that it agrees with detailed experiments. In our axisymmetric case, we choose coordinates whose origin is at the centre of mass of the molecule (somewhere between the C and O atoms) and orient our coordinates so that the z-axis lies along the molecule’s axis of symmetry. Then Ix = Iy = I, say, whilst Iz is diﬀerent. In fact, since the centre of mass of the C and O atoms lie along the z-axis, the molecule’s moment of inertia around this axis is negligible, Iz ≪I (see ﬁgure 8). We can thus rewrite our Hamiltonian as H = 1 2 J2 I + J2 z 1 Iz −1 I . (5.19) – 60 – z x y − − − + + + Figure 8: A simple model of carbon monoxide. By symmetry, Ix = Iy whilst Iz is much smaller. The virtue of expressing H in terms of J2 and Jz is that our knowledge of the spectrum of these operators immediately allows us to write down the spectrum of this Hamiltonian: the state \|j, m⟩is an energy eigenstate with Ejm = ℏ2 2 j(j + 1) I + m2 1 Iz −1 I , (5.20) with \|m\| ≤j. Since Iz ≪I for our diatomic molecule, the coeﬃcient of m2 is very much greater than that of j(j+1), so states with m ̸= 0 will only occur very far above the ground state. Consequently, the low lying states have energies of the form Ej = j(j + 1)ℏ2 2I (5.21) for some j. As we saw in the previous section, only discrete values of j are allowed, so the energy levels are quantized. One can excite a CO molecule, causing it to rotate faster, only by supplying a deﬁnite amount of energy. Similarly, for the molecule to relax down to a state of lower angular momentum, it must emit a quantized lump of energy. Carbon monoxide is a signiﬁcantly dipolar molecule. The carbon atom has a smaller share of the binding electrons that does the oxygen, with the result that it is positively charged while the oxygen atom gains negative charge. In Maxwell’s theory of electrody-namics, a rotating electric dipole is expected to emit electromagnetic radiation. Because we’re in the quantum regime, this radiation emerges as photons which, as we’ll see later in the chapter, can add or carry away only one unit ℏof angular momentum. Thus the energies of the photons that can be emitted or absorbed by a rotating dipolar molecule are Eγ = ±(Ej −Ej−1) = ±jℏ2 I . (5.22) Using the relation E = ℏω, the angular frequencies in the rotation spectrum of the molecule are ωj = jℏ I (5.23) In the case of 12CO, 2πℏ/I evaluates to a frequency ν ∼113.1724 GHz and spectral lines occur at multiples of this frequency. In the classical limit of large j, the molecule’s total – 61 – angular momentum \|J\| ≈jℏ. This is related to the angular frequency Ωat which the molecule rotates by \|J\| = IΩ. Comparing to (5.23) we see that in the classical limit ωj≫1 = Ω, so the frequency of the emitted radiation is just the frequency at which the molecule rotates. Measurements of the radiation at 113.1724 GHz provide one of the two most important probes of interstellar gas48. In denser, cooler regions, hydrogen atoms combine to form H2 molecules which are bisymmetric and do not have an electric dipole moment when they rotate. Consequently, these molecules, which together with similarly uncommunicative Helium make up the vast majority of cold interstellar gas, lack readily observable spectral lines. Astronomers are thus obliged to study the interstellar medium through the rotation spectrum of the few parts in 106 of CO it contains. 5.3 Spin In section 5.3 we saw that the orbital angular momentum operator L and spin operator S each obeyed an identical algebra to that of the rotation generators, [Li, Lj] = iℏ X k ϵijkLk [Li, L2] = 0 [Si, Sj] = iℏ X k ϵijkSk [Si, S2] = 0 (5.24) as well as [Si, Lj] = 0. Since the algebra of the Js was all we used to deduce their spectra, it follows immediately that (L2, Lz) and (S2, Sz) have the same possible spectra as we found for (J2, Jz). It’s traditional to label eigenstates of (L2, Lz) as \|ℓ, m⟩and those of (S2, Sz) as \|s, σ⟩, where ℓand s correspond to the eigenvalues of L2 and S2, respectively, and m and σ label the eigenvalues of Lz and Sz. (Unfortunately, it’s traditional to label the eigenvalue of both Jz and Lz by the same letter m, even though they mean diﬀerent things and may take diﬀerent values in any given state. Which is meant is usually clear from the context.) 5.3.1 Large Rotations Our claim that the possible values of (j, m) are j ∈{0, 1/2, 1, 3/2, . . .} and then m ∈{−j, −j + 1, . . . , j −1, j} (5.25) was based on examining the algebra [Ji, Jj] = iℏP k ϵijkJk, and likewise for (ℓ, mℓ) or (s, σ). In turn, this algebra originally came from considering the behaviour of objects under very small rotations. We now check whether these spectra are also compatible with large rotations. Suppose we rotate the state \|j, m⟩through an amount α around the z-axis. Then \|j, m⟩→U(αˆ z)\|j, m⟩= e−iαJz/ℏ\|j, m⟩= e−imα\|j, m⟩. (5.26) since it is an eigenstate of Jz. Rotations through 2π around any axis return us to our starting point, so are equivalent to no rotation. Thus, for U(α) to be a homomorphism 48The other key probe is the hyperﬁne line of atomic hydrogen that will be discussed in chapter 8. – 62 – from SO(3) to the group of unitary operators on H, we must have U(2π ˆ α) = 1H. In particular, when ˆ α = ˆ k, from above we must have e−2πim = 1. This is true if m is an integer, but not if it is an odd-half-integer. Since m is an integer or (odd) half-integer iﬀj is, we conclude that odd half-integer values of j are in fact not compatible with the behaviour of objects under large rotations: they’re ruled out by our basic requirement that U(α) represents the action of SO(3) on H. The fact that global properties of rotations rule out some of the eigenvalues allowed by the rotation algebra is an example of a phenomenon familiar from IB Methods. The behaviour of function in the neighbourhood of a point may be governed by some diﬀerential equation. The associated diﬀerential operator (if it’s linear) typically has a large spectrum, which is cut down by boundary conditions or periodicity conditions. For example, on any open set U ⊂R the linear operator −id/dx has eigenfunctions eikx for any k ∈C. However, if we know that globally ψ : S1 →C so is periodic, then k must be quantized in units of the circle’s radius. The only diﬀerence in our case is that the non-trivial algebra of the rotation generators already restricted the possible eigenvalues of J2 and Jz to be quantized in units of ℏ. Global properties of the rotation group still remove some of the eigenvalues that were allowed locally. More succinctly, states where j ∈N0 + 1 2 do form representations of the rotation algebra so(3), but not of the rotation group SO(3). In fact, we’ll see that our arguments above have been rather too hasty. 5.3.2 The Stern–Gerlach Experiment Quantization of angular momentum in units of ℏwas originally proposed by Bohr & Som-merfeld as a means by which the stability of atomic orbits could be understood; we’ve now seen how this quantization arises automatically in the full mathematical framework of quantum mechanics. However, back in 1922 it still seemed very mysterious, so Stern and Gerlach designed and conducted experiments to check whether angular momentum is really quantized in Nature. In the Stern–Gerlach experiments, a beam of uncharged atoms of the same type is passed through a region of slowly varying magnetic ﬁeld. If the atoms have mass M, the Hamiltonian for this process is49 H = P2 2M −µ · B (5.27) where B is the applied magnetic ﬁeld, and µ is known as the magnetic dipole moment of the atom. Ultimately, the reason the atom can be treated as a magnetic dipole is because of detailed properties of the distribution of its electrons. It would take us too far aﬁeld to explain this precisely, but the dipole moment arises because the atom has an orientation: a perfectly spherically symmetric atom cannot have any dipole moment as there is no preferred direction for the ‘north’ or ‘south’ poles. Thus the dipole moment is 49The potential energy resulting from coupling a magnetic dipole to an applied magnetic ﬁeld is similar to the (perhaps more familiar) energy −p · E of an electric dipole p in an applied electric ﬁeld E. Note that the atoms carry no net electric charge, so there’s no Lorentz force. – 63 – Figure 9: A schematic picture of the Stern–Gerlach experiment, borrowed from the As-tronomy Cast site. only non-zero for atoms that transform non-trivially under S. If the atom is in the spin s representation, we can write µ = (µ/ℏs) S. Choosing our coordinates so that the z-axis points along the direction of the magnetic ﬁeld, the Hamiltonian becomes H = P2 2M −µ ℏsBSz , (5.28) where B = \|B\|. The equations of motion (obtained, say, using the Heisenberg picture) tell us that the expectation values of position and momentum in any state \|ψ⟩obey d dt⟨X⟩= ⟨P⟩ M and d dt⟨P⟩= µ ℏs⟨(∇B) Sz⟩. (5.29) The second of these equations is analogous to the classical F = −∇V . We see that the force experienced by any given atom depends on its value of Sz. For a spin s particle, these are ℏ{−s, −s + 1, . . . , s −1, s} and in particular can be of either sign. If an atom is in the state \|s, s⟩with all its spin aligned along the direction of B, then it will experience a force pushing it in the direction of increasing magnetic ﬁeld. On the other hand, those atoms in the state \|s, −s⟩will be pushed in the direction of decreasing B, while atoms with σ = 0 (when s is an integer) are unaﬀected. In total, when an initial beam of atoms in which the spins are randomly aligned passes through the region of magnetic ﬁeld, it will be split into 2s + 1 diﬀerent trajectories. Thus, if the atoms are perfectly spherical, they will pass through unaﬀected, whereas if they have s = 1 the beam will be split into three, corresponding to the three possible eigenvalues {ℏ, 0, ℏ}, if they have s = 2 the beam will split into ﬁve, and so on. Finally, if the atoms have very large spin (and so a very deﬁnite orientation in space) the beam will be split into so many paths that we can no longer distinguish the individual paths, seeing instead a broad smear. This is the same result we’d expect to ﬁnd if angular momentum is not quantised, where the amount by which an atom is deﬂected would depend smoothly on its orientation with respect to B. Stern and Gerlach’s original experiment in fact used silver atoms. Their magnetic was controlled by a dial. Initially, as B = 0 the atoms all passed straight through. As they – 64 – turned up the magnetic ﬁeld, the beam split – into two separate paths. As B was further increased, the separation between these two beams increased, but no further splitting was observed. Thus, not only is angular momentum quantized as Bohr & Sommerfeld had predicted, but silver atoms have spin s = 1 2! 5.3.3 Spinors and Projective Representations This section lies beyond the Schedules and is non-examinable. The Stern–Gerlach experiment shows that, despite our arguments, half–integer values of s actually arise in Nature. In fact, the chemical properties of the elements and the structure of the periodic table, together with properties of materials such as metals, conductors and insulators depends crucially on particles having half-integer values of spin. These values are in conﬂict with our current mathematical formalism, so we must have made a mistake, imposing too strong a condition that ruled out the possibility of half-integer spins. The error lay in our claim that we needed to represent the action a transformation group G on Hilbert space H, rather than just on projective Hilbert space PH. (Recall that states which diﬀer only by an overall constant – which does not need to have modulus 1 provided we use the general form (2.69) of the Born rule – yield the same results in all experiments. Thus physical systems are represented by states in projective Hilbert space.) Projectively, it’s enough to require U(g2) ◦U(g1) = eiφ(g2,g1) U(g2 · g1) (5.30) rather than (4.7), where φ(g1, g2) is a real phase. This phase does not aﬀect which ray in H the state lies in, so leaves the physics unchanged. The operator algebra is associative, so we must have U(g3) ◦(U(g2) ◦U(g1)) = (U(g3) ◦U(g2)) ◦U(g1) (5.31) which implies the phases obey φ(g2, g1) + φ(g3, g2 · g1) = φ(g3, g2) + φ(g3 · g2, g1) . (5.32) Phase factors eıφ(g2,g1) obeying this condition are known as cocycles on the group G. One possible solution of (5.32) is to take φ(g2, g1) = β(g2 · g1) −β(g2) −β(g1) . (5.33) for some arbitrary (smooth) function β : G →R. This solution is ‘trivial’, because if φ(g2, g1) takes this form, then can deﬁne a new unitary transformation operator U ′(g) = eiβ(g)U(g) which obeys our original condition (4.7). By agreeing to work with the new operator, the phases never arise. The interesting question is whether there are other, non-trivial solutions to (5.32). – 65 – A theorem in group cohomology50 states that it’s possible for non-trivial projective representations to arise for groups that are not simply connected51. In fact, this is the case for SO(3). The topology of the rotation group can be seen by viewing each rotation as parametrised by a vector α. If we allow the axis of rotation ˆ α to point in any direction, then rotations with \|α\| ∈(0, π) are uniquely speciﬁed. However, when rotating through π we get the same rotation whether rotating around ˆ α or −ˆ α. Thus, we can picture the space of all rotations as a solid, three dimensional ball \|α\| ≤π, but with antipodal points on the surface \|α\| = π identiﬁed. This description show that SO(3) contains smooth, closed paths, beginning and ending at the identity rotation (the origin of the 3-ball) that cannot be continuously shrunk to a point. For example, consider paths which start and end at the identity rotation, i.e. the centre of the sphere. Figure ?? shows a loop which is contractible; it can obviously be shrunk to a point. On the other hand, ﬁgure ?? shows a loop for which the angle of rotation starts at zero, smoothly increasing to π at the point A. At this point it reaches the boundary and reappears at the antipodal point A′, before continuing to increase to 2π back at the identity. It should be intuitively clear that this loop cannot be shrunk to a point whilst keeping both its ends ﬁxed at the identity. Next, consider the loop in ﬁgure ??, along which the angle of rotation increases from 0 to 4π, reaching the boundary of the ball twice, once at A, reappearing at A′ and then again at B reappearing at B′. By moving the second antipodal pair B and B′ (as shown in ﬁgure ??) the section of the path between A′ and B can be pulled across the boundary, smoothly deforming the loop back to the situation in ﬁgure ??. Thus, after two complete rotations the situation becomes simple once more. In general, the angle along any closed path must increase by an integer multiple of 2π. The resulting loop will be contractible if this integer is even, but non-contractible if it is odd. The topological properties of paths in G become important when we consider the behaviour of U(g) as the group element g varies around a loop L, beginning and ending at some ﬁxed g0. It is quite possible for U(g) to change smoothly with g along L in such a way that the operators at each end of the loop do not coincide: they may diﬀer by a phase U(g0) L − →αLU(g0) where the phase αL depends on the loop. This is consistent with the projective homomor-phism (5.30). To investigate the possibilities for αL, let’s specialize to the case in which the operators act on a Hilbert space of ﬁnite dimension N, so that each U(g) can be regarded as an N ×N unitary matrix. We can use up some of the freedom in (5.30) by requiring that each such matrix has unit determinant. This does not ﬁx things completely, but the residual freedom 50Unfortunately, I won’t prove this here. If you’re interested, you can ﬁnd a proof (given in the context of quantum mechanics) in either Weinberg’s The Quantum Theory of Fields, vol. 1 (chapter 2, appendix B), or else in Hall’s Quantum Theory for Mathematicians. 51There’s one other possibility: that the Lie algebra g contains central elements (vectors e ∈g s.t. [e, g] = 0) that cannot be removed by a redeﬁnition of the generators. This case is not relevant for us. – 66 – is discrete: det U(g) = +1 for all g ∈G ⇒ eiNφ(g1,g2) = 1 If G is simply connected, meaning that any closed loop is contractible, then the determinant condition implies αL = 1 for all loops L. This follows straightforwardly using continuity: αL must be an Nth root of unity, but if it varies continuously as we vary our loop L and if all such loops are contractible to the single point g0, then we must have the same matrix at the beginning and ending of each loop, so αL = 1. If G is not simply connected, the same argument still shows that αL = 1 for any contractible L, but there may also be non-contractible loops with αL ̸= 1. In this case, we can at least deduce that αL = αL′ whenever the loops L and L′ are in the same homology class, meaning that they can be smoothly deformed into one another. Furthermore, if loops L and L′ are traversed successively, then U(g0) L →αLU(g0) L′ →αLαL′U(g0), so there are self-consistency constraints. We can now give a unique deﬁnition of U(g) for all g in a simply connected Lie group G. Recalling that U(e) = idH, we deﬁne U(g0) by choosing any path from the identity e to g0 and demanding that U(g) changes smoothly along this path. The values along the path are unique by the determinant and continuity conditions, but the end result U(g0) is also unique, because by traversing them in opposite directions, any two paths e →g0 can be combined to form a closed loop at g0. This loop is contractible since our group is simply connected. With this deﬁnition, continuity also ensures that all cocylces are equal to one, so we have a genuine representation (not just a projective representation) of G. Carrying out the same construction when G is not simply connected, we’ll encounter paths from e to g0 which cannot be smoothly deformed into one another. Thus, starting from U(e) = idH, in general we obtain diﬀerent values for U(g0) depending on the path we take. In this way we’re forced to consider multi-valued functions on G (just as when deﬁning a continuous square root in the complex plane). The ambiguity, or multi-valuedness, in U(g0) can be resolved only by keeping track of the path we used to reach g0, just as for the complex square root we must keep track of how many times we’ve encircled the origin to be sure which branch we’re on. Such a multi-valued deﬁnition inevitably means that non-trivial cocyles appear. As we saw above, the rotation group G = SO(3) is not simply connected, but there rae just two topological classes of loops, depending on whether the net angle of rotation is an even or odd multiple of 2π. Any loop l in the ﬁrst class is contractible and so has αL = 1. Any loop L′ in the second class is non-contractible, but if we traverse L′ twice it becomes contractible again. Thus α2 L′ = 1 and αL′ = ±1. The ﬁnite-dimensional spaces Hj on which the rotation operators act are nothing but the multiplets of total angular momentum j(j + 1)ℏ2, with a basis {\|j, m⟩} where m ∈{−j, −j + 1, . . . , j −1, j} Thus dim Hj = 2j + 1. From the determinant condition (αL)2j+1 = 1 for any loop, but from our topological considerations αL = ±1. If j is an integer then 2j + 1 is odd and it – 67 – follows that αL = 1 for all loops, contractible or not. However, if j is an odd half-integer then 2j + 1 is even and αL = −1 is possible for non-contractible loops. This is exactly the behaviour we found above: under the rotation operator U(αˆ z) = e−iαJz/ℏ, a generic state in the j multiplet transforms as \|ψ⟩= j X m=−j cm\|j, m⟩ 7→ U(α)\|ψ⟩= j X m=−j cm e−iαm\|j, m⟩. (5.34) Any state with j (and hence all m) odd-half-integral changes sign under a rotation through an odd multiple of 2π. The discussion of this section shows that the origin of this unex-pected sign is the non-trivial topology of the rotation group – in our new terminology, we have a projective representation of SO(3). These projective representations play an important role in physics, and are often called spinors. We have, ﬁnally, obtained the correct mathematical framework in which to describe rotations and angular momentum in quantum mechanics. The history of the subject devel-oped rather diﬀerently to the exposition we’ve given here. By the 1920s, careful studies of atomic spectroscopy had revealed that many spectral lines were in fact doubled, composed of two lines, very close in frequency. In 1924 (even before Schr¨ odinger published his famous equation) Pauli proposed that this doubling indicated that, in addition to the quantum numbers n, ℓ, m labelling their energy levels, electrons possessed a further quantum number that took just two values. A year later, Uhlenbeck & Goudsmit suggested that this could be associated to some form of internal angular momentum. Their idea was initially treated with suspicion. If one wished to suppose the electron was a small, rotating sphere, then for it to have the needed angular momentum ℏ/2 and yet keep its radius small enough so that its ﬁnite size would not have been detected by experiment52, the surface of the sphere would need to be travelling faster than light. Nevertheless, the Stern–Gerlach experiment showed that particles could indeed have spin, contributing to the Hamiltonian in the presence of a magnetic ﬁeld in just the same way as would a classical spinning magnetic dipole. Heisenberg, Jordan and C. G. Darwin53 then showed that the internal spin of the electron exactly accounted for the ﬁne splitting of spectral lines that had puzzled Pauli, as we’ll see in chapter 8.1.1. We now understand that spin is an intrinsic property of fundamental particles: the Hilbert space of a fundamental particle is not simply Hspat ∼ = L2(R3, d3x) describing its spatial wavefunction, but rather a tensor product Hspat⊗Hs. However we may excite, crash into, or generally interfere with the motion of an electron or W boson, for as long as they remain electrons and W bosons, their spin will always be s = 1 2 and s = 1, respectively. 5.3.4 Spin Matrices Since σ ∈{−s, −s + 1, . . . , s −1, s}, states of deﬁnite total spin s can be described by a ﬁnite dimensional Hilbert space Hs ∼ = C2s+1. As always, once we pick a basis on H we can 52To date, no experiment has ever detected a ﬁnite size, or any other non-trivial internal spatial structure in the electron. 53Charles Galton Darwin, grandson of Charles Robert Darwin. – 68 – describe the action of linear operators such as S explicitly in terms of matrices. Let’s now carry this out the ﬁrst few spin representations, working in the basis {\|s, σ⟩} of eigenstates of Sz. The simplest case is s = 0, for which the only possible value of σ is also zero. This state obeys e−iα·S/ℏ\|0, 0⟩= \|0, 0⟩for any α. Hence a spin-zero object, like a perfect sphere, is completely unchanged under any rotation. In view of this, spin-zero particles are known as scalar particles. The discovery of the Higgs boson, announced on 4th July 2012 at CERN in Geneva, was the ﬁrst time a fundamental scalar particle had been observed in Nature. The next case is s = 1 2. Electrons, neutrinos and quarks are fundamental particles of spin 1 2, whilst protons, neutrons and the silver atoms used by Stern & Gerlach are examples of composite spin-half particles. When s = 1 2, σ can only take one of the two values ± 1 2. For shorthand, we let \|↑⟩denote the state \|s, σ⟩= \| 1 2, 1 2⟩and \|↓⟩denote \| 1 2, −1 2⟩. A generic state of a spin-half system can thus be expanded as \|ψ⟩= a\|↑⟩+ b\|↓⟩ (5.35) in this basis, where a, b ∈C and \|a\|2 + \|b\|2 = 1 so that \|ψ⟩is correctly normalised. In this basis, we can write the spin operators themselves as the 2 × 2 matrices Sx = ⟨↑\|Sx\|↑⟩⟨↑\|Sx\|↓⟩ ⟨↓\|Sx\|↑⟩⟨↓\|Sx\|↓⟩ ! , Sy = ⟨↑\|Sy\|↑⟩⟨↑\|Sy\|↓⟩ ⟨↓\|Sy\|↑⟩⟨↓\|Sy\|↓⟩ ! , Sz = ⟨↑\|Sz\|↑⟩⟨↑\|Sz\|↓⟩ ⟨↓\|Sz\|↑⟩⟨↓\|Sz\|↓⟩ ! . (5.36) Because \| ↑⟩and \| ↓⟩are eigenstates of Sz, evaluating Sz in this basis is immediate. To also evaluate Sx and Sy, we note that Sx = (S+ + S−)/2 and Sy = (S+ −S−)/2i where S± are the spin raising and lowering operators deﬁned just as for J±. Using (5.16) with j = s = 1 2 gives S+\|↓⟩= ℏ\|↑⟩and S−\|↑⟩= ℏ\|↓⟩. In this way, we obtain Sx = ℏ 2 0 1 1 0 ! , Sy = ℏ 2 0 −i i 0 ! , Sz = ℏ 2 1 0 0 −1 ! . (5.37) The coeﬃcients of ℏ/2 here are known as Pauli matrices and usually written54 as (σx, σy, σz). Thus, for s = 1/2, we can write S = ℏ/2 σ. Proceeding to spin-one, we ﬁnd three possible values σ ∈{−1, 0, 1}. Thus the spin-one Hilbert space is three (complex) dimensional, and when s = 1 we can represent each component of S by a 3 × 3 matrix. In this case, the spin raising and lowering operators S± act as S+\| −1⟩= √ 2ℏ\|0⟩, S+\|0⟩= √ 2ℏ\| + 1⟩, S−\| + 1⟩= √ 2ℏ\|0⟩, S−\|0⟩= √ 2ℏ\| −1⟩, (5.38) 54Do note confuse this vector of Pauli matrices with the eigenvalue label of Sz. – 69 – as follows from (5.16) with j = s = 1. Using these results, one can check that Sx = ℏ √ 2    0 1 0 1 0 1 0 1 0   , Sy = ℏ √ 2    0 −i 0 i 0 −i 0 i 0   , Sz =    1 0 0 0 0 0 0 0 −1   . (5.39) Sadly, these matrices don’t have any special name. Just as for spin-half, we can use this representation to express the state \|1, θ⟩in which a measurement of n·S along some axis n is certain to yield +ℏ. Examples of fundamental particles with spin-one are the somewhat unimaginatively named W and Z bosons55. These are responsible for the weak interactions that, among other things, allows two Hydrogen nuclei to fuse into Deuterium, powering nuclear fusion in the core of stars. In just the same way, we can represent each component of the spin operator by a (2s + 1) × (2s + 1) matrix for any ﬁnite s. Since our basis is adapted to Sz, the matrix representation of Sz will be Sz = ℏdiag(s, s −1, . . . , −s + 1, −s) . (5.40) Matrices for Sx and Sy may be constructed using the spin raising and lowering operators as above. Since S± only change the z-component of the spin by 1 unit, these matrices are very sparse. One ﬁnds that they have non-zero entries only along the subleading diagonals, given by (Sx)σ′σ = ℏ 2 α(σ) δσ′−1,σ + α(σ −1) δσ′+1,σ (Sy)σ′σ = ℏ 2i −α(σ) δσ′−1,σ + α(σ −1) δσ′+1,σ , (5.41) where α(σ) = p (s −σ)(s + σ + 1). Note that, whatever the value of s, we always have three matrices (Sx, Sy, Sz) describing reorientations around the three independent axes in R3. Note also that each of these matrices is indeed traceless, as required by iℏϵijktrH(Sk) = trH([Si, Sj]) = 0 as we said at the beginning of the chapter. In elementary particle physics (and also in Tripos questions), one rarely encounters spins higher than 1. Nonetheless, it’s interesting to consider the limit s ≫1 of very high spins to see how our classical intuition emerges. For example, and electric motor that is roughly 1 cm in diameter and weighs about 10 gm might spin at ∼100 revolutions per second. Its intrinsic angular momentum is then ∼103 kg m2 s−1 ≈1031ℏ. The classical world thus involves huge values of s! Let’s now show that, when s ≫1, there is very little uncertainty in the direction of a systems spin. Suppose our system is in the state \|s, s⟩so that its spin is maximally aligned with the z-axis. Let’s compute ⟨s, s\|n · S\|s, s⟩where n = (sin θ, 0, cos θ). That is, we want to know how much spin we expect to measure along the direction in the xz-plane, inclined 55Although the photon carries one unit of ℏin intrinsic angular momentum, it has only two possible states corresponding to left- or right-circular polarized light. Because the photon is massless, one needs to consider representations of the Lorentz group, rather than the spatial rotation group, in order to describe it accurately. We won’t do this in this course. – 70 – at angle θ to the z-axis. Classically, this would be just ℏs cos θ, the projection of s onto this axis. Quantum mechanically, we have ⟨s, s\|n · S\|s, s⟩= sin θ ⟨s, s\|Sx\|s, s⟩+ cos θ ⟨s, s\|Sz\|s, s⟩= ℏs cos θ , (5.42) using the fact that Sx = (S+ + S−)/2. Thus, on average the quantum result agrees with the classical intuition. This holds for any value of s, but now let’s ask what the uncertainty in this result is. We compute ⟨s, s\|(n · S)2\|s, s⟩= sin2 θ ⟨s, s\|S2 x\|s, s⟩+ sin θ cos θ ⟨s, s\|SxSz + SzSx\|s, s⟩+ cos2 θ ⟨s, s\|S2 z\|s, s⟩ = 1 4 sin2 θ⟨s, s\|S+S−\|s, s⟩+ ℏ2s2 cos2 θ = ℏ2 s 2 sin2 θ + s2 cos θ , (5.43) with all other terms vanishing. Consequently p ⟨(n · S)2⟩−⟨n · S⟩2 = rs 2 ℏ\| sin θ\| (5.44) and so, in the classical limit of large s, the uncertainty is small (∼ p 1/s) compared to ⟨n · S⟩. 5.3.5 Paramagnetic Resonance and MRI Scanners In the presence of an external magnetic ﬁeld, a classical magnetic dipole µ experiences a torque ∂L ∂t = µ × B . (5.45) The dipole will thus turn until it aligns itself along the direction of the ﬁeld, minimizing its energy. This is familiar from a compass. However, suppose the dipole is already spin-ning around its centre of mass, such that µ = γL where the constant γ is known as the gyromagnetic ratio. Then instead one ﬁnds that the torque causes the dipole to precess around B. As in our discussion of the Stern-Gerlach experiment, particles such as a proton or electon do have magnetic dipole moments µ = (2µ/ℏ) S proportional to their spin. We’ll see that in quantum mechanics, this spin does indeed precess around the direction of an applied magnetic ﬁeld. This is the basis of MRI scanners, which have become an enormously important diagnostic tool for both chemistry and medicine. We can understand the basic principles of an MRI machine by using our spin matrices. Unlike the beam of silver atoms in the Stern-Gerlach experiment, here the protons are not free to move, because they’re held in place by the electromagnetic binding forces of a complex molecule, and this molecule is also held in a (roughly) ﬁxed place in our cells. Thus we take the Hamiltonian to be H = −µ · B = −2µB ℏSz , (5.46) – 71 – with no kinetic term. Again, we’ve chosen the direction of the magnetic ﬁeld to deﬁne the ˆ z axis. In particular, for a spin-1 2 particle such as a proton, the eigenvalues of this Hamiltonian are E± = ∓µB, where B = \|B. Initially, we cannot expect the protons in our body to have their spins already aligned along ˆ B. Suppose instead that some proton has its spin aligned along some axis n = (0, sin θ, cos θ) inclined at angle θ to ˆ B. That is, we consider a proton in the state \|θ↑⟩ deﬁned by n · S \|θ↑⟩= ℏ 2 \|θ↑⟩. (5.47) We can always choose to expand this state in terms of our basis {\| ↑⟩, \| ↓⟩}, representing states of deﬁnite spin along ˆ z, as \|θ↑⟩= a\|↑⟩+ b\|↓⟩ (5.48) where \|a\|2 + \|b\|2 = 1. In terms of our matrices (5.37) for spin-1 2, the eigenvalue equa-tion (5.47) becomes ℏ 2 sin θ 0 −i i 0 ! a b ! + ℏ 2 cos θ 1 0 0 −1 ! a b ! = ℏ 2 cos θ −i sin θ i sin θ −cos θ ! a b ! = ℏ 2 a b ! (5.49) Solving this eigenvalue problem and the normalisation condition yields a = cos θ 2 and b = i sin θ 2 (up to a possible phase), so a proton whose spin is aligned along the n axis is in state \|θ↑⟩= cos θ 2 \|↑⟩+ i sin θ 2 \|↓⟩ (5.50) Note that \|θ↑⟩has the expected behaviour at θ = 0 and θ = π, and that it yields −\|↑⟩as θ is continuously increased to 2π. (It’s straightforward to generalise this example to the case of a state with spin aligned along an arbitrary axis n.) Applying the time evolution operator U(t) = e−iHt/ℏwith the Hamiltonian (5.46), we ﬁnd that at time t the proton’s state has evolved to \|ψ(t)⟩= U(t)\|θ↑⟩= cos θ 2 e−iHt/ℏ\|↑⟩+ i sin θ 2 e−iHt/ℏ\|↓⟩ = cos θ 2 e−iωt/2 \|↑⟩+ i sin θ 2 e+iωt/2 \|↓⟩ (5.51) where ω = 2µB is the Larmor frequency. One can check that this state is an eigenstate of the spin operator aligned along the axis n(t) = (sin θ sin ωt, sin θ cos ωt, cos θ), so that at time t, the proton’s spin is deﬁnitely aligned along n(t). Consequently, as time passes, the spin of any proton will precess around the direction of B with frequency ω that is independent of the angle of inclination θ – i.e. independent of the proton’s initial orientation (provided it was not pointing exactly along ˆ B in the ﬁrst place). This is exactly the same behaviour as we found classically. When a material that contains chemically bound hydrogen atoms is immersed in a strong magnetic ﬁeld, over time the protons will emit radiation (not accounted for by our – 72 – Figure 10: An image of a cross-section of the head, produced by an MRI scan. Diﬀerent regions contain diﬀerent concentration of organic molecules, so react more or less strongly to the resonant magnetic ﬁeld. MRI scanners are one of the most important diagnostic tools in modern medicine. above Hamiltonian) so as to sit in the ground state. Thus, eventually the precession will cease and most of the protons’ spins will be aligned along the direction of B. Now suppose that, in addition to the static external B ﬁeld, we apply a small addi-tional magnetic ﬁeld b cos(ωt) that varies at the same Larmor frequency ω = 2µB. The Hamiltonian felt by the protons will then be H = −µ · (B + b cos(ωt) ) = −2µ ℏ B b cos(ωt) b cos(ωt) −B ! (5.52) if the new ﬁeld is in the ˆ x-direction. The TDSE for this system is thus ˙ a ˙ b ! = −i B b cos(ωt) b cos(ωt) −B ! a b ! (5.53) varies with frequency ω = 2µB/ℏ(which corresponds to radio frequencies for µ the dipole moment of the proton and B the strength of a typical magnetic ﬁeld available in hospitals). This radiation has just the right energy to excite these protons into the state where their spin is aligned against B. Consequently, such radiation is readily absorbed by the sample, whereas radiation at nearby frequencies is not. As we have seen, interference between the two states causes the spin to precess around the direction of B, and this precessing magnetic moment couples resonantly to the applied radiation ﬁeld. MRI scanners are typically tuned to determine the concentration of a single type of atom, usually Hydrogen. However, in a complex molecule, not every Hydrogen nucleus – 73 – (proton) feels the same magnetic ﬁeld, because of additional contributions from the elec-trons that bind the atoms together in the molecule. For example, in methanol (CH3OH) the magnetic ﬁeld experienced by the proton that is attached to the oxygen atom diﬀers from those experienced by the protons attached to the carbon atom. Also, of the 3 pro-tons bound to the carbon atom, one is on the opposite side of carbon from oxygen, while the other two lie between the carbon and oxygen atoms. The ﬁrst thus feels a diﬀerent magnetic ﬁeld to the other two. Now, the frequency ω of precession is proportional to the strength of the magnetic ﬁeld at the location of the proton, so for any ﬁxed strength of applied magnetic ﬁeld, methanol has three diﬀerent resonant frequencies. Clues to the chemical structure of a substance can thus be obtained by determining the frequencies at which magnetic resonance occurs in a given imposed ﬁeld. If we choose B to have a spatial gradient, then only a thin slice of our sample material will have ω tuned to its resonant frequency, so we excite transitions to higher energy levels only in this thin slice. Varying this ﬁeld in an orthogonal direction as the nuclear spins decay back down to the ground state allows us to recover three dimensional images. 5.4 Orbital Angular Momentum The topological considerations that allowed us to admit half-integer values of s do not apply to ℓ. To understand this, recall from section 4.3.1 that L could be interpreted as the generator of circular translations — transformations that translate a state around a circle in space, without adjusting its orientation. Unlike the space S3/Z2 of rotations, in R3 the space of such circular paths is contractible (see ﬁgure ??). Consequently, translation around a circular path always leaves our state unchanged. In particular, we must have e−2πiˆ z·L/ℏ\|ℓ, m⟩= e−2πim\|ℓ, m⟩= \|ℓ, m⟩, (5.54) so that m ∈Z and hence ℓ∈N0. 5.4.1 Spherical Harmonics The commutation relations of the Ls are exactly the same as those of the Ss, so for any ﬁnite ℓwe could choose to represent the orbital angular momentum operators by the same (2ℓ+ 1) × (2ℓ+ 1) matrices as we obtained above. However, in practical applications we’re usually interested in states whose orbital angular momentum quantum number ℓ may change, perhaps as a result of the particle being excited from one energy level to another. Thus it’s more convenient to use a formalism that allows us to treat all values of ℓ simultaneously. Furthermore, we often want to know about L at the same time as knowing about linear momentum P or position X, and we have seen that the [Xi, Pj] commutation relations do not have any ﬁnite dimensional representation and we can only represent them as operators acting on (wave)functions. Let’s now see how to reconstruct eigenfunctions of orbital angular momentum – the Legendre polynomials and spherical harmonics you met in IB – from the operator formalism. In the position representation, the orbital angular momentum operators become L = X × P = −iℏx × ∇ (5.55) – 74 – so that in particular Lz = −iℏ x ∂ ∂y −y ∂ ∂x . (5.56) In terms of spherical polar coordinates we have (x, y, z) = (r sin θ cos φ, r sin θ sin φ, r cos θ), so ∂ ∂φ = ∂x ∂φ ∂ ∂x + ∂y ∂φ ∂ ∂y + ∂z ∂φ ∂ ∂z = −y ∂ ∂x + x ∂ ∂y (5.57) so that Lz = −iℏ∂/∂φ. Thus, using these coordinates, the eigenvalue equation for Lz becomes ⟨x\|Lz\|ℓ, m⟩= −iℏ∂ ∂φ⟨x\|ℓ, m⟩= mℏ⟨x\|ℓ, m⟩ (5.58) or −i∂ψℓ,m(x)/∂φ = mψℓ,m(x), where ψℓ,m(x) = ⟨x\|ℓ, m⟩is the position space wavefunc-tion. This is solved by ψℓ,m(x) = K(r, θ) eimφ (5.59) for some function K(r, θ). Since m ∈Z, ψℓ,m is a single–valued function of the azimuthal angle φ. This is often given as a further reason why only integer values of m (and hence ℓ) should be allowed. A straightforward, though somewhat tedious calculation shows that the raising and lowering operators L± = Lx ± iLy = ±ℏe±iφ ∂ ∂θ ± i cot θ ∂ ∂φ (5.60) in the position representation. The condition L+ψℓ,ℓ= 0 then ﬁxes ψℓ,ℓ(x) = R(r) sinℓθ eiℓφ . (5.61) Applying the lowering operators one ﬁnds that all the other ψℓ,m(x)s are of the form ψℓ,m(x) = R(r) Ym ℓ(θ, φ) (5.62) where the spherical harmonic Ym ℓ(θ, φ) = (−1)m s (2ℓ+ 1) 4π (ℓ−m)! (ℓ+ m)! Pm ℓ(cos θ) eimφ (5.63) is given in terms of the associated Legendre polynomial Pm ℓ(x) = 1 2ℓℓ!(1 −x2)m/2 dℓ+m dxℓ+m (x2 −1)ℓ. (5.64) In particular, the spherical harmonics with m = 0 are proportional to the ordinary Legendre polynomial Y0 ℓ(θ) = r 2ℓ+ 1 4π Pℓ(cos θ) . (5.65) which is a odd or even polynomial in cos θ, according to whether ℓis odd or even. In particular, the Pm ℓs are only single–valued as functions on S2 when ℓis an integer, providing – 75 – Figure 11: The ﬁrst few spherical harmonics Ym ℓ(θ, φ). Each row contains pictures of the 2ℓ+ 1 spherical harmonics at ﬁxed ℓ, with m increasing from −ℓto ℓalong each row. another reason why the half–integer values allowed by general considerations of the algebra must in fact be discarded for L. We won’t be concerned with the detailed form of these spherical harmonics, though you may wish to note the orthogonality condition Z S2 Ym′ ℓ′ (θ, φ) Ym ℓ(θ, φ) sin θ dθ dφ = δℓℓ′ δmm′ (5.66) and the fact that r2∇2Ym ℓ(θ, φ) = −ℓ(ℓ+ 1) Ym ℓ(θ, φ) (5.67) where ∇2 is the Laplacian. Furthermore, since L is invariant under parity, Π−1LΠ = +L, it follows that so too are the raising and lowering operators L±. Therefore, all states in a given ℓmultiplet have the same parity. To determine what that parity is, note that in spherical polar coordinates the transformation x 7→−x becomes (r, θ, φ) 7→(r, θ −π, φ + π) (5.68) so in particular cos θ 7→−cos θ. Since Pℓ(−cos θ) = (−1)ℓPℓ(cos θ), we see that the parity of Y0 ℓis odd or even, according to whether ℓis odd or even, and hence Ym ℓ(θ −π, φ + π) = (−1)ℓYm ℓ(θ, φ) (5.69) for all spherical harmonics with a given value of ℓ. – 76 – It’s occasionally useful to have an alternative form of the spherical harmonics. Consider the polynomial in R3 ψ(x) = 3 X i1,i2,...,iℓ=1 ψi1i2...iℓxi1xi2 · · · xiℓ (5.70) that is homogeneous of degree ℓ. The coeﬃcients ψi1i2...iℓ∈C are necessarily totally symmetric in their indices, and the polynomial is harmonic, i.e. ∇2ψ(x) = 0 if ψi1i2...iℓare traceless on any pair of indices. Finally, notice that so far we’ve said nothing about the radial proﬁle of the wavefunc-tion, R(r). Since this function is certainly spherically symmetric, the rotation generators cannot tell us anything about it and to determine R(r) we’d need further information, such as the Hamiltonian. – 77 – 6 Addition of Angular Momentum Back in section 2.3.1, we understood how to describe composite systems using the tensor product of the Hilbert spaces of the individual subsystems. However, in many circum-stances, the basis of the tensor product formed by taking all possible pairs of basis ele-ments from the individual subspaces is not the most convenient. A key case is when the subspaces transform under the action of a group G. We’d like to understand the eﬀect of a G-transformation on the combined space, and the ‘obvious’ tensor product basis usually obscures this. In this chapter, we’ll see how this works for the case of the rotation group G = SO(3). Speciﬁcally, suppose two subsystems are each in states of deﬁnite total angular momentum – meaning mathematically that they transform in irreducible representations of SO(3) – with total angular momentum quantum numbers j1 and j2, respectively. We’d like to express the tensor product of the subsystems in terms of a sum of Hilbert spaces, Hj1 ⊗Hj2 = M j Hj , (6.1) where each Hj describes a state of the whole system with deﬁnite total angular momentum quantum number j56. Physically, to understand this decomposition is to understand how to add the angular momentum of the two subsystems so as to ﬁnd the possible values of the combined angular momentum of the whole system. For example, in a Hydrogen atom, both the proton and electron carry angular momentum ℏ/2 by virtue of their spins, and further angular momentum may be present depending on the electron’s orbit around the proton. If we wish to treat the atom as a whole, then we’ll be concerned with the angular momentum of the combined system rather than that of the electron and proton individually. An even more familiar example is your bike: the total angular momentum comes from a combination of the back and front wheels which can be independent (at least in principle, though I don’t recommend trying this while you’re riding along!). Our second task in this chapter is to ﬁll in a gap left in our knowledge: While we know how to use the raising and lowering operators J± to realign a given amount of an-gular momentum along or away from some axis, we’ve not yet learnt how to perform a mathematical operation that changes the total angular momentum of our state. In sec-tion 6.2.1, we’ll understand that, as well as states, operators themselves can carry angular momentum. Applying such an operator to a state yields a new state whose total angular momentum can diﬀer from that of the original state. We’ll illustrate this using various physical examples, including radiative transitions induced by an electric dipole moment, and the special ‘dynamical’ symmetries present in the Coulomb and harmonic oscillator potentials. 56In quantum mechanics we’re concerned with Hilbert space, but in fact the inner product plays very little role in this story. The whole of this section carries over more generally to the case of representations of a general Lie group G, where we decompose the tensor product Rj1 ⊗Rj2 of two irreducible representations (vector spaces) Rj1 and Rj2 in terms of a sum L j Rj of further irreducible representations. You can learn more by taking the Part II Representation Theory course next term (and more still in the Part III courses). – 78 – Figure 12: Two gyroscopes, of individual angular momenta j1 and j2, may be aligned relative to one another so that their total angular momentum ranges between j1 + j2 and \|j1 −j2\|. For ﬁxed relative alignment, the total angular momentum may be chosen to be aligned along any axis. 6.1 Combining the Angular Momenta of Two States Suppose we have two subsystems (‘gyros’) enclosed in a box, where the ﬁrst system has total angular momentum quantum number j1 and the second has total angular momentum quantum number j2. As we saw in section 5.1, a basis of states of the ﬁrst system is {\|j1, m1⟩} where m1 ∈{−j1, −j1 + 1, . . . , j1} (6.2) while the second system may be described in terms of the basis {\|j2, m2⟩} where m2 ∈{−j2, −j2 + 1, . . . , j2} . (6.3) From the general discussion of section 2.3.1, the state of the combined system can be expressed as \|ψ⟩= j1 X m1=−j1 j2 X m2=−j2 cm1m2\|j1, m1⟩⊗\|j2, m2⟩ (6.4) with some coeﬃcients cm1m2. We can choose m1 and m2 independently, so there are a total of (2j1 + 1)(2j2 + 1) states here – this is the dimension of the tensor product Hj1 ⊗Hj2. We want to understand how the states (6.4) behave under rotations; that is, we’d like to understand which linear combinations in (6.4) correspond to a deﬁnite amount of angular momentum for the system as a whole. Let’s ﬁrst consider the corresponding classical situation, where we may wish to know the combined angular momentum of two gryroscopes. Although the total angular momen-tum of the individual components is ﬁxed, that of the whole system is variable because it depends on the relative orientation of the two gyros: if they are aligned with eachother, the whole system might be expected to have total angular momentum labelled by j1 + j2, while if the two subsystems are aligned exactly against eachother, then you may expect the system as a whole to have total angular momentum \|j1−j2\|. (See ﬁgure 12.) It’s important – 79 – to realise that we’re not saying anything at all about how the individual subsystems may or may not be coupled to one another dynamically – that is, we’re not assuming any form of interaction between them in the Hamiltonian. Rather, we’re just considering the diﬀerent relative alignments of their angular momenta that are possible in principle. Quantum mechanically, the angular momentum operator for the combined system is J = (J1 ⊗1Hj2) + (1Hj1 ⊗J2) (6.5) where J1 and J2 are the angular momentum operators for the two subsystems. It follows that J2 = (J2 1 ⊗1Hj2) + (1Hj1 ⊗J2 2) + 2(J1 ⊗J2) (6.6) where in the ﬁnal term we take the scalar product of the two spin operators over their spatial vector indices. We’ll often abuse notation by writing J = J1 + J2 and J2 = J2 1 + J2 2 + 2J1 · J2 , (6.7) with the tensor product and identity operators being understood. We can rewrite the operator (6.6) in a way that allows us to understand its action on states \|j1, m1⟩\|j2, m2⟩, which form our basis in (6.4) (we’ve again dropped the ⊗symbol). Using the fact that Jx = J+ + J− 2 and Jy = J+ −J− 2i we have 2J1 · J2 = 2(J1xJ2x + J1yJ2y + J1zJ2z) = 2 J1+ + J1− 2 J2+ + J2− 2 + J1+ −J1− 2i J2+ −J2− 2i + J1zJ2z = J1+J2−+ J1−J2+ + 2J1zJ2z . (6.8) Using this to eliminate J1 · J2 from (6.6) allows us to write the total angular momentum operator as J2 = J2 1 + J2 2 + J1+J2−+ J1−J2+ + 2J1zJ2z , (6.9) where we now understand how each of the terms on the rhs act on any state of the form \|j1, m1⟩\|j2, m2⟩. Let’s consider the action of this operator on states of the whole system. We’ll start by examining the state \|j1, j1⟩\|j2, j2⟩in which both gyros are maximally aligned with the ˆ z-axis. Since Jz = J1z + J2z we have Jz\|j1, j1⟩\|j2, j2⟩= (j1 + j2)ℏ\|j1, j1⟩\|j2, j2⟩ (6.10) so indeed this state is an eigenstate of Jz with the expected eigenvalue. Also, using (6.9) we have J2\|j1, j1⟩\|j2, j2⟩= J2 1 + J2 2 + J1+J2−+ J1−J2+ + 2J1zJ2z \|j1, j1⟩\|j2, j2⟩ = (j1(j1 + 1) + j2(j2 + 1) + 2j1j2) ℏ2\|j1, j1⟩\|j2, j2⟩ = (j1 + j2)(j1 + j2 + 1)ℏ2\|j1, j1⟩\|j2, j2⟩ (6.11) – 80 – where we’ve used the fact that \|j1, j1⟩\|j2, j2⟩is annihilated by both J1+ and J2+. Thus, setting j = j1 + j2, we may write \|j, j⟩= \|j1, j1⟩\|j2, j2⟩ (6.12) since \|j1, j1⟩\|j2, j2⟩satisﬁes both the deﬁning equations for a state of total angular momen-tum labelled by j1 + j2, with it all aligned along ˆ z. (This is a highest weight state for j = j1 + j2.) Now that we’ve found one mutual eigenstate of the combined J2 and Jz, we may easily construct others by applying the lowering operator J−= J1−+J2−. Acting on the left and using equation (5.16) we have J−\|j, j⟩= ℏ p j(j + 1) −j(j −1)\|j, j −1⟩= ℏ p 2j\|j, j −1⟩. (6.13) On the other hand, applying J−= J1−+ J2−to the rhs of (6.12) gives (J1−+ J2−)\|j1, j1⟩\|j2, j2⟩ = ℏ hp j1(j1 + 1) −j1(j1 −1)\|j1, j1 −1⟩\|j2, j2⟩+ p j2(j2 + 1) −j2(j2 −1)\|j1, j1⟩\|j2, j2 −1⟩ i = ℏ p 2j1\|j1, j1 −1⟩\|j2, j2⟩+ ℏ p 2j2\|j1, j1⟩\|j2, j2 −1⟩. (6.14) Comparing the two sides we learn that \|j, j −1⟩= s j1 j \|j1, j1 −1⟩\|j2, j2⟩+ s j2 j \|j1, j1⟩\|j2, j2 −1⟩. (6.15) Note that the lhs is an eigenstate of Jz with eigenvalue (j −1)ℏ, and indeed the rhs is an eigenstate of J1z + J2z with eigenvalue (j1 + j2 −1)ℏ. A further application of J−to the lhs and J1−+ J2−to the rhs would produce an expression for \|j, j −2⟩and so on. Since [J2, J−] = 0, all the states we produce by acting on \|j, j⟩with J−have total angular momentum quantum number j = j1 + j2. This corresponds to the individual angular momenta of the subsystems always being aligned with eachother, with the diﬀerent \|j, m⟩telling us about how closely there mutual direction of alignment coincides with the z-axis. It’s perfectly possible for the individual subsystems to not line up with eachother. In such a conﬁguration the net total angular momentum of the combined system will be less than the maximum value j1 + j2. Let’s seek an expression for the state \|j −1, j −1⟩ in which the total angular momentum of the whole system is just less than the maximum possible, but where this angular momentum still points along the ˆ z-axis. It’s trivial to verify that any simple state \|j1, m1⟩\|j2, m2⟩is an eigenstate of Jz = J1z + J2z, with eigenvalue (m1 + m2)ℏ. We require m1 + m2 = j −1 = j1 + j2 −1, so either (m1, m2) = (j1 −1, j2) or (m1, m2) = (j1, j2 −1). Also, the state \|j −1, j −1⟩must be orthogonal to the state \|j, j −1⟩we found in (6.15) since they are each eigenstates of the Hermitian operator J2 with distinct eigenvalues. Therefore we must have \|j −1, j −1⟩= s j2 j \|j1, j1 −1⟩\|j2, j2⟩− s j1 j \|j1, j1⟩\|j2, j2 −1⟩. (6.16) – 81 – Figure 13: Decomposing the tensor product of two irreducible representations into a sum of irreducible representations, drawn as semicircles. The picture on the left shows states obtained by adding a system with j1 = 2 to one with j2 = 1, while that on the right is for j1 = 1 and j2 = 1/2. with a relative sign between the two terms. Again, this is a highest weight state, now with j = j1 + j2 −1. Given this state, we may now proceed to construct all the states \|j −1, m⟩ with −j + 1 ≤m ≤j −1 by applying the lowering operator J−. Figure 13 helps us to organise our results. States of the combined system with well– deﬁned angular momentum are marked by dots. The radius of each semicircle is propor-tional to √j′, where j′(j′ + 1)ℏ2 is the state’s eigenvalue J2. The height of a state above the centre of the semicircles is proportional to m. The outer semicircle thus contains all states with j′ = j1 + j2; going inwards we meet the states with j′ = j1 + j2 −1, then those with j′ = j1 + j2 −2 and so on. To construct \|j −2, j −2⟩we note that it must be in the span of {\|j1, j1 −2⟩\|j2, j2⟩, \|j1, j1 −1⟩\|j2, j2 −1⟩, \|j1, j1⟩\|j2, j2 −2⟩}, and that \|j, j −2⟩and \|j −1, j −2⟩also lie in the span of these states. Having constructed both \|j, j −2⟩and \|j −1, j −2⟩, the state \|j −2, j −2⟩must be the unique orthogonal combination. From our classical considerations, it’s reasonable to conjecture that the smallest pos-sible total angular momentum quantum number of the whole system is \|j1 −j2\|, coming when the two gyros are aligned against one another. Let’s check this by working out the total number of states the conjecture leads to. Without loss, we can assume j1 ≥j2. If the combined system has j values running from j1 + j2 to j1 −j2, then we count a total of j1+j2 X j=j1−j2 (2j + 1) = 2   j1+j2 X j=j1−j2 j  + (2j2 + 1) = (2j1)(2j2 + 1) + (2j2 + 1) = (2j1 + 1)(2j2 + 1) , (6.17) in agreement with the dimension of the tensor product space we found earlier using the ‘obvious’ tensor product basis in (6.4). We’ve thus accounted for all the possible states of the system. The numbers Cj,m(j1, m1; j2, m2) = ⟨j, m\| (\|j1, m1⟩⊗\|j2, m2⟩) (6.18) – 82 – are known as Clebsch–Gordan coeﬃcients. Physically, they represent the amplitude that, when the total system is in state \|j, m⟩, we will ﬁnd the subsystems to be in states \|j1, m1⟩ and \|j2, m2⟩when we peer inside the box. For example, from equation (6.15) we see that C3,2(2, 1; 1, 1) = p 2/3, so if we know our box contains a gyro of spin 2 and a gyro of spin 1, if the box is in state \|3, 2⟩then there’s a 2/3 chance that the second gyro has its spin maximally aligned with the z-axis with the ﬁrst signiﬁcantly inclined, but only a 1/3 chance that it is the ﬁrst gyro that is maximally aligned with the ˆ z-axis. Let’s take a look at a few examples to familiarise ourselves with the above general formalism. 6.1.1 j ⊗0 = j First, the trivial case: if one of the subsystems (say the second) has j2 = 0, then it always lies in \|j2, m2⟩= \|0, 0⟩. Hence the tensor product is trivial H = Hj1 ⊗H0 = Hj1 ⊗C ∼ = Hj1 because we have no choice about the state of the second subsystem. The states {\|j, m⟩= \|j1, m1⟩\|0, 0⟩} therefore form a basis of the full system, and are immediately a single irreducible repre-sentation with j = j1. 6.1.2 1 2 ⊗1 2 = 1 ⊕0 The ﬁrst non–trivial case is when j1 = j2 = 1/2. Physically this is relevant, e.g. to the case of the ground state of the Hydrogen atom where to understand the spin of the whole atom we must combine the spins of both the proton and the electron. From above, we have \|1, 1⟩H = \|↑⟩e \|↑⟩p (6.19) where the subscripts refer to Hydrogen, the electon and the proton, respectively. In this state, the spins of the electron and proton are aligned with eachother, and they each have maximal possible spin along the ˆ z-axis. Applying J−H = J−e + J−p we obtain \|1, 0⟩H = 1 √ 2 (\|↑⟩e \|↓⟩p + \|↓⟩e \|↑⟩p) , (6.20) while applying J−H a second time gives \|1, −1⟩H = \|↓⟩e \|↓⟩p . (6.21) Since both the electron and proton are in the \| ↓⟩state here, any further applications of either lowering operator will annihilate the rhs, in agreement with the action of the total lowering operator J−H on the state \|1, −1⟩H of the whole atom. Let me point out a perhaps surprising feature of this multiplet. In state (6.20), we are certain to ﬁnd that the z-component of the spins of the electron and proton are ‘antiparal-lel’. This may seem surprising given that the atom is still in a spin-1 state, corresponding to – 83 – the spins of the subsystems being aligned. The resolution of the paradox is that while the z-components are indeed antiparallel, the components in the xy-plane are aligned, although their precise direction is unknown to us. Similarly, in both the \|1, 1⟩and \|1, −1⟩states, while the z-components of the electron and proton spins were aligned, their components in the xy-plane were in fact antiparallel. The poor alignment in the xy-plane accounts for the fact that p ⟨J2⟩= √ 2ℏfor the atom, which is less than the sum of p ⟨J2⟩= p 3/4 ℏ for the electron and proton individually. The remaining state of the atom is \|0, 0⟩, in which the atom has no angular momentum. This should be orthogonal to (6.20), so we ﬁnd \|0, 0⟩H = 1 √ 2 (\|↑⟩e \|↓⟩p −\|↓⟩e \|↑⟩p) . (6.22) The change in sign on the rhs of this equation ensures that the electron and proton spins are antiparallel in the xy-plane as well as in the z-direction. In fact, one can check that this state may also be written as \|0, 0⟩H = −1 √ 2 (\|↑x⟩e \|↓x⟩p −\|↓x⟩e \|↑x⟩p) (6.23) where \|↑x⟩and \|↓x⟩are eigenstates of Jx. Similarly, \|1, 0⟩H = 1 √ 2 (\|↑x⟩e \|↑x⟩p −\|↓x⟩e \|↓x⟩p) (6.24) as one can check straightforwardly. Altogether, combining two states each with angular momentum 1/2, we’ve found a triplet of states with total angular momentum j = 1: \|1, 1⟩= \|↑⟩\|↑⟩, \|1, 0⟩= 1 √ 2 (\|↑⟩\|↓⟩+ \|↓⟩\|↑⟩) , \|1, −1⟩= \|↓⟩\|↓⟩ (6.25) and a singlet with total angular momentum j = 0: \|0, 0⟩= 1 √ 2 (\|↑⟩\|↓⟩−\|↓⟩\|↑⟩) . (6.26) Note that each state in the triplet is symmetric under exchange of the angular momenta of the two subsytems, while the singlet state is antisymmetric under this exchange. 6.1.3 1 ⊗1 2 = 3 2 ⊕1 2 If our subsystems have j1 = 1 and j2 = 1 2, then the total system can have j = 3 2 or j = 1 2. Let’s start as always with the highest state 3 2, 3 2 = \|1, 1⟩\|↑⟩. (6.27) Applying the lowering operator, we ﬁnd successively the states 3 2, 1 2 = r 2 3 \|1, 0⟩\|↑⟩+ r 1 3 \|1, 1⟩\|↓⟩, 3 2, −1 2 = r 1 3 \|1, −1⟩\|↑⟩+ r 2 3 \|1, 0⟩\|↓⟩, 3 2, −3 2 = \|1, −1⟩\|↓⟩ (6.28) – 84 – Figure 14: Classically, the sum of two vectors lies on a sphere centred on the end of one of these vectors, at a location determined by their relative orientation. which complete the j = 3 2 multiplet. The remaining states are 1 2, 1 2 = r 1 3 \|1, 0⟩\|↑⟩− r 2 3 \|1, 1⟩\|↓⟩ and 1 2, −1 2 = r 2 3 \|1, −1⟩\|↑⟩− r 1 3 \|1, 0⟩\|↓⟩. (6.29) The ﬁrst of these is obtained by requiring it to be orthogonal to \| 3 2, 1 2⟩, and the second follows by applying the lowering operator to the ﬁrst. 6.1.4 The Classical Limit We should expect to recover a classical picture when we combine two systems each with large amounts of total angular momentum. Let’s see how this occurs. Classically, if we add two angular momentum vectors j1 and j2 then the resultant vector has magnitude j2 = (j1 + j2)2 = j2 1 + j2 2 + 2j1 · j2 . (6.30) If we know nothing about the relative orientation of j1 and j2, then all points on a sphere of radius \|j2\| centred on the end of j1 are equally likely. Consequently, the probability dP that j1 and j2 have relative alignment in the range (θ, θ + dθ) is proportional to the area of the band shown in ﬁgure 14. The magnitudes of j1 and j2 are ﬁxed, so dj2 = 2\|j1\|\|j2\| d cos θ. Hence the probability of this alignment is dP = 2π sin θ dθ 4π = \|j\| d\|j\| 2\|j1\|\|j2\| (6.31) and hence dP/d\|j\| = \|j\|/2\|j1\|\|j2\|. In the quantum case, suppose j1, j2 ≫1. Then the fraction of states in the combined system with some amount j of angular momentum is 2j + 1 (2j1 + 1)(2j2 + 1) ≈ j 2j1j2 (6.32) provided \|j1 −j2\| ≤j ≤j1 + j2. Thus, if we know nothing about the original state of the subsystems, the probability that the combined system has total angular momentum j agrees with the classical probability computed above. – 85 – 6.2 Angular Momentum of Operators Suppose that V is a vector operator, so that in particular U −1(α)VU(α) = R(α)V under rotations, or [Ji, Vj] = iℏ X k ϵijkVk (6.33) inﬁnitesimally. We deﬁne the spherical components of V by V +1 = −1 √ 2(Vx + iVy) , V −1 = 1 √ 2i(Vx −iVy) , V 0 = Vz (6.34) Then the commutation relations (6.33) are equivalent to = mℏV m [J±, V m] = ℏ p 2 −m(m ± 1) V m±1 . (6.35) 6.2.1 The Wigner–Eckart Theorem 6.2.2 Dipole Moment Transitions 6.2.3 The Runge–Lenz Vector in Hydrogen For a spinless particle of mass57 M in a generic central potential, the Hamiltonian takes the form H = P2 2M + V (\|X\|) . (6.36) By deﬁnition, any such Hamiltonian is rotationally invariant, so it follows that [L, H] = 0 and [L2, H] = 0 . (6.37) Since we also have [Li, L2] = 0 we can ﬁnd a complete set of simultaneous eigenstates of H, L2 and Lz, so we can label our states as \|n, ℓ, m⟩. The Hamiltonian does not involve Lz except through L2, so the energy level En,ℓ,m of such a state must certainly be independent of m. However, the Hamiltonian does involve L2 through the kinetic operator, so we do not generically expect the energy levels to be independent of ℓ. Physically, if we give our particle more or less angular momentum, it will whizz round faster or slower. Typically, we’d expect this to change the energy. Nonetheless, it occasionally happens that a Hamiltonian is invariant under transfor-mations beyond those inherited from the transformations of the space R3 in which the quantum system lives. If the algebra of the corresponding generators closes, meaning that the commutator of any pair of generators is equal to another generator (perhaps including the Hamiltonian), then we say we have a dynamical symmetry. In such cases, it may be possible to change the total angular momentum of a state without changing its energy, so that states of diﬀerent orbital angular momentum quantum number ℓcan be degenerate. Hamiltonians with dynamical symmetry are very rare: for single–particle quantum systems moving on R3 with a central potential, the only cases are the harmonic oscillator 57We denote the mass with a capital M so as to avoid any confusion with the z angular momentum quantum number m. – 86 – and motion in a Coulomb potential. However, these two cases are so important that it’s worth considering them from this point of view. The treatment below closely follows section 4.8 of Weinberg – I think I’ll modify it when I write the Wigner-Eckart part The ﬁrst case we’ll study is for the Coulomb potential, relevant e.g. to the hydrogen atom. Classically, conservation of angular momentum in any central potential implies that the motion is conﬁned to a plane, but the Kepler problem has a further conserved quantity known as the Runge–Lenz vector r, given by r = −e2 x \|x\| + 1 M p × L (6.38) for a particle of mass M. Conservation of this vector implies that classical bound states have closed orbits; the motion is not just in a plane, but follows a ﬁxed ellipse. (The observation of a gradual shift in Mercury’s orbit – the precession of its perihelion – was one of the early conﬁrmations of General Relativity’s modiﬁcations to Newton’s Laws of Gravity.) Now let’s consider the quantum mechanical case. We construct the Runge–Lenz oper-ator R = −e2 \|X\|X + 1 2M (P × L −L × P) = −e2 \|X\|X + 1 M P × L −iℏ M P . (6.39) Classically, the two terms in brackets on the ﬁrst line of (6.39) are equal; we’ve written it this way to ensure that R† = R. Since R is a vector operator built from X and P, it follows that its commutation relations with L are [Li, Rj] = iℏ X k ϵijkRk . (6.40) so that it transforms as a vector under rotations. Therefore, the Wigner-Eckart theorem says that ⟨n′, ℓ′, m′\|Rk 1\|n, ℓ, m⟩= Cℓ′,m′(1, k′; ℓ, m) ⟨n′∥R∥n⟩ (6.41) Furthermore, a somewhat tedious calculation (which I’ll spare you, and which I don’t expect you to reproduce) shows that [H, R] = 0 so the Runge-Lenz vector is conserved. Thus the reduced matrix element ⟨n′∥R∥n⟩vanishes unless n′ = n. The orbital angular momentum L = X × P is orthogonal to each of the three terms in this expression, so R · L = L · R = 0. We also have L2 = (X × P) · L = X · (P × L) = (P × L) · X + 3 X i,j,k=1 ϵijk [Xi, PjLk] = (P × L) · X + 2iℏP · X , (6.42) – 87 – as well as P · (P × L) = 0 , (P × L) · P = 2iℏP2 and (P × L)2 = P2 L2 (6.43) where we’ve been careful to keep track of operator orders. Using these results, we ﬁnd that the length-squared of the Runge–Lenz operator is given by R2 = Z2e4 + 2H µ L2 + ℏ2 . (6.44) The point of this expression is that it relates R to H; since we know the spectrum of L2, we can ﬁnd the eigenvalues of H if we can ﬁnd the eigenvalues of R. A rather longwinded calculation (that I won’t expect you to reproduce) shows that the components of the Runge-Lenz operator obey the commutation relations [Ri, Rj] = −2iℏ µ H X k ϵijkLk , (6.45) where we note that H commutes with L. We now introduce the operators A± = 1 2 L ± r µ −2H R (6.46) built from the orbital angular momentum operator, the Runge-Lenz operator and the Hamiltonian, where we recall that functions (such as the square root) of an operator are deﬁned as in (2.42). The point of introducing these strange-looking combinations is that commutation relations (6.40) & (6.45) show that the A± obey the algebra [A±i, A±j] = iℏ X k ϵijkA±k and [A±, A∓] = 0 , (6.47) which we recognise58 as two independent copies of the angular momentum algebra so(3). In particular, just as for the total angular momentum, the eigenvalues of A2 ± take the form a±(a± +1)ℏ2 where a± ∈{0, 1/2, 1, 3/2, . . .}, with the eigenvalues of any component of A± running from −ℏa± to +ℏa± in steps of ℏ. Since [A±, H] = 0 we can ﬁnd simultaneous eigenstates of H, A2 ± and, say, A+z and A−z. However, taking the square of‘(6.46) and using the fact that L · R = R · L = 0, A2 + = A2 −= 1 4 L2 + µ 2H R2 = −Z2e4µ 8H −ℏ2 4 . (6.48) In particular, we cannot choose these eigenvalues of A2 + and A2 −independently, but must have a+ = a−. Calling the common eigenvalue a, an simultaneous eigenstate of H and A2 ± obeys −Z2e4µ 8E = a(a + 1) + 1 4 ℏ2 = ℏ2 4 (2a + 1)2 . (6.49) 58For bound states of the Coulomb potential, the spectrum of H is negative-deﬁnite, so the operators A± are indeed Hermitian when acting on such states. At energies above the ionisation threshold, the square root in (6.46) means that A± can no longer be treated as Hermitian, and this algebra is then better described as sl(2). – 88 – For a any non-negative half integer, 2a + 1 ∈{1, 2, 3, . . .}, so the energy levels can be written En = −Z2e4µ 2ℏ2n2 where n ∈N . (6.50) These are the energy levels of hydrogenic atoms that you obtained in IB QM. The eigenval-ues of A+z and A−z can be chosen independently, and each run from −aℏto aℏin steps of ℏ. Thus the degeneracy of the state with energy En is (2a + 1)2 = n2. Again, this is bigger than the degeneracy of a generic central potential, and the degenerate states transform in representations of the enhanced so(3)×so(3) symmetry algebra. The diagonal part of this, generated by L = A+ + A−corresponds to spatial rotations. Incidentally, it’s useful to note that the energy levels of a hydrogenic atom can be written as En = −Z2α2 2n2 mc2 (6.51) where the dimensionless number α = e2 4πϵ0ℏc (6.52) is known as the ﬁne structure constant. Experimentally, α ≈1/137. The signiﬁcance of this is that, at least for small atomic number Z, \|En/mc2\| ≪1 providing a posteriori justiﬁcation of our use of non–relativistic quantum mechanics to study the hydrogen atom. 6.2.4 Enhanced Symmetry of the 3d Isotropic Harmonic Oscillator As a second example, consider an isotropic 3d harmonic oscillator, with Hamiltonian H = P2 2M + 1 2Mω2X2 (6.53) as in section 3.3. The fact that H is quadratic in X as well as in P means it has an enhanced symmetry beyond the obvious SO(3) rotational invariance. We can see this from the form H = ℏω X i A† iAi + 3 2 ! (6.54) in terms of raising and lowering operators. Written this way, H is clearly invariant under the transformations Ai 7→A′ i 3 X j=1 uij Aj A† i 7→(A′ i)† = 3 X k=1 A† k (u†)ki (6.55) for any 3 × 3 unitary matrix u. These matrices must be unitary if the new A′† is indeed to be the Hermitian conjugate of the new A′. It’s important to realise that, although they are unitary, the matrices uij do not act on the Hilbert space of the harmonic oscillator, but just on the spatial indices of the vector operators A and A†. They thus mix X and P together59. 59For those taking Integrable Systems: classically these are symmetries of phase space that do not come from symmetries of space itself, and reﬂect the fact that the harmonic oscillator is (super-)integrable. – 89 – Note also that the new raising and lowering operators obey the same commutation relations as the original ones. A 3×3 unitary matrix has 9 independent real components, whereas a rotation in three dimensions is completely speciﬁed by only 3 independent Euler angles, so the U(3) sym-metry of this Hamiltonian is much bigger than the SO(3) rotational invariance of a generic central potential. The fact that the Hamiltonian is invariant under the U(3) transforma-tions (6.55) implies that there should be a total of nine conserved quantities corresponding to the nine generators of this symmetry. It’s easy to check that these generators form the tensor operator T with components Tij = A† iAj. We have [H, T] = 0 and hence each component Tij is conserved. To understand these operators, we decompose T into its irreducible parts as in section 6.2.1, obtaining Tij = A† · A δij 3 + A† iAj −A† jAi 2 + " A† iAj + A† jAi 2 −A† · A δij 3 # . (6.56) The trace A† · A is (up to a constant) just the Hamiltonian itself, which is certainly conserved. Recalling the deﬁnition (3.33) of the raising and lowering operators, the anti-symmetric part can be combined into a vector with components X j,k ϵijkA† jAk = 1 2Mℏω X j,k ϵijk(MωXj −iPj)(MωXk + iPk) = i 2ℏ X j,k ϵijk (XjPk −PjXk) = i ℏLi , (6.57) where in the last equality we used the fact ϵijk(XjPk −PjXk) = 2ϵijkXjPk, with the ϵ-symbol ensuring that the components of X and P commute. Thus L = −iℏ(A† × A) , (6.58) so that the vector part of T is nothing but the usual orbital angular momentum operator for the 3d oscillator. It is no surprise that this is conserved. The remaining ﬁve conserved quantities — the symmetric, traceless part (Tij +Tji)/2− P i Tii/3 of T — generate transformations that mix X and P, appropriately rescaled. They’re less familiar, as they’re special to the simple harmonic oscillator. The ﬁve com-ponents of this traceless symmetric part correspond to the spin-2 part of the operator T We can see this degeneracy by examining the excited states of the oscillator. Just as in 1d, these may be obtained by acting on \|0⟩with any component of the raising operators A†. Since the ground state has energy 3ℏω/2 and each application of a raising operator increases the energy by one unit of ℏω, each of the states A† iN · · · A† i2A† i1\|0⟩ (6.59) has the same energy (N + 3/2)ℏω, where we can choose any {i1, i2, . . . , iN} ∈{1, 2, 3}. We often call the (correctly normalised) such state \|nx, ny, nz⟩, where nx is the total number – 90 – of times we’ve applied the x-component of A†, and similarly for ny and nz. Of course, N = nx + ny + nz. Since the components of the raising operators all commute with one another, we can consider the indices i1, . . . , iN in (6.59) to be symmetrized. Consequently, the degeneracy of the Nth energy level is the same as the number of independent components of a rank N tensor in 3 dimensions, which is (N + 1)(N + 2)/2 . (Think ‘stars & bars’, or the number of ways to partition N into three non-negative integers (nx, ny, nx).) This is the larger degeneracy that we promised. It’s easy to derive this degeneracy in number of ways; our derivation here makes clear that the degenerate energy states transform in representations of the larger su(3) algebra. 6.3 Angular Momentum of the Isotropic Oscillator We can ﬁnd the spectrum of Hℓusing raising and lowering operators in a similar fashion to our treatment of the full H above. We introduce Aℓ= 1 √2µℏω iPr −(ℓ+ 1)ℏ R + µωR (6.60) and a short computation using [R, Pr] = iℏshows that Hℓ= ℏω A† ℓAℓ+ ℓ+ 3 2 . (6.61) This is very similar to the expression (3.35) above, but now we ﬁnd the commutation relations [Aℓ, A† ℓ] = 1 + (ℓ+ 1)ℏ µωR2 = Hℓ+1 −Hℓ ℏω + 1 (6.62) in place of (3.34) for the raising and lowering operators in Cartesian coordinates. In consequence, the commutator of Hℓwith Aℓis [Aℓ, Hℓ] = ℏω[Aℓ, A† ℓAℓ] = ℏω[Aℓ, A† ℓ]Aℓ= (Hℓ+1 −Hℓ+ ℏω)Aℓ, (6.63) or equivalently Hℓ+1Aℓ−AℓHℓ= −ℏωAℓ. (6.64) This is very close to showing that Aℓis a lowering operator, but the presence of Hℓ+1 means that the algebra does not close unless we consider all values of ℓ. To understand the implications of this, let \|E(ℓ)⟩be an eigenstate of Hℓ, with Hℓ\|E(ℓ)⟩= E\|E(ℓ)⟩where we emphasise that the energies depend on the total orbital angular momentum quantum number ℓ. Then Hℓ+1(Aℓ\|E(ℓ)⟩) = (E −ℏω)Aℓ\|E(ℓ)⟩, (6.65) which implies that Aℓ\|E(ℓ)⟩is an eigenstate of Hℓ+1 with eigenvalue E −ℏω. The Hamil-tonian Hℓ+1 is the one appropriate for the radial part of wavefunctions with total angular momentum labelled by ℓ+ 1, so applying Aℓto \|E(ℓ)⟩has created a state with lower en-ergy, but with a radial wavefunction corresponding to a state of greater orbital angular momentum. – 91 – Figure 15: The action of the operators Aℓand A† ℓon the states of the isotropic three dimensional oscillator. By a now familiar argument, in any Hℓeigenstate we have Eℓ ℏω −ℓ−3 2 = E(ℓ) Hℓ ℏω −3 2 E(ℓ) = ⟨E(ℓ)\|A† ℓAℓ\|E(ℓ)⟩= ∥Aℓ\|E(ℓ)⟩∥2 ≥0 . (6.66) Thus, for a ﬁxed given energy E the maximum orbital angular momentum our state can have is ℓmax = E ℏω −3 2 . (6.67) As ℓmax is a non–negative integer (since it is a possible value for ℓat given E) we see that, in agreement with the previous subsection, the ground state energy is 3ℏω/2 which occurs when ℓmax = 0, meaning that this ground state is necessarily spherically symmetric. For any given energy, the state \|E(ℓmax)⟩which saturates the bound (6.66) obeys Aℓmax\|E(ℓmax)⟩= 0 and has the maximum possible total angular momentum out of all the states with this energy. It is thus the quantum equivalent of a circular orbit. In the position representation we have 0 = ⟨r\|Aℓmax\|E(ℓmax)⟩= 1 √2µℏω ∂ ∂r + 1 r −(ℓmax + 1) r + mωr ℏ ⟨r\|E(ℓmax)⟩ (6.68) where we use \|r⟩to denote a state that is deﬁnitely located at radius r. This equation is solved by ⟨r\|E(ℓmax)⟩= C rℓmax e−r2/4r2 0 (6.69) where r0 = p ℏ/2µω and C is a constant. Note that the exponential here is just the product of the exponentials we’d expect for Cartesian oscillators. This wavefunction varies with r, so quantum mechanically even a ‘circular’ orbit has some radial kinetic energy. In the limit of large ℓmax in which classical mechanics applies, the radial kinetic energy is negligible compared to the tangential kinetic energy. The full wavefunction is ⟨r\|(\|E(ℓmax)⟩⊗\|ℓmax, m⟩) and so also involves the spherical harmonic Ym ℓmax(θ, φ). Bearing in mind that d3x = r2 sin θ dφ dθ dr, the radial probability density P(r) for this state is P(r) ∼r2ℓmax+2 e−r2/2r2 0. For r/r0 ≪√2ℓmax + 2 this rises as r2ℓmax+2 and it falls as the – 92 – Gaussian takes over when r/r0 > √2ℓmax + 2. The rms deviation in the radial location is ∼r0, which is a small fraction of the expected radial location ⟨R⟩when ℓmax is large. We can obtain the radial wavefunctions of more eccentric orbits using A† ℓ. Taking the adjoint of (6.64) we obtain HℓA† ℓ−A† ℓHℓ+1 = ℏω A† ℓ. (6.70) Acting with this on \|E(ℓ+ 1)⟩we have Hℓ(A† ℓ\|E(ℓ+ 1)⟩) = (E + ℏω) (A† ℓ\|E(ℓ+ 1)⟩) (6.71) so that A† ℓ\|E(ℓ+1)⟩is a state of increased energy, whose radial wavefunction is appropriate for orbital angular momentum ℓ. Thus A† ℓ\|E(ℓ+1)⟩= c\|E(ℓ)+ℏω⟩for some constant c. The radial wavefunctions of the eccentric orbits follow by writing this equation in the position representation, though we won’t perform this calculation here. The way the energy raising / angular momentum lowering operators A† ℓand the energy lowering / angular momentum raising operators Aℓact on the states is summarised in ﬁgure 15. In the previous section we found that there were (N +2)(N +1)/2 degenerate states in the Nth excited level of the isotropic oscillator, each with energy E = (N + 3/2)ℏω. From the point of view of the present section, with energy E = (N + 3/2)ℏω we have ℓmax = N and this degeneracy arises as ℓmax X ℓ=0 ℓ X m=−ℓ 1 = ℓmax X ℓ=0 (2ℓ+ 1) = ℓmax(ℓmax + 1) = (N + 1)(N + 2) 2 (6.72) since there are 2ℓ+ 1 possible states with total orbital angular momentum ℓ. 6.4 The Radial Hamiltonian for the Coulomb Potential For some purposes, it’s useful to study hydrogen from a point of view that emphasises the fact that the potential is central, so energy eigenstates may also be chosen to have deﬁnite angular momentum. In the ﬁrst problem set you showed that the kinetic part of the Hamiltonian could be written as T = 1 2µ P 2 r + L2 R2 , (6.73) where R2 = X · X and Pr = 1 2( ˆ X · P + P · ˆ X) is the radial momentum. Thus, when acting on an orbital angular momentum eigenstate \|ℓ, m⟩, the Hamiltonian of a hydrogenic atom with atomic number60 Z reduces to Hℓ= P 2 r 2µ + ℓ(ℓ+ 1)ℏ2 2µR2 − Ze2 4πϵ0R . (6.74) As in IA Dynamics & Relativity, this is just what we’d ﬁnd for a particle moving in one dimension under the inﬂuence of the eﬀective potential Veﬀ(R) = ℓ(ℓ+ 1)ℏ2 2µR2 − Ze2 4πϵ0R . (6.75) 60The atomic number is the number of protons in the atom’s nucleus. – 93 – Hℓgoverns the oscillations of the particle around the minima of this potential. Note that since Hℓdepends only on the radius R and radial momentum Pr, in position space its eigenstates will determine the radial dependence RE(r) = ⟨r\|E(ℓ)⟩of the full wavefunction. A basis of eigenstates \|E, ℓ, m⟩of the full Hamiltonian H can be chosen to be (tensor) products of the eigenstates \|E(ℓ)⟩of Hℓand the eigenstates \|ℓ, m⟩of L2 and Lz, whose position space wavefunction Ym ℓ(θ, φ) = ⟨ˆ r\|ℓ, m⟩is a spherical harmonic, as in section 5.4.1. All three terms in Hℓmust have the same dimensions, and in particular the ratio of the two terms in Veﬀmust be dimensionless. Thus ℏ2 µR2 × 4πϵ0R e2 = 4πϵ0ℏ2 µe2 × 1 R is dimensionless. (6.76) It follows that the Bohr radius a0 ≡4πϵ0ℏ2 µe2 (6.77) is the characteristic length scale of the hydrogen atom. We have Hℓ= ℏ2 2µ P 2 r ℏ2 + ℓ(ℓ+ 1) R2 −2Z a0R (6.78) in terms of a0. Let’s deﬁne a new operator Aℓby Aℓ= a0 √ 2 i ℏPr −ℓ+ 1 R + Z (ℓ+ 1)a0 . (6.79) Then one ﬁnds A† ℓAℓ= a2 0 2 −i ℏPr −ℓ+ 1 R + Z (ℓ+ 1)a0 i ℏPr −ℓ+ 1 R + Z (ℓ+ 1)a0 = a2 0 2 ( P 2 r ℏ2 + Z (ℓ+ 1)a0 −ℓ+ 1 R 2 + iℓ+ 1 ℏ Pr, 1 R ) . (6.80) Using the commutator [Pr, R] = −iℏthis gives A† ℓAℓ= a2 0 2 P 2 r ℏ2 + Z2 (ℓ+ 1)2a2 0 + (ℓ+ 1)2 R2 −2Z a0R −i ℏ ℓ+ 1 R [Pr, R] = a2 0 2 P 2 r ℏ2 + Z2 (ℓ+ 1)2a2 0 + ℓ(ℓ+ 1) R2 −2Z a0R = a2 0µ ℏ2 Hℓ+ Z2 2(ℓ+ 1)2 . (6.81) Had we instead evaluated AℓA† ℓthe only diﬀerence would have been the sign in front of the commutator, so we would instead ﬁnd AℓA† ℓ= a2 0µ ℏ2 Hℓ+1 + Z2 2(ℓ+ 1)2 (6.82) – 94 – which now involves the Hamiltonian appropriate for states whose orbital angular momen-tum is given by ℓ+ 1. Rearranging (6.81) allows us to write the radial Hamiltonian as Hℓ= ℏ2 µa2 0 A† ℓAℓ− Z2 2(ℓ+ 1)2 (6.83) while taking the diﬀerence of (6.81) and (6.82) gives the commutation relations h Aℓ, A† ℓ i = a2 0µ ℏ2 (Hℓ+1 −Hℓ) . (6.84) The ﬁnal result we need is the commutation relation [Aℓ, Hℓ] = ℏ2 µa2 0 h Aℓ, A† ℓAℓ i = ℏ2 µa2 0 h Aℓ, A† ℓ i Aℓ= (Hℓ+1 −Hℓ) Aℓ, (6.85) where the ﬁnal step uses (6.84). Cancelling the HℓAℓterm on both sides, this simpliﬁes to AℓHℓ= Hℓ+1Aℓ, (6.86) and we also have A† ℓHℓ+1 = HℓA† ℓby taking the Hermitian conjugate. Now let’s understand these operators. Let \|E, ℓ⟩be an eigenstate of Hℓand L2 with Hℓ\|E, ℓ⟩= E\|E, ℓ⟩and L2\|E, ℓ⟩= ℓ(ℓ+ 1)ℏ2 \|E, ℓ⟩. Then Hℓ+1 (Aℓ\|E, ℓ⟩) = AℓHℓ\|E, ℓ⟩= E (Aℓ\|E, ℓ⟩) (6.87) so that Aℓ\|E, ℓ⟩is an eigenstate of Hℓ+1 with the same energy E. Thus, acting on \|E, ℓ⟩, Aℓ creates a new state with the same energy, but where the eﬀective potential corresponds to the angular momentum quantum number ℓ+1. Acting on the new state Aℓ\|E, ℓ⟩with Aℓ+1 again creates a new state with the same energy, but even more orbital angular momentum labelled by ℓ+ 2, and so on. We already know that our energy levels are not inﬁnitely degenerate, so this process must eventually terminate. That is, just as in the harmonic oscillator, there must be a maximum value ℓmax such that Aℓmax\|E, ℓmax⟩= 0 (6.88) and again, by analogy with the classical case where a circular orbit has the highest possible orbital angular momentum for ﬁxed energy, we can view this state as the quantum analogue of a circular orbit. Taking the norm and using equation (6.81) we have 0 = ∥Aℓmax\|E, ℓmax⟩∥2 = ⟨E, ℓmax\|A† ℓmaxAℓmax\|E, ℓmax⟩= a2 0µ ℏ2 E + Z2 2(ℓmax + 1)2 , (6.89) or equivalently E = −Z2ℏ2 2µa2 0 1 n2 (6.90) where n ≡ℓmax + 1 labels the energy level. This agrees with the spectrum of the Hamil-tonian computed earlier using the Runge–Lenz vector (or indeed in IB QM). The virtue – 95 – of the Runge–Lenz derivation is that it makes transparent the role of the additional, dy-namical symmetry of Hydrogen, explaining why the energy levels fall into multiplets of so(3) × so(3). The virtue of the present derivation is that it makes transparent how the degeneracy arises because, aside from simply rotating our system, we can trade radial ki-netic energy and Coulomb potential for diﬀerent orbital kinetic energy whilst keeping the total energy constant. – 96 – 7 Identical Particles One of the most striking facts that helps us understand the Universe is that, apart from details of their motion, all electrons are identical. This fact, though convenient, has no ex-planation within Quantum Mechanics. In Quantum Field Theory, one learns that particles are simply quantised modes of excitation of a ﬁeld. We thus replace the vast number of electrons our Universe contains with a single quantum ﬁeld that has hugely many excita-tions. From this perspective, the fact that all elementary particles of the same species are identical is more or less a tautology, no more surprising than the fact that one factor of x in the monomial xn is just the same as another. 7.1 Bosons and Fermions Consider a system consisting of N particles, individually described by some state in a Hilbert space Ha where a = 1, . . . , N, and suppose \|αa⟩∈Ha gives a complete speciﬁcation of the state of the ath particle, so αa collectively labels all the quantum numbers describing the particle’s energy, orbital angular momentum, spin etc.. In the case that the particles are distinguishable – meaning they are each from diﬀerent species, such as an electron, a proton or a neutron – a complete set of states of the full system is then labelled by linear combinations of states \|α1; α2; . . . ; αN⟩= \|α1⟩\|α2⟩· · · \|αN⟩∈ N O a=1 Ha . (7.1) in accordance with the discussion of section 2.3.1. However, suppose particles 1 and 2 are indistinguishable; for example, they might both be electrons. The state (7.1) thus describes a situation where there is one electron with quantum numbers given by α1 and another whose same properties are labelled by α2. Since both electrons are identical, it can make no diﬀerence which is which. Consequently, there can be no physical diﬀerence between this state and the state in which these two particles are exchanged. This does not imply that the two states are identical, but does imply that they can only diﬀer by a phase: \|α2; α1; . . . ⟩= eiφ\|α1; α2; . . . ⟩, (7.2) where φ is independent61 of the particular values of the labels αa, but may depend on the species of particle we exchange. If we exchange the pair of particles once again we ﬁnd \|α1; α2; . . . ⟩= eiφ\|α2; α1; . . . ⟩= e2iφ\|α1; α2; . . . ⟩ (7.3) 61There’s a (non-examinable!) exception to this in the case of two spatial dimensions, where the phase may depend on the homotopy class of the path the particles take whilst they are being exchanged. In 2 + 1 dimensions, these paths can become braided (tangled) and the multi–particle state transforms in a representation of the braid group, ﬁrst studied in the maths literature by Emile Artin. This leads to the possibility of anyonic statistics, which play an important role in some condensed matter systems. Fortunately, we’re interested in quantum mechanics on R3 and in higher dimensions it turns out the paths can always be disentangled, so anyons do not arise. – 97 – using the fact that the phase is independent of the particular values of the quantum numbers αa. Thus e2iφ = 1, so eiφ = ±1 . (7.4) The choice of sign depends only on the species of the particle. Identical particles which are symmetric under exchange are called62 bosons, while those that are antisymmetric are said to be fermions. If all N of the particles in our system are identical, then the argument immediately extends to say that the state must be either completely symmetric or completely antisym-metric under exchange of any pair, which case being determined by whether the particles are bosons or fermions. That is, for bosons the state of the system is actually described by a vector \|α1; α2; . . . ; αN⟩∈SymNH ⊂ N O H , (7.5) whereas for fermions the state lies in the antisymmetric part \|α1; α2; . . . ; αN⟩∈ N ^ H ⊂ N O H . (7.6) As a simple example, a pair of identical fermions with quantum numbers α1 and α2 live in the antisymmetric state \|ψ⟩fermi = 1 √ 2(\|α1; α2⟩−\|α2; α1⟩) , (7.7) whereas the system would have to be described by the symmetric combination \|ψ⟩bose = 1 √ 2(\|α1; α2⟩+ \|α2; α1⟩) (7.8) if instead these particles were identical bosons. A further fact that also comes from quantum ﬁeld theory is that particles with integer spin are always bosons, whilst particles with (odd) half–integer spin are always fermions. Among the elementary particles, examples of bosons thus include the W and Z bosons (hence the name!) and the photon, while examples of fermions include electrons, neutrinos and quarks. You’ll learn about how this connection between spin and statistics arises if you take the Part III QFT course. The above considerations apply to composite particles as well as to elementary ones. When we exchange a pair of identical composite particles, we exchange all of their con-stituents. Thus, if the composites are made up from an odd number of fermions and any number of bosons, upon exchange we’ll acquire an odd number of minus signs, so the whole state must be antisymmetric. For example, a proton consists of three (valence) quarks and many gluons, so is a fermion. So too is a neutron. On the other hand, composites consist-ing of an even number of fermions and any number of bosons must be symmetric under 62They’re named after Satyendra Nath Bose and Enrico Fermi, respectively, who ﬁrst studied their properties. – 98 – exchange with an identical composite. For example, a hydrogen atom consists of a proton and an electron, each of which are fermions, so the composite Hydrogen atom is itself a boson. This behaviour is consistent with addition of angular momentum. When we add the angular momenta of an odd number of particles each with odd half–integer spin, the total angular momentum is also an odd half–integer, whereas when we combine the angular momenta of an even number of particles each with odd half–integer spin, the total angular momentum is an integer. Even without quantum ﬁeld theory, it would be impossible for all integer spin particles to be fermions, because a composite consisting of an even number of such integer spin particles would also have integer spin, but would be a boson. 7.1.1 Pauli’s Exclusion Principle The fact that the state of identical fermions must be totally antisymmetric has an im-mediate, striking consequence: no two fermions can be put in exactly the same state simultaneously. This fact is known as Pauli’s exclusion principle. As an example, since we must always have ψσ,σ′(x, x′) = −ψσ′,σ(x′, x), if both fermions happen to have the same Sz eigenvalue we ﬁnd ψσ,σ(x, x′) = −ψσ,σ(x′, x). Hence the spatial wavefunction of the system necessarily vanishes at x = x′ so there is zero probability that the two particles are located at the same place. More generally, the state \|Ψ⟩of N identical fermions is represented in terms of the determinants ⟨α1, α2, . . . , αN\|Ψ⟩= 1 √ N! ψ1(α1) ψ1(α2) · · · ψ1(αN) ψ2(α1) ψ2(α2) · · · ψ2(αN) . . . . . . ... . . . ψN(α1) ψN(α2) · · · ψN(αN) (7.9) that in this context are known as Slater determinants. Again, \|α1, α2, . . . , αn⟩represents a state in which some fermion has quantum numbers α1, another has α2 and so on, where the labels αi denote a complete set of quantum numbers for single particle states, including (say) their momentum, spin-z value, etc., while ψ1(α1) = ⟨α1\|ψ1⟩is the amplitude for the ‘ﬁrst’ fermion to have quantum numbers α1. If αi = αj for any i ̸= j, then two columns of this determinant coincide. The amplitude ⟨. . . , αi, . . . , αi, . . . \|Ψ⟩thus vanishes identically, so there is zero probability of any two fermions being found in an identical state. It’s worth stressing that we’ve learned only that the state of a pair of identical fermions must be antisymmetric under exchange of every one of their quantum numbers. The Hilbert space of a single such spin-s fermion is itself a tensor product Hfermion = Hspat ⊗C2s+1, including a factor that describes the fermion’s spatial wavefunction (including details of its likely position, orbital angular momentum etc.) and a separate factor Hspin ∼ = C2s+1 spanned by the possible spin states {\|s⟩, \|s −1⟩, . . . , \| −s⟩} of this spin-s particle. We must exchange both the spin and spatial parts of the state in order to ﬁnd a physically equivalent state. – 99 – In the example of our electron pair, ψσ,σ′(x, x′) is certainly equivalent to ψσ′,σ(x′, x), since both correspond to the electron at x having spin σ and the electron found at x′ having spin σ′. However, antisymmetry of the total state does not imply any relation between the states ψσ,σ′(x, x′) and ψσ′,σ(x, x′): these are distinct physical possibilities because the electron at x has diﬀerent spin in the two cases. We saw in section 6.1.2 that the spin wavefunctions of the two electrons could be combined into a triplet of spin-1 states \|1, 1⟩= \|↑⟩\|↑⟩, \|1, 0⟩= 1 √ 2 (\|↑⟩\|↓⟩+ \|↓⟩\|↑⟩) , \|1, −1⟩= \|↓⟩\|↓⟩ (7.10) which are symmetric, and a single antisymmetric state \|0, 0⟩= 1 √ 2 (\|↑⟩\|↓⟩−\|↓⟩\|↑⟩) (7.11) of spin-zero. Because the overall state of the electron pair must be antisymmetric, the spatial wavefunction must be antisymmetric if the electron pair is in any of the spin-1 states, while the spatial wavefunction must be symmetric if the electron pair is in the spin-0 state. Thus, for a given spatial wavefunction ψ(x, x′) the pair has four possible states, and the states 1 √ 2 ψ(x, x′) −ψ(x′, x) ⊗    \|1, 1⟩ \|1, 0⟩ \|1, −1⟩    and 1 √ 2 ψ(x, x′) + ψ(x′, x) ⊗\|0, 0⟩(7.12) span a basis of the two–electron Hilbert space. In general, as always, an electron pair can be in a superposition of the four states in (7.12), with diﬀerent spatial wavefunctions in each term of the superposition. Simi-larly, an N-fermion system may be in state \|Ψ⟩that is a superposition of diﬀerent states \|α1, α2, . . . , αN⟩, with diﬀerent (distinct) values of the αis, where each of the ampli-tudes ⟨α1, α2, . . . , αN\|Ψ⟩are given in terms of the single–particle states by Slater determi-nants (7.9). 7.1.2 The Periodic Table This section closely follows section 4.5 of Weinberg. One of the outstanding successes of the early years of quantum mechanics was an understanding of the structure of the Periodic Table of the Elements. The exact dynamics of a heavy atom, with large atomic number Z, is very complicated because each electron not only feels an attractive Coulomb potential −Ze2/r from the Z protons in the nucleus, but also a repulsive electromagnetic interaction from all the other electrons. However, it turns out that a reasonable approximation – known as the Hartree approximation – is to imagine that each electron moves in a (roughly) central potential V (r) created by the net eﬀect of the protons and the other electrons, while neglecting the fact that the response of any given electron to this potential will itself modify the potential. Near the nucleus, V (r) ≈−Ze2/r as the electron feels the full strength of the attraction of the protons, while – 100 – at large distances V (r) ≈−e2/r since the protons’ charge is shielded by the (Z −1) other electrons. As for any central potential, [L2, H] = 0 and [L, H] = 0 so we can label the single– electron states by their values of ℓand m, together with another integer n labelling the energy levels. However, because V (r) is not precisely 1/r, unlike the pure Coulomb po-tential, states with diﬀerent ℓdo not have identical energy. Because V (r) ∼1/r near the nucleus, the single–electron wavefunctions behave as rℓfor small r, just as in Hydrogen. This mean that states with high values of ℓare less likely to be found at small r where the potential is deepest, and so typically have slightly larger energy than those of lower ℓ, compared to the pure Coulomb case. Numerical calculations show that the states of roughly equal energy are 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 7p, . . . with energy increasing as one proceeds down the list. Here, the number corresponds to the label n, while s, p, d, f, . . . stand for sharp, principal, diﬀuse, faint, . . . and simply correspond to ℓ= 0, 1, 2, 3, . . . . This historic notation is standard in atomic spectroscopy, which is still of great importance in the pharmaceutical industry. Since electrons have spin-1 2 they are fermions. Pauli’s exclusion principle prevents the electrons in a heavy atom from all burying down into the lowest energy 1s state, and forces them to gradually ﬁll up the higher energy levels. There are 2(2ℓ+ 1) distinct states with orbital angular momentum ℓ, with the extra factor of 2 coming from the two possible spin states of the electron. Thus, the number of possible states in each line of the above table is 2, 2 + 6 = 8, 2 + 6 = 8, 2 + 10 + 6 = 18, 2 + 10 + 6 = 18 and 2 + 14 + 10 + 6 = 32, etc. The ﬁrst two elements, hydrogen and helium, have electrons just in the ground state 1s. The 8 elements from lithium to neon have electrons in the n = 1 and n = 2 states, while the next 8 elements from sodium to argon have electrons in states with n = 1, 2, 3. The ﬁrst excited states of hydrogen and helium are obtained by promoting (one of) their 1s electrons up to the 2s state, and lie 10.2eV and 19.8eV above the corresponding ground states. These energy diﬀerences correspond to the frequency of UV radiation, so this transition cannot be stimulated by optical frequency radiation. By constrast, the ﬁrst excited state of lithium is obtained by promoting a 2s electron to a 2p state, and lies a mere 1.85eV above its ground state. This corresponds to the frequency of rather deep red. Elements that lie beyond helium play an important role in astronomical measurements, even though they are present only in trace amounts compared to hydrogen and helium, because their absorption spectra contain lines at easily observed optical frequencies. The chemical properties of an element are largely determined by the number of elec-– 101 – trons in its highest energy level, since these are least tightly bound. Atoms with no electrons outside ﬁlled energy levels are particularly stable chemically. These elements are called no-ble gases and include helium (Z = 2), neon (Z = 2 + 8 = 10), argon (Z = 2 + 8 + 8 = 18), krypton (Z = 2 + 8 + 8 + 18 = 36), xenon (Z = 2 + 8 + 8 + 18 + 18 = 54) and radon (Z = 2 + 8 + 8 + 18 + 18 + 32 = 86). Elements with either a few more or few less electrons than required to ﬁll a shell have their chemical properties determined by this number, known as the valence and counted positive for extra electrons and negative for fewer. If there is just one electron in the highest energy level, then this electron is easily stripped away, so the element is chemically reactive. These are the alkali metals and include lithium (Z = 2+1 = 3), sodium (Z = 2+8+1 = 11), potassium (Z = 2 + 8 + 8 + 1 = 19) etc. If a large number of such atoms combine to form a solid crystal, then it is energetically favourable for the valence electrons to be shared throughout the solid rather than clinging to their original atom. These electrons thus form a sort of ﬂuid that is free to ﬂow within the crystal when stimulated by a small external force. This gives the crystal high electrical and high thermal conductivity, making it a metal. (You’ll study this in much more detail if you take the Applications of Quantum Mechanics course next term.) Atoms with two electrons more than the noble gases are also chemically reactive, though not as reactive as the alkalis. They are known as the alkaline metals and include beryllium (Z = 2 + 2 = 4), magnesium (Z = 2 + 8 + 2 = 12), calcium (Z = 2 + 8 + 8 + 2 = 20) etc. On the other hand, atoms with one electron missing from their highest energy level tend to strongly attract other electrons and so are chemically reactive non–metals, often reacting violently when brought into contact with metals. These elements are called halogens and include ﬂuorine (Z = 2 + 8 −1 = 9), chlorine (Z = 2 + 8 + 8 −1 = 17), bromine (Z = 2 + 8 + 8 + 18 −1 = 35), iodine (Z = 2 + 8 + 8 + 18 + 18 −1 = 53) etc. Elements with two electrons missing from their highest energy level include oxygen (Z = 2 + 8 −2 = 8), sulfur (Z = 2 + 8 + 8 −2 = 16), etc. These elements are again chemically reactive, though less so than the halogens. The inclusion of the 4f and 5f states in the sixth and seventh energy levels, respec-tively, are responsible for the long sequence of rare earths in the middle of the periodic table. Numerical calculations show that wavefunctions of the 2(2 · 3 + 1) = 14 diﬀerent 4f states have small probability to lie outside the wavefunctions of the two 6s states, de-spite having slightly higher energy. Consequently, these elements are all rather similar chemically. The same is true of the 14 diﬀerent 5f states compared to the wavefunctions of the two 7s states. The 14 elements in which the 4f states are being ﬁlled run from lanthanum (Z = 2 + 8 + 8 + 18 + 18 + 2 + 1 = 57) to ytterbium (Z = 70) and are known as lanthanides, while the next chemically similar sequence are the actinides, running from actinium (Z = 2 + 8 + 8 + 18 + 18 + 32 + 2 + 1 = 89) to nobelium (Z = 102). Beyond this point the nuclei themselves become so unstable that the element tends to undergo radioactive decay before it has chance to participate in any chemical reaction. – 102 – Figure 16: The Periodic Table of the Elements. For formatting reasons, the lanthanides and actinides are traditionally shown below the rest of the table. (Figure from Wikimedia.) – 103 – 7.1.3 White Dwarfs, Neutron Stars and Supernovae 7.2 Exchange and Parity in the Centre of Momentum Frame Let’s now consider the eﬀects of exchanging two identical particles on their spatial wave-function. Letting X1, P1 and X2, P2 denote the position and momentum operators of the two particles, exchanging 1 ↔2 implies that Xcom = X1 + X2 2 7→ X2 + X1 2 = Xcom Pcom = P1 + P2 7→ P2 + P1 = Pcom , (7.13) whilst Xrel = X1 −X2 7→ X2 −X1 = −Xrel Prel = P1 −P2 2 7→ P2 −P1 2 = −Prel . (7.14) Thus, exchange acts trivially on the centre-of-momentum coordinates, but acts on the relative coordinates just like a parity transformation. Since Ym ℓ(−x) = (−1)ℓYm ℓ(x) under parity, we see that if two identical particles have relative orbital angular momentum ℓ, the spatial part of the wavefunction will be either symmetric or antisymmetric under exchange according to whether ℓis odd or even. The behaviour of the entire state under exchange is determined by whether the particles in question are bosons or fermions. In the case of identical bosons, the spins must be combined to form a symmetric overall spin state when ℓis even, or an antisymmetric overall spin state when ℓis odd so as to ensure the overall state is always symmetric under exchange. For fermions, the opposite holds so as to ensure overall antisymmetry. if the neutrons are produced in a state where their relative orbital angular momentum is ℓin the centre of momentum frame, their spatial wavefunction acquires a factor of (−1)ℓ under exchange of the two neutrons. 7.2.1 Identical Particles and Inelastic Collisions The requirement that the state describing N identical bosons/fermions be totally symmet-ric/antisymmetric under exchange of any pair of identical particles has important conse-quences in collision processes where such particles are created or destroyed. For example, consider an exotic type of ‘atom’ consisting of a spin-0 pion (denoted by π−) bound electromagnetically to a deuterium nucleus (denoted by D+ – itself consisting of a proton and a neutron, bound together by the strong nuclear force). The bound state energy levels of this atom due to the electromagnetic Coulomb attraction between the π− and D+ have the same form as in Hydrogen and, in particular, the ground state is \|1, 0, 0⟩, an s-wave having ℓ= 0. Because the pion is much heavier than the electron, the Bohr radius of the D+π−‘atom’ is much smaller than that of Hydrogen and the π−wavefunction closely hugs the D+ nucleus. The D+ is known to have spin-1 whilst the π−is spin zero, so the 1s ground state of the D+π−‘atom’ has total angular momentum j = 1. Now, as well as their electromagnetic interactions, the π−and D+ also interact via the short-range strong nuclear force. This causes the ‘atom’ to be unstable, with the π−rapidly – 104 – being absorbed by the D+, causing the system to disintegrate into a pair of neutrons (each denoted N): π−+ D+ − →N + N . (7.15) (In particle physics terminology, processes such as this, where diﬀerent types of particles appear in the initial and ﬁnal states, are known as inelastic. An elastic process is one in which the initial & ﬁnal states contain the same particles.) Neutrons are fermions with spin-1 2, so the ﬁnal state must be antisymmetric under their exchange. One possibility is for their spins to combine into one of the triplet \|↑⟩\|↑⟩, 1 √ 2 (\|↑⟩\|↓⟩+ \|↓⟩\|↑⟩) , \|↓⟩↓⟩ of symmetric spin-1 states, combined with a spatial wavefunction that is antisymmetric under exchange. From above, this will be the case if the relative orbital angular momentum of the two neutrons is odd. The other possibility is for the spins to form the antisymmetric spin-0 state 1 √ 2 (\|↑⟩\|↓⟩−\|↓⟩\|↑⟩) combined with a relative spatial wavefunction of even ℓ. Total angular momentum is conserved, so the spin and orbital angular momentum of the ﬁnal state must combine to give j = 1 as for the initial ground state of the π−D+. In section 6 we learnt that when we combine systems with angular momenta ℓand s, the combined system could have total angular momentum j ∈{ℓ+ s, ℓ+ s −1, . . . \|ℓ−s\|} depending on the relative alignment of the two subsystems. Thus, in the case that the neutrons spins combine to the antisymmetric spin-0 state, then j = ℓ. However, since this case requires ℓeven for fermionic statistics, we see it is ruled out. The remaining case is that the neutrons spins combine to give net spin-1, so that j ∈{ℓ+ 1, ℓ, ℓ−1} . Since fermionic statistics here require that ℓis odd, the only possibility to get j = 1 is if ℓ= 1 and the spin and orbital angular momenta of the neutron pair are neither perfectly aligned nor perfectly anti-aligned. Thus the ﬁnal state has j = ℓ= s = 1. These considerations also help us to determine the intrinsic parity of the pion. Recall from section 4.6 that the parity operator Π acts on a state \|x⟩as Π\|x⟩= η\|x⟩ (7.16) where the value of η ∈{+1, −1} is known as the intrinsic parity that, like spin, depends on the type of particle that the state \|x⟩represents. More generally, if \|x1, κ1; x2, κ2; · · · ; xN, κN⟩ describes an N-particle state in which a particle of type κa is deﬁnitely located at xa, then from (??) the parity operator acts as Π\|x1, κ1; x2, κ2; · · · ; xN, κN⟩= η1η2 · · · ηN \| −x1, κ1; −x2, κ2; · · · ; −xN, κN⟩. (7.17) – 105 – where the ηk = ±1 is the intrinsic parity of the kth particle. (This holds whether or not the particles are distinguishable.) Like the spin of fundamental particles, these intrinsic parities are independent of any details of the spatial wavefunction. Intrinsic parities thus have no eﬀect in elastic processes, where the same particles are present in both the initial and ﬁnal states. The intrinsic parity of a particle can often be determined by examining inelastic processes in which the particle participates. In the example π−D+ →NN above, equating the parities of the initial and ﬁnal states gives ηπ ηD = (−1)ℓη2 N = −1 , (7.18) since the initial atomic state \|1, 0, 0⟩has ℓ= 0 and hence no parity other than the intrinsic parities of the pion and deuteron. Provided parity is conserved by the strong nuclear interactions causing this decay, we conclude that the π−and D+ must have opposite intrinsic parity. Now, the D+ is predominantly an s-wave bound state of a proton and neutron, so ηD = (−1)ℓηP ηN = ηP ηN, and furthermore the proton and neutron can always be chosen to have the same intrinsic parity since they are related by an ‘isospin’ symmetry63. Thus ηD = +1 and hence the pion must have intrinsic parity ηπ = −1. As we mentioned in section 4.5.1, one of the great surprises of particle physics came in the 1950s. Cosmic rays were found to contain various types of particles, including two similar of similar mass called the τ + and the θ+. Both particles decay quickly, the predominant channels being θ+ − →π+ + π0 and τ + − →π+ + π+ + π−. (7.19) The angular distribution of the pions in the ﬁnal states could be observed in a cloud chamber or bubble chamber, and studying these patterns showed that the pions were produced in an ℓ= 0 state in both cases. Since ηπ = −1 (irrespective of the electric charge of the pion, for the same isospin reason as above), the θ+ should have intrinsic parity ηθ = η2 π = +1, while the τ + should have ητ = −1. However, as measurements improved the masses and lifetimes of the two particles became indistinguishable. The puzzle was resolved in 1956 when Lee and Yang proposed that the two particles were in fact one and the same, but that parity was not conserved in their decay process. Their proposal was largely ignored until Lee persuaded his colleague, Madame Chien–Shiung Wu, to test it using the β–decay of a certain isotope of cobalt. Wu’s experiments showed that parity is indeed violated in the weak interactions. The fact that x →−x is not a symmetry of Nature can be accommodated in QFT, though the deep reason for it remains mysterious64. 63The Standard Model has an (approximate) ‘internal’ SU(2) symmetry known as isospin that rotates protons into neutrons; they are diﬀerent states of the same nucleon, somewhat like the two diﬀerent spin states \|↑⟩, \|↓⟩of the same electron. 64There are various natural ways for parity violation to originate in string theory, but needless to say, none of them have been veriﬁed experimentally. – 106 – 8 Perturbation Theory I: Time Independent Case We’ve now come about as far as we can (in this course) relying purely on symmetry principles. The dynamics of systems of genuine physical interest is rarely simple enough to hope that we can solve it exactly, just using the general constraints of symmetry. In these circumstances we need to make some sort of approximation, treating our system as being ‘close to’ some other system whose dynamics is suﬃciently simple to be controllable. That is, we treat the diﬀerence between our actual system, whose experimental properties we care about, and our model system, whose description is simple enough that we can handle it, as a perturbation. The whole art of a theoretical physicist lies in striking this balance; if one’s model system is too simplistic, it might not provide a reasonable guide to the behaviour of the real case65, while if the model is overly complicated it may itself prove impossible to understand. There’s nothing inherently quantum mechanical about the need to approximate a com-plicated system by a simpler one, but, fortunately, perturbation theory in quantum me-chanics often turns out to be considerably easier than in classical dynamics, largely because of the vector space nature of H. In this chapter and the next, we’ll study various techniques to handle quantum mechanical perturbation theory, beginning with the simplest cases. 8.1 An Analytic Expansion Let H be the Hamiltonian of the experimental system we wish to understand, and H0 be the Hamiltonian of our model system whose eigenstates and eigenvalues we already know. We hope that ∆H = H −H0 is in some sense ‘small’, so that it may be treated as a perturbation. More speciﬁcally, we look for a parameter — let’s call it λ — that our true Hamiltonian depends on such that at λ = 0, H = H0. For λ ∈[0, 1], deﬁne Hλ = H0 + λ∆H . (8.1) We can think of Hλ as the Hamiltonian of an apparatus that is equipped with a dial that allows us to vary λ. At λ = 0 the system is our model case, and at λ = 1 it’s the case of genuine interest. We now seek the eigenstates \|Eλ⟩of Hλ. This may look as though we’ve made the problem even harder — we now need to ﬁnd the eigenstates not just of our model and experimental systems, but of a 1-parameter family of interpolating systems. Our key assumption is that since Hλ depends analytically on λ, so too do its eigenstates. In essence, this amounts to the assumption that small changes in the system will lead to only small changes in the outcome. Every mountain climber knows that this assumption can be dreadfully false and we’ll see that it can easily fail in QM too, but for now let’s see where it takes us. 65Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer,“I have the solution, but it works only in the case of spherical cows in a vacuum”. – 107 – If indeed \|Eλ⟩depends analytically in λ, then we can expand it as \|Eλ⟩= \|α⟩+ λ\|β⟩+ λ2\|γ⟩+ · · · (8.2) and similarly expand the eigenvalues E(λ) = E(0) + λE(1) + λ2E(2) + · · · . (8.3) Plugging these expansions into the deﬁning equation Hλ\|Eλ⟩= E(λ)\|Eλ⟩we obtain (H0 + λ∆H) \|α⟩+ λ\|β⟩+ λ2\|γ⟩+ · · · = E(0) + λE(1) + λ2E(2) + · · · \|α⟩+ λ\|β⟩+ λ2\|γ⟩+ · · · . (8.4) Since we require this to hold as λ varies, it must hold for each power of λ separately. Thus we ﬁnd an inﬁnite system of equations H0\|α⟩= E(0)\|α⟩, H0\|β⟩+ ∆H\|α⟩= E(0)\|β⟩+ E(1)\|α⟩, H0\|γ⟩+ ∆H\|β⟩= E(0)\|γ⟩+ E(1)\|β⟩+ E(0)\|γ⟩, . . . (8.5) and so on. The ﬁrst of these equations simply states that \|α⟩is an eigenstate of the model Hamil-tonian H0 with eigenvalue E(0). This is not surprising; under our analyticity assumptions the terms of O(λ0) are all that would survive when the dial is set to λ = 0, which is indeed the model system. Henceforth, we relabel \|α⟩→\|n⟩and E(0) →En to reﬂect this under-standing. (The notation \|n⟩is intended to imply the nth energy eigenstate of the model Hamiltonian H0, with eigenvalue En; we will distinguish the higher-order corrections to these states with superscripts.) To determine the ﬁrst-order correction E(1) n to the nth energy level of the unperturbed Hamiltonian, we contract the second of equations (8.5) with ⟨α\| ≡⟨n\| to ﬁnd ⟨n\|H0\|β⟩+ ⟨n\|∆H\|n⟩= En⟨n\|β⟩+ E(1) n (8.6) Since (as for any Hamiltonian) the model Hamiltonian is Hermitian, ⟨n\|H0\| β ⟩= ⟨β \|H0\|n⟩= En ⟨n\| β ⟩, (8.7) so in the case that the unperturbed state is \|n⟩, equation (8.6) becomes E(1) n = ⟨n\|∆H\|n⟩. (8.8) In other words, to ﬁrst order in λ, the change in the energy of our system as we move away from the model system is given by the expectation value ⟨∆H⟩n of the change in the Hamiltonian when the system is in its original state \|n⟩. – 108 – To ﬁnd the perturbed state, to ﬁrst order in λ we must understand \|β⟩. We can expand \|β⟩in the complete set {\|n⟩} of eigenstates of the original system as \|β⟩= X n bn\|n⟩. (8.9) Using this expression in the second of equations (8.5) and contracting with ⟨m\| (where \|m⟩̸= \|n⟩, the initial state we’re perturbing) gives (En −Em) bm = ⟨m\|∆H\|n⟩ (8.10) and so, provided Em ̸= En, bm = ⟨m\|∆H\|n⟩ En −Em . (8.11) This will hold provided the energy levels of our model system are non-degenerate; we’ll examine how to handle the more general case including degeneracy in section 8.2. In the non-degenerate case, equation (8.11) determines all the expansion coeﬃcients bm in \|β⟩ except bn. Fortunately, one can argue that bn = 0 from the requirement that \|Eλ⟩remains correctly normalised — I’ll leave this as an exercise, but the essential idea is that if we move a point on a unit sphere, then to ﬁrst order r · δr = 0 so that the variation is only non–zero in directions orthogonal to the original vector. With bn = 0 we have \|β⟩= X m̸=n ⟨m\|∆H\|n⟩ En −Em \|m⟩ (8.12) as the ﬁrst-order perturbation of the state when the unperturbed state is \|n⟩. We can also examine the second-order perturbation, E(2) n , of the nth energy level. To do so, contract the third of equations (8.5) with ⟨n\|. Using the facts that ⟨n\|H0\|γ⟩= En⟨n\|γ⟩ and ⟨n\|β⟩= 0 we have E(2) n = ⟨n\|∆H\|β⟩= X m̸=n ⟨n\|∆H\|m⟩⟨m\|∆H\|n⟩ En −Em = X m̸=n \|⟨n\|∆H\|m⟩\|2 En −Em , (8.13) using our expression (8.12) for \|β⟩. We could go on to higher order in λ, next ﬁnding \|γ⟩in terms of the original states {\|n⟩} and then ﬁnding the third-order energy shift E(3) n etc., but in practice the summations become increasingly messy and we hope (!) that the ﬁrst few terms already provide a good guide to the behaviour of the system near λ = 1. (In high–energy quantum ﬁeld theory, modern experiments typically cost many millions of dollars so it’s especially important to have extremely accurate theoretical predictions to compare to. Consequently, there’s a whole industry of people whose life’s work is to compute higher and higher order terms in perturbation series such as these.) Fortunately, in the Tripos you’ll never be asked to do anything beyond 2nd order. – 109 – Combining our results shows that, to second order in λ, the energy levels of the per-turbed system are given by En(λ) = En + λ⟨n\|∆H\|n⟩+ λ2 X m̸=n \|⟨n\|∆H\|m⟩\|2 En −Em + O(λ3) (8.14) where En are the energies of our model system. Recall that this expression – much beloved of Tripos Examiners – is derived under the assumptions i) that the new energies E(λ) and new states \|Eλ⟩are analytic at λ = 0 and ii) that the model system is non-degenerate so Em = En iﬀ\|m⟩= \|n⟩. Let’s now take a look at the use of this formula in a number of examples. 8.1.1 Fine Structure of Hydrogen Our treatment of the hydrogen atom in the last chapter assumed the electron was moving non–relativistically in the Coulomb ﬁeld of the proton. Since \|E/µc2\| = α2/2n2 ≪1, non–relativistic quantum mechanics should indeed be a good approximation. Nonetheless, better agreement with experiment is obtained by describing the electron using the rela-tivistic Dirac equation66. The energy levels obtained by assuming non-relativistic motion in a pure Coulomb potential are often called the gross structure of the hydrogen atom, whilst the small relativistic corrections to these energies implied by the Dirac equation are known as ﬁne structure. In section 6.2.3, we understood that gross structure energies are independent of ℓbecause of an enhanced symmetry of the pure Coulomb potential, generated by the Runge-Lenz operator. We thus expect that relativistic corrections will lift the degeneracy among states of diﬀerent ℓ. The Dirac equation itself is beyond the scope of this course, but some of its conse-quences are easy to understand in perturbation theory. Firstly, expanding the relativistic dispersion relation around the non-relativistic limit gives E = p p2c2 + µ2c4 ≈µc2 + p2 2µ − p4 8µ3c2 + · · · . (8.15) This shows that relativistic eﬀects alter the kinetic energy by ∆H = −p4 8µ3c2 (8.16) at ﬁrst order. From (8.8), the ﬁrst-order shift in the energy of state \|nℓ, m⟩of hydrogen due to this modiﬁed kinetic term is thus E(1) n,l = ⟨n, ℓ, m\|∆H\|n, ℓ, m⟩. (8.17) (In making this claim, we should note that our perturbation (8.16) is rotationally invariant, so in particular [Lz, ∆H] = 0 and [L2, ∆H] = 0. Therefore ⟨n, ℓ′, m′\|∆H\|n, ℓ, n⟩= 0 unless 66A better approximation still — in precise agreement with the most accurate measurements ever per-formed in any branch of science — comes from quantum ﬁeld theory. – 110 – ℓ′ = ℓand m′ = m. Thus our perturbation does not mix degenerate states of the gross structure, so non-degenerate perturbation theory is suﬃcient.) To compute (8.17), ﬁrst write ∆H = −(H0 −V (r))2/2µc2 where H0 is the gross structure Hamiltonian and V (r) the usual Coulomb potential. Then E(1) n,l = − 1 2µc2 (En)2 −2En⟨V (r)⟩+ ⟨V (r)2⟩ , (8.18) where En = −1 2µc2α2/n2 is the original energy of our state and ⟨V (r)⟩= ⟨n, ℓ, m\|V (r)\|n, ℓ, m⟩. The virial theorem for the 1/r potential tells us that 2⟨T⟩= −⟨V ⟩, so ⟨V ⟩= 2En. Next, to compute ⟨V 2⟩, observe that this 1/r2 term just modiﬁes the eﬀective potential due to the electron’s orbital angular momentum. Consider the Hamiltonian for the radial part of the Hydrogen atom, with an eﬀective potential Veﬀ(r) = ℏ2 2µ ℓ(ℓ+ 1) r2 + α r2 − e2 4πϵ0r = ℏ2 2µ ℓ′(ℓ′ + 1) r2 − e2 4πϵ0r . (8.19) Here we’ve introduced ℓ′ to absorb the α/r2 term into the angular momentum contribution. Of course ℓ′ does not correspond to any actual angular momentum, it’s just a trick to help us compute ⟨V 2⟩. With this eﬀective potential, repeating the calculations of section 6.4 would lead to just the same energy levels as before, but now in terms of ℓ′. That is, in place of (6.90) we’d ﬁnd E(ℓ′) = −1 2µc2α2 1 (ℓ′ + 1)2 . (8.20) using our new (non-integer) ℓ′. This is the exact result for a hydrogen atom with an additional 1/r2 term in its potential. However, in our perturbative context, it’s only appropriate to keep the answer accurate to ﬁrst order – there may be other eﬀects we haven’t yet accounted for (such as expanding (8.15) further) that contribute at higher order. Expanding ℓ′ around ℓwe ﬁnd E(ℓ′) = En + 1 2µc2 α4 n3(ℓ+ 1 2) + · · · (8.21) where n = ℓ+ 1. Combining this with the other terms in (8.18) gives E1 n,ℓ= −1 2µc2 n ℓ+ 1 2 −3 4 ! α4 n4 (8.22) as the ﬁrst-order change in the energy of state \|n, ℓ, m⟩due to the relativistic correction to kinetic energy. Notice that this eﬀect is suppressed by an additional power of α2 ∼v2/c2 compared to the gross structure, and that the degeneracy among states of diﬀerent ℓis lifted, as we expected. There’s a second consequence of the Dirac equation which comes in at the same order as the above kinetic terms. Due to its spin, the electron has a magnetic dipole moment m = −e 2µS . (8.23) – 111 – When placed in a magnetic ﬁeld, this dipole has energy U = −m · B. Relativistically, the electron in a Hydrogen atom does experience a magnetic ﬁeld because it moves through the electric ﬁeld E produced by the proton. If the electron has velocity v = p/(µγ), then classically the Lorentz transformation of E produces a magnetic ﬁeld B = γ c2 v × E = 1 µc2 p × eˆ r 4πϵ0r2 = − e 4πϵ0µc2 1 r3 L (8.24) EXPLAIN THOMAS PRECESSION? Consequently, in QM the Hamiltonian of hydrogen receives a further ﬁne-structure correction HSO = −m · B = αℏ3 2µ2c 1 r3 S · L (8.25) known as spin–orbit coupling. Since the coeﬃcient of the operator S · L is positive, spin– orbit coupling lowers the total energy when the spin and orbital angular momentum are antiparallel. In particular, S · L annihilates any state of hydrogen in which ℓ= 0, since then all components of L act trivially. As an important special case, spin–orbit coupling does not aﬀect the ground state. However, excited states with ℓ̸= 0 do generically feel the eﬀects of spin–orbit coupling. We ﬁrst compute the eﬀect of acting on our electron states with L · S. To do this, clearly we must include a speciﬁcation of the electron’s spin. This could be done using the basis \|n, ℓ, m⟩⊗\|↑⟩ and \|n, ℓ, m⟩⊗\|↓⟩, but it turns out to be more convenient to instead combine the electron’s orbital and spin angular momenta into J = L + S and label states by67 \|n, j, mj; ℓ⟩where j = ℓ+ 1 2 or j = ℓ−1 2 are the two possible values for the combined angular momentum, and \|mj\| ≤j. To make use of this, we write S · L = 1 2 J2 −L2 −S2 . (8.26) Thus S · L\|n, j, mj; ℓ⟩= ℏ2 2 j(j + 1) −ℓ(ℓ+ 1) −3 4 \|n, j, mj; ℓ⟩ = ℏ2 2 ( ℓ\|n, j, mj; ℓ⟩ when j = ℓ+ 1 2 −(ℓ+ 1) \|n, j, mj; ℓ⟩ when j = ℓ−1 2 . (8.27) Consequently, according to our general expression (8.8), the ﬁrst-order change in the energy of the \|n, j, mj : ℓ⟩state of hydrogen due to spin–orbit coupling is E1 n,j;ℓ= − 1 4µ2c2 e2ℏ2 4πϵ0 ( ℓ −(ℓ+ 1) ) 1 r3 n,j;ℓ , (8.28) 67Since [J, L2] = 0, it’s possible to ﬁnd a basis of simultaneous eigenstates of J2, Jz and L2. In this basis, we do not necessarily know the z-components of the electron’s orbital and spin angular momenta separately. – 112 – with the factor of ℓor −(ℓ+ 1) chosen depending on whether j = ℓ± 1 2. To compute the expectation value ⟨1/r3⟩n,j;ℓwe could just perform the integral Z R3 \|ψn,ℓ,m(x)\|2 1 r3 d3x . However, there’s a nifty trick that allows us to short-circuit the evaluation of this integral. Recall that for states of deﬁnite ℓ, the Hamiltonian for the unperturbed atom can be written Hℓ= −P 2 r 2µ + ℏ2ℓ(ℓ+ 1) 2µr2 − e2 4πϵ0r (8.29) where Pr = ( ˆ X · P + P · ˆ X)/2 is the radial momentum, given by −iℏ ∂ ∂r + 1 r in the position representation. Now, the expectation value ⟨[Pr, Hℓ]⟩= 0 when evaluated in any normalizable eigenstate of Hℓ. In our case, the commutator evaluates to [Pr, Hℓ] = −iℏ −ℏ2ℓ(ℓ+ 1) µr3 + e2 4πϵ0r2 (8.30) and therefore, provided ℓ̸= 0, 1 r3 n,j;ℓ = 1 a0 1 ℓ(ℓ+ 1) 1 r2 n,j;ℓ = 1 a3 0 1 ℓ(ℓ+ 1 2)(ℓ+ 1) 1 n2 (8.31) where a0 = ℏ/(αµc) is the Bohr radius and the second equality follows from our previous result (8.21) for ⟨1/r2⟩. Putting this result together with the other factors in (8.28) shows that the ﬁrst-order energy shifts due to spin–orbit coupling are E1 n,j;ℓ= +1 2Z2α4µc2 1 2n3(ℓ+ 1 2)(ℓ+ 1) (8.32a) when j = ℓ+ 1 2, whereas when j = ℓ−1 2 we have E1 n,j;ℓ= −1 2α4µc2 1 2n3ℓ(ℓ+ 1 2) . (8.32b) (We also recall that E1 n,j;ℓ= 0 if ℓ= 0.) As anticipated, states whose spin and orbital angu-lar momenta are anti-aligned (j = ℓ−1 2) have lower energy and so are more tightly bound than the pure Coulomb interaction, whereas if the spin and orbital angular momentum are aligned with eachother (j = ℓ+ 1 2) the state is less tightly bound. Notice also that the deviation from a pure Coulombic interaction partially lifts the degeneracy between states of diﬀerent ℓ. Equations (8.32a)-(8.32b) show that the spin–orbit contribution to the energy is sup-pressed by a factor of α2 compared to the gross structure energy −1 2µα2c2/n2. This is the same suppression as we found for the P4 correction to the kinetic term, so it does not – 113 – really make sense to consider these two eﬀects on the energy separately. Combining (8.22) with (8.32a) & (8.32b), we ﬁnd the net energy levels En,j;ℓ= −1 2µα2c2 " 1 n2 −α2 n3 3 4n − 1 j + 1 2 ! + · · · # (8.33) incorporating ﬁne structure correct to order α4. This formula holds regardless of whether j = ℓ+ 1 2 or j = ℓ−1 2. Even better, although the spin–orbit term only contributes only when ℓ̸= 0, it turns out that there is another term68 that contributes only when ℓdoes equal zero! Even better, this remaining term has the eﬀect that (8.33) is the correct ﬁne structure energy levels even for states with ℓ= 0 where j = 1 2 just from the spin. Proceeding down the periodic table, if we attach a single electron to an atomic nu-cleus containing Z protons, the bare Coulomb interaction increases by a factor of Z. The formula (8.33) remains valid provided we replace α ∼e2 by Zα. Hydrogen has Z = 1, so in the n = 1 ground state the ﬁne structure contribution is suppressed by a factor of α2 ∼1/(137)2 compared to the gross structure, justifying our treatment of the ﬁne struc-ture as a small perturbation. However, for heavier elements with higher Z, the suppression is only by (Zα)2 and the small value of α can be negated by a suﬃciently large Z. In particular, from (8.33) we see that the energy diﬀerence between the states with principal quantum number n but j = ℓ± 1 2 is En,ℓ+ 1 2 ;ℓ−En,ℓ−1 2 ;ℓ= 1 2µc2 1 n3 Z4α4 ℓ(ℓ+ 1) . (8.34) This gives the splitting between the two 2p states of Hydrogen to be 4.53 × 10−5 eV, in excellent agreement with the measured value of 4.54 × 10−5 eV. These splittings reach around 10% of the gross energy as one reaches the middle of the periodic table, making perturbative treatments less reliable. 8.1.2 Hyperﬁne Structure of Hydrogen A proton is a spin-1 2 particle, so like the electron it has a magnetic dipole moment mp proportional to its spin. A magnetic dipole mp placed at the origin generates its own magnetic ﬁeld B = 2µ0 3 mp δ3(x) + µ0 4πr3 (3(mp · ˆ r)ˆ r −mp) , which is (roughly) the ﬁeld you saw traced out by iron ﬁlings when ﬁrst playing with magnets. The dipole moment me of the electron sits inside this ﬁeld, leading to a new contribution Hhfs = −me · B (8.35) 68This remaining relativistic eﬀect is known as the Darwin term and gives a contribution απℏ3 2µ2c δ3(x. The origin of this term is rather subtle, and properly lies within QFT rather than QM. Because of the δ-function, it only aﬀects states which have non-zero probability to be found at r = 0. The only such states are those with ℓ= 0, since all others are prevented from reaching the origin by the eﬀective potential ℓ(ℓ+ 1)ℏ2/(2mr2). – 114 – depending on both dipoles. We’ll just investigate the eﬀect of this perturbation on the ℓ= 0 states of hydrogen, where it turns out that only the ﬁrst term ∼δ3(r) in B has non-vanishing contribution. For these ℓ= 0 states, one ﬁnds Hhfs = 4 3 m M α4µc2 1 n3ℏ2 S · I (8.36) where I is the spin operator for the proton, labelled using a diﬀerent letter just to avoid confusion with the electron spin operator S. The splittings in energy levels caused by this perturbation are suppressed not just by a factor of α4, but by an additional factor of the electron-to-proton mass ratio m/M ≈ 1/1836. The splittings caused by accounting for nuclear spin are thus much smaller than those of the ﬁne structure considered above, and the perturbation (8.35) is known as the hyperﬁne contribution to the energy levels. We can compute the eigenvalues in much the same way as for the spin–orbit contribution to ﬁne structure. Let F = I + S denote the total spin of the proton and electron. Then S · I = 1 2 F2 −I2 −S2 = ℏ2 2 f(f + 1) −3 2 . (8.37) This is −3ℏ2/4 when the proton and electron spins are antiparallel so f = 0, and +ℏ2/4 when f = 1. Plugging these into (8.36) shows that the diﬀerence between the energies for the n = 1 ground state of hydrogen is En=1;f=+1 −En=1;f=0 = 4 3 m M α4µc2 ≈5.88 × 10−6eV , (8.38) corresponding to a frequency of 1.4GHz and a wavelength of 21cm. Thus transitions between the energy levels split by the hyperﬁne structure causes the hydrogen atom to have a microwave emission line. The 21cm line provides the most powerful way of tracing diﬀuse gas in interstellar and intergalactic space. This wavelength is much longer than the size of typical specks of dust, so 21cm radiation can propagate with little absorption right through clouds of dust and gas that do absorb visible light. It was radiation at 1.4GHz that ﬁrst revealed the large-scale structure of our galaxy. The line is intrinsically very narrow, so the temperature and radial (with the Earth as origin) velocity of the hydrogen that emitted the radiation can be accurately measured from the Doppler shift and broadening of the observed spectral line. The hyperﬁne structure of hydrogen leading to this 21cm emission line was ﬁrst pre-dicted theoretically in 1944 by the Dutch physicist H.C. van de Hulst as part of his doctoral thesis, carried out in Utrecht which was then under Nazi occupation. 21cm radiation from our galaxy was ﬁrst detected experimentally in 1951 by three groups working indepen-dently in the USA, Australia and the Netherlands. The Dutch group used a German radio antenna left over from the war. 8.1.3 The Ground State of Helium After hydrogen, helium is the next most abundant element, making up about a quarter of the ordinary matter in the Universe. Much of this helium was created during the period of nucleosynthesis, a tiny fraction of a second after the Big Bang. – 115 – Let’s use perturbation theory to ﬁnd an estimate of the ground state energy of helium. Working in the centre of mass frame and treating the nucleus as stationary at the origin, the (gross structure) Hamiltonian is H = P2 1 2m + P2 2 2m − 2e2 4πϵ0\|X1\| − 2e2 4πϵ0\|X2\| + e2 4πϵ0 1 \|X1 −X2\| (8.39) where (X1, X2) are the position operators and (P1, P2) the momentum operators for the two electrons. The terms describing the electrons’ kinetic energy and attraction to the nucleus are just the sum of those for Hydrogen, except with e2 replaced by 2e2 since there are two protons in the nucleus of helium. We’ll take these terms to be the model Hamiltonian H0, treating the remaining electron-electron repulsion as a perturbation. Before proceeding, we should ask what is the small, dimensionless parameter in which we’re performing our perturbative expansion. One might hope that, as for the ﬁne structure of hydrogen, this is the ﬁne structure constant α = e2/ℏc ≈1/137. However, it’s clear that this cannot be the case here, because both the electron–electron interaction and the electron–proton interactions both scale as e2 ∼α. Furthermore, the virial theorem says that in any state \|Ψ⟩, for the 1/r hydrogenic Coulomb potential, 2⟨T⟩Ψ = −⟨V ⟩Ψ so the kinetic terms in the hydrogenic term are likewise of order α. In fact, we’ll perform a perturbation in the inverse atomic number, taking λ = 2/Z and treating this as a continuous parameter. We’re interested in values λ ∈(0, 1] corresponding to Z ∈(∞, 2]. Note that when Z ≫1, corresponding to a nucleus with a large positive charge, the two electrons’ attraction to this nucleus should certainly overwhelm their mutual repulsion. In the unperturbed Hamiltonian, single–electron states are just the usual hydrogenic states \|n, ℓ, m⟩with energy −4 × 1 2mc2α2 1 n2 , where the factor of 4 arises because each electron’s attraction to the helium nucleus is twice as strong as it was for hydrogen. Since electrons are fermions, the total state of the neutral helium atom must be antisymmetric under their exchange. In particular, the ground state is \|Ψ0⟩= \|1, 0, 0⟩⊗\|1, 0, 0⟩⊗1 √ 2 (\|↑⟩\|↓⟩−\|↓⟩\|↑⟩) (8.40) where the ﬁnal factor is the antisymmetric spin state of the two spin-1 2 electrons. (Note that, at the level of the gross structure of helium, the Hamiltonian (8.39) is independent of the spins of the two electrons, so we are free to seek simultaneous energy and spin eigenstates.) This state is an eigenstate of the model Hamiltonian H0 with energy E1 = −4α2 2 mc2 −4α2 2 mc2 = −4α2mc2 ≈−108.8 eV , (8.41) being the sum of the energies of the two electrons individually. We now take account of the electron–electron repulsion. To use perturbation theory, we set λ = 2 Z and ∆H = e2 4πϵ0 1 \|X1 −X2\| = αℏc \|X1 −X2\| . (8.42) – 116 – Our ∆H is chosen so that i) it is independent of the expansion parameter and ii) λ∆is precisely the electron–electron potential when λ = 1, appropriate for helium. Following the general results above, the ﬁrst order shift in the ground state energy is ⟨Ψ0\|∆\|Ψ0⟩= Z ⟨Ψ0\|x′ 1, x′ 2⟩⟨x′ 1, x′ 2\|∆H(X1, X2)\|x1, x2⟩⟨x1, x2\|Ψ0⟩d3x′ 1 d3x′ 2 d3x1 d3x2 = Z Ψ0(x1, x2) αℏc \|x1 −x2\| Ψ0(x1, x2) d3x1 d3x2 = Z αℏc \|x1 −x2\| \|ψ100(x1)\|2 \|ψ100(x2)\|2 d3x1 d3x2 , (8.43) Here ψ100(x) are the single–electron n = 1 wavefunctions ψ100(x) = 1 √π 2 a2 3 2 e−\|x\|/a2 , (8.44) where the length scale a2 = ℏ 2αmc is half the Bohr radius in hydrogen. In going to the last line of (8.43), we’ve used the fact that the interaction ∆is independent of spin, and the spin states in (8.40) have norm 1. Performing the integral is straightforward but somewhat tedious69, and isn’t the sort of thing I’m going to ask you to reproduce in an exam. One ﬁnds ⟨Ψ0\|∆H\|Ψ0⟩= 5 4α2mc2 (8.45) As we suspected, this perturbation is also of order α2 just like the leading–order term. Including the factor of λ = 2/Z, to ﬁrst–order in perturbation theory the ground state of helium is E1(1/Z)Z=2 = −4α2mc2 1 −5 16 2 Z + · · · Z=2 ≈−74.8 eV . (8.46) 69In case you’re curious: Choose the z-axis of x2 to be aligned with whatever direction x1 is. Then \|x1 −x2\| = p \|r2 1 + r2 2 −2r1r2 cos θ2\|, independent of φ2. So ⟨Ψ0\|∆H\|Ψ0⟩= 4αℏc a3 2 Z \|ψ100(x1)\|2 "Z r2 2 sin θ2 e−2r2/a2 p \|r2 1 + r2 2 −2r1r2 cos θ2\| dr2 dθ2 # d3x1 . The integral in square brackets can be done using Z π 0 sin θ2 dθ2 p \|r2 1 + r2 2 −2r1r2 cos θ2\| = 1 r1r2 Z π 0 d dθ2 q \|r2 1 + r2 2 −2r1r2 cos θ2\| dθ2 = ( 2/r1 when r1 > r2 2/r2 when r1 < r2 . Thus the radial integral dr2 must be broken into two regions. Letting ρ1 := 2r1/a2 be a rescaled radial coordinate, we have ⟨Ψ0\|∆\|Ψ0⟩= 8αℏc a3 2 Z \|ψ100(x1)\|2 Z r1 0 r2 2 r1 e−2r2/a2 dr2 + Z ∞ r1 r2 e−2r2/a2 dr2 d3x1 = 4αℏc a2 Z \|ψ100(x1)\|2 2 −e−ρ1(2 + ρ1) ρ1 d3x1 = 2αℏc a2 Z ∞ 0 ρ1 e−ρ1 2 −e−ρ1(2 + ρ1) dρ1 = 5αℏc 4a2 . This is the value that was used in the text. And no, I’m not going to derive the second–order result for you. – 117 – Figure 17: First ionisation energies of the ground states of the ﬁrst few elements. Figure by Agung Karjono at Wikipedia. This is in signiﬁcantly better agreement with the experimental value of ≈−79.0 eV than our zeroth–order estimate (8.41). Including also the second–order term70 leads to E1(1/Z) = −4α2mc2 1 −5 8 1 Z + 25 256 1 Z2 + O(1/Z3) (8.47) giving E1(1/2) ≈−77.5 eV. Given the crudity of our expansion this is in remarkably close agreement with the experimental value. In heavy atoms, the ground state energy itself is diﬃcult to measure experimentally since it involves computing the energy required to strip oﬀall the electrons from the nucleus. Fortunately, for the same reason it’s also rarely the directly relevant quantity. Instead, chemists and spectroscopists are usually more interested in the ﬁrst ionisation energy, deﬁned to be the energy required to liberate a single electron from the atom. Hydrogen only has one electron, so its ﬁrst ionization energy is just the 13.6 eV which we must supply to eject this electron from the ground state of the atom. In the case of helium, once one electron is stripped away the remaining electron sees the full force of the Coulomb potential from the Z = 2 nucleus, so will have binding energy −54.4 eV. The ﬁrst ionisation energy is thus the diﬀerence ≈(−54.4 + 79.0) eV = 24.6 eV. This is signiﬁcantly greater than the ionisation energy of hydrogen, and in fact helium has the greatest ﬁrst ionisation energy of any of the elements, reﬂecting its status as the ﬁrst noble gas (see ﬁgure 17). 8.1.4 The Quadratic Stark Eﬀect We’ll now consider a diﬀerent type of perturbation, caused not by relativistic corrections to the Hamiltonian of the isolated atom, but by allowing the atom to interact with some external system. 70We could calculate this using the result (8.13) for the second-order energy shift. I’ll just quote the answer. – 118 – Suppose a hydrogen atom is placed in an external electric ﬁeld E = −∇φ. Most electric ﬁelds we can generate in the laboratory are small compared to the strength of the Coulomb ﬁeld felt by the electron due to the nucleus71, so perturbation theory should provide a good estimate of the energy shifts such an external ﬁeld induces. By deﬁnition of the electrostatic potential Φ, the ﬁeld changes the energy of the atom by δE = e(φ(xp) −φ(xe)) where xp and xe are the locations of the proton and electron. We assume the external ﬁeld varies only slowly over scales of order the Bohr radius, so δE ≈−e r · ∇φ = −e r · E (8.48) where r = xe −xp. Let’s choose the direction of the electric ﬁeld to deﬁne the z-axis, and set E = \|E\| so as to avoid confusion with the energy. Thus E = E ˆ z and the eﬀect of the external electric ﬁeld is to add a new term to the Hamiltonian H0 of the unperturbed atom: H = H0 −e EX3 (8.49) where X3 is the position operator for the z-coordinate of the electron relative to the proton. In this section, we’ll just consider the eﬀect of this electric ﬁeld on the ground state \|n, ℓ, m⟩= \|1, 0, 0⟩of the hydrogen atom. This state is non-degenerate for the unperturbed Hamiltonian (and our perturbation is blind to spin), so again non-degenerate perturbation theory will suﬃce. From (8.8) the ﬁrst order change in the ground state energy is E(1) 1 = −e E ⟨1, 0, 0\|X3\|1, 0, 0⟩, (8.50) However, applying the parity operator Π and noting that Π−1X3Π = −X3 we have ⟨1, 0, 0\|X3\|1, 0, 0⟩= −⟨1, 0, 0\|Π−1X3Π\|1, 0, 0⟩= −⟨1, 0, 0\|X3\|1, 0, 0⟩, (8.51) where the last equality uses the fact that Π\|1, 0, 0⟩= \|1, 0, 0⟩since the ground state wave-function is spherically symmetric. Thus ⟨1, 0, 0\|X3\|1, 0, 0⟩= 0 and parity symmetry ensures that, to ﬁrst order in the external electric ﬁeld, the ground state energy is unaﬀected. To see a non-trivial eﬀect, we must go one order further. The second order shift in the ground state energy is given by (8.13) in general. In our case, the sum is to be taken over all states of hydrogen except the ground state, so E(2) 1 = e2E2 ∞ X n=2 X ℓ<n ℓ X m=−ℓ \|⟨1, 0, 0\|X3\|n, ℓ, m⟩\|2 E1 −En . (8.52) In the second problem set you proved that ⟨n′, ℓ′, m′\|X\|n, ℓ, m⟩= 0 unless \|ℓ′ −ℓ\| = 1 . Applying this to our case, ⟨1, 0, 0\|X3\|n, ℓ, m⟩= 0 unless ℓ= 1, which greatly simpliﬁes the sums in (8.52). Furthermore, [Lz, X3] = 0 so X3\|n, 0, 0⟩is an eigenstate of Lz with 71With modern high intensity lasers, electric ﬁelds comparable to the nuclear Coulomb potential can be generated. The behaviour of a hydrogen atom in such a laser must be studied by other methods. – 119 – eigenvalue zero and hence it is orthogonal to \|n, 1, m⟩whenever m ̸= 0. In summary, to this order the electric ﬁeld causes the ground state \|1, 0, 0⟩to mix only with states \|n, 1, 0⟩. Thus the only non-vanishing terms in (8.52) are E(2) 1 = e2E2 ∞ X n=2 \|⟨1, 0, 0\|X3\|n, 1, 0⟩\|2 E1 −En (8.53) involving just a single sum. This is the second-order change in the ground state energy due to the presence of the external electric ﬁeld. This change in the ground state energy is known as the quadratic Stark eﬀect, after it’s discoverer and since it comes in at order E2. There’s a good physical reason why the change in energy comes in at this order: the hydrogen atom is electrically neutral and the unperturbed ground state \|1, 0, 0⟩is spherically symmetric, so the atom in this state has no electric dipole moment. In response to the applied electric ﬁeld, the electronic wavefunction changes by an amount that is proportional to the coeﬃcients bm, which are themselves proportional to E. In other words, the applied electric ﬁeld E polarizes the atom, generating a dipole moment p ∝E. The energy of any dipole p in an electric ﬁeld is Udip = −p · E and since the dipole moment itself is ∼E, for the ground state of hydrogen Udip ∼E2. Higher energy states \|n, ℓ, m⟩can have a permanent dipole moment due to the electron’s elliptical orbits, but to study this we need degenerate perturbation theory. 8.2 Degenerate Perturbation Theory Our derivation of the coeﬃcients bm = ⟨m\|∆\|n⟩ En −Em (8.54) of the ﬁrst-order shift in the state breaks down if there are states \|m⟩that have the same energy as the state \|n⟩that we’re attempting to perturb. Indeed, naively applying (8.54) in this case appears to show that the small perturbation can cause an extremely dramatic shift in the state of the system. There’s nothing particularly surprising about this. Consider a marble lying in the bottom of a bowl at the minimum of the gravitational potential. If we tilt the bowl slightly, the marble will move a little, resettling in the bottom of the tilted bowl as it adjusts to the new minimum of the potential. However, if the marble initially lies at rest on a smooth table, so that it initially has the same energy no matter where on the table it lies, then tilting the table even very slightly will lead to a large change in the marble’s location. To handle this degenerate situation, we ﬁrst observe that the only states that will acquire a signiﬁcant amplitude as a result of the perturbation are those that are initially degenerate with the original state. In other words, to good approximation, the state to which the system changes will be a linear combination of those having the same zeroth– order energy as our initial state. In many situations, there are only a ﬁnite number of these. We then diagonalise the matrix ⟨r\|∆\|s⟩of the perturbing Hamiltonian in the subspace – 120 – spanned by these degenerate states. Since this subspace was completely degenerate wrt H0, in this subspace the eigenstates of the perturbation ∆are in fact eigenstates of the full Hamiltonian H0 + ∆. Let’s see how this works in more detail. Suppose V ⊂H is an N–dimensional subspace with H0\|ψ⟩= EV \|ψ⟩for all \|ψ⟩∈V . For r = 1, . . . , N, we let {\|r⟩} be an orthonormal basis of V and deﬁne PV to be the projection operator PV : H →V . We can write this projection operator as PV = N X r=1 \|r⟩⟨r\| (8.55) in terms of our orthonormal basis. Similarly, we let P⊥= 1 −PV denote the projection onto V ⊥, where V ⊥= { \|χ⟩∈H : ⟨ψ\|χ⟩= 0 ∀\|ψ⟩∈V } . (8.56) Note that, as projection operators, P 2 V = PV and P 2 ⊥= P⊥and that PV P⊥= P⊥PV = 0. Also, we have [H0, PV ] = 0 and [H0, P⊥] = 0 (8.57) since V was deﬁned in terms of a degenerate subspace of H0. Now let’s consider a perturbed Hamiltonian Hλ = H0 + λ∆. Any eigenstate \|ψλ⟩of the full Hamiltonian obeys 0 = (H0 −E(λ) + λ∆)\|ψλ⟩ = (H0 −E(λ) + λ∆)(PV + P⊥)\|ψλ⟩ = (EV −E(λ) + λ∆)PV \|ψλ⟩+ (H0 −E(λ) + λ∆)P⊥\|ψλ⟩ (8.58) where in the second line we’ve separated out the terms in V from those in V ⊥. Acting on the left with either PV or P⊥and using (8.57), we obtain the two equations 0 = (EV −E(λ) + λ PV ∆) PV \|ψλ⟩+ λ PV ∆P⊥\|ψλ⟩ (8.59a) and 0 = λP⊥∆PV \|ψλ⟩+ (H0 −E(λ) + λ P⊥∆) P⊥\|ψλ⟩, (8.59b) in which we have separated out the eﬀects of the perturbation within V and its complement. Suppose now that we’re perturbing a state \|ψ0⟩that, in the absence of the perturbation, lies in V . We write \|ψλ⟩= \|α⟩+ λ\|β⟩+ λ2\|γ⟩+ · · · E(λ) = E(0) + λE(1) + λ2E(2) + · · · (8.60) just as before, noting that P⊥\|ψλ⟩is necessarily at least ﬁrst–order in λ. To zeroth order, equation (8.59a) tells us that E(0) = EV as expected. At ﬁrst order, this equation becomes (−E(1) + PV ∆)\|α⟩= 0 or equivalently PV ∆PV \|α⟩= E(1)\|α⟩. (8.61) This is a remarkable equation! It tells us that, if our assumption that the perturbation ∆causes only a small change in the energies and states of the system is to hold and if – 121 – our zeroth-order state \|ψ0⟩lies in some degenerate subspace, then \|ψ0⟩can’t be just any eigenstate of H0, but must also be an eigenstate of PV ∆PV – the perturbation, restricted to the degenerate subspace V . In other words, in degenerate systems, states that are stable against small perturbations are those that are already eigenstates of the perturbation. If we started with some other \|ψ0⟩∈V that was not an eigenstate of ∆, then as soon as the perturbation is turned on, the state would rapidly change into a ∆eigenstate72. This large response to a small change in H would invalidate our use of perturbation theory; it’s the quantum analogue of the fact that the location of a marble on a perfectly ﬂat table is extremely sensitive to any tilt. Let’s now suppose that we’ve chosen our basis {\|r⟩} of V in such a way that PV ∆Pv\|r⟩= E(1) r \|⟩, so that \|r⟩is indeed an eigenstate of the perturbation with some eigenvalue we’ll call E(1) r . Then choosing our initial state \|α⟩to be \|r⟩, by (8.61) the ﬁrst order shift in its energy is E(1) r = ⟨r\|PV ∆PV \|r⟩= ⟨r\|∆\|r⟩. (8.62) This is exactly what we would have found in the non-degenerate case, but here it’s very im-portant that we’re considering the eﬀects of perturbations when we start with an eigenstate of ∆. In fact, since \|r⟩is an eigenstate of both H0 and ∆, we have Hλ\|r⟩= (H0 + λ∆)\|r⟩= (EV + λE(1) r )\|r⟩ (8.63) so that in the subspace V we’ve solved the perturbed spectrum exactly. In other words, within V we’re not really doing perturbation theory at all! Fortunately, in problems of interest dim(V ) is typically rather small and ﬁnding the exact spectrum of the full Hamil-tonian Hλ within this ﬁnite dimensional subspace is usually much easier than ﬁnding the full spectrum. As we’ve seen, degeneracy in the energy spectrum often arises as a result of some symmetry. If this symmetry is dynamical, such as the enhanced U(d) symmetry of the d-dimensional harmonic oscillator, or the SU(2)×SU(2) symmetry the Runge-Lenz vector implies for the gross structure of hydrogen, it will almost inevitably by broken by a pertur-bation. Even symmetries such as rotations that are inherited from transformations of space may be broken in the presence of external ﬁelds. For example, although the ground state of an atom is typically spherically symmetric, the laboratory in which one conducts an ex-periment likely won’t be and any small stray electric and magnetic ﬁelds in the laboratory will perturb the atom away from pure spherical symmetry. For this reason, perturbations typically lift degeneracy. As we’ve seen, even very small perturbations can have a dramatic eﬀect in immediately singling out a preferred set of states in an otherwise degenerate sys-tem; we’ll return to this point in section 10.1.3 when trying to understand ‘collapse of the wavefunction’ in quantum mechanics. If you take the Applications of Quantum Mechanics course next term, you’ll see that the lifting of degeneracy is also important in giving crys-tals their electronic properties: when a large number of identical metal atoms are brought 72To justify this, in particular to understand what happens “as a perturbation is turned on”, we’ll need to consider time-dependent perturbation theory. See chapter 9. – 122 – together, ignoring the inter-atomic interactions, all the valence electrons in the diﬀerent atoms will be degenerate. The small eﬀect of the potential from nearby atoms breaks this degeneracy and makes these valence electrons prefer to delocalise throughout the crystal. 8.2.1 The Linear Stark Eﬀect As an application of degenerate perturbation theory, let’s again consider the Stark eﬀect but now for the 2s state of our atom. This state is degenerate with the three73 2p states, so here our degenate subspace V = span{\|2, 0, 0⟩, \|2, 1, 1⟩, \|2, 1, 0⟩, \|2, 1, −1⟩} . (8.64) As before, the electric ﬁeld induces a perturbation is74 eEX3 and we ﬁnd a diagonalise this perturbation within V . Parity implies ⟨2, 0, 0\|X3\|2, 0, 0⟩= 0 and ⟨2, 1, m\|X3\|2, 1, m′⟩= 0 ∀m, m′ ∈{1, 0, −1} , (8.65) while since [Lz, X3] = 0 we also have ⟨2, 0, 0\|X3\|2, 1, ±1⟩= 0. Thus the matrix elements of ∆in the basis (8.64) are ∆\|V = eE      0 0 a 0 0 0 0 0 a∗0 0 0 0 0 0 0     , (8.66) where a = ⟨2, 0, 0\|X3\|2, 1, 0⟩= 2π Z ψ2,0,0(r) ψ2,1,0(r, θ) r cos θ r2d(cos θ) = −3a0 , (8.67) with a0 the Bohr radius. The eigenvalues of ∆in this subspace are thus {3eEa0, 0, 0, −3eEa0} with corresponding eigenstates 1 √ 2 (\|2, 0, 0⟩−\|2, 1, 0⟩) , \|2, 1, 1⟩, \|2, 1, −1⟩ and 1 √ 2 (\|2, 0, 0⟩+ \|2, 1, 0⟩) . (8.68) In the presence of the electric ﬁeld, the n = 2 state of lowest energy is (\|2, 0, 0⟩+\|2, 1, 0⟩)/ √ 2 and we conclude that even for a tiny electric ﬁeld, the n = 2 states will jump to this preferred state. From our discussion of the quadratic Stark eﬀect above, we know that a change in energy ∝E requires the dipole moment D of an atom to be independent of E. Since this n = 2 state has a ﬁrs-order energy shift of −3eEa0, we conclude that the n = 2 states of hydrogen do indeed have a permanent electric dipole moment of magnitude 3ea0. Classically, for an electron in an elliptical orbit, this result is expected because the electron 73Since the applied electric ﬁeld does not coupled to spin, we ignore the eﬀect of spin in this discussion – at this order, the Stark eﬀect will not lift the degeneracy of the gross structure of hydrogen wrt the electron’s spin. 74A reminder: we deﬁne the z-direction to be the direction of the electric ﬁeld, and X3 is the corresponding operator. – 123 – would spend more time at the apocentre (the point of its orbit furthest from the atom’s centre of mass) than at the pericentre (the point nearest to the centre of mass). If electron’s orbit was a perfect Keplerian ellipse, the atom would have a permanent electric dipole moment aligned parallel to the orbit’s major axis. However, any small deviation from the 1/r potential will cause the major axis of the ellipse to precess, and the time–averaged dipole moment would vanish. For hydrogen, the potential does diﬀer from 1/r, though only very slightly due to the eﬀects of ﬁne structure. Hence even a weak external ﬁeld can prevent precession and give rise to a steady electric dipole. Quantum mechanically, in the presence of the electric ﬁeld the lowest energy n = 2 state (\|2, 0, 0⟩+ \|2, 1, 0⟩)/ √ 2 is no longer an eigenstate of L2, because the ﬁeld applies torque to the atom, changing it’s total orbital angular momentum. However, this lowest energy state is still an eigenstate of Lz with eigenvalue zero. Recalling that we took the z-direction to be the direction of the applied ﬁeld, we see that the angular momentum is perpendicular to E, as expected from the classical picture of an ellipse with major axis aligned along E. Note also that the electric ﬁeld does not lift the degeneracy between the \|2, 1, 1⟩and \|2, 1, −1⟩states. These two states have their orbital angular momentum maximally aligned or anti-aligned with E; classically, they correspond to orbits conﬁned to a plane perpendicular to E. 8.3 Does Perturbation Theory Converge? We began our study of perturbation theory by assuming that the states and energy eigen-values of the full Hamiltonian depended analytically on a dimensionless parameter λ con-trolling the perturbation. Even when the individual coeﬃcients of powers of λ are ﬁnite, this is often not the case because the inﬁnite perturbative series itself may fail to converge, or may converge only for some range of λ. The issue is that we really have an expansion in λ ⟨m\|∆\|n⟩ Em −En where m ̸= n , and the coeﬃcients of λ may grow too rapidly. Heuristically, the condition for convergence is thus that the typical energy splitting \|⟨m\|∆\|n⟩\| induced by the perturbation should be much smaller than the initial energy diﬀerence Em −En. However, a detailed criterion is often hard to come by since the higher terms in the perturbation expansion involve complicated sums (or integrals) over many diﬀerent intermediate states. To illustrate this in a simple context, let’s consider in turn the following three pertur-bations of a 1d harmonic oscillator potential H = P 2 2m + 1 2mω2X2 +        −λm ω2x0X + 1 2λm ω2X2 +λϵX4 (8.69) where x0 and ϵ are constants. Of course, the ﬁrst two can be solved exactly — we’d never really use perturbation theory to study them. – 124 – In the ﬁrst case, we have H = P 2 2m + 1 2m ω2(X −λx0)2 −λ2 2 mω2x2 0 (8.70) from which we easily see that the exact energies are En(λ) = n + 1 2 ℏω −λ2 2 mω2x2 0 , (8.71) with corresponding position space wavefunction ⟨x\|nλ⟩= ⟨x −λx0\|n⟩, just a translation of the usual harmonic oscillator wavefunction \|n⟩. If we instead tackled this problem using perturbation theory, we’d ﬁnd En(λ) = En −λmω2x0⟨n\|X\|n⟩+ λ2m2ω4x2 0 X k̸=n \|⟨k\|X\|n⟩\|2 (n −k)ℏω + O(λ3) = n + 1 2 ℏω −λ2 2 mω2x2 0 + O(λ3) . (8.72) To obtain this result, we note that the ﬁrst–order term vanishes (e.g. by parity), while since X is a linear combination of creation and annihilation operators A† and A, the only \|k⟩= \|n + 1⟩and \|k⟩= \|n −1⟩terms can contribute to the sum in the second-order term. Going further, we’d ﬁnd that there are no higher corrections in λ – the O(λ3) terms are in fact zero – though this is not easy to see directly. Thus, in this case, the perturbative result converges to the exact result, and the radius of convergence is inﬁnite. This reﬂects the fact that the perturbation −λm ω2x0X didn’t really change the character of the original Hamiltonian. No matter how large λ is, for large enough x the perturbation remains negligible. Turning to the second case, it’s again immediate that the exact energy levels are En(λ) = n + 1 2 ℏω(λ) (8.73) where ω(λ) = ω √ 1 + λ is the modiﬁed frequency. This has a branch cut starting at λ = −1, so the energy is only analytic in the disc \|λ\| < 1. Again using perturbation theory, we ﬁnd En(λ) = En + λ 2 mω2⟨n\|X2\|n⟩+ O(λ3) = n + 1 2 ℏω 1 + λ 2 −λ2 8 + O(λ3) (8.74) agreeing to this order with the Taylor expansion of the exact answer. Continuing further, we’d ﬁnd that this Taylor series does indeed converge provided \|λ\| < 1, and that it then converges to the exact answer. The physical reason why the perturbation series diverges when \|λ\| ≥1 is simply that if λ = −1, the ‘perturbation’ has completely cancelled the original harmonic oscillator potential, so we’re no longer studying a system that can be treated as a harmonic oscillator in the ﬁrst instance. Once λ < −1 the harmonic oscillator – 125 – potential is turned upside down, and we do not expect our system to possess any stable bound states. Finally, consider the case H = HSHO + λϵX4 . (8.75) I do not know whether this model has been solved exactly, but it can be treated perturba-tively. After a fair amount of non-trivial calculation75 one obtains the series E0(λ) = ℏω + ∞ X n=1 (λϵ)nan (8.76) for the ground state energy including the quartic interaction, where the coeﬃcients behave as an = (−1)n+1√ 6 π3/2 3n Γ(n + 1 2) 1 −95 72 1 n + O(n−2) . (8.77) On account of the Γ-function, these grow faster than factorially with n, so the series (8.76) has zero radius of convergence in λ! Once again, this is easy to see from the form of the perturbed Hamiltonian: even though we may only care about λ > 0, our assumption that the perturbation expansion is analytic in λ at λ = 0 means that, if it converges, it will do so for a disc λ ∈D ⊂C. For any λ ∈R<0, the Hamiltonian of the quartic oscillator is unbounded below, so there cannot be any stable bound states that are analytic in λ at λ = 0. Let me comment that even when perturbative series do not converge, they may still provide very useful information as an asymptotic series. You’ll learn far more about these if you take the Part II course on Asymptotic Methods next term, but brieﬂy, we say a series SN(λ) = PN n=0 anλn is asymptotic to an exact function S(λ) as λ →0+ (written SN(λ) ∼S(λ) as λ →0+) if lim λ→0+ 1 λN S(λ) − N X n=0 anλn = 0 . (8.78) In other words, if we just include a ﬁxed number N of terms in our series, then for small enough λ ≥0 these ﬁrst N terms diﬀer from the exact answer by less that ϵλN for any ϵ > 0 (so the diﬀerence is o(N)). However, if we instead try to ﬁx λ and improve our accuracy by including more and more terms in the series, then an asymptotic series will eventually diverge. Most of the perturbative series one meets in the quantum world (including most Feynman diagram expansions of Quantum Field Theory) are only asymptotic series. Just as in our toy examples above, the radius of convergence of such series is often associated with interesting physics. 75I don’t expect you to reproduce it – if you’re curious you can ﬁnd the details in Bender, C. and Wu, T.T., Anharmonic Oscillator II: A Study of Perturbation Theory in Large Order, Phys. Rev. D7, 1620-1636 (1973). – 126 – 9 Perturbation Theory II: Time Dependent Case In this chapter we’ll study perturbation theory in the case that the perturbation varies in time. Unlike the time–independent case, where we mostly wanted to know the bound state energy levels of the system including the perturbation, in the time–dependent case our interest is usually in the rate at which a quantum system changes its state in response to the presence of the perturbation. 9.1 The Interaction Picture Consider the Hamiltonian H(t) = H0 + ∆S(t) (9.1) where H0 is a time-independent Hamiltonian whose spectrum and eigenstates we under-stand, and ∆S(t) is a perturbation that we now allow to depend on time. (The footnote on ∆S(t) is to remind us that this operator has genuine, explicit time dependence even in the Schr¨ odinger picture.) The idea is that H0 models a system in quiescent state, and we want to understand how eigenstates of H0 respond as they start to notice the perturbation. In the absence of the time-dependent perturbation, the time evolution operator would be U0(t) = e−iH0t/ℏas appropriate for the model Hamiltonian H0. We deﬁne the interaction picture state \|ψI(t)⟩as \|ψI(t)⟩= U −1 0 (t)\|ψS(t)⟩, (9.2) where \|ψS(t)⟩is the Schr¨ odinger picture state evolving in accordance with the TDSE for the full Hamiltonian (9.1). Thus, in the absence of ∆S(t), we’d simply have \|ψI(t)⟩= U −1 0 (t) U0(t)\|ψ(0)⟩= \|ψ(0)⟩so the state \|ψI(0)⟩would in fact be independent of time. The presence of the perturbation means that our states do not evolve purely according to U0(t), so \|ψI(t)⟩has non-trivial time dependence. To calculate the evolution of \|ψI(t)⟩, we diﬀerentiate iℏ∂ ∂t\|ψI(t)⟩= −H0 eiH0t/ℏ\|ψS(t)⟩+ eiH0t/ℏiℏ∂ ∂t\|ψS(t)⟩ = −H0 eiH0t/ℏ\|ψS(t)⟩+ eiH0t/ℏ(H0 + ∆(t))\|ψS(t)⟩ = U −1 0 (t) ∆(t) U0(t)\|ψI(t)⟩, (9.3) where we’ve used the full TDSE to evaluate ∂\|ψS⟩/∂t, and in the last line we’ve written our state back in terms of the interaction picture. Similarly, the expectation value of an operator AS in the Schr¨ odinger picture becomes ⟨ψS(t)\|AS\|ψS(t)⟩= ⟨ψI(t)\|U −1 0 (t)ASU0(t)\|ψI(t)⟩ (9.4) in terms of the interaction picture states, so we deﬁne the interaction picture operator AI(t) by AI(t) = U −1 0 (t)ASU0(t) (9.5) again using just the evolution operators of the unperturbed, rather than full, Hamiltonian. Inﬁnitesimally, this is d dtAI(t) = i ℏ[H0, AI(t)] + U −1 0 (t)∂AI(t) ∂t U0(t) (9.6) – 127 – where the ﬁnal term comes from the explicit time dependence present even in the Schr¨ odinger picture operator. In this sense, the interaction picture is ‘halfway between’ the Schr¨ odinger picture, in which states evolve and operators are time independent, and the Heisenberg pic-ture, in which states are always taken at their initial value \|ψ(0)⟩, but operators evolve according to the full Hamiltonian. In the interaction picture, states evolve by just the time-dependent perturbation as iℏ∂ ∂t\|ψI(t)⟩= ∆I(t)\|ψI(t)⟩ (9.7) where ∆I(t) = U −1 0 (t)∆(t)U0(t) is the perturbation in the interaction picture, whilst oper-ators also evolve, but only via the model Hamiltonian. It will turn out to be useful to introduce an interaction picture time evolution operator UI(t) deﬁned so that \|ψI(t)⟩= UI(t)\|ψI(0)⟩ for any initial state \|ψI(0)⟩∈H . (9.8) From (9.3) we ﬁnd that UI(t) satisﬁes the diﬀerential equation iℏ∂ ∂tUI(t) = ∆I(t) UI(t) . (9.9) Note that since the operator ∆I(t) itself depends on time, we cannot immediately integrate this to write UI(t) in terms of the exponential of ∆I(t): in particular it is generally not correct to claim UI(t) ? = exp −i ℏ Z t 0 ∆I(t′) dt′ , (9.10) because since the operator ∆I(t) itself now depends on time, in general [∆I(t), ∆I(t′)]′ ̸= 0 and in expanding the exponential as a power series, we’d have to commute the operators at diﬀerent times through one another. So far our considerations have been exact, but to make progress we must now approx-imate. To do so, it’ll be convenient to rewrite (9.9) as an integral equation76 UI(t) = 1 −i ℏ Z t 0 ∆I(t′) UI(t′) dt′ (9.11) in which the operator UI(t) we wish to understand also appears on the rhs inside the integral. We now replace the UI(t′) in the integral (9.11) by its value according to its own equation, obtaining UI(t) = 1 −i ℏ Z t 0 ∆I(t1) dt1 + −i ℏ 2 Z t 0 ∆I(t1) Z t1 0 ∆I(t2) UI(t2) dt2 dt1 . (9.12) The term containing UI(t) on the rhs now has two explicit powers of the perturbation ∆(t), so may be hoped to be of lower order in our perturbative treatment. Iterating this procedure gives UI(t) = ∞ X n=0 −i ℏ n Z t 0 dt1 · · · Z tn−1 0 dtn ∆I(t1) ∆I(t2) · · · ∆I(tn) (9.13) 76To obtain this equation, we’ve just integrated (??) using the initial condition UI(0) = 1. – 128 – in general. Note that the operators ∆(ti) at diﬀerent times do not necessarily commute — they appear in this expression in a time–ordered sequence, with the integrals taken over the range 0 ≤tn ≤tn−1 ≤· · · ≤t1 ≤t . Instead of the na¨ ıve expression (9.10), we often write the series (9.13) as UI(t) = T exp −i ℏ Z t 0 ∆I(t′) dt′ (9.14) for short, where the time–ordered exponential 77 T exp(· · · ) is deﬁned by the series (9.13). It’s a good exercise to check that this time–ordered exponential is equal to the usual exponential in the case that [∆(t1), ∆(t2)] = 0 for all t1 and t2. The interaction picture perturbation is ∆I(t) = U −1 0 (t)∆S(t) U0(t), so we can also write (9.13) as UI(t) = ∞ X n=0 −i ℏ n t Z 0 dt1 · · · tn−1 Z 0 dtn U −1 0 (t1)∆S(t1) U0(t1−t2) ∆S(t2) · · · U0(tn−1−tn) ∆S(tn) U0(tn) (9.15) using the fact that U −1 0 (ti)U0(ti−1) = U0(ti−1 −ti) . In this form we see that the perturba-tive expansion of UI(t) treats the eﬀect of the perturbation as a sequence of impulses: we evolve according to H0 for some time tn ≥0 then feel the eﬀect ∆S(tn) of the perturbation just at this time, before evolving according to H0 for a further time (tn−1 −tn) ≥0 and feeling the next impulse from the perturbation, and so on. Finally, we integrate over all the possible intermediate times at which the eﬀects of the perturbation were felt, and sum over how many times it acted. This method, closely related to Green’s functions in pdes, is also the basis of Feynman diagrams in QFT, where H0 is usually taken to be a free Hamiltonian. Now let’s put these ideas to work. Suppose that at t = 0 we’re in the state \|ψ(0)⟩= P n an(0)\|n⟩, described in terms of some superposition of the basis {\|n⟩} of eigenstates of the unperturbed Hamiltonian. Then a later time t the interaction picture state will be \|ψI(t)⟩= UI(t)\|ψ(0)⟩and can again be described by a superposition P n an(t)\|n⟩. Contracting with a state ⟨k\| gives ak(t) = ⟨k\| UI(t)\|ψI(0)⟩ (9.16) in the interaction picture. Thus, using (9.13) we have ak(t) ≈ak(0) −i ℏ X n an(0) Z t 0 ⟨k\| ∆I(t′) \|n⟩dt′ = ak(0) −i ℏ X n an(0) Z t 0 ⟨k\| U −1 0 (t′)∆S(t′) U0(t′) \|n⟩dt′ = ak(0) −i ℏ X n an(0) Z t 0 ei(Ek−En)t′/ℏ⟨k\|∆S(t′)\|n⟩dt′ , (9.17) 77Time–ordered exponentials were introduced by Freeman Dyson and play a prominent role in QFT. They’re also familiar in diﬀerential geometry in the context of computing the holonomy of a connection around a closed path. (This is not a coincidence, but it would take us too far aﬁeld to explain here.) – 129 – to ﬁrst non-trivial order in ∆, where we used the fact that H0\|n⟩= En\|n⟩. In particular, if at t = 0 we begin in an eigenstate \|m⟩of H0, so that an(0) = ( 1 if n = m 0 else , (9.18) then the amplitude for our perturbation to caused the system to make a transition into a diﬀerent H0 eigenstate \|k⟩̸= \|m⟩is ak(t) = −i ℏ Z t 0 ei(Ek−Em)t′/ℏ⟨k\|∆S(t′)\|m⟩dt′ (9.19) to ﬁrst order in the perturbation. Henceforth, we’ll drop the subscript on the Schr¨ odinger picture perturbation – note that this is the form in which the perturbation enters the original Hamiltonian. 9.2 Fermi’s Golden Rule To go further, we must specify how the perturbation ∆(t) actually depends on time. We’ll ﬁrst consider small perturbations of the form ∆(t) = ∆e−iωt + ∆† e+iωt (9.20) for some time–independent operator ∆. Thus our Schr¨ odinger picture perturbation ∆(t) oscillates with ﬁxed frequency ω. Note that ∆(t) should be Hermitian, as it’s a term in the full Hamiltonian, so we’ve included both an e−iωt and e+iωt term, and wlog we may take ω > 0. We’ll see later that these two terms are responsible for diﬀerent physics. We call such perturbations monochromatic in anticipation of the case that they describe an applied electromagnetic ﬁeld, corresponding to light of frequency ω. Performing the time integral in (9.19) in this monochromatic case is trivial and yields ak(t) = ⟨k\|∆\|m⟩ Ek −Em −ℏω h ei(Ek−Em−ℏω)t/ℏ−1 i + ⟨k\|∆†\|m⟩ Ek −Em + ℏω h ei(Ek−Em+ℏω)t/ℏ−1 i . (9.21) Each of these terms vanishes at t = 0 and then grow linearly in t for times small compared to either \|Ek −Em −ℏω\|−1 or \|Ek −Em + ℏω\|−1, respectively, where the oscillatory nature of the exponentials reveals itself. Now suppose the ﬁnal state \|k⟩into which we are making a transition has energy very close to Em+ℏω, so that in particular Ek > Em corresponding to the atom absorbing energy from the perturbation. In this case, the timescale \|Ek −Em − ℏω\|−1 is very long and the ﬁrst term on the rhs of (9.21) grows for a long time. On the other hand, if the ﬁnal state has Ek very close to Em −ℏω, corresponding to emission where the atom loses energy to the perturbation, then it is the second term which grows for a long time. In the case of absorption, the small denominator of the ﬁrst term means that this term will dominate at late times. Thus ak(t) ≈ ⟨k\|∆\|m⟩ Ek −Em −ℏω h ei(Ek−Em−ℏω)t/ℏ−1 i (9.22) – 130 – Figure 18: As t →∞, the function sin2(xt)/x2t →πδ(x). for times large compared to \|Ek −Em + ℏω\|−1, so that the probability that the system has made a transition to the kth eigenstate of H0 is \|ak(t)\|2 ≈ 4 \|⟨k\|∆\|m⟩\|2 (Ek −Em −ℏω)2 sin2 (Ek −Em −ℏω)t 2ℏ . (9.23) correct to order \|∆\|2. Now, for x ̸= 0 we have lim t→∞ sin2(xt) x2t ≤lim t→∞ 1 x2t = 0 , (9.24) whilst limx→0 sin2(xt)/x2t = t which diverges if we then send t →∞. The integral of this function is Z R sin2(xt) x2t dx = π , (9.25) independent of t, so we see that lim t→∞ sin2(xt) x2t = π δ(x) (9.26) (see ﬁgure 18). This shows that at late times, the transition rate is Γ(m →k) = lim t→∞ ∂ ∂t\|ak(t)\|2 ≈2π ℏ\|⟨k\|∆\|m⟩\|2 δ(Ek −Em −ℏω) (9.27a) in the case that the system absorbs energy from the perturbation. In the opposite case that the system loses energy back into the perturbation, we have Ek < Em and the corresponding rate of transition is Γ(m →k) ≈2π ℏ\|⟨m\|∆\|k⟩\|2 δ(Ek −Em + ℏω) . (9.27b) with an opposite sign in the δ-function. These results were essentially obtained by Dirac and proved to be so useful in describing atomic transitions that Fermi dubbed them ‘golden rules’. The name stuck and now (9.27a) & (9.27b) are known as Fermi’s golden rule. – 131 – The δ-function in (9.27a) of course says that the perturbation must supply the energy diﬀerence Ek −Em between the two energy levels of our atomic transition. This δ-function has the eﬀect that, in practice, a truly monochromatic perturbation cannot cause transi-tions between two discrete energy levels of a bound system such as an atom, because the frequency of the perturbation would need to be perfectly tuned to the energy diﬀerence78. There are thus two diﬀerent classes of circumstances in which we can use Fermi’s golden rule. On the one hand, we may be interested not in transitions to some speciﬁc ﬁnal state \|k⟩, but to any one of a range of eigenstates of H0. A typical example would be transitions between an initial bound state of an atom to any of the continuum of positive-energy (non-bound) states. In these circumstances, we let ρ(Ek) δEk denote the number of (ﬁnal) states with energy in the interval (Ek, Ek + δEk). The function ρ(Ek) — which needs to be calculated in each case — is known as the density of states; note that it must have dimensions of 1/(energy) in order for ρ(Ek) δEk to be a number of states. The total rate of transition from \|m⟩is then Z Γ(m →k) ρ(Ek) dEk = 2π ℏ Z \|⟨k\|∆\|m⟩\|2 δ(Ek −Em −ℏω) ρ(Ek) dEk = 2π ℏ\|⟨Em + ℏω\|∆\|m⟩\|2 ρ(Em + ℏω) . (9.28) We see that we always make transitions through an energy of exactly ℏω, with the density of states telling us how many states of energy Em + ℏω there actually are. The second set of circumstances is when our perturbation is not purely monochromatic, but contains a range of diﬀerent frequencies. In this case we can make transitions between two bound states, because there is always some perturbation of exactly the right frequency ω = Ek −Em/ℏeven though the rhs takes discrete values as k varies over diﬀerent bound states as the ﬁnal state. To handle this situation, we decompose the perturbation as a Fourier transform ∆(t) = Z R ∆(ω) e−iωt dt . (9.29) Everything goes through as above, but we now ﬁnd the probability of being in state \|k⟩at time t is \|ak(t)\|2 = i ℏ 2 (9.30) 9.2.1 Ionization by Monochromatic Light A common example of the use of Fermi’s golden rule is to calculate the rate at which an applied monochromatic electromagnetic ﬁeld can ionize an atom, say hydrogen, stripping oﬀthe electron from a bound state into the continuum of ionized states. We’ll assume that the electromagnetic ﬁeld itself may be treated classically – this assumption is valid e.g. in a laser, or at the focus of the antenna of a radio telescope where the ﬁeld is large. 78At times that are of order \|Ek −Em −ℏω\|−1, before our replacement of sin2(xt)/x2t by a δ-function was valid, no tuning was needed. However, in this case the probability Pk(t) does not continue to increase with time once the perturbation is ended, and there is no overall transition rate. – 132 – In the vacuum, the electromagnetic ﬁeld is divergence free and is entirely generated by Faraday’s Law ∇× E = −∂B/∂t. The whole electromagnetic ﬁeld can be described by a vector potential A via B = ∇× A and E = −1 c ∂A ∂t . (9.31) In the monochromatic case, Faraday’s equation is solved by A(x, t) = ϵ cos(k · x −ωt) , (9.32) where ϵ is a constant vector describing the polarization of the wave and k is the wavevector. From the vacuum Maxwell equation ∇· E = 0, we learn that k · ϵ = 0 , (9.33) saying that the wave is transverse. The coupling of an elementary particle of charge q to such a vector potential is given by the minimal coupling rule P →P −qA(X, t) (9.34) in the Hamiltonian79, so H = (P −qA)2 2µ + V (X) = H0 −q 2µ (P · A + A · P) + q2A2 2µ . (9.35) In our case of atomic transitions, an electron has charge q = −e so to ﬁrst–order in e the perturbing Hamiltonian is ∆(t) = e 2µ (P · ϵ cos(k · X −ωt) + cos(k · X −ωt) ϵ · P) . (9.36) The component ϵ · P of the momentum operator in the direction of the polarization of the electromagnetic wave commutes with cos(k · X −ωt) by the transversality condition (9.33) (recall that ϵ itself is a constant vector). Thus ∆(t) = e µϵ · P cos(k · X −ωt) = e 2µϵ · P ei(k·X−ωt) + e−i(k·X−ωt) , (9.37) simplifying the perturbation. Suppose that electromagnetic wavelength is much larger than the Bohr radius of the atom. This is a good approximation provided the atomic transition occurs between states that are separated in energy by much less than αµc2, as will be the case for waves with frequencies that are less than those of soft X-rays. In this approximation, k · X ≪1 for 79This is called minimal coupling because more complicated modiﬁcations are possible if the charged particle is not elementary, or in other more complicated circumstances. You’ll learn more about this in the Part II Classical Dynamics course, or if you take AQM next term. – 133 – all locations in the atom or molecule at which there is signiﬁcant probability of ﬁnding the electron. To lowest order, we can thus approximate ∆(t) ≈e 2µϵ · P e−iωt + e+iωt . (9.38) We’ve retained the full time–dependence, because large values of t cannot be ruled out in the way we excluded large values of the electron’s position. Finally, since the momentum operator occurs in H0 only in the kinetic term, we can write this perturbation as ∆(t) = ie 2ℏ[H0, ϵ · X] e−iωt + e+iωt (9.39) The purpose of rewriting the perturbation in this way is that we know how the unperturbed Hamiltonian H0 acts on the original eigenstates. Note that the perturbation is proportional to the dipole operator D = −e X whose matrix elements between the initial and ﬁnal states we will need to take. This result, known as the dipole approximation, arose from our approximation of the vector potential as constant over the scale of the atom. Since the electric ﬁeld E ∝ω ϵ we can interpret the transition as occuring due to the interaction between the applied electric ﬁeld and this dipole. We must now calculate the matrix element ⟨f\|∆\|i⟩= ie 2ℏ(Ef −Ei) ⟨f\|ϵ · X\|i⟩. Suppose that the hydrogen atom is initially in the ground state so that \|i⟩= \|100⟩, with electronic wavefunction ⟨x\|100⟩= 1 √ πa3 e−r/a , (9.40) where a is the Bohr radius. If the energy of the ejected electron is suﬃciently great, we can ignore its Coulomb attraction back to the ion and treat the ﬁnal electron as a free particle of momentum ℏk, so \|f⟩= \|k⟩with energy E = ℏ2k2/2µ and wavefunction ⟨x\|k⟩= 1 (2πℏ)3/2 eik·x (9.41) as in (4.35). Thus the matrix element we seek is determined by the integral ⟨k\|ϵ · X\|100⟩= 1 (2πℏ)3/2(πa3)1/2 Z R3 e−ik·x ϵ · x e−r/a d3x . (9.42) Performing this integral is rather messy80; when the dust settles one ﬁnds ⟨k\|ϵ · X\|100⟩= 8 √ 2 i π a7/2 ℏ3/2(1 + k2a2)3 ϵ · k . (9.43) 80If you insist: First note that for any function f(r) of the radius, Z e−ik·x f(r) d3x = 4π k Z ∞ 0 r f(r) sin kr dr with r = \|x\| and k = \|k\|. Diﬀerentiating this result along ϵ · ∂/∂k gives −i Z e−ik·x ϵ · x e−r/a d3x = 4π k3 ϵ · k Z ∞ 0 r e−r/a (−sin kr + kr cos kr) dr = 32πa5 (1 + k2a2)3 ϵ · k , where the ﬁnal radial integral is done by parts. – 134 – The fact that this result had to be proportional to ϵ·k is clear: the only direction in which the expectation value ⟨k\|X\|100⟩can point is that of k. We deﬁne the diﬀerential rate dΓ\|100⟩→\|k⟩by dΓ\|100⟩→\|k⟩= 2π ℏ\|⟨k\|∆\|100⟩\|2 δ(Ep −E100 −ℏω) p2dp dΩ (9.44) where dΩ= sin θ dθ dφ. For a free particle of mass µ we have p2dp = p2(dEp/dp)−1dEp = µp dEp, so dΓ\|100⟩→\|p⟩ dΩ = 2π ℏ\|⟨p\|∆\|100⟩\|2 δ(Ep −E100 −ℏω) µp dEp = 26µp3 π e2ϵ2 ℏ3 a7(Ep −E100)2 (1 + k2a2)6 cos2θ δ(Ep −E100 −ℏω) dEp , (9.45) where ϵ = \|ϵ\|. On the support of the δ-function we can replace Ep −E100 by ℏω, and the remaining eﬀect of the δ–function is to soak up the dEp integral. Since we’re assuming the energy of the ﬁnal state electron is so great that we can neglect it’s attraction back to the proton, we must have k2 ≫1/a2. Thus we simplify our diﬀerential rate (9.45) to dΓ\|100⟩→\|p⟩ dΩ ≈28 π µℏ2 k9a5 e2E2 cos2θ . (9.46) where E = ϵω/2 is the magnitude of the electric ﬁeld and ℏ2k2 = 2µEk is frozen to 2µ(ℏω + E100) ≈2µℏω. 9.2.2 Absorption and Stimulated Emission In our derivation of Fermi’s golden rule (9.28), we took the frequency ω of the perturbation to be ﬁxed and assumed a continuum of ﬁnal states. Now let’s consider the opposite case of transitions into a discrete set of ﬁnal states, such as the diﬀerent bound state energy levels of an atom, but where we have a continuous range of perturbing frequencies. This case is relevant e.g. when our atom is immersed not in a monochromatic source such as a laser, but in a thermal bath of radiation. We’ll assume that, while the actual value of the perturbation experienced by our system may be subject to rapid ﬂuctuations, the statistical properties of these ﬂuctuations remain constant in time so that ⟨k\|∆(t1)\|m⟩⟨m\|∆(t2)\|k⟩= fkm(t1 −t2) (9.47) where the bar indicates the average over ﬂuctuations. We can write the function fkm(t1−t2) as a Fourier transform fkm(t1 −t2) = Z Fkm(ω) e−iω(t1−t2) dω (9.48) where Fkm(ω) represents the average strength of the frequency-ω contribution of our per-turbation. From equation (9.19) for the case that an(0) = δnm (so that the system is initially in state \|m⟩), we have to ﬁrst order \|ak(t)\|2 = 1 ℏ2 Z t 0 Z t 0 ⟨k\|∆(t1)\|m⟩⟨m\|∆(t2)\|k⟩ei(Ek−Em)(t1−t2)/ℏdt1 dt2 . (9.49) – 135 – Averaging over the ﬂuctuations, equation (9.47) gives \|ak(t)\|2 = 1 ℏ2 Z Fkm(ω) Z t 0 Z t 0 ei(Ek−Em−ℏω)(t1−t2)/ℏdt1 dt2 dω = 1 ℏ2 Z Fkm(ω) Z t 0 ei(Ek−Em−ℏω)t1/ℏdt1 2 dω = 4 Z Fkm(ω) sin2((Ek −Em −ℏω)t/2ℏ) (Ek −Em −ℏω)2 dω . (9.50) Just as before, for large times we can replace sin2(xt)/x2 →tδ(x) in this integral. Thus we ﬁnd the rate of transitions Γ(\|m⟩→\|k⟩) = \|ak(t)\|2 t = 2π ℏ2 Fkm ((Ek −Em)/ℏ) (9.51) in the case that the states \|m⟩and \|k⟩are discrete but we have a continuous range of perturbing frequencies. The classic application of the above results is to an atom in a bath of radiation. As before, we’ll assume the radiation has wavelengths ≫than the typical atomic size, so that in the dipole approximation the Hamiltonian may be taken to be H = H0 + e X r E(t) · Xr , (9.52) where H0 is the Hamiltonian of the unperturbed atom, E(t) is the ﬂuctuating electric ﬁeld due to the radiation and Xr is the position operator for the rth electron in the atom. As in (9.47) take the ﬂuctuating electric ﬁeld to be correlated as Ei(t1) Ej(t2) = δij Z R P(ω) e−iω(t1−t2) dω . (9.53) The presence of δij on the rhs means that ﬂuctuations in diﬀerent directions are uncorre-lated, whilst ﬂuctuations that are aligned are correlated equally no matter their direction. This just says that there is no preferred direction for the electric ﬁeld and holds e.g. for a thermal bath of radiation. We can understand the meaning of the function P(ω) as follows. As you’ll learn if you’re taking the Part II Electrodynamics course, the energy density of an electromagnetic ﬁeld is ρ(x, t) = 1 8π E2(x, t) + B2(x, t) (9.54) and in a purely radiative (source–free) ﬁeld E2 = B2 so ρ = E2/4π. In our case, equa-tion (9.53) gives ρ = 1 4πE2(t) = 3 4π Z R P(ω) dω (9.55) so 3P(ω)/2π represents the average energy density in the radiation (at any time) due to frequencies in the range (\|ω\|, \|ω\| + d\|ω\|). – 136 – From (9.47) we have that perturbation itself is correlated as ⟨k\|∆(t1)\|m⟩⟨m\|∆(t2)\|k⟩= e2 X r ⟨k\|Xr\|m⟩ 2 Z R P(ω) e−iω(t1−t2) dω (9.56) where \|n⟩and \|m⟩are states of the unperturbed atom and the mod-square includes the Euclidean inner product (dot product) of the two ⟨k\|Xr\|m⟩matrix elements. Thus Fkm(ω) = e2 X r ⟨k\|Xr\|m⟩ 2 P(ω) (9.57) and the rate at which the radiation stimulates the atom to make a transition from state \|m⟩to a state \|k⟩ Γ(\|m⟩→\|k⟩) = 2πe2 ℏ2 X r ⟨k\|Xr\|m⟩ 2 P(ωkm) (9.58) where ωkm = (Ek −Em)/ℏ. Now, if the ﬂuctuating electric ﬁeld is real, then the power function must obey P(ω) = P(−ω) and P(ω) = P∗(ω). In particular, the rate Γ(\|m⟩→\|k⟩) in (9.58) is an even function of ωkm and consequently, for ﬁxed \|Ek −Em\|, this rate is the same whether Ek > Em or Ek < Em. For Ek > Em, the atom absorbs energy from the radiation, exciting to a higher energy level, whilst if Ek < Em the atom emits energy back into the radiation as it decays to a lower energy state. The important result we’ve obtained is that the rate for stimulated emission is the same as the rate of absorption. 9.2.3 Spontaneous Emission Our treatment above calculated the rate at which an atom would decay (or excite) due to being immersed in a bath of radiation. However, we currently cannot understand decays of an isolated atom: if the atom is prepared in any eigenstate of H0, quantum mechanics says that in the absence of a perturbation it will just stay there. Remarkably, Einstein was able to relate the rate of spontaneous decay of an atom to the rate of stimulated decay that we calculated above. Einstein deﬁned Am→k(ω) as the rate at which an atom would spontaneously decay from a state of energy Em to a state of lower energy Ek, where ω = (Em −Ek)/ℏis the energy of the emitted photon. If the atom is immersed in a bath of photons with ρ(ω) dω modes of frequency in the range (ω, ω +dω), we let ρ(ω) Bk→m(ω) denote the rate at which energy is absorbed from the radiation bath, exciting the atom from \|k⟩→\|m⟩. Similarly, let ρ(ω) Bm→k denote the rate of decay of the excited atom due to stimulation by the presence of the radiation bath. We calculated values for Bm→k and Bk→m above, but let’s pretend that (like Einstein) we do not know this yet. In thermodynamic equilibrium these rates must balance, so if there are nk atoms in state \|k⟩and nm in state \|m⟩we must have nm (Am→k(ω) + ρ(ω) Bm→k(ω)) = nk ρ(ω) Bk→m(ω) . (9.59) – 137 – We now borrow two results you’ll derive in the Part II Statistical Mechanics course, next term. First, when a gas of atoms is in equilibrium at temperature T, the relative numbers of atoms in states with energies Em and Ek is given by the Boltzmann distribution nk nm = e−Ek/kBT e−Em/kBT = eℏω/kBT , (9.60) where kB ≈1.38 × 10−23 JK−1 is Boltzmann’s constant. Furthermore, if radiation is in equilibrium at the same temperature then the density of states at frequency ω is ρ(ω) = ℏω3 π2c3 1 eℏω/kBT −1 , (9.61) which is essentially the Bose–Einstein distribution. Using these in (9.59) gives Am→k(ω) = ℏω3 π2c3 1 eℏω/kBT −1 eℏω/kBT Bk→m(ω) −Bm→k(ω) . (9.62) However, Am→k is supposed to be the rate of spontaneous emission of radiation from the atom, so it cannot depend on any properties of the radiation bath. In particular, it must be independent of the temperature T. This is possible iﬀ Bk→m(ω) = Bm→k(ω) (9.63) so that the rates of stimulated emission and absorption agree. In this case, we have Am→k(ω) = ℏω3 π2c3 Bm→k(ω) (9.64) so that knowing the rate Bk→m(ω) at which a classical electromagnetic wave of frequency ω = (Em−Ek)/ℏis absorbed by an atom, we can also calculate the rate Am→k(ω) at which the atom spontaneously decays, even in the absence of radiation. Now, in equation (9.58) we found from a ﬁrst–principles calculation that Γ(\|m⟩→\|k⟩) = 2πe2 ℏ2 X r ⟨k\|Xr\|m⟩ 2 P(ωkm) = (2πe)2 3ℏ2 X r ⟨k\|Xr\|m⟩ 2 ρ(ωkm) (9.65) where in the second line we’ve used the fact that P(ω) = 2πρ(ω)/3 as obtained in (9.55). Consequently, for our atom Einstein’s B coeﬃcient is Bk→m(ω) = Bm→k(ω) = (2πe)2 3ℏ2 X r ⟨k\|Xr\|m⟩ 2 . (9.66) for stimulated emission or absorption. From Einstein’s statistical argument we thus learn that Am→k(ω) = 4e2ω3 3ℏc3 X r ⟨k\|Xr\|m⟩ 2 (9.67) – 138 – is the rate at which an excited atom must spontaneously decay, even in the absence of any applied radiation. Ingenious though it is, there’s something puzzling about this calculation of the rate of spontaneous decay. Where does our quantum mechanical calculation – stating that isolated excited atoms do not decay – go wrong? The answer is that while we treated the energy levels of the atom fully quantum mechanically, the electromagnetic ﬁeld itself was treated classically. Spontaneous emission is only possible once one also quantizes the electromagnetic ﬁeld, since it is ﬂuctuations in the zero–point value of this ﬁeld that allow the atom to emit a photon and decay; heuristically, the ﬂuctuating value of the quantum electromagnetic ﬁeld – even in the vacuum – ‘tickles’ the excited atom prompting the decay. Treating the EM ﬁeld classically is appropriate if the radiation comes from a high–intensity laser, where the energy density of the ﬁeld is so high that the terms B(ω)ρ(ω) dominate in (9.59). By contrast, emission of light from the atoms in a humble candle is an inherently quantum phenomenon, occurring purely by spontaneous emission once the ﬂame is lit — candlelight shines even in ambient darkness. Einstein gave his statistical argument in 1916, when Bohr’s model of the atom (the ﬁrst thing you met in 1B QM) was still the deepest understanding physicists had of quantum theory. This was at the same time as he was revolutionising our understanding of gravity, space and time, which obviously wasn’t enough to keep him fully occupied. The quantum treatment of stimulated emission given in the previous section is due to Dirac in 1926, the same year as Schr¨ odinger ﬁrst published his famous equation. A ﬁrst–principles calculation of the spontaneous emission rate Am→k, requiring the quantization of the electromagnetic ﬁeld, was also given by Dirac just one year later. Things move fast. Dirac’s 1927 results agreed with those we’ve found via Einstein’s argument. The necessity also to quantize the electromagnetic ﬁeld heralded the arrival of Quantum Field Theory. Let’s use these results to calculate the typical lifetime of an excited state of an isolated atom. When the radiation density ρ(ω) is very small, the number of atoms nm in the excited state obeys ∂nm ∂t = −Am→k nm (9.68) so the atom decays exponentially with a characteristic timescale ∼A−1 m→k. For hydrogen or an alkali atom, typically the outermost electron changes its energy level in the transition, so we can drop the sum in (9.67) to ﬁnd Am→k(ω) = 4e2ω3 3ℏc3 \|⟨k\|X\|m⟩\|2 (9.69) where X is the position of the outermost electron. Unless ⟨k\|X\|m⟩vanishes for some reason (we’ll consider possible reasons momentarily) we’d expect this matrix element to be of the typical size a0 characteristic of the atom, so \|⟨k\|X\|m⟩\|2 ∼a2 0. Hence the characteristic lifetime of the excited atomic state is roughly τ = A−1 m→k ∼ 3ℏc3 4e2ω3a2 0 = 3c3µ 4ℏω3a0 (9.70) – 139 – where µ is the mass of the electron. Optical light has a frequency of order ω ∼1015 Hz, so the timescale τ is around 107 times longer than the timescale ℏ/E200 associated to the energy E200 of the ﬁrst excited state of hydrogen. Thus around 107 oscillations of the hydrogen atom occur before it radiates a photon, decaying down into the ground state. 9.2.4 Selection Rules In the previous sections we’ve found that, in the dipole approximation that the wavelength of any radiation is large compared to the scale of the atom, transition amplitudes in hydro-genic atoms are proportional to the matrix elements ⟨n′, ℓ′, m′\|X\|n, ℓ, m⟩. We now consider a set of conditions that are necessary if this matrix element is not to vanish. Firstly, in the dipole approximation the perturbing operator is independent of any details of the electron’s spin, so the initial and ﬁnal spin states must be the same. (We’ve been tacitly assuming this by suppressing any mention of these spins thus far.) Next, the parity operator Π acts as81 Π\|n, ℓ, m⟩= (−1)ℓ\|n, ℓ, m⟩ and Π−1XΠ = −X , (9.71) so since Π−1 = Π we have ⟨n′, ℓ′, m′\|X\|n, ℓ, m⟩= ⟨n′, ℓ′, m′\|Π Π−1XΠ Π−1\|n, ℓ, m⟩= (−1)ℓ′+ℓ+1⟨n′, ℓ′, m′\|X\|n, ℓ, m⟩. (9.72) Consequently, parity imposes ⟨n′, ℓ′, m′\|X\|n, ℓ, m⟩= 0 unless ℓ+ ℓ′ ∈odd (9.73) as we saw in section 4.6. This is known as a selection rule on allowed transitions. We can obtain further selection rules by considering angular momentum. In the second problem set you showed that [L2, [L2, X]] = 2ℏ2(L2X + XL2) (9.74) and hence that (β −β′)2 −2(β + β′) ⟨n′, ℓ′, m′\|X\|n, ℓ, m⟩= 0 (9.75) where β = ℓ(ℓ+ 1) and similarly for β′. Thus the matrix element ⟨n′, ℓ′, m′\|X\|n, ℓ, m⟩ vanishes unless (β −β′)2 = 2(β + β′), which implies \|ℓ−ℓ′\| = 1 or ℓ= ℓ′ = 0. The parity selection rule already rules out this latter case, so we have ⟨n′, ℓ′, m′\|X\|n, ℓ, m⟩= 0 unless \|ℓ−ℓ′\| = 1 , (9.76) strengthening our previous rule (9.73). A ﬁnal selection rule comes from noticing that, since X is a vector operator independent of spin, [Ji, Xj] = [Li, Xj] = iℏP k ϵijkXk. In particular, if we deﬁne X± = X1 ± iX2 (9.77) 81The electron has intrinsic parity ηe = +1, but our result (9.73) would hold even if it didn’t as the electron appears in both the initial and ﬁnal states. – 140 – then [J3, X3] = 0 and [L3, X±] = ±ℏX± . (9.78) We therefore ﬁnd 0 = ⟨n, ℓ′, m′\|[L3, X3]\|n, ℓ, m⟩= (m′ −m)ℏ⟨n′, ℓ′, m′\|X3\|n, ℓ, m⟩ (9.79a) and, since X±\|n, ℓ, m⟩is an eigenstate of L3 with eigenvalue (m ± 1)ℏ, 0 = (m′ −m ± 1)ℏ⟨n′, ℓ′, m′\|X±\|n, ℓ, m⟩. (9.79b) Combining these two we have the further selection rule ⟨n′, ℓ′, m′\|X\|n, ℓ, m⟩= 0 unless \|m −m′\| ≤1 , (9.80) as you also derived in the second problem sheet. More speciﬁcally, for transitions with m′ = m only X3 can contribute, so the matrix element ⟨n′, ℓ′, m′\|X\|n, ℓ, m⟩points in the ˆ z direction. On the other hand, transitions with \|m′−m\| = 1 cannot receive any contribution from X3, so the matrix element lies in the xy-plane and in fact ⟨n′, ℓ′, m + 1\|X\|n, ℓ, m⟩∝    1 −i 0    while ⟨n′, ℓ′, m + 1\|X\|n, ℓ, m⟩∝    1 +i 0   . (9.81) Finally, we remark that these selection rules were derived under the assumption that the transition is just due to the dipole approximation where the perturbation involves matrix elements of the dipole moment operator eX. The eﬀects of both higher orders in perturbation theory, and inclusion of interactions beyond those of an electric ﬁeld with wavelength λ ≫a0 mean that transitions that are ‘forbidden’ according to the above selection rules may in fact occur. They will typically do so, however, at a much smaller rate. – 141 – 10 Interpreting Quantum Mechanics In this ﬁnal chapter, I want to explore the process of measurement in quantum mechanics. 10.1 The Density Operator To get started, we must ﬁrst come clean about an unrealistic feature of our treatment so far. Up to this point, we’ve assumed that we know the precise quantum state our system is in. For macroscopic objects this assumption is totally implausible; we can’t possibly hope to discover the quantum state of the ∼1023 C atoms in a diamond, or the ∼105 atoms in a protein molecule. In fact, even in purely classical systems there’s always some uncertainty in our knowledge of the system. In reality, we cannot know the momentum of a single particle with inﬁnite precision as all our measurements will be subject to some experimental error. To handle this situation in quantum mechanics, we need a way to describe systems even when we’re not sure which state they’re in. Suppose we know only that our system is in one of the states {\|α⟩}, and that the probability it is in state \|α⟩is pα. It’s important to be clear that we’re not saying that system is in state \|φ⟩= X α √pα\|α⟩, (10.1) because \|φ⟩itself is a well–deﬁned quantum state. Rather, we’re admitting that we don’t know the true state of the system, which could be any of the states {\|α⟩}. Indeed, these states do not need to form a complete set, and do not even need to be orthogonal, although we will take them each to be correctly normalized ⟨α\|α⟩= 1. We deﬁne the density operator ρ : H →H by ρ = X α pα\|α⟩⟨α\| (10.2) where the pα are the probabilities introduced above. Thus ρ projects onto state \|α⟩with probability pα. The density operator has the following three properties: First, since prob-abilities are real ρ† = ρ, so the density operator is Hermitian. Second, trHρ = 1 (10.3) since the probabilities sum to one, and third ⟨χ\|ρ\|χ⟩≥0 for all \|χ⟩∈H (10.4) since probabilities are non-negative. We often write this property as ρ ≥0 for short. In fact, these three properties can be taken to be the deﬁning properties of a density operator, in the sense that any operator obeying these three properties is the density operator for some system To see this.... The deﬁnition (10.2) of the density operator is reminiscent of the deﬁnition Q = P j qj\|qj⟩⟨qj\| of an arbitrary operator in terms of its eigenstates. However, ρ should not be – 142 – considered as corresponding to an observable, because the pα are subjective rather than objective: they have nothing to do with the state of the system itself, rather they just quantity our knowledge or ignorance of that state. For example, if all our records of the results of measurements become lost, our values of the pα will change, but the physical system itself will not. By contrast, the spectrum {qj} of Q is determined by the Laws of Nature, independent of our knowledge of them. Note that, even though the states {\|α⟩} may not form a complete set or even be orthogonal, we’re always free to expand ρ in terms of some complete basis {\|i⟩}, as ρ = X i,j \|i⟩⟨i\| ρ \|j⟩⟨j\| = X i,j,α \|i⟩pα⟨i\|α⟩⟨α\|j⟩⟨j\| = X i,j ρij\|i⟩⟨j\| (10.5) where ρij = P α pα⟨i\|α⟩⟨α\|j⟩are the matrix elements of ρ in this basis. The density operator allows us to obtain the expectation value of any observable of our system, accounting for both the quantum and classical uncertainties. To see this, let Q be a Hermitian operator corresponding some observable, and let {\|qi⟩} be a orthonormal basis of Q-eigenstates82. Now consider the quantity Q deﬁned by Q = trH(ρ Q) . (10.6) Using the \|qi⟩basis to perform the trace, we have Q = X i ⟨qi\| ρ Q\|qi⟩= X α,i pα⟨qi\|α⟩⟨α\|Q\|qi⟩ = X α pα X i qi\|⟨qi\|α⟩\|2 ! = X α pα ⟨α\|Q\|α⟩. (10.7) This expression combines the quantum expectation value of Q in the state \|α⟩(which may not be an eigenstate of Q), together with our lack of knowledge of the system’s state, represented by the pαs. Rethnk through this argument In the Schr¨ odinger picture, the time evolution of ρ is given by iℏdρ dt = iℏ X n pn ∂\|n⟩ ∂t ⟨n\| + \|n⟩∂⟨n\| ∂t = X n pn (H\|n⟩⟨n\| −\|n⟩⟨n\|H) = [H, ρ ] . (10.8) This is the quantum analogue of Liouville’s equation dρ/dt = {H, ρ} in Classical Dynamics, which governs the time evolution of a probability density ρ on phase space. To obtain the time evolution of an arbitrary expectation value (of a quantity that has no explicitt time dependence in the Schr¨ odinger picture) we use (10.8) to ﬁnd iℏd dttrH(ρQ) = iℏ X a ⟨a\|dρ dt Q\|a⟩= X a ⟨a\|(Hρ −ρH)Q\|a⟩= trH(ρ [Q, H]) (10.9) 82Our discussion here is adapted to the case of dim(H) ﬁnite, but is can be extended to inﬁnite dimensional Hilbert spaces. – 143 – where the last equality uses the cyclicity of the trace. We know from before that the rate of change of the expectation value of Q in any pure quantum state is given by the expectation value of [Q, H]/iℏ. Equation (10.9) states that – even when our knowledge of the quantum state is imprecise – the expected rate of change of Q is the appropriately weighted average of the rates of change of Q for each of the possible states of the system. The density operator and the operators for the Hamiltonian and other observables encapsulate a complete, self–contained theory of quantum dynamics. If we have incomplete knowledge of the system’s quantum state, then use of this formalism is mandatory. If we do happen to know that our system is initially in the precise quantum state \|ψ⟩, we can still use this apparatus by setting ρ = \|ψ⟩⟨ψ\|, rather than using the TDSE, though in this case use of the density operator is optional. 10.1.1 Von Neumann Entropy When our knowledge of the state is incomplete, we say that the system is in an impure or mixed state, and correspondingly we refer to a regular quantum state \|ψ⟩as pure. This terminology is somewhat misleading, because it is really just our knowledge that is incomplete — the system itself is presumably in some perfectly well–deﬁned quantum state, it’s just that we don’t know which one. If we’re deﬁnitely in some state \|ψ⟩, so the system is pure, then ρ = \|ψ⟩⟨ψ\|. For such pure states it is easy to see that ρn = ρ using the fact that \|ψ⟩is normalized. If our density operator has pm = 0.99999999 for some state \|m⟩, with the remaining probability spread in some way among other states, then although our knowledge of the system’s state is imperfect, the eﬀects of the impurity are likely to be negligible. On the other hand, if the density operator has pn ≤10−20 for all states \|n⟩so that a tremendous number of states are equally probable, then our knowledge of the true quantum state of a system is very poor indeed. We would like to have a way to quantify how much knowledge, or information, about a state we have once the probability distribution {pi} has been speciﬁed. To achieve this, deﬁne the von Neumann entropy S(ρ) associated to a density operator by S(ρ) = −trH(ρ ln ρ) , (10.10) where we recall that ln ρ = P j ln(ρj)\|qj⟩⟨qj\| where . Thus we have −trH(ρ ln ρ) = − X n ⟨n\| X i pi\|i⟩⟨i\| !  X j ln(pj) \|j⟩⟨j\|  \|n⟩ = − X n pn ln(pn) (10.11) in terms of the probabilities appearing in the density operator. In this form, it’s easy to see that S(ρ) ≥0 with S(ρ) = 0 iﬀρ describes a pure state, where only one of the pns is non-zero (and hence equal to 1) as we have complete certainty about which state our system is in. Also easy to see is that the maximum value of S(ρ) is ln dim(H), which is attained iﬀ ρ = ρmax = 1H/dim(H). To see this, use the method of Lagrange multipliers to impose the – 144 – constraint trHρ = 1 and vary S(ρ) −λ (trH(ρ) −1) with respect to the probabilities and Lagrange multiplier λ. At an extremum, 0 = −trH δρ ln ρ + ρρ−1δρ + λδρ 0 = δλ (trH(ρ) −1) . (10.12) In the ﬁrst line, we’ve used the fact that tr(ρ ρ−1δρ) = tr(ρ δρ ρ−1) inside the trace, so the order of the variation in the logarithm doesn’t matter. These equations must hold for arbitrary variations δρ and δλ, so the ﬁrst tells us that, at an extremum of the von Neumann entropy, ρ = ρmax = e−λ−11H . (10.13) for some constant e−λ−1. Taking the trace, the second equation ﬁxes the constant of proportionality so that ρmax = 1 dim(H) 1H , (10.14) with the corresponding maximum entropy being S(ρmax) = −trH (ρmax ln ρmax) = −trH(1H) dim(H) ln(dim(H)−1) = ln dim(H) . (10.15) The form of ρmax shows the entropy is maximised when all states are equally likely – meaning we have no idea about which state our system is actually in. One of the main uses of the density operator and von Neumann entropy is in Quantum Statistical Mechanics. As an example, suppose we wish to extremize the entropy subject to both trH(ρ) = 1 and trH(ρH) = U, saying that we know our system has a ﬁxed average energy U. Then using two Lagrange multipliers λ and β, at an extremum we have 0 = δ [S(ρ) −λ (tr(ρ) −1) −β (tr(ρH) −U)] (10.16) which gives the three conditions 0 = −trH [δρ (ln ρ + 1 + βH + λ)] 0 = δλ (tr(ρ) −1) 0 = δβ (trH(ρH) −U) (10.17) Since these must hold for arbitrary variations, the ﬁrst equation gives ρ = e−βHe−λ−1 . (10.18) at a maximum of S(ρ) with ﬁxed energy. The second two conditions just enforce our constraints: to ensure trH(ρ) = 1 we must set eλ+1 = Z(β) where the constant Z(β) = trH(e−βH) . (10.19) – 145 – is known as the partition function of our system. Thus, in a state of maximum entropy for ﬁxed average energy, the density operator takes the form ρ = 1 Z(β)e−βH = 1 Z(β) X n e−βEn\|En⟩⟨En\| , (10.20) where in the ﬁnal expression we have inserted a complete set of H eigenstates. This form of density operator is known as the Gibbs distribution, It plays a fundamental role in quantum statistical mechanics, as you’ll see next term. β is usually denoted 1/ kBT where T is called the temperature and kB the Boltzmann constant. For ﬁxed average energy U of the system, the temperature is determined by the constraint trH(ρH) = U. In other words, the temperature T is determined by the average energy of the system. You’ll work much more with the density operator and entropy if you take the Part II Statistical Mechanics course next term. 10.1.2 Reduced density operators If our system comprises two (or more) identiﬁable subsystems A and B, then H = HA⊗HB so the full Hilbert space is the tensor product of the Hilbert spaces of the subsystems. Let’s suppose A describes the system we’re really interested in, whilst B is the ‘environment’. That is, B describes the quantum state of everything in the Universe except our immediate object of study A. Of course, we can’t hope to know the precise quantum state of B. We now show that, even if A starts in a pure state, once it has entangled itself with B, it will be in an impure state. Let {\|i⟩} denote a complete set of orthonormal states for A and {\|r⟩} a complete orthonormal set for B, so that {\|i⟩⊗\|r⟩} is a basis of HA ⊗HB. As in (10.5) we can expand the density operator for the entire system in terms of this basis as ρAB = X ij rs ρir,js \|i⟩⊗\|r⟩⟨j\|⊗⟨s\| . (10.21) Now consider an observable property purely of subsystem A, represented on the full Hilbert space by some Hermitian operator Q ⊗1B. The expectation value of Q is Q = trHA⊗HB(ρAB Q ⊗1B) = X k t ⟨k\|⊗⟨t\|  X ij,rs \|i⟩⊗\|r⟩ρir js ⟨j\|⊗⟨s\|  (Q ⊗1B) \|k⟩⊗\|t⟩ = X ij ⟨j\|Q\|i⟩ X r ρir,jr ! , (10.22) where in going to the last line we’ve used the orthonormality of the bases. Comparing this expression to (10.6) suggests we deﬁne a reduced density operator ρA of the subsystem A by ρA = trHB(ρAB) , (10.23) taking the trace only over subsystem B. Then ρA = X r ⟨r\|ρAB\|r⟩= X ij \|A; i⟩ X r ρir,jr ! ⟨A; j\| , (10.24) – 146 – where the second equality uses our expression (10.21) for ρAB. We can write our expres-sion (10.22) for Q in terms of the reduced density operator as Q = X i ⟨i\|ρAQ\|i⟩= trHA(ρAQ) . (10.25) That is, for any operator of the form Q ⊗1B we have trHA⊗HB(ρAB Q ⊗1B) = trHA(ρAQ) (10.26) in terms of the reduced density operator. The reduced density operator enables us to obtain expectation values of subsystem A’s observables without bothering about the states of B. For an example of a situation in which the reduced density operator is useful, consider an atom placed in a low-intensity ﬁeld of radiation. Every so often, a photon (quanta of electromagnetic radiation) may come along and scatter oﬀthe atom, or be absorbed by the atom into an excited state which subsequently decays, re-emitting the photon, or even cause the atom to be temporarily ionized. If we wish to keep track of the whole system, then as more and more photons interact with the atom, we need to use a larger and larger Hilbert space encompassing further and further tensor products of the Hilbert spaces of individual photons. However, if our interest is just in the state of the atom, it’s enough to keep track of the atom’s reduced density operator, which refers solely to the Hilbert space of the atom. 10.1.3 Decoherence We now show a very important result: Even if a quantum system A is initially pure, meaning we know exactly which state it’s in, interactions with an unmonitored environment B generically cause A to become impure. Suppose both A and B start in pure quantum states \|φ(0)⟩and \|χ(0)⟩, respectively. Initially then, ρAB(0) = \|ψ(0)⟩⟨ψ(0)\| = (\|φ(0)⟩⊗\|χ(0)⟩) (\|φ(0)⟩⊗\|χ(0)⟩) (10.27) and the whole system will evolve unitarily in time via the operator UAB(t) built from the Hamiltonian of the full system. If the full Hamiltonian does not couple A and B, so that H = HA ⊗I + I ⊗HB, then UAB(t) = e−iHt/ℏ= e−i(HA+HB)t/ℏ= e−iHAt/ℏ⊗e−iHBt/ℏ= UA(t) ⊗UB(t) , (10.28) using the fact that [HA, HB] = 0 since they act on diﬀerent Hilbert spaces. Thus, in this case \|ψ(t)⟩= UAB(t)\|ψ(0)⟩= UA(t)\|φ(0)⟩⊗UB(t)\|χ(0)⟩= \|φ(t)⟩⊗\|χ(t)⟩ (10.29) and the density operator at time t will be ρAB(t) = UAB(t)\|ψ(0)⟩⟨ψ(0)\|U −1 AB(t) = (\|φ(t)⟩⊗\|χ(t)⟩)(⟨φ(t)\| ⊗⟨χ(t)\|) . (10.30) – 147 – Consequently, tracing over HB we have that the reduced density operator ρA(t) = trHBρAB(t) = \|φ(t)⟩⟨φ(t)\| (10.31) at time t. In other words, if A starts in a pure state and it does not interact with the environment then it will remain in a pure state. In every realistic case, subsystems are coupled to eachother — however weak, there is some term HAB in the Hamiltonian that is not diagonal with respect to the splitting HA ⊗HB. In the presence of an interaction term HAB, the time evolution operator is not generically a product of the time evolution operators of the two subsystems, and states of A and B will typically become entangled. Let’s demonstrate this assertion in the simplest case that A and B each have only two possible states. Suppose that at time t they have evolved into the entangled state \|ψ(t)⟩= 1 √ 2 (\|A; 0⟩\|B; 0⟩+ \|A; 1⟩\|B; 1⟩) . (10.32) In this \|ψ(t)⟩, the states of A and B are correlated. Before making any measurements, there is a probability 1/2 that A is in either of its two states. However, if we discover (by performing some measurement) that B is in state \|B; 0⟩then A must certainly be in state \|A; 0⟩, whilst if we learn that B is in state \|B; 1⟩, A must be in state \|A; 1⟩. However, suppose we don’t keep track of the state of B. Then we should trace over the states in HB, giving ρA = 1 2 X a=1,2 ⟨B; a\| (\|A; 0⟩\|B; 0⟩+ \|A; 1⟩\|B; 1⟩)(⟨A; 0\|⟨B; 0\| + ⟨A; 1\|⟨B; 1\|) \|B; a⟩ = 1 2 (\|A; 0⟩⟨A; 0\| + \|A; 1⟩⟨A; 1\|) , (10.33) which is the density operator of an impure state. In tracing over HB we’ve lost the infor-mation carried by the correlations between the two subsystems, so A now we only know that A is equally likely to be in either state. Thus, even when the whole system is in a pure state, if we only measure part of it, the state can appear to be impure. As another example, system A could represent a hydrogen atom which we monitor closely, whilst system B could describe an electromagnetic ﬁeld to which the atom couples, but which we do not measure precisely. If our atom starts in its ﬁrst excited state and the electromagnetic ﬁeld in its ground state then initially, the atom, ﬁeld and whole system are all in pure states. After some time, coupling between the atom and ﬁeld means the whole system will evolve into a pure but entangled state \|ψ(t)⟩= a0(t)\|A; 0⟩\|F; 1⟩+ a1(t)\|A; 1⟩\|F; 0⟩ (10.34) where \|A; n⟩represents the nth excited state of the atom, while \|F; n⟩is the state of the ﬁeld when it contains n photons of frequency associated with transitions between the atom’s ground and ﬁrst excited state. If we fail to monitor the electromagnetic ﬁeld, then we must describe the atom by its reduced density operator ρA = \|a0(t)\|2\|A; 0⟩⟨A; 0\| + \|a1(t)\|2\|A; 1⟩⟨A; 1\| . (10.35) – 148 – Figure 19: Strong subadditivity of the von Neumann entropy. This density operator shows that the atom is now in an impure state. In general, interactions between an experimental system and the wider environment mean that the state of the whole Universe rapidly becomes entangled. Since we don’t keep track of all the details of the environment, sooner or later we’re obliged to describe our experimental system by its reduced density operator, which will be impure. The tendency for subsystems to evolve from pure quantum states to impure states through interactions with the environment is known as quantum decoherence. Trying to isolate a system from the environment so as to prevent it from becoming impure is one of the main challenges to be overcome in building a practical quantum computer. Give some proofs of these statements using monotonicity & subadditivity of entropy of a reduced density operator Perhaps the most important properties of entropy come from considering it’s behaviour on subsystems. If H = HA ⊗HB then S(ρAB) ≤S(ρA) + S(ρB) (10.36) where ρA and ρB are the reduced density matrices for the two subsystems. The equality is saturated if and only if the two subsystems are uncorrelated (unentangled) so that ρAB = ρA ⊗ρB. This property is known as subadditivity and you’ll explore it further in the ﬁnal problem sheet. It tells us that the entropy of a whole is no greater than the sum of entropies of its parts. It follows from subadditivity that if H = HA ⊗HB ⊗HC then S(ρABC) ≤S(ρA) + S(ρBC) ≤S(ρA) + S(ρB) + S(ρC) (10.37) but in fact something stronger is true: we have S(ρABC) + S(ρB) ≤S(ρAB) + S(ρBC) (10.38) which is known as strong subadditivity and was proved in 1973 by Elliott Lieb and Mary Beth Ruskai. To interpret it, we consider AB and BC are each subsystems of ABC, with AB ∩BC = B. Strong subadditivity states that the entropy of the whole is no greater than the entropies of the overlapping subsystems AB and BC, even when the entropy of the overlap B is removed. – 149 – 10.2 Quantum Mechanics or Hidden Variables? 10.2.1 The EPR Gedankenexperiment Einstein was never happy with the probabilistic nature of quantum mechanics. In the EPR thought experiment, an electron and a positron83 are produced in a state with net spin 0, perhaps by the decay of some nucleus from a spin-0 excited state to a lower spin-0 state. The electron and positron travel in opposite directions, each carrying the same amount of momentum. At some distance from the decaying nucleus Alice detects the electron and measures the component of its spin in a direction a of her choice. Since electrons have spin-1 2, Alice inevitably discovers either +ℏ/2 or −ℏ/2. Meanwhile Bob, who is sitting at a similar distance on the opposite side of the nucleus, detects the positron and measures its spin in some direction b of his choice. We are free to choose the z-axis to be aligned with Alice’s direction a. Since the electron–positron system has combined spin zero, it must be in the state \|ψ⟩= 1 √ 2 (\|↑⟩\|↓⟩−\|↓⟩\|↑⟩) (10.39) that entangles the separate spins of the electron and positron. According to the Copen-hagen interpretation, when Alice measures +ℏ/2 for the electron spin, the system collapses into the state \|ψ′⟩= \|↑⟩\|↓⟩. (10.40) Thus, whilst before Alice’s measurement the amplitude for the positron to have spin +ℏ/2 along a was 1/2, after she has measured the electron spin, there is no chance that the positron also has spin up along the same axis. The state of the positron corresponding to deﬁnitely having spin +ℏ/2 along the b-axis is \|↑b⟩= cos θ 2 e−iφ/2\|↑⟩+ sin θ 2 eiφ/2\|↓⟩ (10.41) as we found in the second problem sheet, where θ = cos−1(a · b). Given that after Al-ice’s measurement the positron is certainly in state \| ↓⟩, it follows from (10.41) that the probability Bob measures spin up along b is \|⟨↑b \|↓⟩\|2 = sin2 θ 2 . (10.42) In particular, there is only a small probability he would ﬁnd spin-up along a direction closely aligned with Alice’s choice a. We’ve supposed that Alice measures ﬁrst, but if the electron and positron are far apart when the measurements are made, a light signal sent to Bob by Alice when she makes her measurement would not have arrived at Bob by the time he makes his measurement, and vice versa. In these circumstances, relativity tells us that the order in which the measurements appear to be made depend on the velocity of the observer who is judging 83The positron is the antiparticle of the electron, predicted in the relativistic theory by Dirac’s equation. It has the same mass and spin as an electron, but opposite sign electric charge. – 150 – the matter. Consequently, for consistency the predictions of quantum mechanics must be independent of who is supposed to make the ﬁrst measurement and thus collapse the state. This condition is satisﬁed by the above discussion, since the ﬁnal probability depends only on a · b and is thus symmetric between Alice and Bob. What bothered EPR is that after Alice’s measurement there is a direction (a) along which Bob can never ﬁnd +ℏ/2 for the positron’s spin, and this direction depends on what exactly Alice chooses to measure. This fact seems to imply that the positron somehow ‘knows’ what Alice measured for the electron, and the collapse of the entangled wavefunc-tion 1 √ 2 (\|↑⟩\|↓⟩−\|↓⟩\|↑⟩) − → \|↑⟩\|↓⟩ apparently conﬁrms this suspicion. Since relativity forbids news of Alice’s work on the electron from inﬂuencing the positron at the time of Bob’s measurement, EPR argued that the required information must have travelled out in the form of a hidden variable which was correlated at the time of the nuclear decay with a matching hidden variable in the electron. These hidden variables would then explain the probabilistic nature of quantum mechanics — QM would contain no uncertainties once replaced by a ‘better’ theory taking into account these hidden variables. 10.2.2 Bell’s Inequality Remarkably, Bell was able to show that the predictions of any theory of hidden variables are in conﬂict with the predictions of quantum mechanics. Suppose we assume that the result of measuring the electron’s spin in the a-direction is completely determined by the values taken by hidden variables in addition to a. That is, if we knew these variables, we could predict with certainty the result of measuring the component of the electron’s spin along any direction a; Alice is only uncertain of the outcome because she does not yet know the values of the hidden variables. We take these variables to be the components of some n-dimensional vector v. Thus the result of measuring the electron’s spin along a is a function σe(a, v) that takes values ±ℏ/2 only. Similarly, the result of measuring the positron’s spin along b is some function σp(b, v). We have σe(a, v) + σp(a, v) = 0 (10.43) by conservation of angular momentum. Let’s suppose that v has a probability distribution ρ(v), such that the probability dP that v lies in the inﬁnitesimal volume dnv is dP = ρ(v) dnv . (10.44) Then the expectation value of interest is ⟨σe(a, v) σp(b, v)⟩= Z ρ(v) σe(a, v) σp(b, v) dnv = − Z ρ(v) σe(a, v) σe(b, v) dnv . (10.45) – 151 – Now suppose Bob sometimes measures the spin of the positron parallel to b′ rather than b. The fact that σp(b, v)2 = ℏ2 4 allows us to write ⟨σe(a, v) σp(b, v)⟩−⟨σe(a, v) σp(b′, v)⟩= − Z ρ(v) σe(a, v) [σp(b, v) −σp(b′, v)] dnv = − Z ρ(v) σe(a, v) σe(b, v) 1 −4 ℏ2 σe(b, v) σe(b′, v) dnv . (10.46) The expression [1 −4σe(b, v)σe(b′, v)] is non–negative, while the product σe(a, v) σe(b, v) ﬂuctuates between ± ℏ2 4 . Hence we obtain the bound ⟨σe(a, v) σp(b, v)⟩−⟨σe(a, v) σp(b′, v)⟩ ≤ℏ2 4 Z ρ(v) 1 −4 ℏ2 σe(b, v) σe(b′, v) dnv = ℏ2 4 −⟨σe(b, v) σp(b′, v)⟩ (10.47) This is Bell’s inequality. It must hold for any three unit vectors a, b and b′ if the proba-bilistic nature of QM really comes from some underlying hidden variables. Bell’s inequality can be tested experimentally as follows: for a large number of trials, Alice measures the electron’s spine along a while Bob measures the positonr’s spin along b in half these trials, and along b′ in the other half. From the results of these trials, the lhs of (10.47) can be estimated. The rhs is estimated from a new series of trials in which Alice measures the electron’s spin along b and Bob measures the positron’s spin along b′. If hidden variables exist, the results will obey Bell’s inequality. We now show that quantum mechanics itself violates Bell’s inequality. Since the ran-dom variables σe(a) and σp(b) can each take only the values ±ℏ/2 there are just four cases to consider. We ﬁnd ⟨σe(a)σp(b)⟩ = ℏ2 4 [PA(↑a)PB(↑b \| ↓a) −PA(↑a)PB(↓b \| ↓a) −PA(↓a)PB(↑b \| ↑a) + PA(↓a)PB(↓b \| ↑a)] , (10.48) where PB(↑b \| ↑a) is the probability Bob measures the positron’s spin to be +ℏ/2 along b, given that Alice it’s spin must be down along a as a result of the collapse of wavefunction after Alice’s measurement. Since nothing is known about the orientation of the electron before Alice makes her measurement, quantum mechanics gives PA(↑a) = PA(↓a) = 1 2 (10.49) Above, we calculated that PB(↑b \| ↑a) = sin2 θ 2 and PB(↓b \| ↑a) = cos2 θ 2 (10.50) – 152 – and similarly for PB(↓b \| ↓a) and PB(↑b \| ↓a). Thus we have ⟨σe(a) σp(b)⟩= ℏ2 4 sin2 θ 2 −cos2 θ 2 = −ℏ2 4 cos θ = −ℏ2 4 a · b (10.51) according to quantum mechanics. Using this correlation in either side of Bell’s inequality we ﬁnd LHS = ℏ2 4 \|a · (b −b′)\| whereas RHS = ℏ2 4 (1 −b · b′) (10.52) In particular, if a · b = 0 and b′ = b cos α + a sin α we ﬁnd LHS = ℏ2 4 \| sin α\| whereas RHS = ℏ2 4 (1 −cos α) (10.53) and it is easy to see that Bell’s inequality is violated for all α ̸= 0, π/2. The predictions of quantum mechanics are thus inconsistent with the existence of hidden variables. Experi-ments84 have been carried out to determine whether Nature herself obeys Bell’s inequalities or violates them — and quantum mechanics triumphs. 84Freedman, S. & Clauser, J., Experimental Test of Local Hidden Variable Theories, Phys. Rev. Lett. 28, 938 (1972). Freedman & Clauser’s experiment used a modiﬁed version of Bell’s inequalities using the polarization states of photons, rather than the spins of electrons. See also Aspect, A. & Roger, G. Experimental Realization of the Einstein–Podolski–Rosen–Bohm Gedankenexperiment: A New Violation of Bell’s Inequalities, Phys. Rev. Lett. 49, 91 (1982). – 153 –
4659	https://www.khanacademy.org/math/math3/x5549cc1686316ba5:rationals/x5549cc1686316ba5:common-factors/a/simplifying-rational-expressions-advanced	Simplifying rational expressions (advanced) (article) \| Khan Academy Skip to main content If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. 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Skip to lesson content Integrated math 3 Course: Integrated math 3>Unit 13 Lesson 1: Cancelling common factors Reducing rational expressions to lowest terms Intro to rational expressions Reducing rational expressions to lowest terms Simplifying rational expressions: common monomial factors Reduce rational expressions to lowest terms: Error analysis Simplifying rational expressions: common binomial factors Simplifying rational expressions: opposite common binomial factors Simplifying rational expressions (advanced) Reduce rational expressions to lowest terms Simplifying rational expressions: grouping Simplifying rational expressions: higher degree terms Simplifying rational expressions: two variables Simplify rational expressions (advanced) Math> Integrated math 3> Rational functions> Cancelling common factors © 2025 Khan Academy Terms of usePrivacy PolicyCookie NoticeAccessibility Statement Simplifying rational expressions (advanced) Google Classroom Microsoft Teams Have you learned the basics of rational expression simplification? Great! Now gain more experience with some trickier examples. What you should be familiar with before taking this lesson A rational expression is a ratio of two polynomials. A rational expression is considered simplified if the numerator and denominator have no factors in common. If this is new to you, we recommend that you check out our intro to simplifying rational expressions. What you will learn in this lesson In this lesson, you will practice simplifying more complicated rational expressions. Let's look at two examples, and then you can try some problems! Example 1: Simplifying 10 x 3 2 x 2−18 x‍ Step 1: Factor the numerator and denominator Here it is important to notice that while the numerator is a monomial, we can factor this as well. 10 x 3 2 x 2−18 x=2⋅5⋅x⋅x 2 2⋅x⋅(x−9)‍ Step 2: List restricted values From the factored form, we see that x≠0‍ and x≠9‍. Step 3: Cancel common factors 2⋅5⋅x⋅x 2 2⋅x⋅(x−9)=2⋅5⋅x⋅x 2 2⋅x⋅(x−9)=5 x 2 x−9‍ Step 4: Final answer We write the simplified form as follows: 5 x 2 x−9‍ for x≠0‍ Why do we require x≠0? Recall that the original expression requires x≠0,9‍. The simplified expression must have the same undefined values. Because of this, we must note that x≠0‍. We do not need to note that x≠9‍, since this is understood from the expression. Main takeaway In this example, we see that sometimes we will have to factor monomials in order to simplify a rational expression. Check your understanding 1) Simplify 6 x 2 12 x 4−9 x 3‍. Choose 1 answer: Choose 1 answer: (Choice A) 2 x 2−3 2 x‍ A 2 x 2−3 2 x‍ (Choice B) 2 x(4 x−3)‍ B 2 x(4 x−3)‍ (Choice C) 1 2 x 2−9 x 3‍ C 1 2 x 2−9 x 3‍ Check I need help! Step 1: Factor the numerator and denominator 6 x 2 12 x 4−9 x 3=3⋅2⋅x 2 3 x 3(4 x−3)‍ Step 2: List restricted values From the factored form, we see that x≠0‍ and x≠3 4‍. Step 3: Cancel common factors 3⋅2⋅x 2 3 x 3(4 x−3)=3⋅2⋅x 2 3⋅x⋅x 2(4 x−3)=3⋅2⋅x 2 3⋅x⋅x 2(4 x−3)=2 x(4 x−3)‍ Step 4: Final answer 2 x(4 x−3)‍ Notice that the original requires x≠0,3 4‍. The simplified expression has the same requirements. We don't need to restrict any additional values. Example 2: Simplifying (3−x)(x−1)(x−3)(x+1)‍ Step 1: Factor the numerator and denominator While it does not appear that there are any common factors, x−3‍ and 3−x‍ are related. In fact, we can factor −1‍ out of the numerator to reveal a common factor of x−3‍. =(3−x)(x−1)(x−3)(x+1)=−1(−3+x)(x−1)(x−3)(x+1)=−1(x−3)(x−1)(x−3)(x+1)Commutativity‍ Step 2: List restricted values From the factored form, we see that x≠3‍ and x≠−1‍. Step 3: Cancel common factors =−1(x−3)(x−1)(x−3)(x+1)=−1(x−3)(x−1)(x−3)(x+1)=−1(x−1)x+1=1−x x+1‍ The last step of multiplying the −1‍ into the numerator wasn't necessary, but it is common to do so. Step 4: Final answer We write the simplified form as follows: 1−x x+1‍ for x≠3‍ Why do we require x≠3? Recall that the original expression requires x≠3,−1‍. The simplified expression must have the same undefined values. Because of this, we must note that x≠3‍. We do not need to note that x≠−1‍, since this is understood from the expression. Main takeaway The factors x−3‍ and 3−x‍ are opposites since −1⋅(x−3)=3−x‍. In this example, we saw that these factors canceled, but that a factor of −1‍ was added. In other words, the factors x−3‍ and 3−x‍canceled to-1‍. In general opposite factors a−b‍ and b−a‍ will cancel to −1‍ provided that a≠b‍. Check your understanding 2) Simplify (x−2)(x−5)(2−x)(x+5)‍. Choose 1 answer: Choose 1 answer: (Choice A) 1‍ A 1‍ (Choice B) −1‍ B −1‍ (Choice C) x−5 x+5‍ for x≠2‍ C x−5 x+5‍ for x≠2‍ (Choice D) 5−x x+5‍ for x≠2‍ D 5−x x+5‍ for x≠2‍ Check I need help! Step 1: Factor the numerator and denominator The numerator and denominator are already in factored form. (x−2)(x−5)(2−x)(x+5)‍ Step 2: List restricted values From the factored form, we see that x≠2‍ and x≠−5‍. Step 3: Cancel opposite factors We can cancel (x−2)‍ and (2−x)‍ to a factor of −1‍. (x−2)(x−5)(2−x)(x+5)=−1(x−2)(x−5)(2−x)(x+5)=−1(x−5)x+5=5−x x+5‍ Step 4: Final answer We write the simplified form as follows: 5−x x+5‍ for x≠2‍ 3) Simplify 15−10 x 8 x 3−12 x 2‍. for x≠‍ Check I need help! Step 1: Factor the numerator and denominator 15−10 x 8 x 3−12 x 2=5(3−2 x)4 x 2(2 x−3)‍ Step 2: List restricted values From the factored form, we see that x≠0‍ and x≠3 2‍. Step 3: Cancel opposite factors We can cancel (3−2 x)‍ and (2 x−3)‍ to a factor of −1‍. 5(3−2 x)4 x 2(2 x−3)=5⋅(−1)(3−2 x)4 x 2(2 x−3)=−5 4 x 2‍ Step 4: Final answer We write the simplified form as follows: −5 4 x 2‍ for x≠3 2‍ Let's try some more problems 4) Simplify 3 x 15 x 2−6 x‍. Choose 1 answer: Choose 1 answer: (Choice A) 5 x−2‍ A 5 x−2‍ (Choice B) 1 5 x−2‍ for x≠0‍ B 1 5 x−2‍ for x≠0‍ (Choice C) 1 5 x 2−6‍ for x≠0‍ C 1 5 x 2−6‍ for x≠0‍ Check I need help! Step 1: Factor the numerator and denominator 3 x 15 x 2−6 x=3 x 3 x(5 x−2)‍ Step 2: List restricted values From the factored form, we see that x≠0‍ and x≠2 5‍. Step 3: Cancel common factors 3 x 3 x(5 x−2)=3 x 3 x(5 x−2)=1 5 x−2‍ Note that if the numerator cancels completely, a factor of 1‍ still remains. Step 4: Final answer We write the simplified form as follows: 1 5 x−2‍ for x≠0‍ 5) Simplify 3 x 3−15 x 2+12 x 3 x−3‍. for x≠‍ Check I need help! Step 1: Factor the numerator and denominator Notice that there is a common factor of 3 x‍ in the numerator. We can factor this out, then factor the resulting trinomial. 3 x 3−15 x 2+12 x 3 x−3=3 x(x 2−5 x+4)3(x−1)=3 x(x−4)(x−1)3(x−1)‍ Step 2: List restricted values From the factored form, we see that x≠1‍. Step 3: Cancel common factors 3⋅x(x−4)(x−1)3⋅(x−1)=3⋅x(x−4)(x−1)3⋅(x−1)=x(x−4)‍ Step 4: Final answer We write the simplified form as follows: x(x−4)‍ for x≠1‍ 6) Simplify 6 x 2−12 x 6 x−3 x 2‍. Choose 1 answer: Choose 1 answer: (Choice A) −2‍, ‍ for x≠0‍ and x≠2‍ A −2‍, ‍ for x≠0‍ and x≠2‍ (Choice B) 2‍, ‍ for x≠0‍ and x≠2‍ B 2‍, ‍ for x≠0‍ and x≠2‍ (Choice C) x−4 x‍, ‍for x≠0‍ C x−4 x‍, ‍for x≠0‍ (Choice D) x 2−4 x‍, ‍ for x≠2‍ D x 2−4 x‍, ‍ for x≠2‍ Check I need help! Step 1: Factor the numerator and denominator Notice that there is a common factor of 6 x‍ in the numerator and 3 x‍ in the denominator. 6 x 2−12 x 6 x−3 x 2=6 x(x−2)3 x(2−x)‍ Step 2: List restricted values From the factored form, we see that x≠0‍ and x≠2‍. Step 3: Cancel common factors and opposite factors Notice that we can write 6 x‍ as 2⋅3 x‍. Then we can cancel 3 x‍ from both the numerator and the denominator. 6 x(x−2)3 x(2−x)=2⋅3 x⋅(x−2)3 x⋅(2−x)=2⋅3 x⋅(x−2)3 x⋅(2−x)=2⋅(x−2)(2−x)‍ Next we can cancel the opposite factors (x−2)‍ and (2−x)‍ to −1‍. 6 x(x−2)3 x(2−x)=2⋅(x−2)(2−x)=2⋅(−1)(x−2)(2−x)=−2‍ Step 4: Final answer We write the simplified form as follows: −2‍ for x≠0,2‍ Skip to end of discussions Questions Tips & Thanks Want to join the conversation? Log in Sort by: Top Voted khare9770 8 years ago Posted 8 years ago. Direct link to khare9770's post “3x+1/x : express as ratio...” more 3x+1/x : express as rational expression Answer Button navigates to signup page •Comment Button navigates to signup page (6 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Fred Haynes 5 years ago Posted 5 years ago. Direct link to Fred Haynes's post “I just finished an the Pr...” more I just finished an the Practice Problems in the next exercise. To get to the part I don't understand I will simply ask this question which was the last section of the problem. The final answer was -(z+11)/(z-4) then it was multiplied out to z+11/4-z. Why was the answer -(z+11)/(z-4) not correct. Thanks Answer Button navigates to signup page •1 comment Comment on Fred Haynes's post “I just finished an the Pr...” (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Kim Seidel 5 years ago Posted 5 years ago. Direct link to Kim Seidel's post “Your answer is correct. ...” more Your answer is correct. For some reason, KA is setup so it doesn't except some small variations in correct answers. You may want to report it as a problem within the exercise so maybe they'll change it. 1 comment Comment on Kim Seidel's post “Your answer is correct. ...” (5 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Maria Nia Teodorova Tzoneva 2 years ago Posted 2 years ago. Direct link to Maria Nia Teodorova Tzoneva's post “8. Perform the indicated ...” more 8. Perform the indicated operation and simplify: (26x+5)–(−4x2–13x+5) I've tried everything but nothing is working Answer Button navigates to signup page •2 comments Comment on Maria Nia Teodorova Tzoneva's post “8. Perform the indicated ...” (2 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Kim Seidel 2 years ago Posted 2 years ago. Direct link to Kim Seidel's post “Start by distributing the...” more Start by distributing the "-" or "-1" that is in front of the 2nd set of parentheses. This changes all the signs on the 3 terms. 26x+5+4x^2+13x-5 Then combine like terms to get: 4x^2+39x Comment Button navigates to signup page (2 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Igor Skoldin 8 years ago Posted 8 years ago. Direct link to Igor Skoldin's post “What would happen if in t...” more What would happen if in the Example 2 we multiply a term in the denominator (instead of a term in the nominator) to get a common factor? (3-x)(x-1)/(x-3)(x+1) = (3-x)(x-1)/(-1)(3-x)(x+1) # multiply (x-3) by -1 and get the common factor of (3-x) = (x-1)/(1+x) # cancel out (3-x) and multiply (x+1) by -1 in the denominator So we get (x-1)/(1+x) for x ≠ 3 (and x ≠ -1, which is implied by the expression). But we should have gotten (1-x)/(1+x), which is not exactly the same. Actually, it is equivalent to (x-1)(-1)/(1+x). Did I make a mistake or why doesn't it work? Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer robertalkemade 9 years ago Posted 9 years ago. Direct link to robertalkemade's post “in example 4 3x/(15x^2...” more in example 4 3x/(15x^2-6x) shouldnt answer = 1/(5x-2) x<>0 AND X<>6/15 Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Batyr Seid 16 days ago Posted 16 days ago. Direct link to Batyr Seid's post “Why just we multiply only...” more Why just we multiply only the numerator not both numerator and denominator? Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer zaraenay 2 years ago Posted 2 years ago. Direct link to zaraenay's post “how do you write a ration...” more how do you write a rational expression? Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Cynthia Scott Koklas 8 years ago Posted 8 years ago. Direct link to Cynthia Scott Koklas's post “You need to correct the a...” more You need to correct the answer that is checkmarked on Example 2 when you click the "Check" link. I picked the answer 5-x/x+5 as the answer to the problem but when I clicked the "Check" box your system put a check mark next to x-5/x+5 which is the wrong answer. I verified this by going to your "I need help" link and the answer I picked is given as the right answer. #6 also checks the wrong answer when you click it. Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer wburnett 9 years ago Posted 9 years ago. Direct link to wburnett's post “In example 3, you didn't ...” more In example 3, you didn't include x=0 as an invalid value for x. Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Deanna 8 years ago Posted 8 years ago. Direct link to Deanna's post “how do you simplify 2-(5/...” more how do you simplify 2-(5/6)/(1/4) Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Kim Seidel 8 years ago Posted 8 years ago. Direct link to Kim Seidel's post “Follow PEMDAS rules. 1) ...” more Follow PEMDAS rules. 1) Divide the 2 fractions 2) Then subtract the result from 2 Give it a try. Comment back if you get stuck and tell me what you tried. 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4660	https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems/Problem_20?srsltid=AfmBOorjfb83Fxkz-gs21r97HKmXeD8-BnUsS1PUGY0eMxMgZ118V2pk	Art of Problem Solving 2025 AMC 8 Problems/Problem 20 - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki 2025 AMC 8 Problems/Problem 20 Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search 2025 AMC 8 Problems/Problem 20 Contents [hide] 1 Problem 20 2 Solution 1 3 Solution 2 (Estimation) 4 Solution 3 Speed 5 Solution 4 (Basically Video Solution) 6 Video Solution by Pi Academy 6.1 Video Solution 7 Video Solution 2 by SpreadTheMathLove 8 Video Solution (A Clever Explanation You’ll Get Instantly) 9 Video Solution by Thinking Feet 10 A Concise Video Solution in 56 Seconds by Dr. Xue's Math School 11 See Also Problem 20 Sarika, Dev, and Rajiv are sharing a large block of cheese. They take turns cutting off half of what remains and eating it: first Sarika eats half of the cheese, then Dev eats half of the remaining half, then Rajiv eats half of what remains, then back to Sarika, and so on. They stop when the cheese is too small to see. About what fraction of the original block of cheese does Sarika eat in total? Solution 1 Let the total amount of cheese be . We will track the amount of cheese Sarika eats throughout the process. First Round: Sarika eats half of the total cheese, so she eats: Second Round: Dev eats half of what remains after Sarika's turn, which is: Third Round: Rajiv eats half of the remaining cheese after Dev’s turn, which is: At the end of the first round, the total cheese eaten is: We observe that Sarika’s consumption follows a geometric sequence. In the first round, she eats , and in subsequent rounds, she eats half of what remains from the previous round. This gives the following series for Sarika’s total consumption: This is a geometric series with first term and common ratio . The sum of this infinite geometric series is given by the formula: where is the first term and is the common ratio. Substituting and : Thus, Sarika eats of the original block of cheese. The correct answer is: ~ aoum Solution 2 (Estimation) Sarika eats of the original cheese, and then because the others eat and , she eats next, and then , and then so on. Since the values later are going to be too small to make a huge difference, we can use these values. She ate . We can replace the with a for now, so , which simplifies to around . Since there is a little bit more of the cheese to be accounted for, the amount that she eats will be around . ~Sigmacuber Solution 3 Speed Her first bites will be and . Everything else won't matter, so the answer will be closest to , . ~Moonwatcher22 Solution 4 (Basically Video Solution) Let Sarika eat a total of units of cheese. We can then see that Dev will eat a total of units and Rajiv will eat units. Our desired answer is , or . ~Irfans123 Video Solution by Pi Academy Video Solution Key Idea: Let be the fraction eaten by Sarika. Then Dev eats and Rajiv eats . Hence so solving for we get . Video Link: Video Solution 2 by SpreadTheMathLove Video Solution (A Clever Explanation You’ll Get Instantly) ~hsnacademy Video Solution by Thinking Feet A Concise Video Solution in 56 Seconds by Dr. Xue's Math School Work smarter, work more efficiently, not just harder. See Also 2025 AMC 8 (Problems • Answer Key • Resources) Preceded by Problem 19Followed by Problem 21 1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25 All AJHSME/AMC 8 Problems and Solutions These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 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4661	https://hal.science/hal-01280786/document	HAL Id: hal-01280786 Submitted on 1 Mar 2016 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL , est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. The sum of digits function in finite fields Cécile Dartyge, András Sárközy To cite this version: Cécile Dartyge, András Sárközy. The sum of digits function in finite fields. Proceedings of the American Mathematical Society, 2013, 141 (12), pp.4119-4124. 10.1090/S0002-9939-2013-11801-0. hal-01280786 (7/5/2012, 16h15) The sum of digits function in finite fields C´ecile Dart yge (Nancy) and Andr´ as S´ ark¨ ozy (Budap est) ∗ Abstract. We define and study certain sum of digits function in the con text of finite fields. We giv e the num ber of polynomial values of Fq with a fixed sum of digits. We also state a result for the sum of digits of polynomial values with generator argumen ts. In tro duction Let g ∈ N be fixed with g > 2. If n ∈ N, then represen ting n in the num ber system to base g: (1·1) n = r−1 X j=0 cj gj , 0 6 cj 6 g − 1, cr > 1, we write (1·2) S(n) = r−1 X j=0 cj . Man y pap ers ha ve been written on the connection bet ween this sum of digits function S(n) and the arithmetic prop erties of n (for example , , , , , , , ). In particular, Mauduit and S´ ark¨ ozy studied the arithmetic structure of the in tegers whose sum of digits is fixed, while Mauduit and Riv at , obtained some asymptotic form ulae for the num ber of squares and also for the num ber of primes whose sum of digits is ev en resp ect. odd. In this pap er our goal is to study the analogs of some of these problems in finite fields. Indeed, let p be a prime num ber, q = pr with r > 2, and consider the field Fq . Let B = {a1, . . . , ar} be a basis of the linear vector space formed by Fq over Fp, i. e. , let a1, a2, . . . , ar be linearly indep endan t over Fp. Then ev ery x ∈ Fq has a unique represen tation (1·3) x = r X j=1 cj aj with cj ∈ Fp. Write (1·4) sB(x) = r X j=1 cj . An imp ortan t sp ecial case is when the basis B consists of the first r powers of a generator of F∗ q : B = {a1, a2, . . . , ar} = {1, z, z2, . . . , zr−1}. 2010 Mathematics Sub ject Classification: Primary 11A63; Secondary 11L99. Key words and phrases: sum of digits function, finite fields, character sums, generators, primitiv e elemen ts. ∗Researc hpartially supp orted bythe Hungarian National Foundation for Scien tific Researc h, Gran ts K72731 and K100291 and the Agence Nationale de la Rec herc he, gran tANR-10-BLAN 0103 MUNUM. 2 C´ecile Dartyge and Andr´as S´ark¨ozy Then (1 ·3) becomes (1·5) x = r X j=1 cj zj−1. (1 ·4) and (1 ·5) are of the same form as (1 ·1) and (1 ·2), th us we ma y consider (1 ·3) as the finite field analog of the represen tation (1 ·1), and we ma y call c1, . . . , cr in (1 ·3) as “digits”, and sB(x) can be called as “sum of digit ” function. If we consider the generators (or primitiv e elemen ts) as finite fields analogs of primitiv e ro ots of Fp, then we end up with the finite fields analogs of some problems men tioned ab ove : Ho w man y squares are there in Fq with a fixed sum of digits, and more generally , ho w man y values f (x) of a polynomial f ha ve a fixed sum of digits? Ho w man y generators of F∗ q whose sum of digits is a fixed value? In this pap er our goal is to study these problems. Let c ∈ Fp. We define Qc as the set of the squares of Fq suc h that their sum of digits is equal to c: Qc = n x ∈ Fq : sB(x) = c and ∃y ∈ Fq suc h that y2 = x o . We pro ve the follo wing result : Theorem 1.1. For all c ∈ Fp, we ha ve: Ø Ø Ø\|Qc\| − pr−1 2 Ø Ø Ø 6 √q. Let f ∈ Fq [X] be of degree n with (n, q) = 1. We are no w in terested in the cardinalit y of the sets: D(f, c) = {x ∈ Fq : sB(f (x)) = c}. While Theorem 1.1 can be pro ved elemen tarily , we will need Weil’s Theorem to estimate \|D(f, c)\|: Theorem 1.2. Let f ∈ Fq [X] be of degree n with (n, q) = 1. Then for all c ∈ Fp, we ha ve Ø Ø Ø\|D(f, c)\| − pr−1 Ø Ø Ø 6 (n − 1) √q. The main term of this estimate is larger than the one of Theorem 1.1 because here the values f (x) are tak en with multiplicities. We denote by G the set of the generators (or primitiv e elemen ts) of F∗ q and for c ∈ Fp we consider the sets G(f, c) = {g ∈ G : sB(f (g)) = c}. Theorem 1.3. Let f ∈ Fq [X] be of degree n with (n, q) = 1. Then for all c ∈ Fp we ha ve Ø Ø Ø\|G(f, c)\| − ϕ(q − 1) p Ø Ø Ø 6 (n − 1) τ (q − 1) √q where τ (n) denotes the divisor function. The sum of digits function in finite fields 3 The sum of digits of the squares In this section we pro ve Theorem 1.1. First we supp ose that c ∈ F∗ p . We use the quadratic character to detect the elemen ts of Qc: \|Qc\| = 1 2 X (c1,c 2,... ,c r)∈Fr p c1+c2+··· +cr=c h 1 + γ ≥ rX j=1 cj aj ¥i , where γ is the quadratic character. We replace cr by c − (c1 + · · · + cr−1): \|Qc\| = pr−1 2 + 1 2 X (c1,... ,c r−1)∈Fr−1 p γ ≥ ca r + r−1 X j=1 cj (aj − ar) ¥ . To mak e the cj indep enden t, it is con venien t to switc h to the additiv e characters via the Gaussian sums. We recall that if χ is a multiplicativ e character of F∗ q and ψ is an additiv e character of Fq then the Gaussian sum of χ and ψ is defined by (2·1) G(χ, ψ) = X x∈F∗ q χ(x)ψ(x), (see ). Then we can switc h to additiv e characters with the follo wing form ula for all x ∈ F∗ q : χ(x) = 1 q X ψ G(χ, ψ)ψ(x). Since c 6 = 0, ca r + Pr−1 j=1 cj (aj − ar) ∈ F∗ q for all c1, . . . , cr−1 ∈ Fp. Then we obtain for \|Qc\|: (2·2) \|Qc\| = pr−1 2 + 1 2q X ψ G(γ, ψ) X (c1,... ,c r−1)∈Fr−1 p ψ(ca r) r−1 Y j=1 ψ(cj (aj − ar)) . If ψ(aj ) 6 = ψ(ar), then ψ(aj −ar) is a p-th ro ot of the unit y ; there exists λj ∈ {1, . . . , p− 1} suc h that ψ(aj − ar) = e( λj /p ) with the standard notation e( t) = exp (2 iπt). In this case, X cj∈Fp ψ(cj (aj − ar)) = X cj∈Fp e ≥ cj λj p ¥ = 0. Th us, in the righ t hand side of (2 ·2), the summation over (c1, . . . , cr−1) is not 0 if and only if ψ(aj ) = ψ(ar) for all 1 6 j 6 r − 1. This means that ψ is a power of ψ1 where ψ1 is the additiv e character defined by ψ1(aj ) = e(1 /p ) for all 0 6 j 6 r − 1. Then we ha ve: \|Qc\| = pr−1 2 + 1 2p p−1 X j=0 G(γ, ψj 1 )ψj 1 (c). Next we use the classical fact ( Theorem 5.1) that \|G(χ, ψ)\| 6 √q if (χ, ψ) 6 = (χ0, ψ0) the couple of the trivial multiplicativ e resp ectiv ely additiv e character. We obtain Ø Ø Ø\|Qc\| − pr−1 2 Ø Ø Ø 6 √q 2 . If c = 0 then we ha ve to remo ve the term with c1 = c2 = . . . = cr−1 = 0 in (2 ·2). This giv es an extra error term √q/2 and we obtain Ø Ø Ø\|Q0\| − pr−1 2 Ø Ø Ø 6 √q.4 C´ecile Dartyge and Andr´as S´ark¨ozy The sum of digits of the polynomial values Let f ∈ Fq [X] of degree n suc h that (n, q) = 1. We are no w in terested in the cardinalit y of the sets D(f, c). The character ψ1 defined in the previous section is connected with the sum of digits function sB by the form ula ψ1(x) = e ≥ sB(x) p ¥ . Th us we ha ve: \|D(f, c)\| = 1 p p−1 X h=0 X x∈Fq ψh 1 (f (x))e ≥ −hc p ¥ . The main term is pro vided by h = 0: \|D(f, c)\| = q p + 1 p p−1 X h=1 E(h), with E(h) = X x∈Fq ψh 1 (f (x))e ≥ −hc p ¥ . We will use the follo wing theorem of Weil (, see also Theorem 5.38 p. 223) to obtain an upp er bound for the terms E(h): Theorem 3.1(W eil). Let g ∈ Fq [X] be of degree n > 1 with (n, q) = 1 and ψ a non trivial additiv e character of Fq . Then Ø Ø Ø X x∈Fq ψ(g(x)) Ø Ø Ø 6 (n − 1) √q. By this Theorem, we deduce that \|E(h)\| 6 (n − 1) √q for all 1 6 h 6 p − 1. Th us we obtain: Ø Ø Ø\|D(f, c)\| − q p Ø Ø Ø 6 (n − 1) √q. The sum of digits of polynomial values with primitiv e elemen t argumen ts Lik e the previous section, we consider a polynomial f ∈ Fq [X] of degree n with (n, q) = 1, but we study no w the sets G(f, c). By the same argumen t as in the previous section we ha ve (4·1) \|G(f, c)\| = 1 p p−1 X h=0 X g∈G ψh 1 (f (g))e ≥ −hc p ¥ . By using Weil’s Theorem 3.1 we will deduce the follo wing bound for additiv e character sums with primitiv e elemen t argumen ts. The sum of digits function in finite fields 5 Lemma 4.1. Let g ∈ Fq [X] be of degree n with (n, q) = 1. Let Ψ be a non trivial additiv e character of Fq . Then Ø Ø Ø X g∈G ψ(f (g)) Ø Ø Ø 6 (n − 1) τ (q − 1) √q + ϕ(q − 1) q − 1 . Pr oof. The pro of follo ws the argumen t of Lemma 2.3 of whic h giv es a similar upp er bound for sum with multiplicativ e characters over Fp. Let g0 be a primitiv e ro ot of F∗ q . Then we ha ve: X g∈G ψ(f (g)) = X 16k<q (k,q −1)=1 ψ(f (gk 0 )) . Then lik e in , we use the M¨ obius function to handle the coprimalit y condition and next we remark that gkd 0 is perio dic in k with perio d (q − 1) /d : (4·2) X g∈G ψ(f (g)) = X d\|q−1 μ(d) (q−1) /d X k=1 ψ(f (gkd 0 )) = X d\|q−1 μ(d) d X x∈F∗ q ψ(f (xd)) . When d\|q −1, the degree of f (Xd) is coprime with q and we can apply the Theorem 3.1. This ends the pro of of the Lemma 4.1, the ϕ(q − 1) /(q − 1) term is the con tribution of x = 0 excluded in (4 ·2). It remains to insert Lemma 4.1 in (4 ·1) and this giv es Ø Ø Ø\|G(f, c)\| − ϕ(q − 1) p Ø Ø Ø 6 (n − 1) τ (q − 1) √q + 1. This ends the pro of of Theorem 1.3. References W. D. Banks, A. Conflitti and I. E. Shparlinski, Character sums over in tegers with restricted g-ary digits, Illinois J. Math. (3) 46 (2002), 819-836. C. Dart yge and A. S´ ark¨ ozy , On additiv e decomp ositions of the set of primitiv e ro ots mo dulo p, Monatsh. Math. (to app ear). C. Dart yge and G. Tenen baum, Sommes des chi ff res de multiples d’en tiers, Ann. Inst. Fourier Grenoble/ 55 (2005), 2423-2474. C. Dart yge and G. Tenen baum, Congruences de sommes de chi ff res de valeurs polynomiales, Bull. London Math. So c. 38 (2006), no. 1, 61-69. M. Drmota, C. Mauduit and J. Riv at, The sum of digits function of polynomial sequences, J. London Math. So c. 84 (2011), 81-102. E. Fouvry and C. Mauduit, M´etho des de crible et fonctions sommes des chi ff res, Acta Arith. 77 (1996), 339-351. E. Fouvry and C. Mauduit, Sommes des chi ff res et nom bres presque premiers, Math. Ann. 305 (1996), 571-599. A. O. Gelfond, Sur les nom bres qui on t des propri ´et ´es additiv es et multiplicativ es donn ´ees, Acta Arith. 13 (1967/1968), 259-265. R. Lidl and H. Niederreiter, Finite Fields, Encyclop edia of Mathematics and its Applic ations , vol. 20, Addison-W esley Publishing Compan y, (1983). C. Mauduit and J. Riv at, La somme des chi ff res des carr ´es. Acta Math., vol. 203, 2009, pp. 107-148. C. Mauduit and J. Riv at, Sur un probl `eme de Gelfond : la somme des chi ff res des nom bres premiers. Annals of Math., vol. 171, n 3, 2010, 1591-1646. C. Mauduit and A. S´ ark¨ ozy , On the arithmetic structure of the in tegers whose sum of digit is fixed, Acta Arith. 81 (1997), 145-173. T. Stoll, The sum of digits of polynomial values in arithmetic progressions, Functiones et Appro ximatio, to app ear. A. Weil, Sur les courb es alg ´ebriques et les vari ´et ´es qui s’en d´eduisent , Publ. Inst. Math. Univ. Strasb ourg 7 (1945), Hermann, Paris, (1948). 6 C´ecile Dartyge and Andr´as S´ark¨ozy C´ecile Dart yge Institut ´Elie Cartan Univ ersit ´e de Lorraine,BP 239 54506 Vandœuvre Cedex, France dart yge@iecn.u-nancy .fr Andr´ as S´ ark¨ ozy Departmen t of Algebra and Num ber Theory E¨ otv¨ os Lor´ and Univ ersit y 1117 Budap est, P´ azm´ an y P´eter s´et´ an y 1/C Hungary sark ozy@cs.elte.h u
4662	https://www.studocu.com/en-us/messages/question/2875532/what-is-the-difference-between-unfused-and-fused-tetanus	[Solved] What is the difference between unfused and fused tetanus - Anatomy And Physiology (BIOL 2201) - Studocu Skip to main content Teachers University High School Discovery Sign in Welcome to Studocu Sign in to access study resources Sign in Register Guest user Add your university or school 0 followers 0 Uploads 0 upvotes Home My Library AI Notes Ask AI AI Quiz New Recent What is the difference between unfused and fused tetanus?Anatomy And Physiology (BIOL 2201) My Library Courses You don't have any courses yet. Add Courses Studylists You don't have any Studylists yet. Create a Studylist Community College of Rhode Island Anatomy And Physiology Question What is the difference between unfused and fused tetanus Community College of Rhode Island Anatomy And Physiology Question Anonymous Student 2 years ago What is the difference between unfused and fused tetanus? Like 0 Related documents Physio Ex Exercise 2 Activity 4 Anatomy And Physiology Coursework 93%(173) Physio Ex Exercise 2 Activity 3 Anatomy And Physiology Coursework 93%(193) Physio Ex Exercise 2 Activity 5 Anatomy And Physiology Coursework 98%(158) Physio Ex Exercise 6 Activity 1 Anatomy And Physiology Other 100%(9) PhysioEx 2: Skeletal Muscle Physiology Lab Report & Activity 1 Anatomy And Physiology Assignments 100%(5) PhysioEx Lab Report - Skeletal Muscle Tetanus Activity (Physio AP 2024)Anatomy And Physiology Assignments None Physio Ex Exercise 2 Activity 4 Anatomy And Physiology Coursework None Answer Created with AI 2 years ago Tetanus is a persistent muscular contraction that results when a skeletal muscle's motor neuron generates action potentials at an extremely fast rate. Tetanus may be fused or unfus... Like 0 This is a preview Go Premium and unlock full access to AI trained answers and study materials Optimised for study questions Access all study materials Get Unlimited Downloads Upload Share your documents to unlock Free Trial Get 7 days of free Premium Already Premium?Log in AI answers may contain errors. Please double check important information and use responsibly. Ask a new question Discover more from: Anatomy And Physiology BIOL 2201 Community College of Rhode Island 218 Documents Go to course 3 Physio Ex Exercise 1 Activity 2 Anatomy And Physiology 100%(46) 4 Physio Ex Exercise 2 Activity 5 Anatomy And Physiology 98%(158) 6 Physio Ex Exercise 2 Activity 6 Anatomy And Physiology 97%(135) 11 Physio Ex Exercise 2 Activity 2 Anatomy And Physiology 97%(258) Discover more from: Anatomy And Physiology BIOL 2201Community College of Rhode Island218 Documents Go to course 3 Physio Ex Exercise 1 Activity 2 Anatomy And Physiology 100%(46) 4 Physio Ex Exercise 2 Activity 5 Anatomy And Physiology 98%(158) 6 Physio Ex Exercise 2 Activity 6 Anatomy And Physiology 97%(135) 11 Physio Ex Exercise 2 Activity 2 Anatomy And Physiology 97%(258) 5 Physio Ex Exercise 2 Activity 7 Anatomy And Physiology 96%(158) 6 Physio Ex Exercise 2 Activity 1 Anatomy And Physiology 96%(119) Related Answered Questions 14 days ago Describe the intrinsic factors that control stroke volume. Anatomy And Physiology (BIOL 2201) 14 days ago Describe how the heart alters stroke volume. Anatomy And Physiology (BIOL 2201) 14 days ago Explain what happened to the pump rate when you increased the stroke volume? Why do you think this occurred? How well did the results compare with your prediction? Anatomy And Physiology (BIOL 2201) 17 days ago Describe how the sympathetic and parasympathetic nervous systems work together to regulate heart rate Anatomy And Physiology (BIOL 2201) 17 days ago Explain two ways that the heart can overcome excessive vagal stimulation Anatomy And Physiology (BIOL 2201) 17 days ago Explain the eﬀect that extreme vagus nerve stimulation had on the heart. Anatomy And Physiology (BIOL 2201) Ask AI Home My Library Discovery Discovery Universities High Schools High School Levels Teaching resources Lesson plan generator Test generator Live quiz generator Ask AI English (US) United States Company About us Studocu Premium Academic Integrity Jobs Blog Dutch Website Study Tools All Tools Ask AI AI Notes AI Quiz Generator Notes to Quiz Videos Notes to Audio Infographic Generator Contact & Help F.A.Q. Contact Newsroom Legal Terms Privacy policy Cookie Settings Cookie Statement Copyright & DSA English (US) United States Studocu is not affiliated to or endorsed by any school, college or university. Copyright © 2025 StudeerSnel B.V., Keizersgracht 424-sous, 1016 GC Amsterdam, KVK: 56829787, BTW: NL852321363B01 Cookies give you a personalised experience We’re not talking about the crunchy, tasty kind. These cookies help us keep our website safe, give you a better experience and show more relevant ads. We won’t turn them on unless you accept. Want to know more or adjust your preferences? Reject all Accept all cookies Manage cookies
4663	https://www.dadsworksheets.com/worksheets/combining-like-terms.html	Worksheets Age CalculatorCountdown TimerDegrees to RadiansFactoring CalculatorFraction CalculatorLong Division CalculatorMultiplication CalculatorPercentage CalculatorPrime Factorization CalculatorRoman Numeral ConverterSlope CalculatorAll Tools Graph PaperPlace Value ChartNumber LineMultiplication ChartMultiplication TableProperties of Multiplication ChartExponent Rules ChartPrime Numbers ChartSquare Root ChartCoordinate PlaneTrig IdentitiesMeasurement ChartProbability ChartPrintable Play MoneyFraction ChartRoman Numerals ChartHundreds ChartHandwriting PaperComic Strip TemplateBlank Clock TemplateAll Printables PuzzlesT-Shirts Combining Like Terms Worksheets This collection of algebra worksheets focuses on combining like terms. Expertly crafted, they provide ample practice to hone algebraic skills. Each PDF worksheet includes an answer key, facilitating easy self-assessment and effective learning. Perfect for students aiming to master the fundamental concept of combining like terms in algebra. Combining Like Terms (2 Terms) Combining Like Terms 2 Terms Addition Subtraction V1 Combining Like Terms 2 Terms Addition Subtraction V2 Combining Like Terms 2 Terms Addition Subtraction V3 Combining Like Terms 2 Terms Addition Subtraction V4 Combining Like Terms (3 Terms) Combining Like Terms 3 Terms Addition Subtraction V1 Combining Like Terms 3 Terms Addition Subtraction V2 Combining Like Terms 3 Terms Addition Subtraction V3 Combining Like Terms 3 Terms Addition Subtraction V4 Combining Like Terms 3 Terms Addition Subtraction Multiplication V1 Combining Like Terms 3 Terms Addition Subtraction Multiplication V2 Combining Like Terms 3 Terms Addition Subtraction Multiplication V3 Combining Like Terms 3 Terms Addition Subtraction Multiplication V4 Combining Like Terms (3 Terms) Combining Like Terms 4 Terms Addition Subtraction V1 Combining Like Terms 4 Terms Addition Subtraction V2 Combining Like Terms 4 Terms Addition Subtraction V3 Combining Like Terms 4 Terms Addition Subtraction V4 Combining Like Terms 4 Terms Addition Subtraction Multiplication V1 Combining Like Terms 4 Terms Addition Subtraction Multiplication V2 Combining Like Terms 4 Terms Addition Subtraction Multiplication V3 Combining Like Terms 4 Terms Addition Subtraction Multiplication V4 Combining Like Terms (5 Terms) Combining Like Terms 5 Terms Addition Subtraction V1 Combining Like Terms 5 Terms Addition Subtraction V2 Combining Like Terms 5 Terms Addition Subtraction V3 Combining Like Terms 5 Terms Addition Subtraction V4 Combining Like Terms 5 Terms Addition Subtraction Multiplication V1 Combining Like Terms 5 Terms Addition Subtraction Multiplication V2 Combining Like Terms 5 Terms Addition Subtraction Multiplication V3 Combining Like Terms 5 Terms Addition Subtraction Multiplication V4 Combining Like Terms (6 Terms) Combining Like Terms 6 Terms Addition Subtraction Multiplication V1 Combining Like Terms 6 Terms Addition Subtraction Multiplication V2 Combining Like Terms 6 Terms Addition Subtraction Multiplication V3 Combining Like Terms 6 Terms Addition Subtraction Multiplication V4 Parts of an Algebraic Expression Algebraic expressions are part of a math problem that has some unknown numbers hidden behind letters. This mathematical “phrase” is comprised of numbers, variables, and operations without an equal sign. Let us quickly know the parts of an algebraic expression before we learn how to combine like terms. Refer to the illustration below... Variables: These are unknown numbers represented by letters, typically x, y, or z. The variables in the given expression above would be x and y. Coefficients: These are the numbers that multiply the variables. See our example above. For 2x, the coefficient would be 2. Then for 3y, the coefficient would be 3. Constants: These are the numbers we can see and can be easily recognizable numbers such as, 3, -5, or even 0, but could also be more complicated such as fractions like ¼, or square roots such as √8. The constant in our example above is -10. Terms: A term could be a variable on its own, a constant on its own, or a combination of the two by multiplication or division. The terms above would be 2x, 3y, and -10. Operator: A symbol that tells you what to do (add, subtract, etc.). The operators on our sample illustration are + and -. Understanding Like Terms Before we dive into the mechanics of combining like terms, let's clarify what "like terms" are in algebra. Like terms are terms that have the same variables and corresponding exponents. In simpler terms, they are terms that can be combined or added together because they represent the same kind of quantity. For example, in the expression 4x+5x, these are like terms because they have the same variable: x. You could combine these terms and simplify the expression as 9x. How about unlike terms? These are terms that have different variables, such as 2x and 3y. Because they are different, you can not simplify them any further. If your expression was 2x+3y+5y, that could be simplified as 2x+8y, but that’s as far as you can go. How to Combine Like Terms? Algebra is often seen as a formidable foe by students of mathematics, but it's a critical foundation for more advanced mathematical concepts. One of the fundamental skills in algebra is combining like terms, a process that simplifies complex expressions, making them more manageable and easier to work with. In this guide, we will explore the concept of combining like terms and why it's a crucial skill for any algebra student to master. Addition and Subtraction of Like Terms Here's a step-by-step process of combining like terms involving adding or subtracting terms with the same variables and exponents. Step 1: Identify Like Terms. Begin by identifying the terms in the expression that have the same variables and exponents. 3x + 2y - 5x + 7y Like terms are 3x and -5x, 2y and 7y. Step 2: Add or Subtract: For like terms with addition or subtraction signs between them, simply add or subtract their coefficients. The variable part remains unchanged. Remember to take the sign in front. In the expression 3x + 2y - 5x + 7y, we can combine like terms as follows: 3x - 5x becomes -2x 2y + 7y becomes 9y Step 3: Reassemble: After combining like terms, reassemble the simplified expression. In our example, the simplified expression is -2x + 9y. AAddition, Subtraction, and Multiplication of Like Terms Next, you will learn how to use the distributive property and combine like terms in order to solve more complex equations. This time, it involves adding, subtracting, and multiplying terms that have the same variables and exponents. Here are the steps: Step 1: If you see a parenthesis, with more than one term inside, then distribute first. Take note of how I distribute first before applying the rules for solving equations. 2 (3x + 2y - 5x + 7y) style="padding-top:8px;"Step 2: Rewrite your equations with like terms together. Take the sign in front of each term. Multiply 2 one by one with all the terms inside the parenthesis. 2 (3x + 2y - 5x + 7y) 6x + 4y - 10x + 14y Step 3: Combine like terms. 2 (3x + 2y - 5x + 7y) 6x + 4y - 10x + 14y 6x - 10x becomes -4x 4y + 14y becomes 18y In our example, the simplified expression is -4x + 18y. Tips on Combining Like Terms Here are some tips and tricks to help you effectively combine like terms: Identify like terms: Like terms are terms that have the same variables raised to the same exponents. For example, in the expression, 3x + 2y - 5x - 7y, 3x and -5x are like terms because they both have 'x' as the variable with the same exponent, and 2y and -7y are like terms for the same reason. Group like terms: To combine like terms, group them together. This makes it easier to see which terms can be added or subtracted from each other. Add or subtract coefficients: Once you've grouped like terms together, simply add or subtract the coefficients (the numbers in front of the variables) to combine them. Be sure to pay attention to the signs (+ or -) when combining terms. Example 1: 3x - 5x becomes -2x Example 2: 2y - 7y becomes -5y Keep constants separate: Constants (terms without variables) are also like terms. Combine them separately. For example, in the expression 6x + 3 - 2x - 5, first combine the x-terms (6x and -2x) and then combine the constants (3 and -5). Combined x-terms: 6x - 2x becomes 4x Combined constants: 3 - 5 becomes -2 Watch for zero coefficients: If adding or subtracting like terms results in a coefficient of zero, you can omit that term from the expression since it doesn't affect the overall value. When adding and subtracting numbers it's important to be consistent with positive and negative values. Depending on the magnitude and sign of the values, as well as the operation, your result may be positive or negative. This is a great opportunity to use a number line to locate the first operand and then careful consider the direct the operation and the sign of the second operand will move you to get the final result. When numbers are multiplied or divided, two positive numbers will result in a positive sign. When two negative numbers are multiplied or divided, the result will have a positive sign. When there’s one number with a negative sign and the other with a positive sign, the result will have a negative sign. Use parentheses when necessary: Sometimes, you may need to use parentheses to clarify which terms should be combined first. For example, in the expression 3(2x + 4) - 2(3x - 1), distribute the constants inside the parentheses before combining like terms. 3(2x + 4) - 2(3x - 1) 6x + 12 - 6x + 2 Practice: Combining like terms becomes easier with practice. Work on various algebraic expressions to improve your skills. The Purpose of Combining Like Terms Why do we bother with combining like terms? The primary purpose is to simplify algebraic expressions. By combining like terms, we can reduce the complexity of an expression and make it more concise. This simplification makes the expression easier to understand and paves the way for solving equations and inequalities more efficiently. Like any skill, mastering the art of combining like terms requires practice. Worksheets, exercises, and real-world problems are excellent ways to hone this skill. Whether you're a student tackling algebra for the first time or someone returning to algebraic concepts, remember that combining like terms is a crucial step on your mathematical journey. It's a skill that not only opens doors to more advanced math but also enhances problem-solving abilities in everyday life. It is a fundamental skill that simplifies expressions, aids in solving equations, and has real-world applications. By understanding the concept and practicing regularly, students and learners of all levels can confidently navigate the world of algebra and mathematics. So, embrace the challenge, practice diligently, and master the art of combining like terms to unlock the full potential of algebraic problem-solving. 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4664	https://brilliant.org/wiki/functions-composition/	Function Composition Ram Mohith, Hemang Agarwal, Arron Kau, and Mahindra Jain Gene Keun Chung Calvin Lin Jimin Khim contributed Function composition refers to the pointwise application of one function to another, which produces a third function. When we compose the function f with g, we obtain f∘g. Sometimes, f∘g(x) is also denoted as f(g(x)). Contents Existence of Composite Codomain of g does not lie within Domain of f Iterated Function Composition Existence of Composite To ensure that this is well-defined, we must check that f∘g(x) makes sense. For example, if f(x)=x and g(x)=x−1, then f∘g(0) is not properly defined, because we cannot have −1. As such, the condition that we have to check is that the codomain of g(x) lies within the domain of f(x). In a later section, we will look at the case when this condition is not satisfied. For now, we will make the assumption that the range of g(x) is equal to the domain of f(x), which satisfies this condition. We consider the functions f:Y→Z and g:X→Y, which allows us to define function f∘g:X→Z. Let's look at the following example of function composition, where the functions f:R→R and g:R→R are given by f(x)=x2 and g(x)=x+1. Let's see how we can find certain values of f(g(x)) by plugging in x=1,2,3,4,5: This gives us a pictorial sense of what is happening. Algebraically, we evaluate that f∘g(x)=f(x+1)=(x+1)2=x2+2x+1. We can do the same for g∘f(x) and show that g∘f(x)=g(x2)=x2+1. Observe that f∘g=g∘f, which is a common mistake made. We say that function composition is not commutative. If f={(1,1),(2,3),(3,2),(4,4)} and g={(1,2),(2,1),(3,4),(4,3)}, what is f∘g? We need to find f∘g(x) for x=1,2,3,4: [\begin{align} f \circ g (1) &= f( 2) = 3 \ f \circ g (2) &= f(1) = 1 \ f \circ g (3) &= f( 4) = 4 \ f \circ g(4) &= g(3) = 2. \end{align} ] Thus, f∘g={(1,3),(2,1),(3,4),(4,2)}. □ Consider the functions f:R→R and g:R→R given by f(x)=x2+2x and g(x)=x+1. What is f∘g(x)? We have f∘g(x)=f(x+1)=(x+1)2+2(x+1)=x2+2x+1+2x+2=x2+4x+3. □ The function M(t) is the number of chairs that are produced each month if the factory operates t hours each day. The function P(c) is the monthly profit earned by selling c chairs each month. What is the monthly profit earned by letting the factory operate 8 hours each day? If the factory operates 8 hours each day, the amount of chairs produced is M(8). If M(8) chairs were sold, then the monthly profit is P(M(8)). □ Note: More generally, we know that the monthly profit of operating t hours each day is P(M(t)). Consider the functions f:R+→R+ and g:R+→R+ given by f(x)=x+1 and g(x)=x1. What is f∘g(x) and g∘f(x)? We have [\begin{align} f \circ g(x) &= f \left ( \frac{1}{x} \right) = \frac{1}{x} + 1 = \frac{ x+1} { x} \ g \circ f(x) &= g (x+1) = \frac{ 1}{ x+1}. \ _\square \end{align}] Bonus question: If f:R→R and g:R→R are defined by f(x)=2x2+3 and g(x)=3x−2, then find 1)2)3)f∘g(x)g∘f(x)f∘f(0). We have 1)f∘g(x)2)g∘f(x)3)f∘f(0)=f(g(x))=f(3x−2)=2(3x−2)2+3=2(9x2−12x+4)+3=18x2−24x+11=g(f(x))=g(2x2+3)=3(2x2+3)−2=6x2+7=f(f(0))=f(2(0)+3)=f(3)=2(3)2+3=21. □ Codomain of g does not lie within Domain of f As shown above with the example of g(x)=x−1 and f(x)=x, if the codomain of g does not lie within the domain of f, then the function f(g(x)) might not make sense. However, what we could do is to restrict the domain further such that the range of values of g lies within the domain of f. Simply put, we are getting rid of values which do not make sense: Since f(x) is only defined on the non-negative real numbers, the range of g(x) has to be the non-negative real numbers, or x−1≥0. This implies that x≥1, which is the restricted domain that will allow us to define f(g(x)). Hence, f∘g:[1,∞)→R,f∘g(x)=x−1. Consider the functions f:[0,1]→[1,2] given by f(x)=x+1, and g:R→R given by g(x)=x−1. Define f∘g on a suitable domain. The image of g(x) is all real numbers, which is not a subset of the domain of f(x). Hence, we have to restrict the domain of g(x). We require that 0≤g(x)≤1, or that 0≤x−1≤1, which implies 1≤x≤2. On this restricted domain, we have f∘g(x)=f(x−1)=x−1+1=x. Hence, f∘g:[1,2]→[1,2],f∘g(x)=x. □ Consider the functions f(x)=x1,x=0 and g(x)=x−1. What is f∘g? In this case, since we are not given the domain, we will assume that it is the real numbers. Also, we know that x1 is not defined for x=0, so the domain of f is x=0. We have to restrict the domain of g(x) such that g(x)=0, or that x−1=0. Hence, the domain to use is x=1. On this restricted domain, we have f∘g(x)=f(x−1)=x−11. Hence, f∘g(x)=x−11,x=1. □ Consider the functions f:[2,10]→[1,3] given by f(x)=x−1 and g:R→R given by g(x)=2x−3. What is f∘g? The image of g(x) is all real numbers, which is not a subset of the domain of f(x). Hence, we have to restrict the domain of g(x). We require 2≤g(x)≤10, or equivalently 2≤2x−3≤10, which implies 25≤x≤213. On this restricted domain, we have f∘g(x)=f(2x−3)=2x−3−1=2x−4. Therefore, it follows that f∘g(x)=2x−4, where 25≤x≤213. □ Consider the functions f(x)=x2−3x+21 and g(x)=x+1. What is f∘g? The given function can be rewritten as f(x)=x2−3x+21=(x−1)(x−2)1.(1) From (1), we know that x2−3x+21 is not defined for x=1 and x=2, so the domain of f is x=1,2. We have to restrict the domain of g(x) such that g(x)=1,2, or equivalently x+1=1,2. Hence, the domain to use is x=0,1. On this restricted domain, we have f∘g(x)=f(x+1)=(x+1)2−3(x+1)+21=x2−x1. Hence, f∘g(x)=x2−x1, where x=0,1. □ Consider the functions f(x)=x−41 and g:R→R given by g(x)=x2−12. What is f∘g? The value of x−4 is either positive or zero, and a denominator cannot be zero. Hence, the domain of x for f(x) is x−4≥0,x=4⟹x>4. So, we must restrict the domain of g(x) such that g(x)>4, or equivalently x2−12>4, which implies that the domain to use is x<−4 or x>4. On this restricted domain, we have f∘g(x)=f(x2−12)=x2−12−41=x2−161. Hence, f∘g(x)=x2−161, where x<−4 or x>4. □ Consider the functions f(x)=ln(x−4−1) and g(x)=x1. What is f∘g? The value of x−4−1 is positive and the value of x−4 is either positive or zero. Hence, it follows that the domain of x for f(x) is x−4−1>0 and x−4≥0⟹x>5. So, the range of g(x) is g(x)>5, or x1>5, which implies that the domain to use is x<51. On this restricted domain, we have f∘g(x)=f(x1)=ln(x1−4−1). Hence, f∘g(x)=ln(x1−4−1), where x<51. □ Consider f(x)=x2 and g(x)=2x. Solve the equation f∘g(x)=g∘f(x). We have f∘g(x)g∘f(x)=f(g(x))=f(2x)=(2x)2=22x=g(f(x))=g(x2)=2x2. As f∘g(x)22x2xx2−2xx(x−2)=g∘f(x)=2x2=x2=0=0, we have x=0,2. □ Bonus questions: 1) If f(x)=2x2+3 and g(x)=3x−2, then find g∘f∘f(3). 2) If f(x)=2,g(x)=x2, and h(x)=2x, then find h∘g∘f(x). We have 1) g∘f∘f(3)2) h∘g∘f(x)=g[f(f(3))]=g[f(18+3)]=g[f(21)]=g(2(21)2+2)=g(885)=3(885)−2=2653=h[g(f(x))]=h[g(2)]=h=2(4)=8. □ Iterated Function Composition If the range of a function is a subset of the domain of a function, then we can compose this function with itself. If so, we use f2(x) to denote f∘f(x). More generally, we say that fn(x) is f composed with itself n times, i.e. n timesf∘f∘⋯∘f. A function which satisfies f2(x)=x is called an involution. This means that the function is its own inverse. Consider the function f:R→R given by f(x)=x+1. What is fn(x)? We have [\begin{align} f(x) &= x + 1 \ f^2 (x) &= f( x + 1) = x + 1 + 1 = x + 2\ f^3 (x) &= f(x + 2) = x+2 + 1 = x + 3. \end{align} ] This strongly suggests that fn(x)=x+n, which we can prove using induction. □ Consider the function f(x)=x−1x+1 (x=±1). Then find f∘f∘f(x) and f∘f∘f∘f(x). Given f∘f(x)=x−1x+1, let us first find f∘f(x): f∘f=f(f(x))=f(x−1x+1)=x−1x+1−1x−1x+1+1=x−12x−12x=x. You can see that f(x) is an identity function as f∘f(x)=x: f∘f∘f(x)f∘f∘f∘f(x)=f(xf∘f(x))=f(x)=x−1x+1=f(f(xf∘f))=f(f(x))=f∘f(x)=x. □ In some questions where you have to find multiple composite functions, if you get f∘f(x) as Identity function (which means f∘f(x)=x), you are lucky enough and you need not solve the whole problem. If f∘f(x)=x, then fn(x)=n timesf∘f∘⋯∘f (x)={f(x)xwhen n is an odd numberwhen n is an even number. For example, take the above problem where we got that f∘f(x) is an Identity function. In f∘f∘f(x), f is repeated 3 times or it can be written as f3(x), where 3 is an odd number. So, f∘f∘f(x)=f(x)=x−1x+1. In f∘f∘f∘f(x)=f4(x), 4 is an even number, so f∘f∘f∘f(x)=x. Bonus question : Consider the previous problem. If f(x)=x+1x−1, then find f3(x) and f4(x). Given f(x)=x+1x−1, f∘f(x)=f(x+1x−1)=x+1x−1+1x+1x−1−1=x−1+x+1x−1−x−1=−x1. Here, f∘f(x)=x, so we cannot apply any shortcut and must go through fundamentals: f3(x)f4(x)=f(f(x))=f(−x1)=−x1+1−x1−1=−x−1x+1=f(f3(x))=f(−x−1x+1)=x−1−x−1+1x−1−x−1−1=−x−1+x−1−x−1−x+1=x. □ Consider the function g:R+→R+ given by g(x)=x1. What is gn(x)? We have [\begin{align} g(x) &= \frac{1}{x} \ g^2(x) &= g \left( \frac{1}{x} \right) = \frac{1}{\hspace{3mm} \frac{1}{x}\hspace{3mm} } = x \ g^3 (x) &= g(x) = \frac{ 1}{x}. \end{align} ] Thus, we have gn(x)={x1xn is odd n is even. □ Cite as: Function Composition. Brilliant.org. Retrieved 18:18, August 23, 2025, from
4665	https://www.nature.com/articles/s41467-024-54404-w	Skip to main content Download PDF Article Open access Published: Bias and negative values of COVID-19 vaccine effectiveness estimates from a test-negative design without controlling for prior SARS-CoV-2 infection Ryan E. Wiegand ORCID: orcid.org/0000-0002-9486-18501, Bruce Fireman2, Morgan Najdowski1, Mark W. Tenforde ORCID: orcid.org/0000-0002-8702-83933, Ruth Link-Gelles ORCID: orcid.org/0000-0002-9617-806X1 & … Jill M. Ferdinands3 Nature Communications volume 15, Article number: 10062 (2024) Cite this article 7503 Accesses 3 Citations 16 Altmetric Metrics details Abstract Test-negative designs (TNDs) are used to assess vaccine effectiveness (VE). Protection from infection-induced immunity may confound the association between case and vaccination status, but collecting reliable infection history can be challenging. If vaccinated individuals have less infection-induced protection than unvaccinated individuals, failure to account for infection history could underestimate VE, though the bias is not well understood. We simulated individual-level SARS-CoV-2 infection and COVID-19 vaccination histories and a TND. VE against symptomatic infection and VE against severe disease estimates unadjusted for infection history underestimated VE compared to estimates adjusted for infection history, and unadjusted estimates were more likely to be below 0%, which could lead to an incorrect interpretation that COVID-19 vaccines are harmful. TNDs assessing VE immediately following vaccine rollout introduced the largest bias and potential for negative VE against symptomatic infection. Despite the potential for bias, VE estimates from TNDs without prior infection information are useful because underestimation is rarely more than 8 percentage points. Similar content being viewed by others Impact of unequal testing on vaccine effectiveness estimates across two study designs: a simulation study Article Open access 24 May 2025 The need for a clinical case definition in test-negative design studies estimating vaccine effectiveness Article Open access 12 August 2023 Rapid evaluation of COVID-19 vaccine effectiveness against symptomatic infection with SARS-CoV-2 variants by analysis of genetic distance Article Open access 16 June 2022 Introduction Test-negative designs (TNDs) are an indispensable tool for assessing vaccine effectiveness (VE). TNDs were designed to assess VE against symptomatic infection of seasonal influenza1,2, but have been used to estimate VE against SARS-CoV-2 symptomatic infection3, emergency department or urgent care encounters4, hospitalizations5, invasive mechanical ventilation6, and death7 and to support policy decisions8. A TND can be performed rapidly, at lower cost than other studies, and with reduced confounding from health care seeking behavior compared to other observational study designs1,2. The efficiency and feasibility of a TND comes with many challenges9,10, especially regarding the assumptions of how cases and controls are ascertained; controls should be representative of the source population that yielded the cases11. Protection from infection-induced immunity can present challenges when estimating VE from a TND. Participants’ history of prior SARS-CoV-2 infection has often not been incorporated into VE studies11. For COVID-19 studies, infection history data is not collected due to self-testing, asymptomatic infection, and mild infections not requiring medical attention12. Bias can arise if prior infection status is misclassified13 or not accounted for in models14 and could result in a VE estimate below zero15. Serologic testing has been recommended to correct this bias14 but possesses many challenges, including decreasing sensitivity due to antibody decay16, potential inability to detect past infection in people with a current infection, increased cost, and decreased power17 since over 87% of the US population had detectable SARS-CoV-2 antibodies from infection in October–December of 2023182 (2023)."). Additionally, many people have had multiple prior SARS-CoV-2 infections, and serologic testing does not provide information on the number of total infections nor the time since or variant of the last infection, which are important for understanding the potential impact of past infection on VE. Considering these challenges, we endeavored to assess the bias in VE against symptomatic SARS-CoV-2 infection and severe disease from a TND when prior infection is unaccounted for in analyses. Microsimulations were created based on the COVID-19 pandemic, where each person’s vaccination and infection history was generated up to May 2023, followed by a hypothetical vaccination campaign and TND to estimate VE against symptomatic infection or severe disease. Multiple parameters relating to vaccine and infection protection waning and the TND study design were varied. Results Results from all simulated parameter sets and aggregated estimates are in the Supplementary Excel File. VE against symptomatic infection Per simulated population, the median protection against symptomatic infection at the end of the historical period ranged from 0.26 to 0.51 (where zero was no protection and one was complete protection) and, on aggregate, the distribution of median protection against symptomatic infection was lower when infection protection completely waned by 72 weeks compared to 96 weeks (Supplementary Table 1). The distribution of median protection against symptomatic infection by similar across the number of vaccinations (Supplementary Table 2) but increased with increasing number of infections (Supplementary Table 3). VE against symptomatic infection (Fig. 1, panels a, c, e) in unadjusted models was highest for people 1–2 months since vaccination (VE = 46.3%; CI: 45.6, 47.0) and decreased with more months since vaccination, reaching the lowest at 5–11 and 12 or more months (VE = –1.6%; CI: –1.9, –1.3). VE against symptomatic infection was also lower the more months included in the recent vaccination exposure, the longer time since vaccination, and the fewer number of total vaccination doses. Distributions of estimated VE against symptomatic infection tended to be wide and cover a wide range of VE values, except for exposures with VE against symptomatic infection estimates close to zero which had narrow, unimodal distributions (Supplementary Fig. 1). For each exposure definition, at least 94.6% of simulations possessed a bias of VE against symptomatic infection in unadjusted analyses of 8 percentage points (pp) or less (Fig. 2, panels a, c, e). All vaccination definitions excluding 5 vaccination doses and 3–4 months since vaccination had at least 95.4% of simulations with a bias of VE against symptomatic infection less than 6 pp. Sensitivity analyses utilizing only simulations when the TND happened during or after the vaccination rollout possessed similar results compared to the results from all simulations (Supplementary Fig. 2). Mean bias was at most 5.5 pp for any exposure definition (Supplementary Fig. 3). For the exposure of vaccination in the previous 3 months (Fig. 3), the overall mean bias was −1.4 pp (CI: −1.5,−1.3). Bias was higher when hybrid protection was defined as the greater source of protection boosted by 30% (Bias = −1.7 pp; CI: −1.8,−1.6) and lower when the greater of VP or boosted by 30% of IP or IP boosted by 10% of VP (Bias = −1.1 pp; CI: −1.2, −1.0). The timing of the vaccination rollout and TND also impacted bias. For people vaccinated in the previous 3 months, the largest bias occurred when the vaccination rollout happened immediately before the TND (vaccination rollout in weeks 1–12 and TND in weeks 11–22: Bias = –1.9 pp; CI: −2.1, −1.7; vaccination rollout in weeks 11–22 and TND in weeks 21−32: Bias = −2.2 pp; CI: −2.4, −2.0) and the smallest bias was when the vaccination rollout and TND took place in weeks 1−12 and 21−32, respectively (Bias = −0.9 pp; CI: −1.4, −0.5), though the confidence interval overlapped with multiple other timing combinations and all combinations were over 99% likely to be biased no more than 6 pp. In addition, the timing of the vaccination rollout in relation to the TND influenced VE against symptomatic infection estimates. For people vaccinated in the previous 3 months, VE against symptomatic infection from unadjusted models was nearly 20 pp lower when the vaccination rollout immediately preceded the TND (vaccination rollout in weeks 1−12 and TND in weeks 11−12: VE = 25.4%; CI: 24.6, 26.3; vaccination rollout in weeks 11−22 and TND in weeks 21−32: VE = 26.4%; CI: 25.6, 27.3) compared to when vaccination rollout and TND overlapped (weeks 1–12: VE = 46.0%; CI: 45.5, 47.0; weeks 11–22: VE = 45.1%; CI: 44.4, 45.7; weeks 21–32: VE = 46.2%; CI: 45.5, 47.0) (Fig. 3). Other exposure definitions also attributed the largest differences in bias to the hybrid protection definition and the timing of the vaccination rollout and TND (Supplementary Fig. 4–16), though the waning of vaccine-induced protection also impacted bias and the likelihood of bias being below 6 or 8 pp for multiple exposures (Supplementary Figs. 4, 5, 8–11, 15). The timing of the vaccination rollout and TND was the only factor which contributed to unadjusted VE against symptomatic infection being negative for people vaccinated in the previous 3 months (Fig. 3). Negative VE against symptomatic infection was most likely when the vaccination rollout happened after the TND, which is similar to performing a TND long after a vaccination campaign was completed (0.2% when vaccination rollout in weeks 11–22 and TND in weeks 1–12 and vaccination rollout in weeks 21–32 and TND in weeks 1–12) which was similar to exposures with a long time since vaccination, e.g., in people 12 or more months since vaccination which had at least 40% of VE estimates below zero (Supplementary Fig. 30). VE against severe disease The median protection against severe disease at the end of the historical period ranged from 0.87 to 0.97 and the distribution of median protection was higher when hybrid protection was boosted by 30% compared to when VP was boosted by 30% of IP or IP was boosted by 10% of VP (Supplementary Table 1). The distribution of median protection against severe disease increased with increasing number of vaccinations (Supplementary Table 2) and was lower for those without a prior infection compared to any number of prior infections (Supplementary Table S3). VE against severe disease (Fig. 1, panels b, d, f) in unadjusted models was highest for people 1−2 months since vaccination (VE = 91.1%; CI: 90.8, 91.3) and lowest for people 12 or more months since vaccination (VE = 42.2%; CI: 40.3, 44.1). For recent vaccination definitions, VE against severe disease had a small range from 87.4% (CI: 87.0, 87.7) for vaccination in the last 2 months to 85.9% (CI: 85.6, 86.2) for vaccination in the last 6 months. Unadjusted models were at least 92% likely to underestimate VE against severe disease by at most 8 pp across all vaccination exposures (Fig. 2, panels b, d, f). The likelihood of bias of VE against severe disease being at or below 6 pp was above 98.5% for all recent vaccination exposures, 2 and 3 vaccination doses, and 1–2, 3–4, and 12+ months since vaccination. Mean bias of unadjusted models for VE against severe disease was no greater than 5.1 pp (Supplementary Fig. 3). Overall bias for VE against severe disease for those vaccinated in the previous 3 months (Fig. 4) was −1.8 pp (CI: −2.0, −1.6). Bias for all parameter levels overlapped with those limits, except bias was less when a constant 4% case distribution was assumed (Bias = −2.0 pp; CI: −2.3, −1.7). Bias associated with all parameter levels was less than or equal to 6 pp in at least 99.3% of simulations. Other vaccination exposure definitions (Supplementary Fig. 17–29) also demonstrated differences in bias by variable levels, including by case definition (Supplementary Fig. 17, 19, 22, 25, 28, 29), hybrid protection (Supplementary Fig. 25–28), and vaccine-induced protection definition (Supplementary Fig. 17, 20, 21, 23–29). Discussion These microsimulations suggest that, when many people have experienced at least one prior infection, failure to adjust for infection-induced protection does not dramatically change VE estimates from a TND. On the aggregate, across an array of exposure definitions, VE against symptomatic infection and VE against severe disease were underestimated by less than 8 percentage points in over 99% of simulations for most exposure definitions. Biases of between 6 to 8 percentage points in TNDs have been considered minimal enough to use for vaccine policy making19,20,21,22, and, as has been argued previously, biases toward 0% should not restrict the utility of a VE estimate as a downward biased VE estimate may provide a lower bound13. Though, the aggregated results mask variability between parameter combinations. First, for simulation parameters, the bias of VE against symptomatic infection was impacted by the timing of the vaccination rollout and TND. The association between bias and timing varied by exposure but tended to be lowest when the vaccination rollout and TND were contemporaneous and largest when the vaccination rollout started three months prior to the TND. The increase in bias may be due to increased time since vaccination since vaccination was most likely early in the 12-week vaccination period, indicating vaccine-induced protection waned before the TND. Differential depletion of susceptibles14,23 may also be a factor since vaccination is assumed to offer limited protection against infection and higher protection against severe disease. Second, bias for exposure and parameter combinations was as low as −13.2 pp (CI: −18.1, −8.8). In total, 150 exposure and parameter combinations possessed a bias of less −8 pp out of 10,752 total combinations (1.4%). One hundred of those were from VE against symptomatic disease simulations where the vaccination rollout occurred before the TND period. The most common exposures with a bias of less than −8 pp were 4- or 5-dose VE in 40 parameter combinations. These results suggest recognizing the entire context and all parameters is important to understanding the potential bias. The timing of the vaccination rollout and TND also affected VE against symptomatic infection. VE against symptomatic infection for vaccination in the previous 3 months was ~46% with concurrent vaccination rollout and TND, 27% when rollout immediately preceded the TND, and 38% otherwise. These results suggest an impact for VE against COVID-19 symptomatic infection, potentially of 20 percentage points. Since VE against symptomatic infection wanes quickly, understanding the relative timing of the TND and vaccination rollout is critical for estimating VE for all exposures. VE against symptomatic infection less than zero (negative VE) was more likely for exposure groups with more months since the last vaccination dose or fewer vaccination doses. Waning of vaccination-induced protection is a potential contributor to negative VE estimates24,25. Vaccinated individuals further from their last vaccination dose or with fewer doses have vaccination-induced protection that has completely or near-completely waned, which is likely driving the negative VE estimates in these exposures. This is especially true for symptomatic infection since waning may mean vaccinated individuals can be at a similar or greater risk of a mild outcome with SARS-CoV-2 infection compared to unvaccinated individuals during the TND since unvaccinated individuals are more likely to have a prior SARS-CoV-2 infection compared to vaccinated individuals12, indicating that unvaccinated people are at greater likelihood of protection unaccounted for in unadjusted analyses compared to vaccinated people. As a comparison, VE against severe disease had no lower confidence limits below zero since VE against severe disease is greater than VE against symptomatic infection, and VE against severe disease wanes at a much slower rate than VE against symptomatic infection. Vaccine protection waning and existing infection-induced protection in unvaccinated participants suggest a higher outcome rate may be observed in vaccinated TND participants compared to unvaccinated TND participants, leading to a negative VE estimate. In addition, scenarios where a TND was performed three months after the vaccination rollout had the greatest likelihood of negative VE, further supporting vaccine-induced protection waning as a contributor to negative VE estimates. Bias also can contribute to negative VE15, and we found a positive VE in adjusted analyses of < 6% could be underestimated in unadjusted analyses enough to bias an estimate below zero. Finally, random variation may also play a role and some exposures from individual parameter sets with a VE against symptomatic infection point estimate above 40% had a lower confidence intervals below zero. Therefore, exposure categories further out from the last vaccination possessed a high enough VE estimate to avoid the underestimation from unadjusted models resulting in a negative VE. Our finding that VE estimates unadjusted for prior infection remain reliable and thus can be used to inform policy is especially important as prior infection is challenging to accurately measure. For example, adult VE studies from the US during SARS-CoV-2 Omicron variant circulation found ~15% of included patients with prior documented or self-reported laboratory-confirmed SARS-CoV-2 infection during a period when the vast majority of adults in the US had serological evidence of past infection26,27. A number of factors are likely to contribute to this, including asymptomatic or paucisymptomatic infection28 that does not prompt testing, a lack of clinical testing despite symptomatic illness, receiving a prior positive test for SARS-CoV-2 in settings not captured in the surveillance network such as a different healthcare system and at-home testing29,30, and imperfect accuracy of SARS-CoV-2 diagnostic assays. In addition, while a binary indication of prior infection may be available via serology in some study platforms, infection-induced protection is likely related to the number of prior infections, variant of prior infection(s), and time since prior infection, none of which are indicated via serology or fully captured by electronic health records or self-reporting. The results also suggest that, when measuring VE for recent vaccination exposures, VE against severe disease is more stable than VE against symptomatic infection due to the slower waning of protection against severe disease. For all evaluated durations of the recent vaccination exposure, unadjusted VE against severe disease ranged from 83.9% to 85.9% whereas VE against symptomatic infection ranged from 21.2% to 46.2% indicating that the choice of recent exposure definition had less impact on VE against severe disease compared to VE against symptomatic infection. The lack of a clear function of how infection-induced and vaccine-induced protection combine to become hybrid protection was one of the multiple limitations of these simulations. We utilized published meta-analyses that attempted to characterize the waning effectiveness of vaccines31 and hybrid protection32,33, but we required additional assumptions for our simulations. There is rich information on antibody titer trajectories34,35 but challenges remain for determining the relationship between neutralization titers and protection36. We tried to create realistic simulations that were also succinct and understandable. As a result, we did not incorporate other known sources of bias, such as errors in vaccine registry linkage37 or correlation between COVID-19 and influenza vaccination20. Another major consequence of creating realistic simulations was the true VE was dependent on the population in each simulation. Therefore, bias in these simulations was not based on a true, underlying parameter. We also did not vary the population size, which may affect the uncertainty of bias estimates. In addition, although we found differences in the bias associated with vaccination doses, likely this was attributable to the timing of the last vaccination dose. Fewer vaccination doses were typically associated with a longer duration since the last vaccination dose. Therefore, in this simulation, people with fewer doses possessed less vaccine-induced protection and were more likely to have overall protection levels similar to unvaccinated people. TNDs have been recommended as the most efficient and feasible method for assessing VE38. The effectiveness of vaccinations delivered is based not only on the vaccine formulations and the circulating pathogens but also on the characteristics of the population, including people’s underlying immunity from past infections. Prior SARS-CoV-2 infections, including the number, variant, and timing of past infections, cannot be ascertained with certainty and are more common in unvaccinated compared to vaccinated individuals12. Although VE estimates unadjusted for prior infection are lower than adjusted estimates, the difference was in line with accepted underestimation of VE. Extra care should be taken when performing analyses by number of total vaccine doses as more recent doses have the potential for greater bias when not controlled for past infection and doses further in the past have greater potential to result in negative VE estimates. Ideally, researchers could adjust VE estimates from a TND for prior infection history if data are available, but unadjusted VE estimates from a TND remain useful. Methods Simulation methods A thorough summary of the simulation methods and a full list of microsimulation parameter sets and results are included in the supplementary materials. We created populations of 100,000 people aged 18–49 years without protection against SARS-CoV-2 infection at the beginning of the COVID-19 pandemic (the week of January 19, 2020)39."). Each week until the week of May 7, 2023, we updated each person’s vaccine- and infection-induced protection against SARS-CoV-2 infection based on their most recent infection and vaccine dose since people could accumulate multiple infections and doses over time (Fig. 5). Weekly infection probabilities were derived from aggregated case count data from 60 U.S. jurisdictions39.") divided by 2020 population estimates40. (2021).") (Supplementary Fig. 30). These proportions were increased by a multiplier (Supplementary Fig. 31) to account for underreporting of infections41.") and to reach ~95%–98% of the population acquiring a prior infection by the end of the study period (Supplementary Fig. 32). Individual, weekly probabilities of vaccination receipt utilized U.S. vaccination data, the number of prior doses and prior infection status. Data on vaccination distributions by vaccination dosage and week for U.S. people aged 18–49 years42.") (Supplementary Fig. 33) were fit to probability distributions (Supplementary Fig. 34). For naïve people, we set the probability of obtaining two vaccination doses at 0.70, the probability of a third dose conditional on having two doses at 0.30, and the probability of a fourth dose conditional on having a third dose at 0.1042."). People with a prior SARS-CoV-2 infection have been less likely to initiate or subsequently receive an additional vaccination dose43."),44."),45."),46 9, 119 (2021)."),47."),48."). We assumed people with a prior infection were less likely to receive an additional dose with an odds ratio of 0.52543."),44."),45."),46 9, 119 (2021)."),47."),48."). A person’s protection was based on the most recent week of vaccination and infection, based on product-limit survival curve estimates from the third dose efficacy trial of the BNT162b2 vaccine (Pfizer–BioNTech), where the placebo and vaccine curves started to diverge after approximately one week49, and usage of a one-week lag in recent VE studies5,50,51,52,53. Waning curves were based on trajectories in published literature24,31,32,54,55,56. A week after vaccination, we assumed 90% vaccine-induced protection against infection (VP) prior to Omicron predominance and 70% protection thereafter. VP waned linearly to zero (1) at 48 weeks post-vaccination prior to Omicron predominance and 24 weeks thereafter or (2) at 24 weeks post-vaccination prior to Omicron predominance and 12 weeks thereafter (Supplementary Fig. 35) with variability by person (Supplementary Fig. 36). A week after infection, infection-induced protection (IP) had 90% protection against infection that waned to zero at 96 or 72 weeks (Supplementary Fig. 37) again with variability by person (Supplementary Fig. 38). Hybrid immunity or protection (HP) definitions were taken from meta-analyses of protective effectiveness32,33: (1) the greater of VP or IP was boosted by 30% of the other (Supplementary Fig. 39); or (2) VP was boosted by 30% of IP or IP was boosted by 10% of VP, whichever was greater (Supplementary Fig. 40). Both HP definitions were truncated at 99%. In these simulations, we considered 8 different protection calculations since we simulated each combination of the two VP, two IP, and two HP definitions. Infections were generated from a person’s weekly protection with the function $$\Pr \left({I}_{j,k}\right)=\Pr \left({c}_{k}\right) \left(1-{\psi }_{j,k-1}\right),$$ (1) where the probability of infection for each person ((j)) and week ((k)), (\Pr \left({I}_{j,k}\right)), depended on (\Pr \left({c}_{k}\right)), the case probability in week (k) and person (j)’s protection calculated from the previous week (({\psi }_{j,k-1})). An infection for person (j) in week (k) was generated from a Bernoulli distribution with probability (\Pr \left({I}_{j,k}\right)). A total of 200 populations were generated for each of the 8 protection definition combinations. An example of protection trajectories is included in the supplementary materials (Supplementary Fig. 41). The analytic period consisted of a hypothetical 32-week period beginning immediately after the historical period. Infections, vaccination doses, and protection were generated similarly to the historical period. Parameters were the 8 protection definition combinations, case distribution, vaccination rollout timing, total vaccination coverage, TND timing, and type of outcome (symptomatic infection or severe disease). Four infection distributions were utilized during the analytic period (Supplementary Fig. 42): weekly 2%; weekly 4%; weekly 2% increase to a peak of 4% at weeks 16 and 17 before returning to 2%; and weekly 2% increasing to a peak of 6% at weeks 16 and 17 before returning to 2%. The vaccination rollout happened in weeks 1–12 (before the case peak), weeks 11–22 (during the case peak), or weeks 21–32 (after the case peak) and followed a lognormal distribution with a mean of 1.5 and a standard deviation of 0.5 (Supplementary Fig. 43). Other weeks had a vaccination probability of 0.005. End-of-season vaccination coverage in the analytic period alone was 10% or 25% (Supplementary Fig. 44) to provide parameters above and below the estimated 14% coverage in people aged 18–49 years in the 2023–2024 season57/ (2024)."). The TND for symptomatic infections was implemented in weeks 1–12, weeks 11–22, or weeks 21–32. Since we implemented all possible combinations of vaccination rollout and TND timing, some scenarios involve assessing VE via the TND before the vaccination rollout. These scenarios approximated the situation where VE is assessed long after vaccination has been given. All 32 weeks were used for the TND for severe disease. COVID-19 symptoms were expected in 80% of infected people (Supplementary Fig. 45) and were present only in the week of infection. An uninfected person in week (k) was expected to have COVID-like symptoms with a probability of 0.20 divided by the number of weeks in the TND (Supplementary Fig. 46). For estimating VE against symptomatic infection, all symptomatic people were included in the TND. Diagnostic testing was assumed to have perfect specificity, but sensitivity was 90% during the week of infection and declined thereafter58 (Supplementary Table 4). For estimating VE against severe disease, VP was 90% the week after vaccination and waned to zero after 48 months24,31,56,59. IP against severe disease started at 95% protection the week after infection and waned to zero after 96 months32 (Supplementary Fig. 47). For people with a SARS-CoV-2 infection, the probability of severe disease was $$\Pr \left({S}_{j,k} \| \, {I}_{j,k}=1\right)=\frac{\left(1-{{\psi }^{s}}_{j,k-1}\right)}{\left(1-{\psi }_{j,k-1}\right)},$$ (2) where Sj,k is a severe disease event for person j in week k, and ({{\psi }^{s}}_{j,k-1}) is the protection against severe disease for person j in week k. All people with severe disease were included in the TND with perfect detection. A total of 1000 simulations were run for each parameter set. Each of the 200 populations was utilized five times in each parameter set. Statistical methods Exposures analyzed were vaccination at any time during the analytic period, vaccination in the previous 2 months, vaccination in the previous 3 months, vaccination in the previous 4 months, vaccination in the previous 5 months, vaccination in the previous 6 months, the time since vaccination (unvaccinated as the reference group, 0–2 months, 3–4 months, 5-11 months, and 12 or more months), and the number of doses (unvaccinated as the reference group, 2-dose, 3-dose, 4-dose, or 5-dose) where someone with 5 doses received all available vaccination doses, a person with 4 doses missed one of the available vaccination doses, and so on. Two logistic regression models were fit to each exposure definition. The first model included only the exposure variable (henceforth, the unadjusted model), whereas the second model added categorical time since the last infection (categories were monthly from 1 to 11 months and 12 or more months) and the number of prior infections as a continuous variable (the adjusted model). Odds ratios (OR) from logistic regressions were converted to VE in percentage points by the formula $${\mbox{VE}}=\left(1-{\mbox{OR}}\right) 100.$$ (3) Our primary measure is the difference between the VE estimate from the unadjusted model and the VE estimate from the adjusted model, which we refer to as bias. Bias is defined not in the traditional sense as the deviation from truth, but as the percentage point difference in VE from the unadjusted model and VE from the adjusted model. Bias less than zero indicated VE was underestimated without accounting for prior infection. A small percentage of simulations resulted in small sample sizes and unstable estimates. Details on bias definition and handling of unstable estimates are in the supplementary methods. Results were aggregated by parameter set and exposure and plotted by exposure with ridgeline plots (Supplementary Fig. 2). Simple, random effects meta-regression was used to estimate the expected VE and bias and, for infection outcomes, the percentage of simulations with a negative VE estimate. Separate meta-regressions were run for the unadjusted and adjusted VE estimates. Multivariable meta-regression models were run with simulation parameters to determine the mean VE, bias, and negative VE associated with each parameter level and the 95% confidence intervals. Sensitivity analyses were performed for VE against symptomatic infection by removing scenarios where the TND was implemented before the vaccination rollout. All simulations were performed in R version 4.0.4, and analyses in R version 4.2.4. Reporting summary Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article. Data availability No empirical data were used in the analyses of this manuscript. The simulated data generated for this study can be created using scripts provided as a compressed file in the Supplementary Information. Code availability Programming code used in the project is available as a compressed file (Supplementary Code 1). Please refer to the readme PDF in the compressed file for more information. References Jackson, M. L. & Nelson, J. C. The test-negative design for estimating influenza vaccine effectiveness. Vaccine 31, 2165–2168 (2013). Article PubMed Google Scholar 2. Foppa, I. M., Haber, M., Ferdinands, J. M. & Shay, D. K. The case test-negative design for studies of the effectiveness of influenza vaccine. Vaccine 31, 3104–3109 (2013). Article PubMed Google Scholar 3. Link-Gelles, R. et al. 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Effectiveness of a third dose of BNT162b2 mRNA COVID-19 vaccine in a large US health system: a retrospective cohort study. Lancet Reg. Health Am. 9, 100198 (2022). PubMed PubMed Central Google Scholar Download references Acknowledgements The findings and conclusions in this report are those of the authors and do not necessarily represent the views of the CDC. This study was supported by the Centers for Disease Control and Prevention. B.F.’s time was supported by the Centers for Disease Control and Prevention contract number 75D30120C07765 to Kaiser Foundation Hospitals. We thank clearing officials at CDC and anonymous reviewers for improving this manuscript. Author information Authors and Affiliations Coronavirus and Other Respiratory Viruses Division, Centers for Disease Control and Prevention, Atlanta, GA, USA Ryan E. Wiegand, Morgan Najdowski & Ruth Link-Gelles 2. Kaiser Permanente Vaccine Study Center, Kaiser Permanente Northern California Division of Research, Oakland, CA, USA Bruce Fireman 3. Influenza Division, Centers for Disease Control and Prevention, Atlanta, GA, USA Mark W. Tenforde & Jill M. Ferdinands Authors Ryan E. Wiegand View author publications Search author on:PubMed Google Scholar 2. Bruce Fireman View author publications Search author on:PubMed Google Scholar 3. Morgan Najdowski View author publications Search author on:PubMed Google Scholar 4. Mark W. Tenforde View author publications Search author on:PubMed Google Scholar 5. Ruth Link-Gelles View author publications Search author on:PubMed Google Scholar 6. Jill M. Ferdinands View author publications Search author on:PubMed Google Scholar Contributions R.L.G. conceived the idea. R.E.W. supervised the project, implemented the simulation study, analyzed the results, and prepared the manuscript. R.E.W., B.F., M.N., M.W.T., R.L.G., and J.M.F. contributed to the study design, discussed the results, read the manuscript and supplement, and provided critical feedback. Corresponding author Correspondence to Ryan E. Wiegand. Ethics declarations Competing interests The authors declare that they do not have any commercial or other associations that might pose a conflict of interest. Peer review Peer review information Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available. Additional information Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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To view a copy of this licence, visit Reprints and permissions About this article Cite this article Wiegand, R.E., Fireman, B., Najdowski, M. et al. Bias and negative values of COVID-19 vaccine effectiveness estimates from a test-negative design without controlling for prior SARS-CoV-2 infection. Nat Commun 15, 10062 (2024). Download citation Received: Accepted: Published: DOI: Share this article Anyone you share the following link with will be able to read this content: Provided by the Springer Nature SharedIt content-sharing initiative Subjects Computational models Epidemiology Vaccines Viral infection This article is cited by Effectiveness of a hepatitis E vaccine against medically-attended symptomatic infection in HBsAg-positive adults from a test-negative design study Chunlan Zhuang Xiaohui Liu Ningshao Xia Nature Communications (2025)
4666	https://insight.jci.org/articles/view/151515	JCI Insight - Nuclear membrane ruptures underlie the vascular pathology in a mouse model of Hutchinson-Gilford progeria syndrome Go to The Journal of Clinical Investigation About Editors Consulting Editors For authors Publication ethics Publication alerts by email Transfers Advertising Job board Contact Physician-Scientist Development Current issue Past issues By specialty Back COVID-19 Cardiology Immunology Metabolism Nephrology Oncology Pulmonology All ... 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Email the journal Go to Top Abstract Introduction Results Discussion Methods Author contributions Supplemental material Acknowledgments Footnotes References Version history PDF Metrics See more details Posted by 12 X users 32 readers on Mendeley Article usage Citations to this article (26) Advertisement Research ArticleVascular biology Open Access \| 10.1172/jci.insight.151515 ShareXFacebookLinkedInWeChatBlueskyEmailCopy Link Nuclear membrane ruptures underlie the vascular pathology in a mouse model of Hutchinson-Gilford progeria syndrome Paul H. Kim,1,2Natalie Y. Chen,1,3Patrick J. Heizer,1Yiping Tu,1Thomas A. Weston,1Jared L.-C. Fong,1Navjot Kaur Gill,3Amy C. Rowat,2,3Stephen G. Young,1,4 andLoren G. Fong 1 1 Department of Medicine, 2 Department of Bioengineering, 3 Department of Integrative Biology and Physiology, and 4 Department of Human Genetics, UCLA, Los Angeles, California, USA. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Find articles by Kim, P. in: PubMed \| Google Scholar \| 1 Department of Medicine, 2 Department of Bioengineering, 3 Department of Integrative Biology and Physiology, and 4 Department of Human Genetics, UCLA, Los Angeles, California, USA. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Find articles by Chen, N. in: PubMed \| Google Scholar \| 1 Department of Medicine, 2 Department of Bioengineering, 3 Department of Integrative Biology and Physiology, and 4 Department of Human Genetics, UCLA, Los Angeles, California, USA. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Find articles by Heizer, P. in: PubMed \| Google Scholar 1 Department of Medicine, 2 Department of Bioengineering, 3 Department of Integrative Biology and Physiology, and 4 Department of Human Genetics, UCLA, Los Angeles, California, USA. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Find articles by Tu, Y. in: PubMed \| Google Scholar \| 1 Department of Medicine, 2 Department of Bioengineering, 3 Department of Integrative Biology and Physiology, and 4 Department of Human Genetics, UCLA, Los Angeles, California, USA. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Find articles by Weston, T. in: PubMed \| Google Scholar \| 1 Department of Medicine, 2 Department of Bioengineering, 3 Department of Integrative Biology and Physiology, and 4 Department of Human Genetics, UCLA, Los Angeles, California, USA. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Find articles by Fong, J. in: PubMed \| Google Scholar 1 Department of Medicine, 2 Department of Bioengineering, 3 Department of Integrative Biology and Physiology, and 4 Department of Human Genetics, UCLA, Los Angeles, California, USA. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Find articles by Gill, N. in: PubMed \| Google Scholar 1 Department of Medicine, 2 Department of Bioengineering, 3 Department of Integrative Biology and Physiology, and 4 Department of Human Genetics, UCLA, Los Angeles, California, USA. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Find articles by Rowat, A. in: PubMed \| Google Scholar 1 Department of Medicine, 2 Department of Bioengineering, 3 Department of Integrative Biology and Physiology, and 4 Department of Human Genetics, UCLA, Los Angeles, California, USA. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Find articles by Young, S. in: PubMed \| Google Scholar 1 Department of Medicine, 2 Department of Bioengineering, 3 Department of Integrative Biology and Physiology, and 4 Department of Human Genetics, UCLA, Los Angeles, California, USA. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Find articles by Fong, L. in: PubMed \| Google Scholar Published August 23, 2021 - More info Published in Volume 6, Issue 16 on August 23, 2021 JCI Insight. 2021;6(16):e151515. © 2021 Kim et al. This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit Published August 23, 2021 - Version history Received: May 18, 2021; Accepted: July 1, 2021 View PDF Abstract The mutant nuclear lamin protein (progerin) produced in Hutchinson-Gilford progeria syndrome (HGPS) results in loss of arterial smooth muscle cells (SMCs), but the mechanism has been unclear. We found that progerin induces repetitive nuclear membrane (NM) ruptures, DNA damage, and cell death in cultured SMCs. Reducing lamin B1 expression and exposing cells to mechanical stress — to mirror conditions in the aorta — triggered more frequent NM ruptures. Increasing lamin B1 protein levels had the opposite effect, reducing NM ruptures and improving cell survival. Remarkably, raising lamin B1 levels increased nuclear compliance in cells and was able to offset the increased nuclear stiffness caused by progerin. In mice, lamin B1 expression in aortic SMCs is normally very low, and in mice with a targeted HGPS mutation (Lmna G609G), levels of lamin B1 decrease further with age while progerin levels increase. Those observations suggest that NM ruptures might occur in aortic SMCs in vivo. Indeed, studies in Lmna G609G mice identified NM ruptures in aortic SMCs, along with ultrastructural abnormalities in the cell nucleus that preceded SMC loss. Our studies identify NM ruptures in SMCs as likely causes of vascular pathology in HGPS. Graphical Abstract Introduction Hutchinson-Gilford progeria syndrome (HGPS) is a pediatric progeroid disorder caused by a point mutation in LMNA (the gene for prelamin A and lamin C) that causes aberrant prelamin A splicing and the production of an internally truncated prelamin A protein (progerin) (1, 2). Progerin cannot undergo the proteolytic processing step by ZMPSTE24 that normally converts farnesyl–prelamin A to mature lamin A; thus, progerin retains a farnesyl lipid anchor at its carboxyl terminus (3). In HGPS fibroblasts, progerin accumulates at the nuclear periphery and leads to a high frequency of misshapen nuclei (1, 4). Children with HGPS develop several aging-like phenotypes, but the most devastating is atherosclerosis in the coronary and cerebral arteries, which often leads to death by the mid-teenage years (5). The atherosclerosis in HGPS is unusual in that it occurs in the absence of common risk factors (e.g., hypercholesterolemia, smoking, and diabetes mellitus) (6). Histopathologic studies uncovered another atypical feature of the arterial disease in HGPS: the loss of medial smooth muscle cells (SMCs) (7, 8). The loss of SMCs in large arteries, accompanied by thickening of the adventitia, is also observed in transgenic and gene-targeted mouse models of HGPS (9–11). The severity of the arterial disease depends on levels of progerin expression; the arterial pathology is more severe — and the onset is earlier — in mice that have 2 targeted HGPS mutant alleles as compared with only 1 (12, 13). The arterial disease is less severe in HGPS mice treated with a farnesyltransferase inhibitor (14), an antisense oligonucleotide that alters RNA splicing and reduces progerin expression (11), or by inhibiting N-acetyltransferase 10 (15), endoplasmic reticulum stress (16), MMP13 (17), or ICMT (18). Mechanical stress is relevant to the vascular pathology in HGPS. In cultured SMCs that express progerin, mechanical stretching increases DNA damage and cell death (12). Disrupting the linker of the nucleoskeleton and cytoskeleton (LINC) complex, which reduces force transmission to the nucleus (19, 20), reduces DNA damage and increases cell survival. Also, in gene-targeted mice expressing progerin (Lmna G609G–knock-in mice), the loss of medial SMCs is most pronounced in segments of the aorta that are subjected to high hemodynamic stress (e.g., inner curvature of the ascending thoracic aorta) (12, 21). Of note, disrupting the LINC complex in the SMCs of Lmna G609G mice markedly reduces arterial pathology (12). In Lmna G609G mice, nuclear abnormalities and cell death are widespread in medial SMCs but absent in intimal endothelial cells (12), even though both cell types are subjected to pulsatile hemodynamic forces. One potential clue to this discrepancy relates to differences in the composition of the nuclear lamina. In both WT and Lmna G609G mice, lamin B1 expression is very low in medial SMCs but is robust in endothelial cells (12). Lamin B1, like progerin, is tethered to the inner nuclear membrane (NM) by a farnesyl lipid anchor (22), but whether the physiologically low levels of lamin B1 in medial SMCs render cells more susceptible to progerin toxicity is unknown. Knocking down lamin B1 expression in cultured osteosarcoma cells was reported to trigger NM ruptures (23), but the relevance of that finding to aortic SMCs in vivo is open to question because plating cells on plastic dishes increases susceptibility to NM ruptures (24, 25). NM ruptures have been observed in cortical neurons of lamin B1–deficient mouse embryos (26), but the relevance of that finding to arterial SMCs is also questionable because embryonic neurons, unlike arterial SMCs, do not express lamin A or lamin C (27, 28), 2 nuclear lamins that are crucial for the integrity of the nuclear envelope. We believe that the low levels of lamin B1 in arterial SMCs are normally well tolerated because SMCs produce high levels of lamin A and lamin C (12, 29). However, we suspected that the low levels of lamin B1, combined with progerin expression, could be relevant to the vascular pathology of HGPS. Specifically, we hypothesized that progerin expression in SMCs, particularly when the cells are subjected to mechanical stress, could reduce NM integrity and trigger NM ruptures, resulting in reduced cell survival. We also hypothesized that increasing lamin B1 expression in progerin-expressing SMCs (so as to model the situation in intimal endothelial cells) would render SMCs less susceptible to NM ruptures. In addressing our hypotheses, we were keenly aware that cell culture studies are an imperfect model for arterial SMCs and HGPS pathology. Therefore, we also examined, in Lmna G609G mice, whether progerin triggers NM ruptures in arterial SMCs but not in endothelial cells, and whether the onset of NM ruptures in SMCs correspond temporally to the emergence of the hallmark arterial pathology of HGPS. Results Progerin expression causes NM ruptures in cultured SMCs. We generated aortic SMCs expressing nuclear-targeted GFP (Nuc-GFP) and doxycycline-inducible (Dox-inducible) constructs for WT prelamin A (PreA-SMC) or progerin (Prog-SMC). Dox levels were adjusted to achieve lamin A (or progerin) levels similar to those in aortas of WT mice (Lmna+/+) or mice with a targeted HGPS mutation (Lmna G609G/+) (Figure 1A). Both lamin A and progerin were located in the cell nucleus (Supplemental Figure 1A; supplemental material available online with this article; After 2 days, progerin increased the frequency of misshapen nuclei in both live SMCs (Figure 1B) and fixed cells (Supplemental Figure 1B). NM ruptures, evident from the escape of a nuclear-targeted fluorescent protein into the cytoplasm (23, 30, 31) (Figure 1C), were detected by fluorescence microscopy. In the absence of Dox, the frequency of NM ruptures was low in PreA-SMCs and Prog-SMCs (~3%) (Figure 1D). In the presence of Dox, 17% of Prog-SMCs exhibited a NM rupture, whereas only approximately 3% of PreA-SMCs had a rupture. In Prog-SMCs, the frequency of ruptures increased with higher levels of progerin expression (Supplemental Figure 1D). Figure 1 Progerin expression causes NM ruptures in cultured SMCs. (A) Western blot analysis of SMCs expressing prelamin A (PreA) and progerin (Prog), and in the aorta and kidney of WT and Lmna G609G/+ mice. The bar graph shows the expression of lamin A plus progerin, relative to total protein (Supplemental Figure 1C). (B) Bar graph showing that progerin increases the frequency of abnormally shaped nuclei in SMCs, as judged by live-cell microscopy (mean ± SEM, n = 4 experiments; Student’s t test, P< 0.001). (C) Illustration depicting a NM rupture, defined as the escape of nuclear-targeted GFP (Nuc-GFP; green) into the cytoplasm (tan). (D) Bar graph showing that progerin causes NM ruptures in SMCs (mean ± SEM, n = 3 experiments; Student’s t test, P< 0.01). NM ruptures were measured in fixed cells 48 hours after adding Dox (blue). (E) Fluorescence microscopy images of live SMCs expressing progerin and Nuc-GFP (green) at 1-hour intervals. A DIC image (grey) is superimposed. The yellow arrow points to a cell followed for 14 hours by time-lapse microscopy (see Supplemental Video 1; images were captured every 10 minutes). The cell had several NM ruptures and eventually died. Scale bar: 20 μm. (F–I) Characteristics of NM ruptures in PreA-SMC and Progerin-SMC derived from time-lapse microscopy studies. Bar graphs show the percentage of cells with a NM rupture; the number of NM ruptures per cell; the percentage of cells with a NM rupture that die; and the percentage of cells with a NM rupture and abnormal-shaped nuclei (mean ± SEM, n = 4 experiments; Student’s t test, P< 0.02, P< 0.002, P< 0.001). The results from the individual experiments are shown in Supplemental Figure 1E. To define the dynamics of NM ruptures, Prog-SMCs were monitored by time-lapse microscopy (images were taken every 10 minutes for 24 hours). Snapshots of Prog-SMCs at 1-hour intervals are shown in Figure 1E (also see Supplemental Video 1). Our analysis of time-lapse experiments (915 PreA-SMCs and 735 Prog-SMCs) revealed that NM ruptures were more frequent in Prog-SMCs than in PreA-SMCs (18% vs. 2.5%; P< 0.001) (Figure 1F) and often occurred repetitively (Figure 1G) (see Supplemental Videos 2 and 3). Cell death also occurred more frequently in Prog-SMCs with NM ruptures (25/141 cells vs. only 1/21 in PreA-SMCs; P< 0.02) (Figure 1H). NM ruptures were more frequent in Prog-SMCs that had a misshapen nucleus (Figure 1I). The results from individual experiments are shown in Supplemental Figure 1E. Impact of lamin B1, mechanical stress, and progerin farnesylation on NM ruptures. To examine the effects of low lamin B1 expression and mechanical stress on NM ruptures — so as to model conditions in the aorta — Prog-SMCs and PreA-SMCs were treated with a Lmnb1 siRNA (or a control siRNA), plated on collagen-coated polydimethylsiloxane (PDMS) membranes, and subjected to either uniaxial stretching (2 mm, 0.5 Hz for 2 hours) or static conditions. Western blots documented reduced lamin B1 levels in Lmnb1 siRNA–treated cells (Supplemental Figure 2A). The lower levels of lamin B1 expression increased NM ruptures in Prog-SMCs by 58% in static conditions (P< 0.05) and 40% in stretch conditions (P< 0.02) (Figure 2A) but had no effect on ruptures in PreA-SMCs (Supplemental Figure 2B). Stretching increased NM ruptures in both untreated Prog-SMCs (1.6-fold vs. static Prog-SMCs; P< 0.02) and Lmnb1 siRNA–treated Prog-SMCs (2.3-fold vs. static Prog-SMCs; P< 0.0001). The expression of a nonfarnesylated (nf) progerin (nf-Prog), a progerin that terminated with –SSIM rather than –CSIM, a modification to the C-terminal CaaX signal sequence that blocks progerin farnesylation) (Supplemental Figure 2A), did not trigger NM ruptures in either static or stretched SMCs (Figure 2A). Figure 2 Impact of lamin B1, mechanical stress, and progerin farnesylation on NM ruptures. (A) Bar graph comparing the effects of prelamin A (PreA), progerin (Prog), lamin B1 knockdown (B1 KD), and nf-Prog on NM ruptures in static (white bars) and stretched (black bars) SMCs (mean ± SEM, n = 5 experiments). Differences were compared with static SMCs expressing prelamin A by 2-way ANOVA (P< 0.005, P< 0.0001). The total numbers of cells examined are shown in parentheses above each bar. (B) Bar graph showing transcript levels for genes involved in NM repair (mean ± SEM, n = 3 experiments; Student’s t test, all nonsignificant). (C) Confocal fluorescence microscopy images showing SMCs expressing Nuc-RFP (red) and either GFP-KASH2ext (green) or GFP-KASH2 (green). Scale bar: 20 μm. (D–F) Characterization of NM ruptures in live PreA-SMCs and Prog-SMCs expressing KASH2ext (white bars) or KASH2 (blue bars). Bar graphs show the percentage of cells with a NM rupture; the number of NM ruptures per cell; and the percentage of cells with a NM rupture that die (mean ± SEM, n = 3 experiments; Student’s t test, P< 0.01, P< 0.005). (G) Confocal fluorescence microscopy images showing increased phosphorylated-STING (p-STING; green) in Progerin-SMCs. SMCs were examined in both static (left) and stretched (right) conditions. PreA-SMCs were included as a control. Lap2β staining is shown in red. Scale bar: 20 μm. (H) Bar graph showing the percentage of cells staining positive for p-STING in PreA-SMCs (white bars) and Prog-SMCs (black bars) in static and stretched conditions in a representative experiment. The ratios above each bar show the number of cells positive for p-STING over the total number of cells assessed. The endosomal sorting complex required for transport III (ESCRT-III) protein complex plays a role in repairing NM ruptures (31, 32). Transcript levels for components of the ESCRT-III protein complex (Vps20, Snf7b, Vps24, Vps2, Vps4, and Vps60) were not different in Prog-SMCs and PreA-SMCs (Figure 2B). Finding increased NM ruptures in stretched Prog-SMCs implied that mechanical strain contributes to ruptures. To explore that idea, we expressed a nuclear-targeted red fluorescence protein (RFP) (Nuc-RFP) in SMCs and then disrupted the LINC complex with a GFP-tagged Klarsicht/Anc-1, Syne homology (KASH) domain of nesprin-2 (GFP-KASH2) (33) (Figure 2C). NM ruptures, documented by the escape of Nuc-RFP into the cytoplasm, were reduced by KASH2 expression in Prog-SMCs (compared with KASHext, an inactive KASH2 protein) (Figure 2, D and E). There was a trend of reduced cell death in KASH2-expressing Prog-SMCs but the differences did not achieve statistical significance (P = 0.12) (Figure 2F). KASH2 expression had no effect on NM ruptures in PreA-SMCs (Figure 2, D and E). The results from individual experiments are shown in Supplemental Figure 2C. NM ruptures result in the intermixing of nuclear and cytoplasmic contents (23, 30, 34). To determine if NM ruptures activate the cGAS-STING pathway (32, 35), we assessed phosphorylated-STING by immunofluorescence microscopy. Phosphorylated-STING was detected at a higher frequency in both static and stretched Prog-SMCs (Figure 2, G and H). Lamin B1 reduces progerin’s toxicity and association with NMs. To determine if increasing lamin B1 levels alter the frequency of progerin-induced NM ruptures, we created Prog-SMCs harboring a Dox-inducible lamin B1 construct. Inducing lamin B1 (Figure 3, A and B) did not alter the growth of Prog-SMCs (Figure 3C) but reduced the number of cells with misshapen nuclei (Figure 3D). Induced lamin B1 expression also markedly reduced NM ruptures in Prog-SMCs under both static (61% decrease) and stretched (80% decrease) conditions (Figure 3E). Also, lamin B1 expression resulted in less DNA damage in Prog-SMCs (but not in PreA-SMCs), as judged by H2AXγ levels (Figure 3F). Finally, lamin B1 increased cell survival in stretched Prog-SMCs (Figure 3G). Figure 3 Lamin B1 reduces progerin’s toxicity and association with NMs. (A) Western blot showing Dox-induced expression of mouse lamin B1 in PreA-SMCs and Prog-SMCs. (B) Microscopy images showing lamin B1 expression (red) in Prog-SMCs. Scale bar: 10 μm. (C) Growth curves showing that induced lamin B1 expression (blue circles) does not affect cell growth in control (black lines) or UV-treated SMCs (red lines). Mean ± SEM (n = 3 experiments). (D) Bar graph showing that lamin B1 overexpression (blue) reduces abnormal nuclear shape in Prog-SMCs (mean ± SEM, n = 3 experiments; Student’s t test, P< 0.02). (E) Bar graph comparing the effects of lamin B1 knockdown (B1 KD) and lamin B1 overexpression (B1) on NM ruptures in Prog-SMCs under static (white bars) and stretched (black bars) conditions (mean ± SEM, n = 4 experiments). NM ruptures were compared with static PreA-SMCs by 2-way ANOVA (P< 0.05, P< 0.0001). The total numbers of cells examined are shown in parentheses above each bar. (F) Bar graph showing that lamin B1 reduces H2AX-γ levels in Prog-SMCs (mean ± SEM, n = 3 experiments). Levels were compared with control (Con) Prog-SMCs by ANOVA (P< 0.01). (G) Bar graph showing that lamin B1 reduces cell death in stretched Prog-SMCs (mean ± SEM, n = 3 experiments; Student’s t test, P< 0.05). Dead cells detach from membranes, reducing cell protein on the membranes (12). (H) Western blot showing that lamin B1 expression increases the solubility of progerin. Nuclei from Prog-SMCs were sequentially extracted as described in Methods and the soluble extracts analyzed by western blotting. (I) Bar graph comparing the extraction profiles for progerin in Prog-SMCs (white) and Prog-SMCs plus lamin B1 (blue) (mean ± SEM, n = 3 experiments; Student’s t test, P< 0.02, P< 0.01, P< 0.001). The farnesyl lipid anchor is thought to tether lamin B1 and progerin to the inner NM (36–38). To determine if increased lamin B1 interferes with progerin’s association with the nuclear envelope, Prog-SMC nuclei were sequentially extracted with NaCl, Triton, and urea (39). In the absence of lamin B1 induction (–Dox), approximately 30% of the progerin appeared in the NaCl/Triton fractions, whereas approximately 70% appeared in the urea fractions. With lamin B1 expression (+Dox), 60% of the progerin appeared in the NaCl/Triton fractions, indicating reduced association of progerin with the nuclear envelope (Figure 3, H and I). In control studies, we found, as expected, a high percentage of nf-Prog (~60%) in the NaCl/Triton fractions (Supplemental Figure 3, A and B). Lamin B1 decreases nuclear stiffness. In live-cell imaging studies, we noted that SMC nuclei appeared larger in cells expressing lamin B1 (Figure 4A and Supplemental Video 4). Since nuclear size is determined by chromosomal content, nuclear lamina stiffness, and cytoskeletal elements (i.e., microfilaments and microtubules), we hypothesized that lamin B1 increased nuclear size by decreasing nuclear stiffness. To pursue this, we expressed prelamin A, lamin B1, nf-lamin B1, and lamin B2 in SMCs (Supplemental Figure 4A) and examined the nuclear size in suspension cultures stained with Hoechst at baseline and after compressing the cells with a glass coverslip (Figure 4B). The purpose of the coverslip was to mimic the pressure on the nucleus by cytoskeletal elements and to apply a uniform level of force. The coverslip increased nuclear area in all SMCs, as expected, but the increase was particularly striking in the lamin B1–transduced SMCs (Figure 4C). Increased nuclear area was also observed in fibroblasts transduced with lamin B1 (Supplemental Figure 4, B and C). Figure 4 Lamin B1 decreases nuclear stiffness. (A) Microscopy images of a live SMC (arrow) expressing Nuc-GFP (green) and increased amounts of lamin B1 (see Supplemental Video 4). Scale bar: 20 μm. (B) Microscopy images of Hoechst-stained nuclei in suspended cells expressing lamin B1 (B1), nf-lamin B1 (nf-B1), or lamin B2 (B2) before (left) and after (right) compressing the cells with a glass coverslip. Scale bar: 10 μm. Nuclear lamin expression was induced with Dox (see Supplemental Figure 4A). (C) Inducing lamin B1 expression (+Dox) increases nuclear area (mean ± SD, numbers of cells examined are reported above each bar; Student’s t test, P< 0.001). (D) Lamin B1 (B1) decreases nuclear stiffness (Young’s modulus) in SMCs, as measured by AFM (mean ± SEM, n = 5 experiments). Measurements were compared with noninduced SMCs by ANOVA (P< 0.002, P< 0.0001). (E) Lamin B1 increases nuclear size (spreading) in PreA-SMCs and Progerin-SMCs. Nuclei size was measured as in C (mean ± SD; Student’s t test, P< 0.001). (F) Progerin (Prog) but not nf-Prog increases nuclear stiffness in SMCs (mean ± SEM, n = 5 experiments). Expression data are shown in Supplemental Figure 4F. All measurements were compared with control SMCs (Con) by ANOVA (P< 0.0001). (G) Western blot showing increased expression of lamin B1 (B1) and nf-B1 in SMCs examined in H. (H) Increasing lamin B1 expression but not nf-B1 reduces nuclear stiffness in Prog-SMCs (mean ± SEM, n = 3 experiments). Measurements were compared with SMCs expressing progerin by ANOVA (P< 0.0001). (I) Disrupting the LINC complex with KASH2 (blue bars) reduces nuclear stiffness in Prog-SMCs (mean ± SEM; n = 3 experiments). Nuclear stiffness was compared with cells expressing the inactive mutant KASH2ext (white bars) by the Student’s t test (P< 0.01). We suspected that lamin B1 expression increased nuclear size by decreasing nuclear stiffness. By atomic force microscopy (AFM) measurements over the cell nucleus, lamin B1 expression reduced nuclear stiffness by 23% (P< 0.002), whereas a lamin B1 knockdown increased nuclear stiffness by approximately 70% (P< 0.0001). Increased expression of prelamin A, lamin B2, and nf-lamin B1 had no effect on nuclear stiffness (Figure 4D and Supplemental Figure 4D). The increase in nuclear area in SMCs expressing lamin B1 (induced with 10 ng/mL Dox) was not accompanied by increased expression of nuclear envelope proteins (Supplemental Figure 4E), suggesting that lamin B1 did not increase nuclear size by inducing larger nuclei. Lamin B1 expression also increased nuclear area in cells expressing progerin that were compressed with a glass coverslip (Figure 4E). However, the extent of nuclear spreading was significantly lower in Prog-SMCs than in PreA-SMCs, implying that progerin expression made nuclei stiffer. Indeed, by AFM, progerin increased nuclear stiffness more than 2-fold (P< 0.0001), whereas prelamin A and nf-Prog had no effect (Figure 4F and Supplemental Figure 4F). The increased nuclear stiffness in Prog-SMCs was reduced by expression of lamin B1 and KASH2, but not by nf-lamin B1 or KASH2ext (Figure 4, G–I). Progerin levels increase with age in Lmna G609G/+ mice whereas lamin B1 levels decrease. NM rupture frequency in cultured SMCs depends on the expression levels of progerin and lamin B1. To explore the relevance of these findings in living mice, we began by using western blots to compare progerin and lamin B1 expression in aortas of 4- and 21-week-old Lmna G609G/+ mice (Figure 5A). Progerin levels in the aorta of Lmna G609G/+ mice were approximately 50% higher at 21 weeks than at 4 weeks (Figure 5B), while lamin B1 levels were approximately 50% lower (Figure 5C and Supplemental Figure 5, A–C) (both P< 0.001). Microscopy revealed that the lower levels of lamin B1 were due to reduced expression in medial SMCs (Figure 5D). The age-dependent change in lamin B1 protein levels was accompanied by a 67% decrease in Lmnb1 expression; however, the increase in progerin protein levels could not be explained by changes in gene expression (Figure 5E). At 32 weeks, the levels of progerin protein, relative to lamin B1, stabilized (Figure 5F and Supplemental Figure 5, D and E). Figure 5 Progerin levels increase with age in Lmna G609G/+ mice whereas lamin B1 levels decrease. (A) Western blot comparing the expression of progerin and lamin B1 in the aorta of 4- and 21-week-old Lmna G609G/+ mice. Mouse IDs are shown above each sample. (B and C) Bar graphs showing progerin and lamin B1 levels, relative to tubulin, for the western blot in (A) (mean ± SEM, n = 6 mice/group; Student’s t test, P< 0.001). (D) Immunofluorescence microscopy images of aortic rings showing lamin B1 (green) expression is reduced in SMCs in 21-week-old Lmna G609G/+ mice. The border between the media and adventitia is marked with a white line. Scale bar: 100 μm. (E) Quantitative RT-PCR studies showing Lmna, progerin, and Lmnb1 transcript levels in aortas from 4-week-old (n = 6 mice) and 21-week-old (n = 7 mice) Lmna G609G/+ mice (mean ± SEM; Student’s t test, P< 0.05, P< 0.001. (F) Bar graph comparing the progerin-to-lamin B1 ratio in the aorta from 1-day-old Lmna G609G/+ mice (n = 2), and from 4-, 21-, and 32-week-old Lmna G609G/+ mice (n = 4) for the western blot in Supplemental Figure 5D. Comparisons were made to 4-week-old Lmna G609G/+ mice by ANOVA (P< 0.0001). (G) Western blot comparing the expression of lamin B1 in the aorta from 4- (n = 6) and 21- (n = 5) week-old Lmna+/+ mice. Mouse IDs are shown above each sample. For comparison, a sample from a 21-week-old Lmna G609G/+ mouse is included. (H) Bar graph showing the expression of lamin B1, relative to tubulin, in the western blot in G (mean ± SEM; Student’s t test). (I) Bar graph showing Lmnb1 expression in the aorta of 4- and 21-week-old Lmna+/+ mice (mean ± SEM, n = 4 mice/group; Student’s t test, P< 0.02). The decrease in lamin B1 levels in the aorta from 4–21 weeks was observed only in Lmna G609G/+ mice. In WT mice, lamin B1 levels were not significantly reduced at 21 weeks (Figure 5, G and H) (P = 0.33), despite the fact that Lmnb1 transcript levels had fallen by more than 50% (Figure 5I). NM ruptures and ultrastructural abnormalities in aortic SMCs precede SMC loss in Lmna G609G/G609G mice. Finding high progerin levels but low lamin B1 levels in aortas of Lmna G609G/+ mice suggested that we might find NM ruptures in vivo in aortic SMCs. To explore this possibility, we bred WT and Lmna G609G/G609G mice with a nuclear-targeted tdTomato (Nuc-tdTomato) transgene (26). By fluorescence microscopy, Nuc-tdTomato expression was uniform in the thoracic aorta and was confined to the cell nucleus (Figure 6A). Figure 6 NM ruptures in aortic SMCs precede SMC loss in Lmna G609G/G609G mice. (A) Expression of Nuc-tdTomato in SMCs is uniform in the ascending aorta. Fourteen sequential cross sections through the ascending aorta were collected every 100 μm from a Lmna+/+ mouse. Half of the sections (odd-numbered sections; top row) were imaged at low magnification to visualize the entire cross section. Scale bar: 100 μm. The even-numbered sections (bottom row) were imaged at higher magnification to visualize Nuc-tdTomato (orange) in nuclei stained with DAPI (blue). Scale bar: 20 μm. (B) Confocal fluorescence microscopy images of the ascending aorta from a 14-week-old Lmna+/+ and Lmna G609G/G609G mouse. The boxed regions are shown at higher magnification in the middle and far-right columns. The colored images show DAPI (blue), elastic fibers (green), and Nuc-tdTomato (orange). The yellow arrows (middle column) point to Nuc-tdTomato outside of an SMC nucleus. To help visualize the boundaries of nuclei, the DAPI stain (white) is shown by itself in the far-right column. Scale bars: 20 μm (left column); 10 μm (middle column). (C) H&E-stained sections of the inner ascending aorta from 8-, 10-, and 16-week-old Lmna G609G/G609G mice. The black line spans the medial layer and the yellow line spans the thickened and fibrotic adventitia in the 16-week-old mouse. Scale bar: 20 μm. Note the loss of SMC nuclei in the aorta from the 16-week-old mouse. (D) Confocal fluorescence microscopy images of the ascending, upper descending, and lower descending thoracic aorta from an 8-week-old Lmna G609G/G609G mouse. Boxed regions are analyzed as described in (B). Scale bars: 20 μm (left column); 10 μm (middle column). We used fluorescence microscopy to visualize Nuc-tdTomato in aortas of Lmna G609G/G609G mice at 14 weeks of age. At this time point, loss of medial SMCs is clearly evident (12). We found frequent NM ruptures in medial SMCs of Lmna G609G/G609G mice, as judged by the escape of Nuc-tdTomato into the cytoplasm (Figure 6B). No NM ruptures were observed in aortic endothelial cells consistent with the absence of endothelial cell loss in Lmna G609G/G609G mice. In WT mice, Nuc-tdTomato was confined to the nucleus of SMCs (Figure 6B). We next asked whether the appearance of NM ruptures in the aortic SMCs of Lmna G609G/G609G mice precedes the onset of SMC loss. Loss of SMCs in Lmna G609G/G609G mice was detectable at 10 weeks of age but absent at 8 weeks (Figure 6C). However, at 8 weeks of age, we observed frequent NM ruptures in medial SMCs of Lmna G609G/G609G mice in the ascending aorta and upper descending aorta and less frequent ruptures in the lower descending aorta (Figure 6D and Supplemental Figure 6, A and B). To determine the frequency of NM ruptures in Lmna G609G/G609G mice, NM ruptures in cross sections of the ascending thoracic aorta in 3 8-week-old Lmna+/+ mice and 3 8-week-old Lmna G609G/G609G mice (Figure 7A) were counted and expressed relative to total SMC nuclei. The frequency of NM ruptures in aortic SMCs was significantly higher in Lmna G609G/G609G mice than in Lmna+/+ mice (26.7% vs. 2.0%; P< 0.001) (Figure 7B). NM ruptures were detected in WT mice but they occurred at a very low frequency. No NM ruptures were detected in endothelial cells in either WT or Lmna G609G/G609G mice (Figure 7B). NM ruptures were also quantified in tissue sections of the upper and lower descending aorta (Supplemental Figure 6, C and D). Consistent with the lower frequency of SMC loss in the descending thoracic aorta (12), NM ruptures were less frequent in the upper and lower descending aorta (Figure 7C). No NM ruptures were detected in the heart or liver of Lmna G609G/G609G mice (Figure 7D). Figure 7 NM ruptures are frequent in aortic SMCs but absent in cardiomyocytes and hepatocytes of Lmna G609G/G609G mice. (A) Fluorescence microscopy images of cross sections (10 μm–thick) of the ascending aorta from 3 8-week-old Lmna+/+ (top row) and 3 8-week-old Lmna G609G/G609G (bottom row) mice expressing the Nuc-tdTomato transgene. The mouse IDs are shown in the bottom left-hand corner of each image. Nuc-tdTomato (orange); DAPI (blue). Scale bar: 100 μm. (B) Quantification of NM ruptures in SMCs and endothelial cells (EC) in the ascending aorta of 8-week-old Lmna+/+ and Lmna G609G/G609G mice. The number of NM ruptures is reported as a percentage of total nuclei examined in individual cross sections (A) (mean ± SEM; Student’s t test; P< 0.001). The total number of nuclei scored is shown in parentheses. (C) Quantification of NM ruptures in SMCs in the upper and lower descending aorta of 3 8-week-old Lmna+/+ and Lmna G609G/G609G mice. The number of NM ruptures is reported as a percentage of total nuclei examined in individual cross sections (see Supplemental Figure 6C) (mean ± SEM; Student’s t test; P< 0.001). The total number of nuclei scored is shown in parentheses. (D) Confocal fluorescence microscopy images of the ascending aorta, heart, and liver from an 8-week-old Lmna G609G/G609G mouse. The colored images show DAPI (blue), elastic fibers (green), and Nuc-tdTomato (orange). To assist in visualizing the boundaries of nuclei, the DAPI stain is shown in white (bottom row). The yellow arrow points to Nuc-tdTomato outside of a nucleus in an aortic SMC. Scale bar: 10 μm. Additional studies revealed that NM ruptures were absent in medial SMCs of Lmna G609G/G609G mice at 6 weeks of age (0/7 mice) but were easily detectable at 7 weeks, although at a low frequency (5/5 mice). NM ruptures were also observed in older Lmna G609G/+ mice (Supplemental Figure 7, A and B). The onset of NM ruptures in Lmna G609G/G609G mice coincided with the onset of ultrastructural pathology in SMC nuclei (12), as judged by electron microscopy. At 4 and 6 weeks, the morphology of medial SMCs in Lmna G609G/G609G mice was normal by electron microscopy, indistinguishable from the SMCs in WT mice. By 8 weeks, however, intranuclear tubules (the same structures described as intranuclear vesicles in ref. 12) were detected in aortic SMCs of Lmna G609G/G609G mice (Supplemental Figure 7C). Intimal endothelial cells were entirely normal. By electron microscopy, there were no abnormalities in SMCs of the urinary bladder in 8- and 16-week-old Lmna G609G/G609G mice, nor were there nuclear abnormalities in the heart, quadriceps, or kidney (Supplemental Figure 7D). Discussion The hallmark of large artery lesions in HGPS is loss of medial SMCs (40), but for years the mechanism has been elusive. In the current study, we found that progerin expression in SMCs, combined with physiologically low levels of lamin B1, trigger NM ruptures, abnormal nuclear morphology, and SMC death. Our findings in progerin-transfected cultured SMCs and aortic SMCs of Lmna G609G mice were concordant. In cultured SMCs, progerin expression — at levels matching those in aortas of Lmna G609G/+ mice — results in NM ruptures, DNA damage, activation of the cGAS-STING pathway, and cell death. The frequency of NM ruptures in SMCs increased with increasing levels of progerin expression. Reducing lamin B1 expression in progerin-expressing SMCs (so as to mirror the physiologically low levels of lamin B1 in aortic SMCs) and subjecting SMCs to uniaxial stretching (so as to mimic the pulsatile stretching of the aortic wall) increased the number of NM ruptures. Conversely, increasing lamin B1 expression or interfering with the transmission of cytoskeletal forces to the cell nucleus reduced NM ruptures. De Vos and colleagues (30) previously identified NM ruptures in transiently transfected human fibroblasts expressing EGFP-tagged progerin, but the incidence of NM ruptures in cells transfected with a control plasmid (e.g., EGFP-prelamin A) was not determined. The relevance of our cell culture findings to living animals was established by examining the medial SMCs in aortas of Lmna G609G/G609G mice. In those mice, the expression of progerin in aortic SMCs, combined with very low levels of lamin B1, was accompanied by frequent NM ruptures. The onset of NM ruptures in aortic SMCs was at 7 weeks of age, preceding SMC loss. Our studies identified nuclear lamin composition as a key factor in the vascular pathology of HGPS. The levels of progerin (which triggers NM ruptures) increased with age in the aorta of Lmna G609G/+ mice, whereas levels of lamin B1 (which protects from NM ruptures) decreased. The age-related increase in progerin in the aorta occurred despite decreasing progerin transcript levels, suggesting that progerin is a stable protein with a slow turnover rate. Lamin B1 transcripts declined with age in both WT and Lmna G609G/+ mice, but the levels of lamin B1 protein remained stable in WT mice. The preservation of lamin B1 levels in WT mice in the face of reduced transcript levels is consistent with the fact that lamin B1 has a very long half-life in mouse tissues (41, 42). However, given that lamin B1 is a long-lived protein in mice, why did aortic levels of lamin B1 fall with age in Lmna G609G/+ mice? The most likely explanation, we believe, is that progerin accelerates the turnover of lamin B1. In any case, the age-related changes in nuclear lamin composition in SMCs of Lmna G609G/+ mice — high progerin levels and low lamin B1 levels — appear to be “double trouble” for NM integrity, increasing the risk of NM ruptures. We hypothesize that the high levels of progerin and low levels of lamin B1 in SMCs increase nuclear stiffness, rendering nuclei more susceptible to damage from mechanical forces (i.e., that reduced compliance renders the NMs more susceptible to ruptures). All of the cells in the aorta of Lmna G609G mice (intimal endothelial cells, medial SMCs, and adventitial fibroblasts) express high levels of progerin, and all of these cells are subjected to rhythmic stretching from pulsatile blood flow, but both NM ruptures and cell death are confined to the SMCs. We suspect that differences in nuclear lamin composition explain, at least in part, why endothelial and adventitial cells are protected from NM ruptures and cell death. By immunofluorescence microscopy, lamin B1 levels are very low in the medial SMCs of Lmna G609G mice but high in endothelial cells and adventitial cells. In cell culture studies, increasing lamin B1 expression in progerin-expressing SMCs markedly reduced nuclear shape abnormalities, NM ruptures, DNA damage, and cell death. For this reason, we suspect that the robust expression of lamin B1 in endothelial and adventitial cells protects those cells from the toxicity of progerin (i.e., NM ruptures and cell death). Our sequential nuclear extraction experiments provided insights into how lamin B1 reduces progerin-induced NM ruptures. Progerin and lamin B1 both contain carboxyl-terminal modifications (e.g., farnesylation) that promote interaction with the inner NM (36, 43). The nuclear extraction studies revealed that increased lamin B1 expression reduced progerin’s association with the nuclear envelope, likely by limiting interactions between progerin’s farnesyl lipid with the inner NM. Consistent with this interpretation, the association of nf-Prog with NMs was far weaker than with farnesylated progerin. The reduced association of nf-Prog with NMs likely explains the inability of nf-Prog to trigger NM ruptures. We suspect that increased lamin B1 expression limits NM ruptures in progerin-expressing SMCs in the same way, by competing with progerin for binding sites at the nuclear periphery and reducing progerin’s association with the NMs. In addition to limiting progerin’s association with the NMs, increased lamin B1 expression alters the mechanical properties of the cell nucleus. In the past, others have reported, using micropipette aspiration and nuclear strain measurements, that the cell nucleus in progerin-expressing fibroblasts is stiffer than in WT cells (44–46). In the current studies, we used AFM to quantify nuclear stiffness in progerin-expressing SMCs. Consistent with the earlier reports (44–46), we found increased stiffness of the cell nucleus in progerin-expressing SMCs, but we sought to extend this further to determine if boosting lamin B1 expression had an impact on nuclear compliance. Our AFM results were clear: increasing lamin B1 expression decreased nuclear stiffness in SMCs. Lamin B1’s capacity to decrease nuclear stiffness in SMCs was unique; nf-lamin B1, lamin A, and lamin B2 did not alter nuclear stiffness whereas knocking down lamin B1 expression increased nuclear stiffness. Remarkably, increased lamin B1 expression also decreased nuclear stiffness in progerin-expressing SMCs, indicating that the baseline levels of lamin B1 are an important factor in determining nuclear compliance (47). Our discovery that lamin B1 decreases nuclear stiffness was initiated by studies showing that lamin B1 overexpression led to larger nuclei in adherent SMCs. Since cell spreading and cytoskeletal forces can affect nuclear size in adherent cells, we quantified the nuclear area in nonadherent SMCs that were compressed with a glass coverslip. In those studies, we observed larger nuclei in cells that overexpressed lamin B1. Lamin B1 overexpression also increased the nuclear area in progerin-expressing SMCs, but the effect was less pronounced (reflecting progerin’s ability to increase nuclear stiffness). We observed consistent findings in lamin B1–transduced fibroblasts. In those cells, we showed that lamin B1 overexpression increased nuclear area at the expense of reduced nuclear height. The identification of NM ruptures in aortic SMCs of Lmna G609G mice adds to our understanding of the vascular pathology in HGPS. Because NM ruptures in SMCs precede any evidence of SMC loss, it seems likely that the ruptures contribute to cell death. The expression of progerin in cultured SMCs, at levels matching those in the aorta of the Lmna G609G/+ mice, not only leads to NM ruptures but also to DNA damage and activation of the cGAS–STING pathway. The proposal that NM ruptures lead inexorably, directly or indirectly, to SMC death in Lmna G609G mice is strengthened by recent studies of cortical neurons in the developing brain of lamin B1–deficient mouse embryos (26). In those studies, there was a clear association between NM ruptures and the death of migrating neurons within the cortical plate. Recently, NM ruptures were proposed to contribute to skeletal myopathy in Lmna-deficient mice (48). In those studies, NM ruptures were detected in individual muscle fibers that had been isolated by collagenase treatment and manual pipetting; however, it was unclear whether NM ruptures preceded myopathic changes. Our current studies focused on progerin-expressing SMCs and the arterial pathology in the ascending thoracic aorta — the region with the most severe arterial disease. We would point out, however, that the disease phenotypes in mouse models of progeria (13, 49–52) are not confined to the aorta. We have not yet explored whether NM ruptures contribute to disease phenotypes in other tissues, but based on the current studies we would not be surprised if future studies found NM ruptures in other sites, particularly in tissues where conditions favor NM ruptures (i.e., high progerin, low lamin B1, and high mechanical stress). In summary, our current studies showed that the expression of progerin, but not WT prelamin A, in cultured SMCs triggers NM ruptures, DNA damage, and reduced cell survival. We showed that both high levels of progerin and low levels of lamin B1 increase the number of NM ruptures, which are further increased in cells subjected to mechanical stretching. In aortic SMCs of Lmna G609G/+ mice, lamin B1 levels in SMCs are very low. With age, aortic lamin B1 levels in these mice decline while the levels of progerin increase. These findings, combined with our cell culture studies, led us to suspect that we would find NM ruptures in the aortic SMCs of Lmna G609G mice. Indeed, we observed frequent NM ruptures in the medial SMCs of Lmna G609G mice. Importantly, the onset of NM ruptures and the appearance of abnormal nuclear morphology preceded SMC loss. Methods Mice. Mice with a targeted HGPS mutation (Lmna G609G) (11, 12) were bred with Nuc-tdTomato transgenic mice from The Jackson Laboratory (Stock 023035). The Nuc-tdTomato mouse strain was genotyped by PCR with a mutant forward primer 5′-CCAGGCGGGCCATTTACCGTAAG-3′; a WT forward primer 5′-GGAGCGGGAGAAATGGATATG-3′; and a common reverse primer 5′-AAAGTCGCTCTGAGTTGTTAT-3′ (yielding a 320-bp product for the mutant allele and a 603-bp product for the WT allele). Mice were housed in a specific pathogen-free barrier facility with a 12-hour light/dark cycle. The mice were provided pelleted mouse chow (NIH31), water ad libitum, and nutritional food cups (DietGel, ClearH 2 0) as required for supportive care. Cells. Immortalized mouse aortic SMCs (ATCC, CRL-2797) were cultured in DMEM containing 10% FBS (HyClone), 1× nonessential amino acids, 2 mM glutamine, 1 mM sodium pyruvate, and 0.2 mg/mL G418 (12). For some experiments, SMCs were exposed to UV light (100 mJ/cm 2) with a Stratalinker 2400 (Stratagene). To reduce lamin B1 levels, cells were transfected with 100 nM Lmnb1 siRNA (Ambion, AM16706) using RNAiMAX (Invitrogen) according to manufacturer’s instructions. Dox-inducible expression of nuclear lamins in SMCs. SMCs harboring Dox-inducible pTRIPZ expression vectors for human prelamin A and human progerin have been described previously (12). The same approach was used to generate SMCs expressing Dox-inducible constructs for lamin B1, lamin B2, nf-Prog, and nf-lamin B1 (Supplemental Table 3). The expression plasmids were constructed by InFusion cloning (Takara Bio) of cDNAs into pTRIPZ. The mouse lamin B1 cDNA was amplified from pCMV6-Lmnb1 (Origene, MC219388) with forward primer 5′-TACCGGTCCGGAATTATGGCGACCGCGAC-3′ and reverse primer 5′-ATACTCTAGAGCGGCTCACATAATGGCACA-3′. The human lamin B2 cDNA was amplified from pCMV6-LMNB2 (Origene, SC106163) with forward primer 5′-ATGCGTGGACCTGGAGAAA-3′ and reverse primer 5′-GTAGCCCCTTGAATTTCACATCACGTAGCAGCCTCTTGA-3′. A nf-Prog cDNA was created by changing the C-terminal –CSIM signal sequence in pCMV-XL5-progerin (12) to –SSIM using the QuickChange Lightning mutagenesis kit (Agilent) with forward primer 5′-CAGAGCCCCCAGAACTCCAGCATCATGTAATCT-3′ and reverse primer 5′-AGATTACATGATGCTGGAGTTCTGGGGGCTCTG-3′. A nf-lamin B1 cDNA was created by changing the C-terminal –CAIM signal sequence in pCMV6-Lmnb1 (Origene, MC219388) to –SAIM with forward primer 5′-GTCAGATCGCACCGGATGGCGACCCGCGA-3′ and reverse primer 5′-GGCTCACATAATGGCACTGCTTTTATTGGATGCTC-3′. All plasmids were verified by DNA sequencing. Packaging of lentivirus and cell transduction were performed by the UCLA Vector Core. Transduced cells were selected with 1.5 μg/mL puromycin for 2 weeks; clones were isolated by limiting dilution. Constitutive expression of RFP and nuclear lamins in SMCs. SMCs expressing GFP with a nuclear localization signal (NLS-GFP), KASH2-GFP, and extKASH2-GFP have been described previously (12, 26). SMCs expressing NLS-RFP, prelamin A, and progerin were generated by transducing SMCs with pCDH expression plasmids (Supplemental Table 3). The pCDH-NLS-RFP plasmid was created by excising the GFP cDNA in pCDH-NLS-GFP (26) with Sma I and Not I and replacing it with the RFP cDNA from TurboRFP-H2B (Addgene, 58047) by InFusion cloning. The RFP cDNA was amplified with forward primer 5′-GCGTAAGGTTCCCGGTATGAGCGAGCTGATCAAGGAGA-3′ and reverse primer 5′-CAGATCCTTGCGGCCTTATCTGTGCCCCAGTTTGC-3′. The prelamin A and progerin cDNAs were subcloned into Eco RI and Not I sites of pCDH by In-Fusion cloning. Human prelamin A was amplified with forward primer 5′-GCTAGCGAATTATGGAGACCCCGTCCCAGC-3′ and reverse primer 5′-GATCCTTGCGGCCTTACATGATGCTGCAGT-3′; and human progerin was amplified with forward primer 5′-GCTAGCGAATTATGGAGACCCCGTCCCAGC-3′ and reverse primer 5′-CAGATCCTTGCGGCCTTACATGATGCTGCA-3′. All plasmids were verified by DNA sequencing. Packaging of lentivirus and cell transduction was performed by UCLA’s Vector Core. Transduced cells were selected with 3 μg/mL blasticidin (Thermo Fisher Scientific) for 2 weeks; clones were isolated by limiting dilution. Measurement of NM ruptures in live SMCs. SMCs stably expressing GFP or RFP in the nucleus were seeded into 2-well chamber slides with glass coverslip bottoms (Thermo Fisher Scientific) and cultured in DMEM with 10% FBS (as described earlier). Dox was added to induce nuclear lamin expression and incubated for 24 hours before examination by microscopy. The cell culture chamber slide was mounted into a CO 2 and temperature-controlled stage on a Zeiss LSM 800 confocal laser-scanning microscope controlled by Zen Blue 2.3 software (all from Zeiss). The cells were visualized for 24 hours at 37°C and 5% CO 2 with a Plan-Apochromat 20×/0.8 NA objective. Images of a 3 × 3 tiled field (~1.3 mm 2) were captured every 10 minutes. Differential interface contrast (DIC) was acquired with the transmitted light detector (T-PMT; Zeiss) at the same time as the GFP signal. Composite images of 10 z-sections (1 μm sections) were generated and analyzed for NM ruptures. The number of cells with a NM rupture, defined by the escape of GFP (or RFP) in the cytoplasm, was divided by the total number of cells in a field. Measurement of NM ruptures in stretched SMCs. SMCs stably expressing GFP were seeded onto PDMS membranes and cultured for 48 hours. The membranes were clamped into a custom-built biaxial cell stretching device (12) and stretched 2 mm at 0.5 Hz for 2 hours. The membranes were fixed with 4% paraformaldehyde (PFA; MilliporeSigma) and processed for confocal laser-scanning microscopy. Images from random locations were acquired and scored for NM ruptures. A minimum of 200 cells were scored by 2 trained observers blinded to sample identity. Measurement of NM ruptures in aortic sections. Mice expressing the Nuc-tdTomato transgene were perfused with 3% PFA in PBS and tissues postfixed in the same solution at 4°C. Frozen tissue sections (10 μm–thick) were stained with DAPI and mounted in Antifade (Invitrogen). Images were captured on a Zeiss LSM 800 laser-scanning confocal microscope controlled by Zen Blue 2.3 software using a Plan-Apochromat 20×/0.8 NA objective. Maximum image projections were generated from scans of the tdTomato and DAPI channels, combined with the autofluorescence signal in the GFP channel. The latter was used to identify the elastic fibers in the media layer. A NM rupture was defined as the appearance of tdTomato fluorescence outside of a nucleus, as judged by DAPI staining. If tdTomato fluorescence was present between 2 adjacent nuclei, this was counted as a single NM rupture. NM ruptures were counted and expressed relative to the total number of nuclei examined. Measurement of phosphorylated-STING in static and stretched SMCs. SMCs expressing prelamin A or progerin were seeded onto PDMS membranes and cultured for 48 hours. Membranes were stretched for 3 hours (2 mm at 0.5 Hz), fixed with 4% PFA, and processed for immunocytochemistry. Cells were incubated with antibodies against phosphorylated-STING (Cell Signal Technology) and LAP2β, and bound antibodies detected with species-specific fluorescent-labeled antibodies (Supplemental Table 1). Fluorescence microscopy was performed on a Zeiss LSM 800 laser-scanning microscope. Quantification of cells with positive phosphorylated-STING staining was performed by 2 trained observers blinded to sample identity. A minimum of 200 cells were scored per group. Sequential extraction of SMC nuclei. SMC nuclei were extracted as described by Nmezi and colleagues (38). The sequential extraction of nuclei (with low and high salt, detergent, and urea), and quantification of nuclear lamins in the soluble extracts, provides a measure of the association of nuclear lamins with the nuclear envelope. SMCs in 100 mm tissue culture dishes were collected by scraping into 3 mL of ice-cold PBS and centrifugation at 10,000 g for 30 seconds. The cell pellet was resuspended in 1 mL of lysis buffer (ice-cold PBS, 0.1% IGEPAL from Sigma-Aldrich, and protease inhibitors from Sigma-Aldrich) and the lysate centrifuged at 10,000 g for 1 minute at 4°C. The pellet was resuspended in 250 μL of nuclear isolation buffer (10 mM HEPES pH 7.4; 2 mM MgCl 2; 25 mM KCl; 250 mM sucrose; 1mM DTT; and protease inhibitors), and 50 μL (“nuclei” fraction) was set aside for analysis by western blotting. The remainder was sonicated on ice (five 2-second pulses), centrifuged at 20,000 g for 5 minutes at 4°C, and the supernatant (“sonicate” fraction) collected. The pellet was resuspended in 150 μL of nuclear extraction buffer (20 mM HEPES pH 7.4 and protease inhibitors) containing 0.2 M NaCl, and incubated for 20 minutes with end-over-end rotation. The sample was centrifuged at 200,000 g for 10 minutes at 4°C and the supernatant (“0.2 M NaCl” fraction) collected. The extraction procedure was repeated with 150 μL of 0.5 M NaCl, 2% Triton X100, 4 M urea, and 8 M urea (all in nuclear extraction buffer). Protein extracts were stored at –80°C until analysis. Measurement of nuclear area in nonadherent cells. Dox was added to SMCs to induce nuclear lamin expression, and then stained with Hoechst 33342 (5 μg/mL) for 5 minutes in PBS. The cells were washed 3 times with PBS and detached with trypsin. The cells were resuspended in culture media and diluted to 1 × 10 6 cells per mL. A 5-μL aliquot of stained cells was placed into a glass coverslip-bottom dish (Thermo Fisher Scientific) and allowed to settle for 5 minutes. Images of cell nuclei were recorded with a Zeiss LSM 800 confocal microscope before and after placing a 12 mm coverslip (Fisher Scientific) on top of the cells. The nuclear area was measured with ImageJ (NIH) software. Measuring nuclear stiffness by AFM. SMCs in 35 mm glass coverslip-bottom dishes (World Precision Instruments) were incubated with Dox for 24 hours to induce nuclear lamin expression. Nuclear stiffness (Young’s modulus) in adherent cells was measured on a NanoWizard 4a Bioscience AFM (JPK Instruments) coupled with a Zeiss LSM5 confocal fluorescence microscope (53, 54). A spherical AFM tip with a radius of 500 nm and 0.2 N/m spring constant cantilever (Nanotools, B500-CONT) was placed on top of the cell over the nucleus, identified by staining DNA with Hoechst 33342. Three different areas of the nucleus were sampled with a 2-nN maximum set point. The 3 measurements were used to calculate the average Young’s modulus for a single cell. All measurements were performed at 37° C in HEPES-buffered culture medium. The Young’s modulus was measured in 25 randomly selected cells for each group, and the force curves analyzed with JPK Data Processing software (JPK Instruments). The Young’s modulus was calculated with the Hertz model for a spherical tip and applied to fit the slopes of the approach curve. Statistics. Statistical analyses were performed with GraphPad Prism software. Experimental groups were analyzed by unpaired 2-tailed Student’s t test, or 1-way and 2-way ANOVA with Tukey’s multiple comparisons test. Statistical significance was considered when the P value was less than 0.05. Red circles in bar graphs show the average values of independent experiments or values for individual animals. Study approval. All animal studies were approved by UCLA’s Animal Research Committee. Author contributions PHK, SGY, and LGF designed the research studies. PHK, NYC, YT, PJH, TAW, JLCF, and LGF performed the experiments. NKG and PHK conducted the parallel microfiltration studies. PHK, SGY, and LGF wrote the first draft of the manuscript. PHK, NYC, PJH, YT, TAQ, JLCF, NKG, ACR, SGY, and LGF edited the manuscript. Supplemental material View Supplemental data View Supplemental video 1 View Supplemental video 2 View Supplemental video 3 View Supplemental video 4 Acknowledgments We thank Dino Di Carlo (UCLA) for the use of the plasma cleaner. This work was supported by the National Institutes of Health (AG047192) and the National Center for Advancing Translational Sciences UCLA CTSI (UL1TR001881). Virus production and transduction were performed by the IMTC/UCLA Vector Core, which is supported by CURE/P30 DK041301. Address correspondence to: Stephan G. Young or Loren G. Fong, UCLA Department of Medicine/Division of Cardiology, 650 Charles E. Young Dr. South, A2-237 CHS, Los Angeles, California 90095, USA. Phone: 310.825.4934; Email: sgyoung@mednet.ucla.edu (SGY). Phone: 310.825.4997; Email: lfong@mednet.ucla.edu (LGF). Footnotes Conflict of interest: The authors have declared that no conflict of interest exists. Copyright: © 2021, Kim et al. 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4667	https://www.wyzant.com/resources/answers/557274/use-the-discriminant-to-find-the-gradients-of-the-lines-that-pass-through-t	WYZANT TUTORING Elena S. Use the discriminant to find the gradients of the lines that pass through the point (7,1) and are tangent to the circle x^2 + y^2 = 25. Working out please! Step by step.. Thank you! 1 Expert Answer Paul M. answered • 11/28/18 Learn "how to" do the math and why the "how to" works! I am not sure what is meant by using the discriminant to find the gradient; however, here is my solution to the problem. A point on the circle is (x,sqrt(25-x2) At that point the slope of the tangent line to the circle is -x/sqrt(25-x2). Then the equation of the tangent line is -x/sqrt(25-x2) = [sqrt(25-x2) - 1]/(x-7) Cross multiply -x2 + 7x = 25 - x2 - sqrt(25-x2) => 7x - 25 = sqrt(25-x2) and then 49x2 - 350x + 625 = 25 - x2 =>50x2 -350x + 600 = 0 which has solutions x=3 and x = 4. You now need a figure to see that the points of tangency are (3.4) and (4,-3) from which you can compute the slope of each tangent!, Still looking for help? Get the right answer, fast. Get a free answer to a quick problem. Most questions answered within 4 hours. OR Choose an expert and meet online. No packages or subscriptions, pay only for the time you need. RELATED TOPICS RELATED QUESTIONS Find a is y = 3(a+2)x^2 + 6ax + (4-3a) has no zeros. Answers · 1 Find the gradients of the lines that pass through the point (1,7) and are tangent to the parabola y = (2 - x) (1 + 3x). Answers · 1 The line y = mx + 4 is a tangent to y = 3x^2 + 5x + 7. Find the value of m. Answers · 1 for what values of b is the line y=x+b a tangent to the curve y=2x^2-7x 4? Answers · 1 RECOMMENDED TUTORS Shefali J. Frankie B. Violet C. find an online tutor Download our free app A link to the app was sent to your phone. Get to know us Learn with us Work with us Download our free app Let’s keep in touch Need more help? Learn more about how it works Tutors by Subject Tutors by Location IXL Comprehensive K-12 personalized learning Rosetta Stone Immersive learning for 25 languages Education.com 35,000 worksheets, games, and lesson plans TPT Marketplace for millions of educator-created resources Vocabulary.com Adaptive learning for English vocabulary ABCya Fun educational games for kids SpanishDictionary.com Spanish-English dictionary, translator, and learning Inglés.com Diccionario inglés-español, traductor y sitio de aprendizaje Emmersion Fast and accurate language certification
4668	https://artofproblemsolving.com/wiki/index.php/2013_USAMO_Problems/Problem_4?srsltid=AfmBOorFuU2_NPkhBrXp5llLC8or76jMFcCFdTrLsEe-dpzAhj12tHxa	Art of Problem Solving 2013 USAMO Problems/Problem 4 - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki 2013 USAMO Problems/Problem 4 Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search 2013 USAMO Problems/Problem 4 Problem Find all real numbers satisfying Solution (Cauchy or AM-GM) The key Lemma is: for all . Equality holds when . This is proven easily. by Cauchy. Equality then holds when . Now assume that . Now note that, by the Lemma, . So equality must hold in order for the condition in the problem statement to be met. So and . If we let , then we can easily compute that . Now it remains to check that . But by easy computations, , which is obvious. Also , which is obvious, since . So all solutions are of the form , and all permutations for . Remark: An alternative proof of the key Lemma is the following: By AM-GM, . Now taking the square root of both sides gives the desired. Equality holds when . Solution 2 WLOG, assume that . Let and . Then , and . The equation becomes Rearranging the terms, we have Therefore and Express and in terms of , we have and Easy to check that is the smallest among , and Then , and Let , we have the solutions for as follows: and permutations for all --J.Z. These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
4669	https://www.superteacherworksheets.com/least-common-multiple.html	Log In Become a Member Membership Info Addition (Basic) Addition (Multi-Digit) Algebra & Pre-Algebra Comparing Numbers Daily Math Review (Math Buzz) Division (Basic) Division (Long Division) Hundreds Charts Multiplication (Basic) Multiplication (Multi-Digit) Order of Operations Place Value Skip Counting Telling Time Word Problems (Daily) More Math Worksheets Reading Comprehension Reading Comprehension Pre-K/K Reading Comprehension Grade 1 Reading Comprehension Grade 2 Reading Comprehension Grade 3 Reading Comprehension Grade 4 Reading Comprehension Grade 5 Reading Comprehension Grade 6 Reading Comprehension Reading & Writing Reading Worksheets Cause & Effect Daily ELA Review (ELA Buzz) Fact & Opinion Fix the Sentences Graphic Organizers Synonyms & Antonyms Writing Prompts Writing Story Pictures Writing Worksheets More ELA Worksheets Phonics Consonant Sounds Vowel Sounds Consonant Blends Consonant Digraphs Word Families More Phonics Worksheets Early Literacy Alphabet Build Sentences Sight Word Units Sight Words (Individual) More Early Literacy Grammar Nouns Verbs Adjectives Adverbs Pronouns Punctuation Syllables Subjects and Predicates More Grammar Worksheets Spelling Lists Spelling Grade 1 Spelling Grade 2 Spelling Grade 3 Spelling Grade 4 Spelling Grade 5 Spelling Grade 6 More Spelling Worksheets Chapter Books Bunnicula Charlotte's Web Magic Tree House #1 Boxcar Children More Literacy Units Science Animal (Vertebrate) Groups Butterfly Life Cycle Electricity Human Body Matter (Solid, Liquid, Gas) Simple Machines Space - Solar System Weather More Science Worksheets Social Studies 50 States Explorers Landforms Maps (Geography) Maps (Map Skills) More Social Studies Art & Music Colors Worksheets Coloring Pages Learn to Draw Music Worksheets More Art & Music Holidays Autumn Back to School Johnny Appleseed Halloween More Holiday Worksheets Puzzles & Brain Teasers Brain Teasers Logic: Addition Squares Mystery Graph Pictures Number Detective Lost in the USA More Thinking Puzzles Teacher Helpers Teaching Tools Award Certificates More Teacher Helpers Pre-K and Kindergarten Alphabet (ABCs) Numbers and Counting Shapes (Basic) More Kindergarten Worksheet Generator Word Search Generator Multiple Choice Generator Fill-in-the-Blanks Generator More Generator Tools S.T.W. Full Website Index Help Files Contact Least Common Multiple - LCM Use these worksheets when teaching students to calculate the least common multiple, or LCM, of a group of numbers. LCM of 2 Numbers Least Common Multiple Another worksheet in which students determine the least common multiple for each number pair. 4th through 6th Grades View PDF Least Common Multiple 2 FREE Find the LCM for each pair of numbers. 4th through 6th Grades View PDF Least Common Multiple 3 Step-by-step practice sheet for finding the LCM of 2 numbers. 4th through 6th Grades View PDF Least Common Multiple 4 Students find the LCM for each number set, then answer the short-answer thinking questions. 4th through 6th Grades View PDF Least Common Multiple 5 Another worksheet on least common multiple 4th through 6th Grades View PDF Task Cards: LCM This file contains 15 least common multiple activity cards. Use them for learning centers, peer review, exit slips, classroom scavenger hunts, or on your Smartboard's document camera. 4th through 6th Grades View PDF LCM of 3 Numbers Least Common Multiple: 3 Numbers Find the LCM for each set of three of numbers. 4th through 6th Grades View PDF See Also: Factors and GCF Factor tree worksheets and greatest common factor Multiples On these printable worksheets, students will identify multiples of the given numbers. Fraction Worksheets (Advanced) Simplifying fractions, equivalent fractions, and comparing fractions Multiplication Worksheets Games and worksheets that teach basic multiplication facts Sample Worksheet Images My Account Become a Member Membership Information Site License Information Login Site Information Help / FAQ Full Site Index Privacy Policy Terms of Service Contact Us Useful Links Homepage What's New? Blog Newsletter Generators Social Media Follow Us Not a Member? For complete access to thousands of printable lessons click the button or the link below. Become a Member © 2025 Super Teacher Worksheets
4670	https://etd.ohiolink.edu/acprod/odb_etd/ws/send_file/send?accession=case1307497361&disposition=inline	Application of Volatility Diagrams for Investigating Dry Etching Reactions of Chromium Oxide in LTCSS Processing by Man-Ching Hsieh Submitted in partial fulfillment of the requirements For the degree of Master of Science Thesis adviser: Arthur H. Heuer Department of Materials Science and Engineering CASE WESTERN RESERVE UNIVERSITY August, 2011 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of ________ candidate for the ______degree . (signed)_______ (chair of the committee) _______ ______ ______ ______ ______ (date) ________ We also certify that written approval has been obtained for any proprietary material contained therein. TABLE OF CONTENTS List of Tables ................................................................................................................ 4 List of Figures ...............................................................................................................6 Acknowledgement........................................................................................................10 Abstract ........................................................................................................................11 Chapter 1 Introduction ..............................................................................................12 Chapter 2 Literature Review……………...………...…………................................13 2.1 Low-temperature gas-phase carburization............................................................ 13 2.2 Volatility diagrams …………………………..………………………….……......16 Chapter 3 Calculation Methods…………………...…...……...………...………......18 3.1 Volatility diagrams for Cr-O system...................................................................... 18 3.2 Thermodynamic database...................................................................................... 18 3.3 Construction of volatility diagrams....................................................................... 19 3.4 Equilibria between condensed phases................................................................... 21 3.5 Equilibria between Cr(g) and condensed phases................................................... 21 3.6 Equilibria between CrO(g) and condensed phases................................................ 22 3.7 Equilibria between CrO 2 (g) and condensed phases............................................... 23 3.8 Equilibria between CrO 3 (g) and condensed phases.............................................. 23 3.9 Analysis of the volatility diagram for the Cr-O system......................................... 24 3.10 Isomolar line: Cr-based gas pressure in a nonreactive environment……..……. 25 3.11 Metastable Vapor Pressure Lines ………………….……………………..……..26 1 3.12 Isobaric points: Etching in a reactive environment based on hydrogen gas………………………………………………………………………………..….. 26 3.13 Isobaric lines: Etching in a reactive environment based on mono-atomic hydrogen…………………………………………………..…………………..……. 29 Chapter 4 Volatility diagrams in chlorine-based environments……………………. 44 4.1 Construction of volatility diagram for Cr-Cl system……..……………............ 44 4.2 Etching of Cr 2 O3 due to formation of Cr-based gas species in HCl environment………………………………………...……………………………...... 45 Chapter 5 Etching of Chromium (VI) oxide…...……………...……………............ 58 5.1 Etching of CrO 3 due to formation of Cr-based gas species in HCl environment………………………………………...…………………………...…... 58 5.2 Construction of volatility diagram for Cr-O system in ozone and atomic oxygen environments………...………………………...…………….…………………….... 59 Chapter 6 Conclusions ……..………......................................................................... 70 Chapter 7 Future Work………………………............................................................ 71 Appendix A Cr 2 O3 (c) reduction into CrO(g) in hydrogen environment…...………. 72 Appendix B Cr 2 O3 (c) reduction into CrO(g) in the mono-atomic hydrogen environment……………………………………………………………………...….. 73 Appendix C Constructing volatility diagrams for Chromium - atomic oxygen system.......................................................................................................................... 74 Appendix D Constructing volatility diagrams for Chromium - atomic oxygen system....………………………….............................................................................. 76 2 Appendix E Possibly achieved maximum partial pressure of Cr-Cl gaseous species from CrO 3 (c) in the constant flowing HCl ………………….…...…………………78 Appendix F Possibly achieved maximum partial pressure of Cr-Cl gaseous species from CrO 3 (c) in the constant flowing HCl…………………….....…………………81 Reference…………………..…………………........................................................... 84 3 List of Tables Table I Thermodynamic data utilized for construction of volatility diagrams in the Cr-O system……………………………………………………………………...……... 41 Table II Reactions utilized for plotting volatility diagrams in the Cr-O system in oxygen environment.................................................................................................... 42 Table III Possible etching reactions between Cr-O condensed phases and atomic hydrogen or hydrogen gas........................................................................................... 43 Table IV The chlorine partial pressure from HCl dissociation.................................... 54 Table V Thermodynamic data utilized for construction of volatility diagrams in the Cr-Cl system............................................................................................................... 55 Table VI Reactions utilized for volatility diagrams in the Cr-Cl system.................... 56 Table VII Possible etching reactions between Cr 2O3 (c) and HCl …….……..........…57 Table VIII Possible etching reactions between CrO 3 (c) and HCl …………………...67 Table IX Reactions utilized for plotting volatility diagrams in the Cr-O system in atomic oxygen environment……………………………...…………………………. 68 Table X Reactions utilized for plotting volatility diagrams in the Cr-O system in ozone environment...................................................................................................... 69 Table XI Cr 2O3 (c) reduction into CrO(g) in the hydrogen environment..................... 72 Table XII Cr 2 O3 (c) reduction into CrO(g) by monoatomic hydrogen………..............73 Table XIII Possibly achieved maximum partial pressure of CrCl 2 (g) for Cr 2 O3 (c) in the constant flowing HCl environment …………………………………………...... 78 4 Table XIV Possibly achieved maximum partial pressure of CrCl 3 (g) for Cr 2 O3 (c) in the constant flowing HCl environment …………………………………………...... 79 Table XV Possibly achieved maximum partial pressure of CrCl 4 (g) for Cr 2 O3 (c) in the constant flowing HCl environment ………………………….……….……………...80 Table XVI Possibly achieved maximum partial pressure of CrCl 2 (g) for CrO 3 (c) in the constant flowing HCl environment ………………………………............................ 81 Table XVII Possibly achieved maximum partial pressure of CrCl 3 (g) for CrO 3 (c) in the constant flowing HCl environment ……………………………………….......... 82 Table XVIII Possibly achieved maximum partial pressure of CrCl 4 (g) for CrO 3 (c) in the constant flowing HCl environment ……….……………………………………..83 5 List of Figures Figure 1(a). Volatility diagram for Cr-O system at 373K showing the equilibrium lines between Cr gas and all the condensed phases..............................................................30 Figure 1(b). Volatility diagram for Cr-O system at 373K showing the equilibrium between CrO gas phase and Cr condensed phase....................................................... 30 Figure 1(c). Volatility diagram for Cr-O system at 373K showing the equilibrium between CrO 2 (g) condensed phases............................................................................ 31 Figure 1(d). Volatility diagram for Cr-O system at 373K showing the equilibrium between CrO 3 (g) condensed phases ........................................................................... 31 Figure 2(a). The complete volatility diagram for Cr-O system at 373K ……........... 32 Figure 2(b). Complete volatility diagram for Cr-O system at 473K........................... 33 Figure 2(c). Complete volatility diagram for Cr-O system at 573K…….…..……….33 Figure 2(d). Complete volatility diagram for Cr-O system at 673K........................... 34 Figure 3(a). Master volatility diagram for the Cr-O 2 system...................................... 34 Figure 3(b). Master volatility diagram for the Cr-O 2 system with isomolar line…………………………………………………………………………………...35 Figure 4(a). Complete volatility diagram for the Cr-O system at 373K with metastable extention…………………………………………..…............................................... 35 Figure 4(b). Complete volatility diagram for the Cr-O system at 473K with metastable extention..................................................................................................................... 36 Figure 4(c). Complete volatility diagram for Cr-O system at 573K with metastable extention...................................................................................................................... 36 6 Figure 4(d). Complete volatility diagram for Cr-O system at 673K with metastable extention...................................................................................................................... 37 Figure 5(a) Volatility diagram for the Cr-O system showing the isobaric lines for various hydrogen partial pressures........................................................................ 38 Figure 5(b) Volatility diagram for the Cr-O system showing the isobaric lines for various partial pressures of atomic hydrogen........................................................ 39 Figure 5(c). logP Cr as a function of logP H for isobaric points..................................... 40 Figure 6(a). Complete volatility diagram for Cr-Cl system at 373K......................... 47 Figure 6(b). Complete volatility diagram for Cr-Cl system at 473K........................ 48 Figure 6(c). Complete volatility diagram for Cr-Cl system at 573K........................ 48 Figure 6(d). Complete volatility diagram for Cr-Cl system at 673K.......................... 49 Figure 6(e). Master volatility diagram for the Cr-Cl system...................................... 49 Figure 7(a). Possibly Achieved Maximum CrCl 2 Pressure Over Cr 2 O3 (c) in HCl Environment…………………………………………………………...…..…..……. 50 Figure 7(b). Possibly Achieved Maximum CrCl 3 Pressure Over Cr 2 O3 (c) in HCl Environment……………………………………………………………..…….……. 50 Figure 7(c). Possibly Achieved Maximum CrCl 4 Pressure Over Cr 2 O3 (c) in HCl Environment................................................................................................................ 51 Figure 7(d). Possibly Achieved Maximum CrO 2 Cl 2 Pressure Over Cr 2 O3 (c) in HCl Environment……………………………………...………………............................. 51 Figure 7(e). Possibly Achieved Maximum CrO 2 Cl 2 Pressure Over Cr 2O3 (c) in HCl Environment……………………………...………………………............................. 52 7 Figure 8(a). Cr 2 O3 (c) etched by HCl at 373K............................................................. 52 Figure 8(b). Cr 2 O3 (c) etched by HCl at 473K............................................................ 53 Figure 8(c). Cr 2 O3 (c) etched by HCl at 573K............................................................ 53 Figure 8(d). Cr 2 O3 (c) etched by HCl at 673K............................................................ 54 Figure 9(a). Possibly Achieved Maximum CrCl 2 Pressure Over CrO 3 (c) in HCl Environment……………………………………………………………...………..... 60 Figure 9(b). Possibly Achieved Maximum CrCl 3 Pressure Over CrO 3 (c) in HCl Environment…………………………………………………………………….…... 61 Figure 9(c). Possibly Achieved Maximum CrCl 4 Pressure Over CrO 3 (c) in HCl Environment…………………………………………………………………..…...... 61 Figure 9(d). Possibly Achieved Maximum CrO 2 Cl 2 Pressure Over CrO 3 (c) in HCl Environment................................................................................................................ 62 Figure 10(a). CrO 3 (c) etched by HCl at 373K............................................................ 62 Figure 10(b). CrO 3 (c) etched by HCl at 473K............................................................ 63 Figure 10(c). CrO 3 (c) etched by HCl at 573K............................................................ 63 Figure 10(d). CrO 3 (c) etched by HCl at 673K............................................................ 64 Figure 11. Schematics of the planar APPJ…………………..……………….…….....64 Figure 12. Temperature map of the APPJ’s effluent……………………………........ 65 Figure 13. Volatility diagram for Cr-O system in atomic oxygen environment……………………………………………………………………...…...65 Figure 14. Volatility diagram for Cr-O system in ozone environment….…………... 66 Figure 15(a). Complete volatility diagram for Cr-O system in the atomic oxygen 8 environment at 373K…………………………………............................................... 74 Figure 15(b). Complete volatility diagram for Cr-O system in the atomic oxygen environment at 473K…………………………………............................................... 74 Figure 15(c). Complete volatility diagram for Cr-O system in the atomic oxygen environment at 573K…………………………………............................................... 75 Figure 15(d). Complete volatility diagram for Cr-O system in the atomic oxygen environment at 673K…………………………………............................................... 75 Figure 16(a). Complete volatility diagram for Cr-O system in the ozone environment at 373K…………………………………………........................................................ 86 Figure 16(b). Complete volatility diagram for Cr-O system in the ozone environment at 473K…………………………………………........................................................ 86 Figure 16(c). Complete volatility diagram for Cr-O system in the ozone environment at 573K………………………………………………................................................ 87 Figure 16(d). Complete volatility diagram for Cr-O system in the ozone environment at 673K………………………………………………................................................ 87 9 ACKNOWLEDGMENTS I am heartily thankful to my advisor, Dr. Arthur H. Heuer, whose guidance and support from the initial to the final level enabled me to develop an understanding of the subject and successfully finish my thesis. I would like to thank Dr. Gary M. Michal for his kind help with my detailed calculation work. I am grateful to my committee, Dr. Mark R. DeGuire, for his valuable suggestions and assistance about improving my thesis draft. Last but not least, I offer my regards and blessings to all of those who supported me in any respect during the completion of the project. . 10 Application of Volatility Diagrams for Investigating Dry Etching Reactions of Chromium Oxide in LTCSS Processing Abstract by MAN-CHING HSIEH Low-temperature colossal supersaturation (LTCSS) is a diffusional surface hardening process that provides a uniform and conformal hardened gradient surface on austenitic stainless steels. The treatment retains the austenitic phase. In addition, because parts are treated at low temperature, they do not distort or change dimensions. To make the process work, the naturally forming Cr 2 O3 passive layer on the stainless steel has to be removed to “activate” the surface for carburization. Without this activation step, carbon diffusion into the surface is blocked. This is solved by placing the steel in an atmosphere of gaseous HCl, which removes the Cr 2 O3 from the surface. The passive film reactions with gaseous HCl atmosphere importantly influence the effectiveness of the following diffusion process, and thus we should investigate the detailed reactions during the activation step. The present work makes use of volatility diagrams for understanding all possible etching reactions in the Cr-O and Cr-Cl system. The results show three effective methods for etching Cr 2 O3 (c): (1) atomic hydrogen (0.1-0.0001bar) can etch Cr 2 O3 (c) effectively at 373-673K. (2) hydrogen chloride gas (at around 1bar) can etch Cr 2 O3 (c) effectively only when the temperature is higher than about 473K. (3) CrO 3 (c) can be etched more effectively than Cr 2 O3 (c) in a HCl environment. Thus, we can oxidize Cr 2O3 (c) into CrO 3 (c) by ozone or atomic oxygen and then etch CrO 3 (c) in a HCl environment to achieve a higher etching rate. 11 Chapter 1 Introduction Chromium oxide (Cr 2 O3) is the major compound in the passive film of stainless steel, which prevents stainless steel from further oxidation. Although chromium oxide is desirable for corrosion prevention, during surface hardening processing, it needs to be removed. A typical method of increasing the surface hardness of stainless steel is introducing carbon into the interstitial sites. Since the passive film would block the diffusion of carbon, before the carbon diffusion process, the naturally forming chromium oxide layer on the stainless steel has to be removed to “activate” the surface for carburization. This activation step is achieved by placing the steel in an atmosphere of gaseous HCl, which etches away the chromium oxide from the surface. The reactions between chromium oxide and a gaseous HCl importantly influence the effectiveness of the subsequent carbon diffusion process. Thus, the detailed reactions during the activation step deserve closer investigation. The present work makes use of volatility diagrams for understanding etching reactions in the Cr-O and Cr-Cl system. We investigate the volatility diagrams in the Cr-O and Cr-Cl systems to see the reactions in air and in HCl/Cl 2 atmospheres by thermodynamic calculations and then evaluate the effectiveness of etching reactions by the maximum possible achieved vapor pressure of Cr-Cl gas species. Possible alternative procedures to remove the passive layers are also discussed. 12 Chapter 2 Literature Review 2.1 Low-temperature gas-phase carburization Low-temperature gas-phase carburization, or low-temperature colossal supersaturation (LTCSS), is a carburization process used to generate very high surface interstitial carbon contents in an austenitic stainless steel. The remarkably high carbon content is achieved because the carburization is carried out at temperatures where ‘‘paraequilibrium’’, rather than conventional thermodynamic equilibrium, determines the phase composition: paraequilibrium can be realized under conditions where substitutional solutes such as Cr and Ni are immobile whereas interstitial solutes such as carbon are not 1,2 . This high interstitial carbon content can significantly improve the surface mechanical properties and corrosion resistance of austenitic stainless steel with essentially no loss in ductility. Recently, researchers have investigated many aspects of steels treated via this process, such as fatigue performance, wear resistance and other mechanical properties 3-5 . The carburized case, with a surface hardness three times that of the core and a surface compressive stress greater than 2 GPa, leads to significantly enhanced fatigue performance, due to residual compressive stresses accompanying the low-temperature carburization . The so-called endurance limit (defined as the stress at which the fatigue life is 10 7 cycles) increased from about one-third to about one-half the yield stress (from 200 to 325 MPa). Fractographic investigations revealed that surface stresses change the preferred site of fatigue crack nucleation from the surface for 13 noncarburized samples to the interior for carburized samples 3 . The hardened surface also enhances the wear resistance. Multiple treatments of LTCSS have shown higher values in both micro-indentation and scratch hardness tests under heavier loads. The friction and wear characteristics of Type 316 stainless steel with multiple LTCSS treatments were evaluated in non-lubricated unidirectional sliding (pin-on-disk) against Type 440C stainless steel. While little change was observed on friction behavior, substantial further improvement on wear resistance has been achieved for the multiple treatments 5 . The enhanced mechanical properties arise from the enormous carbon solubility without carbide formation. Using interaction parameters from the latest CALPHAD assessment of the Fe-Cr-Ni-carbon system, researchers have calculated the equilibrium and paraequilibrium carbon solubility in a model Fe-18Cr-12 Ni (wt pct) austenitic steel (essentially a model 316L composition), as well as the carbon solubility in this austenite when paraequilibrium carbide formation occurs (i.e., when carbides form in a partitionless manner). For temperatures in the range 725 to 750 K, the discrepancy between the calculated solubility and experimental solubility arises because the CALPHAD Cr–carbon interaction parameters may be not sufficiently exothermic at the low temperatures used for paraequilibrium carburization. After multiple paraequilibrium carburization cycles, carbide formation can occur. The carbides that form under these conditions do so in a nearly-partitionless manner (there 14 is modest Ni rejection to the austenite/carbide interface) and have an stoichiometry of M 5 C2 1 . Low-temperature gas-phase carburization can yield a single-phase ‘‘case’’ with concentrations of interstitially dissolved carbon exceeding the equilibrium solubility limit by orders of magnitude. Upon prolonged treatment, however, carbides (mostly, M 5 C2 ) can precipitate and degrade the properties. A high-resolution transmission electron microscopy study revealed the precise carbide–austenite orientation relationship, a highly coherent interface, and suggested that precipitation only occurs when (i) the carbon-induced lattice expansion of the austenite has reached a level that substantially reduces volume-misfit stress and (ii) diffusional transport of nickel, chromium, and iron enhanced by structural defects can locally reduce the nickel concentration to the solubility limit of nickel in χ -carbide 6 . Carburization of 316L austenitic stainless steel under paraequilibrium conditions produces an anelastic relaxation peak at 543 K (1.0 Hz), due to a carbon-containing defect with a highly anisotropic strain field. Interstitial solid-solution strengthening theories were used to explain the approximate three-fold increase in hardness 7 . Traditional carburization methods have historically decreased the corrosion resistance of stainless steels. However, LTCSS dramatically improves the localized corrosion resistance of 316L austenitic stainless steel in ambient temperature seawater. 15 In particular, the LTCSS-treated steel increases the seawater breakdown potential by more than 600 mV 8 . 2.2 Volatility diagrams Thermodynamic data of many compounds among certain different temperature ranges are available in the JANAF table 9 , the Metals Handbook 10 , the Appendix of Kubaschewski’s Metallurgical Thermochemistry 11 , and the Thermochemical Properties of Inorganic Substances 12 . Graphical representation of these thermochemical data can be very useful. For example, Ellingham diagrams show the oxidation/ reduction behavior of solid or liquid oxides at various O 2 pressures over a wide range of temperature. Another example is Pourbaix diagrams, also known as a potential/pH diagram, which map out possible stable phases of an aqueous electrochemical system. For reactions between gaseous and condensed phases, a proper graphical display is represented as a volatility diagram: an isothermal plot showing the partial pressure of several gaseous species in equilibrium with one or more condensed phases possible in the system 13 . The isothermal plots in volatility diagrams shows pressure of every gas species, and the dominating gas species can thus be readily identified. The stability of the composition can be examined on the basis of volatility diagrams 14 . Passive and active oxidation can also be explained by the information provided in the volatility diagrams 13,15,16 . The metastable extension lines in the volatility diagram were used to predict 16 the maximum possible etching effect and help design another novel etching procedure 17 . 17 Chapter 3 Calculation Methods 3.1 Volatility Diagrams for the Cr-O System In the volatility diagram for the Cr-O system, the partial pressures of the Cr-based gaseous species are plotted as a function of the oxygen partial pressure at given temperatures. For any given Cr-based gas species, this functional relation, which is a line, can be determined from the thermodynamic equilibrium between the relevant Cr-based solid species and the Cr-based gas species. Reactions involving only gases are ignored. Only reactions between condensed phases and the gas phase are considered, since only the gas products can be taken away in the constant flowing environment to increase the effectiveness of these etching reactions. In the following parts of this chapter, we will take several reactions for examples to show how these functional relations are derived. 3.2 Thermodynamic database Thermochemical data of pure substances were available in JANAF tables and other reference books 18 . The thermodynamic database for this thesis was primarily obtained from the book “Thermochemical Properties of Inorganic Substances”. Rather than using the tabulated data available in the JANAF tables, data sources in this book, which contain functions for the specific heat capacity ( Cp) together with the standard enthalpies and entropies of formation, were utilized (see Table I). The condensed species for the Cr-O system included Cr, Cr 2 O3, and CrO 3 . CrO 3 melt 18 at 471K. That is a very low melting point. However, the phase transformation does not significantly influence the volatility diagram because the Gibbs free energy difference is small between the solid and liquid phases. All the other condensed species are still solid at the temperature lower than 673K. H 2(g) and HCl (g) are included for reasons that will become apparent in the isobaric analysis. The gaseous species for which adequate thermodynamic information was not available were not included. 3.3 Construction of volatility diagrams. The list of relevant reactions for constructing the volatility diagram for the Cr-O system is given in Table II. The equilibrium constant for any reaction at a specified temperature can be obtained from the thermodynamic data given in Table I. Take this reaction for example: The equilibrium constant is given by where ∆G, ∆H, and ∆S are the standard Gibbs free energy, enthalpy, and entropy for this reaction; T is the absolute temperature; R is the universal gas constant; K is the equilibrium constant for this reaction. ∆H and ∆S can be obtained respectively from standard enthalpy of formation ∆Ho and from standard entropy of formation S o , 19 together with specific heat capacity C p (T). The oxygen partial pressure at which Cr (c) is converted to Cr 2O3(C) is defined by The values for log( K) in Table II are given at four temperatures: 373K, 473K, 573K, and 673K. The equilibrium pressures of all the gas species can be determined from the equilibrium constants for each reaction through similar calculation. Then, we plot the logarithm of the Cr-based gaseous species (partial pressures) as a function of the logarithm of oxygen partial pressure at various temperatures. For equilibrium between condensed phases, the equilibrium lines are vertical because there is no Cr-based gas involved in these reactions. For equilibrium between condensed phase and gas phase, we consider the equilibrium between each specific gas species and three condensed phases: Cr, Cr 2 O3 , and CrO 3 . The equilibrium lines between condensed phases and the gas phase are linear in the volatility diagrams. 20 3.4 Equilibria between condensed phases. The first step in the construction of the volatility diagram is to determine the regions that contain the condensed phases. For this purpose, reactions 1 and reaction 2 in Table II are used. The oxygen partial pressure at which Cr (s) is converted into Cr 2 O3(s) is obtained from log( K1) in equation . The oxygen partial pressure at which Cr 2O3(s) is converted into CrO 3(s) are obtained from log( K2) by similar calculations. It is assumed that the condensed species are in their standard states, and thus their activities are unity. Since there are no Cr-based gas species involved in this reaction, reaction 1 and 2 appear as vertical lines on the volatility diagram. For Reaction 1 in the volatility diagram, to the left of the vertical line, Cr (s) is the stable phase, while to the right, Cr 2 O3(s) is the stable phase. Similarly, Reaction 2 delineates the Cr 2 O3 and the CrO 2 phase regions The combination of Reactions 1 and 2 gives the stable condensed phases at the specified temperature under equilibrium conditions in Fig. 1(a) 3.5 Equilibria between Cr (g) and condensed phases Reactions 3-5 involve the equilibrium between Cr (g) and the condensed species in this system, Cr, Cr 2 O3 , and CrO 3 . Reaction 3 describes the vaporization of Cr (c) to Cr (g) , and Cr partial pressure can be obtained from log K3: 21 PCr is independent of the partial pressure of oxygen and hence log PCr plots as a horizontal line in Fig.1(a) at a temperature of 373K. The intersection of this line with the vertical boundary between Cr and Cr 2 O3 results in a triple point. Reaction 4expresses the reduction of Cr 2 O3 to Cr (g) and O 2(g) , and Cr partial pressure can be obtained from log K4:. Here, logP Cr is plotted as a line with slope of –3/4. Similar to Reaction 4, Reaction 5 is plotted as a line with slope of -3/2. For the purpose of etching, the weight loss due to the loss of the volatile Cr-based species should be appreciable. Normally, weight loss in a TGA (thermogravimetric analyzer) can be measured if the vapor pressure of the evaporating species is higher than 10 –8 atm (approximately 10 –3 Pa or 10 -8 bar or 10 –5 Torr). That is, effective etching requires the Cr-based gas pressures to be higher than 10 –8 atm. It is seen from Fig. 1(a), that the vapor pressure of Cr at 373K is far lower than 10 –8 bar. In fact, the volatility diagrams for Cr-O system are just used to investigate the Cr-based species in the oxygen gas environment, not in the etching atmosphere. That is why we see the pressures of all the gas species be far lower than 10 –8 bar, the criterion of effective etching. Later, we will find that the pressures of Cr gas species in the etching environment could be higher than the criterion of effective etching. 3.6 Equilibria between CrO(g) and condensed phases. 22 Reactions 6-8 involve the vaporization of CrO (g) from the condensed species in this system, Cr, Cr 2 O3 and CrO 3 . Reaction 6, which describes the oxidation of Cr (c) to CrO (g) , is plotted as a line with slope of 1/2 in Fig.1(b) at a temperature of 373K. Reaction 7 plots as a line with slope of –1/4 and expresses the vaporization of Cr 2O3(s) to CrO (g) . Similarly, Reaction 8 plots as a line with slope of -1. Fig. 1(b) shows the resulting volatility diagram. As we can expect, the maximum partial pressure of CrO is very low since this is not an etching environment. 3.7 Equilibria between CrO 2 (g) and condensed phases. Reactions 9-11 involve the vaporization of CrO 2(g) from the condensed species in this system, Cr, Cr 2 O3 and CrO 3 . Reaction 9, which describes the oxidation of Cr (c) to CrO 2(g) , is plotted as a line with slope of 1 in Fig.1(c) at a temperature of 373K. Reaction 10 plots as a line with slope of 1/4 and expresses the oxidation of Cr 2 O3(s) to CrO 2 (g). Similarly, Reaction 11 plots as a line with slope of -1/2 and expresses the reduction of CrO 3(C) to CrO 2(g) . Fig 1(c) shows the resulting volatility diagram. As expected, the maximum vapor pressure of CrO 2 that can be achieved is still very low at the temperature considered. 3.8 Equilibria between CrO 3 (g) and condensed phases. Reactions 12-14 involve the vaporization of CrO 3 (g) from the condensed species in this system, Cr, Cr 2 O3 and CrO 3 . Reaction 12, which describes the oxidation of Cr (c) to CrO 3(g) , is plotted as a line with slope of 3/2 in Fig.1(d) at a temperature of 373K. 23 Reaction 13 plots as a line with slope of 3/4 and expresses the oxidation of Cr 2 O3(c) to CrO 3(g) . Reaction 14, which describes the vaporization of CrO 3(c) to CrO 3(g) , is independent of the partial pressure of oxygen and hence plots as a horizontal line in Fig.1(d). Fig. 1(d) shows the constructed volatility diagram for equilibrium between CrO 3(g) and condensed phases. The CrO 3(c) - CrO 3(g) line (Reaction 14) has the highest vapor pressure in this diagram. 3.9 Analysis of the volatility diagram for the Cr-O system. A superposition of Fig. 1(a)-(d) leads to the complete volatility diagram for the Cr-O system at 373K in Fig. 2(a). The green line indicates the equilibrium between Cr(g) and condensed phase; the red line indicates the equilibrium between CrO(g) and condensed phase; the blue line indicates the equilibrium between CrO 2 (g) and condensed phase; the magenta colored line indicates the equilibrium between CrO 3 (g) and condensed phase. The dominant vapor pressure lines corresponding to the highest Cr-based gas pressure at different partial oxygen pressure can be observed clearly in Figure 2(a): the dominant gas species in region I is Cr (g) ; the dominant gas species in region II is CrO 2(g) ; the dominant gas species in region III is CrO 3(g) ; CrO (g) is not the dominant gas species for any region. By the same calculation method, we constructed the volatility diagrams at 473K, 573K, and 673K, shown in Figure 2(b)-(d). The equilibrium lines for all the dominating gas species at different temperatures are shown in Figure 3(a). The dominating Cr-based gas pressures increase with the increase of temperature, but all the Cr-based gas pressures are lower than 10 –8 bar 24 (Figure 3(a)). 3.10 Isomolar line: Cr-based gas pressure in a nonreactive environment. The dashed line in Figure 3(b) is called an isomolar line, which is plotted according to the mass balance criterion. For reaction 4, when 1 mol of condensed Cr 2 O3undergoes vaporization, 2 mol of Cr(g) and 1.5 mol of O 2 (g) are produced at the same time in a nonreactive environment: Assuming ideal gas behavior and equal diffusivity of all gaseous species, the mass balance criterion requires that or This results in a line known as the isomolar line that always has a slope of 1. Intersection of this line with the equilibrium Cr 2 O3(c) -Cr (g) line results in a isomolar point where both the mass balance criterion as well as the thermodynamic equilibrium specified by Reaction 4 are satisfied. The value of this isomolar point may be easily determined by substitution of the mass balance criterion into the relation for the equilibrium constant of Reaction 4 at given temperatures. The isomolar points that fall on the Cr 2 O3(c) -Cr (g) vapor pressure line are marked in Figure 3(b). This isomolar line defines the maximum partial pressure of Cr (g) in a non- reactive 25 environment . That is, Cr creates its own atmosphere, and no etching gases or other reactant gases were added into this system. 3.11 Metastable Vapor Pressure Lines The solid lines in Figure 2 (a)-(d) indicate the thermodynamic equilibrium between different phases. However, the etching process, such as constant flowing conditions, may not be in an equilibrium situation. In the non-equilibrium situation, the metastable extensions of stable equilibrium lines may be useful. We plot the metastable extensions of all the stable equilibrium lines as dashed lines in the Fig. 4(a)-(d), but we should understand that only the extentions to higher pressure are meaningful. In essence, although thermodynamically metastable, the vapor pressure lines may be kinetically stable under reducing conditions in a flowing environment. In a hydrogen gas or hydrogen plasma environment, chromium oxides may be reduced into gas species. This is a possible method to remove the chromium oxides on the stainless steel, so that the passive film can be removed to let the carbon species diffuse into stainless steel. The maximum vapor pressure of Cr (g) over Cr 2 O3(c) for a given hydrogen pressure at a specified temperature can be defined as isobaric points. 3.12 Isobaric points: Etching in a reactive environment based on hydrogen. The isomolar point at a given temperature for the Cr 2 O3(c) -Cr (g) dissociation reaction is no longer applicable in an etching environment. In the presence of hydrogen, the 26 relevant reaction can be considered as: According to this reaction, the mass balance relation can be written as: or The equilibrium constant for this reaction can be written as: The equilibrium constant K(T) can be found in Table III. The activity of Cr 2 O3(c) is unity assuming that it is in its standard state. Hence, Substituting the mass balance relation (equation 12) into above equation gives the partial pressure of Cr (g) at the isobaric point. Equation 15 indicates that at a given temperature the isobaric point is a function of hydrogen pressure. We set hydrogen pressures as 10 bar, 1 bar and 0.1 bar, and then calculated the corresponding log PCr values. We marked these log PCr values as isobaric points on the Cr 2 O3(c) -Cr (g) extension line in the volatility diagram (Figure 5a). As a 27 result, these isobaric points satisfy both equation and equation at the same time. Equation indicates the equilibrium line between Cr(g) and Cr 2 O3 (C); equation indicates the equilibrium between Cr(g) and Cr 2 O3 (c) with a certain fixed hydrogen partial pressure. Isobaric point is actually the intersection between the extended Cr(g)/Cr 2 O3 (C) equilibrium line and the equilibrium line obtained from equation at given partial hydrogen pressure. In a constant flowing environment with a fixed hydrogen pressure, the isobaric point indicates the maximum Cr vapor pressure over Cr 2 O3 (c). As the Cr vapor is taken away with the constant flowing hydrogen gas, Cr 2 O3 (c) may release Cr vapor again and again. However, the isobaric point indicates the equilibrium between fixed hydrogen pressure and Cr vapor at a specified temperature, so that the vapor pressure can not be higher than the value indicated by the isobaric point. Cr 2 O3 (c) may also be reduced to CrO (g). However, CrO(g) is not the dominating gas species in the volatility diagrams. That is, the CrO partial pressure is relatively small. Actually, the generated CrO partial pressure is far lower than 10 -8 bar even in an atomic hydrogen environment. Thus, we put its related reaction in Appendix A and Appendix B and do not discuss it here. In Figure 5(a), the dashed isobaric lines are obtained by connecting the isobaric points of different temperatures at the given hydrogen pressure. All these three isobaric lines in Figure 5(a) are parallel to each other with a positive slope. These isobaric lines will be discussed in section 3.13. 28 3.13 Isobaric lines: Etching in a reactive environment based on atomic hydrogen. Figure 5(a) shows that hydrogen gas cannot etch Cr 2 O3(c) effectively. The free energy of Cr 2 O3 (c) is far lower than the free energy of Cr(g) so that hydrogen gas is not able to reduce Cr 2 O3 (c) into Cr(c). Thus, we consider monoatomic hydrogen, a much more reactive environment: Data for the reactions between chromium oxide and atomic hydrogen are listed in Table III. Similarly, we find the isobaric points at monoatomic hydrogen partial pressure of 0.1 bar, 0.01 bar, 0.001 bar, and 0.0001 bar in Fig. 5(b). The isobaric lines are obtained by joining the isobaric points of different temperatures at the given hydrogen pressure. All these three isobaric lines in Figure 5(b) are parallel to each other, but their slope is negative, in contrast to Fig 5(a). The isobaric points indicate the maximum vapor pressure of Cr (g) over Cr 2 O3(c) for a given monoatomic hydrogen pressure at a specified temperature. From Fig 5(b), we can see that monoatomic hydrogen with a partial pressure between 0.0001 bar and 0.1 bar can etch Cr 2 O3(c) effectively. Experimentally, the volatility diagram is what we can not change, but the atomic hydrogen pressure is what we can control in the etching process. To see the influence of atomic hydrogen pressure clearly, we depicted the isobaric points in Fig 5(b) as a function of atomic hydrogen pressure in Fig 5(c). In the following chapter 4, we will also express the isobaric points as a function of reacting gas pressure (e.g. PHCl ), so that the environmental effects can be seen clearly. 29 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 CrO 3(c) T = 373 K logP Cr (Unit: bar) logP O2(Unit: bar) (1) (2) (3) (4) (5) Cr (c) Cr 2O3(c) Cr (g) Figure 1(a). Volatility diagram for Cr-O system at 373K shows the equilibrium lines between Cr gas and all the condensed phases -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 T = 373 K logP CrO (Unit: bar) logP O2(Unit: bar) (1) (2) (6) (7) (8) Cr (c) Cr 2O3(c) CrO 3(c) CrO (g) Figure 1(b). Volatility diagram for Cr-O system at 373K shows the equilibrium between CrO gas phase and Cr condensed phase. 30 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 T = 373 K logP CrO 2(Unit: bar) logP O2(Unit: bar) (1) (2) (9) (10) (11) Cr (c) Cr 2 O3(c) CrO 3(c) CrO 2(g) Figure 1(c). Volatility diagram for Cr-O system at 373K shows the equilibrium between CrO 2 (g) and the condensed phases. -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 T = 373 K logP CrO 3(Unit: bar) logP O2(Unit: bar) (1) (2) (12) (13) (14) Cr (c) Cr 2O3(c) CrO 3(c) CrO 3(g) Figure 1(d). Volatility diagram for Cr-O system at 373K shows the equilibrium between CrO 3 (g) condensed phases. 31 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 3 4 5logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) 6 7 8Cr (c) Cr 2O3(c) CrO 3(c) 9 10 11 Region II Region III 12 13 14 T = 373 K Region I Figure 2(a). The complete volatility diagram for Cr-O system at 373K is acombination of Figure 1 (a-d) and shows that the dominating Cr-based gas pressures are different in region I, II, and III. In region I, the dominating gas species is Cr gas. In region II, the dominating gas species is CrO 2. In region III, the dominating gas species is CrO 3 . 32 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 3 4 5Cr (c) Cr 2O3(c) CrO 3(c) 6 7 8 9 10 11 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) 12 13 14 T = 473 K Figure 2(b). Complete volatility diagram for Cr-O system at 473K -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 logP O2(Unit: bar) 3 4 5T = 573 K logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) 6 7 8Cr (c) Cr 2 O3(c) CrO 3(c) 9 10 11 12 13 14 Figure 2(c). Complete volatility diagram for Cr-O system at 573K 33 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) 3 4 5 6 7 8 9 10 11 logP O2(Unit: bar) 12 13 14 T = 673 K CrO 3(c) Cr 2O3(c) Cr (c) Figure 2(d). Complete volatility diagram for Cr-O system at 673K -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) logP O2(Unit: bar) logP O2(Unit: bar) logP O2(Unit: bar) 373K 473K 573K 673K Cr (g) CrO 3(g) Cr (g) CrO 2(g) CrO 3(g) Figure 3(a). Master volatility diagram for the Cr-O 2 system. 34 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) Isomolar line for Reaction 4 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) logP O2(Unit: bar) logP O2(Unit: bar) logP O2(Unit: bar) 373K 473K 573K 673K Figure 3(b). Master volatility diagram for the Cr-O 2 system with isomolar line. -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) T = 373 K Figure 4(a). Complete volatility diagram for the Cr-O system at 373K with metastable extention. 35 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 T = 473 K logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) Figure 4(b). Compltet volatility diagram for the Cr-O system at 473K with metastable extention. -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) T = 573 K logP O2(Unit: bar) Figure 4(c). Complete volatility diagram for Cr-O system at 573K with metastable extention 36 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) T = 673 K Figure 4(d). Complete volatility diagram for Cr-O system at 673K with metastable extention 37 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 0 logP O2(Unit: bar) logP O2(Unit: bar) logP O2(Unit: bar) logP O2(Unit: bar) 373K 473K 573K 673K logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) H2= 10 bar logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) H2= 1 bar logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O2(Unit: bar) H2= 0.1 bar Figure 5(a) Volatility diagram for the Cr-O system showing the isobaric lines for various hydrogen partial pressures for the reaction Cr 2 O3 (c) + 3H 2 (g) = 2Cr(g) + 3H 2 O(g). The isobaric lines are obtained by connecting the isobaric points, which indicate the maximum vapor pressure of Cr over Cr 2 O3 (c), using the mass balance criterion, . The isobaric lines have a positive slope. As a result, the Cr pressure for a given pressure of atomic hydrogen increases as the temperature is increased. Exceedingly low Cr pressures over Cr 2 O3 (c) are obtained in this hydrogen environment at low temperatures. Thus, the etch rates for Cr 2 O3 (c) is expected to be low. 38 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 020 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 020 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 020 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 020 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 020 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 020 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 020 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 020 -140 -120 -100 -80 -60 -40 -20 0 20 -140 -120 -100 -80 -60 -40 -20 020 T=473K logP O2(Unit: bar) T=573K logP O2(Unit: bar) logP O2(Unit: bar) T=373K H: 0.1 bar H: 0.01 bar H: 0.001 bar H: 0.0001 bar T=673K logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) Figure 5(b) Volatility diagram for the Cr-O system showing the isobaric lines for various partial pressures of monoatomic hydrogen for the reaction Cr 2 O3 (c)+6H(g)=2Cr(g)+3H 2 O(g). The isobaric lines are obtained in a manner similar to Fig. 5(a) using the same mass balance criterion, . The isobaric lines have a slightly negative slope, which is in contrast to the isobaric lines in Fig. 5(a). As a result, the Cr pressure for a given pressure of atomic hydrogen decreases as the temperature is increased. Extremely high Cr pressures over Cr 2 O3 (c) are obtained in this atomic hydrogen environment at low temperatures. Thus, the etch rates for Cr 2 O3(c) is expected to be high. 39 -4 -3 -2 -1 -2 -1 0123 logP Cr (Unit: bar) logP H (Unit: bar) 373K 473K 573K 673K Figure 5(c) logP Cr as a function of logP H for isobaric points 40 Table I. Thermodynamic data utilized for construction of volitility diagrams in the Cr-O system (T o = 298.15K, P o=0.1MPa) Cp =a+b10 -3 T+c10 6 T-2 +d10 -6 T2 No. Species s o h0 a b c d (JK -1 mol -1 ) (Jmol -1 ) (JK -1 mol -1 ) Ref. 1 Cr(c) 23.640 0 24.514 2.050 -0.180 5.950 2 O2 (g) 205.146 0 29.154 6.477 -0.184 -1.017 3 Cr (g) 174.310 397480 20.983 -1.992 0.038 1.828 4 CrO(g) 239.266 188280 35.422 1.406 -0.402 0 5 Cr 2 O3 (c) 81.170 -1140558 109.650 15.456 0 0 6 CrO 2 (g) 269.240 -75312 52.844 2.753 -0.912 0 7 CrO 3 (c) 73.220 -587015 71.756 87.864 -1.674 0 8 CrO 3 (g) 266.199 -292880 75.709 3.841 -1.854 0 9 CrCl 2 (g) 307.947 -136247 61.116 1.356 -0.343 0 10 CrCl 3 (g) 317.645 -325180 83.345 3.155 -0.736 0 11 CrCl 4 (g) 364.401 -426768 106.433 1.314 -0.950 0 12 H2 (g) 130.679 0 26.882 3.586 0.105 0 13 HCl(g) 186.908 -92307 26.527 4.602 0.109 0 14 O (g) 161.059 249169 21.008 -0.247 0.088 0.071 15 O3 (g) 238.932 142674 54.258 2.004 -1.556 0 16 H2 O(g) 188.824 -241856 34.376 7.841 -0.423 0 17 CrO 2 Cl 2 (g) 329.000 -538623 105.311 1.326 -2.209 0 41 Table II. Reactions utilized for ploting volitility diagrams in the Cr-O system (in oxygen environment). No. Reaction log(K)= -ΔG/(2.303RT) ΔH (Jmol -1 ) 373K 473K 573K 673K 373K 473K 573K 673K 1 2Cr (c) + 3/2 O 2(g) =Cr 2 O3(c) 145.306 111.645 89.768 74.415 -1138425 -1136493 -1134768 -1133215 2 Cr 2 O3(c) +3/2 O 2(g) =2CrO 3(c) -7.952 -8.871 -9.378 -9.644 -32552.9 -28885.1 -23089.5 -15434 3 Cr (c) =Cr (g) -47.765 -36.016 -28.38 -23.023 397177.5 396702.9 396105.7 395397 4 Cr 2 O3(c) =2 Cr (g) +3/2 O 2(g) -240.837 -183.682 -146.533 -120.463 1932780 1929898 1926979 1924009 5 CrO 3(c) = Cr (g) + 3/2 O 2(g) -116.442 -87.405 -68.577 -55.409 982666.5 979391.8 975034.3 969721.5 6 Cr (c) + 1/2 O 2(g) =CrO (g) -20.440 -14.897 -11.303 -8.787 187637.1 186934.5 186167.3 185313.1 7 Cr2O 3(c) =2CrO(g) + 1/2 O 2 (g) -186.186 -141.439 -112.375 -91.991 1513699 1510362 1507102 1503841 8 CrO 3(c) = CrO (g) + O 2(g) -89.117 -66.284 -51.498 -41.173 773126.2 769623.4 765095.9 759637.6 9 Cr (c) + O 2(g) =CrO 2(g) 12.669 10.407 8.920 7.865 -76120.9 -76846.8 -77568.7 -78344.8 10 Cr 2 O3(c) + 1/2 O 2(g) =2CrO 2(g) -119.967 -90.831 -71.928 -58.685 986183.3 982799 979630.3 976525.4 11 CrO 3(c) = CrO 2(g) + 1/2 O 2(g) -56.007 -40.979 -31.274 -24.520 509368.1 505842.1 501359.9 495979.7 12 Cr (c) + 3/2 O 2(g) = CrO 3(g) 37.614 28.915 23.245 19.256 -293742 -294202 -294508 -294790 13 Cr 2 O3(C) +3/2 O 2(g) =2CrO 3(g) -70.076 -53.814 -43.277 -35.902 550941.9 548088 545752.5 543636 14 CrO 3(c) =CrO 3(g) -31.062 -22.471 -16.949 -13.129 291747.4 288486.5 284421 279535 42 43 Table III. Possible etching reactions between Cr-O condensed phases and atomic hydrogen or hydrogen gas No. Reaction log(K)= -ΔG/(2.303RT) ΔH (Jmol -1 ) 373K 473K 573K 673K 373K 473K 573K 673K H1 Cr 2 O 3 (c) + 3H 2(g)= 2Cr (g) + 3H 2O(g) -146.178 -110.624 -87.585 -71.465 1204104.897 1198513.331 1193187.937 1188003.040 H2 Cr 2O 3(c) + 6H(g)=2Cr (g) + 3H 2O(g) 21.462 18.160 15.831 14.073 -107250.499 -116615.460 -125668.986 -134508.975 H3 Cr 2 O 3(c) + H 2(g)=2CrO(g) + H 2O(g) -154.633 -117.086 -92.725 -75.659 1270807.630 1266566.597 1262505.192 1258505.948 H4 Cr 2 O 3(c) + 2H(g)=2CrO(g) + H 2O(g) -98.753 -74.159 -58.253 -47.146 833689.000 828190.000 822886.000 817669.000 H5 CrO 3 (C) + 3H 2(g)=Cr (g) + 3H 2O(g) -21.784 -14.347 -9.630 -6.411 253991.307 248006.668 241243.090 233715.515 H6 CrO 3(C) + 2H 2(g)= CrO(g) + 2H 2O(g) -26.011 -17.578 -12.200 -8.508 287342.673 282033.301 275901.717 268966.969 H7 CrO 3(C) + H 2 (g)=CrO 2(g) + H 2O(g) -24.454 -16.627 -11.626 -8.188 266476.401 262047.024 256762.792 250644.344 Chapter 4 Volatility Diagrams in the Environments of Chlorine-based species 4.1 Construction of volatility diagram for Cr-Cl system. When chromium oxide is etched in the HCl environment, chromium chloride gaseous species are generated. We thus want to investigate which gaseous species of chromium chloride is dominant at various chlorine partial pressures. That would be very clear in the volatility diagrams of the Cr-Cl system. We will also show the dependence of chlorine partial pressure on HCl partial pressure in Table IV. Table V shows the needed thermochemical data of related Cr-Cl species. All the needed reactions for constructing the volatility diagram in the Cr-Cl system are listed Table VI. The method of constructing this volatility diagram is the same as in chapter 3. Figures 6(a)-(d) show the dominant reaction in different regions at 373K, 473K, 573K, and 673K. Figure 6(e) is the combination of Figure 6(a)-(d). The chlorine gas pressure can be generated from HCl dissociation: The equilibrium constant K HCl can be expressed as or According to mass balance relation: 44 Substituting equation into equation results in or From equation , we can calculate chlorine partial pressure from known HCl partial pressure in Table IV. When the HCl partial pressure is around 1 bar, we can see that the chlorine partial pressure generated from HCl dissociation ranges from -13.991 to -8.114 bar at 373-673K in Table IV and corresponds to the CrCl 4 dominating region in the Cr-Cl volatility diagram in Figure 6(e). 4.2 Etching of Cr 2 O3 due to formation of Cr-based gas species in HCl environment. As described in chapter 1, the etching of Cr 2 O3 (c) occurs in the HCl environment in LTCSS processing. In a reactive environment containing hydrogen chloride, the reactions between Cr 2 O3 (c) and hydrogen chloride needs to be taken into account. All these possible etching reactions of Cr 2 O3 (c) with HCl are listed in Table VII. In Table VII, all the reactions between Cr 2 O3 (c) and HCl show negative logK(T) . That is, the free energies of these reactions are positive; they cannot occur spontaneously. However, these reactions may be useful in the constant flowing conditions if the partial pressure of gaseous product is high enough. The partial pressures of gaseous 45 products in these reactions are shown as a function of HCl partial pressure in Figure 7(a)-(e). In the LTCSS etching environment, the HCl partial pressure is usually around 1bar. Again, the effective etching criterion is: Cr-Cl gas species must show partial pressure higher than 10 -8 bar when the constant flowing HCl partial pressure is around 1 bar. Reactions E1, E2 and E3 are the etching of Cr 2 O3 (c). For reaction E1, it meets the effective etching criterion only when temperature is around 673K (Figure 7(a)). Reactions E2 and E3 meet the effective etching criterion only at temperatures above 473K (Figure 7(b)-(c)). We compare the partial pressure of CrCl 2 , CrCl 3 , and CrCl 4 at the same temperature in Figure 8(a)-(d). When the HCl partial pressure is around 1 bar, the partial pressures of CrCl 3 , and CrCl 4 are similar while the partial pressure of CrCl 2 (g) over Cr 2 O3 (c) is lowest among these three Cr-Cl gaseous species. At 373K and 473K, the partial pressures of CrCl 4 is higher than the other two; at 573K, the partial pressures of CrCl 3 and CrCl 4 are similar; at 673K, the partial pressures of CrCl 3is higher than that of CrCl 4 . The formation of CrO 2 Cl 2 (g) from Cr 2 O3 (c) etched with HCl(g) is expected to be a self oxidation-reduction reaction (Reaction E4 and E5) in an absolutely dry environment. If water moisture exists, then reaction E6 may also be expected as one of the formation reactions of CrO 2 Cl 2 (g) from Cr 2 O3 (c) etched with HCl(g). When the HCl partial pressure is around 1 bar, the partial pressures of CrO 2 Cl 2 (g) over Cr 2 O3 (c) are too low to be useful (Figure 7(d)-(e)). 46 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0-80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 CrCl 3(g) CrCl 2(g) CrCl 2(c) Cr (c) CrCl 3(c) logP Cr or logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP Cl 2(Unit: bar) T=373K Cr(g) CrCl 2(g) CrCl 3(g) CrCl 4(g) Figure 6 (a). Complete volatility diagram for Cr-Cl system at 373K 47 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0-80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 CrCl 2(c) Cr (c) CrCl 3(c) logP Cr or logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP Cl 2(Unit: bar) T=473K CrCl 4(g) CrCl 3(g) CrCl 3(g) CrCl 2(g) CrCl 2(g) Cr(g) Figure 6(b). Complete volatility diagram for Cr-Cl system at 473K -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0-80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 CrCl 2(c) Cr (c) CrCl 3(c) logP Cr or logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP Cl 2(Unit: bar) T=573K CrCl 4(g) CrCl 3(g) CrCl 3(g) CrCl 2(g) CrCl 2(g) Cr(g) Figure 6(c). Complete volatility diagram for Cr-Cl system at 573K 48 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0-80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 CrCl 2(c) Cr (c) CrCl 3(c) logP Cr or logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP Cl 2(Unit: bar) T=673K CrCl 4(g) CrCl 3(g) CrCl 3(g) CrCl 2(g) CrCl 2(g) Cr(g) Figure 6(d). Complete volatility diagram for Cr-Cl system at 673K -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0-80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0-80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0-80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0-80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 CrCl 2(c) logP Cr or logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP Cl 2(Unit: bar) T=673K Cr (c) CrCl 3(c) logP Cr or logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP Cl 2(Unit: bar) T=473K logP Cr or logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP Cl 2(Unit: bar) T=573K logP Cr or logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP Cl 2(Unit: bar) T=373K CrCl 4(g) CrCl 3(g) CrCl 3(g) CrCl 2(g) CrCl 2(g) Cr(g) Figure 6(e). Master volatility diagram for the Cr-Cl system. 49 -10 -5 0 5-25 -20 -15 -10 -5 logP CrCl 2(Unit: bar) logP HCl (Unit: bar) 373K 473K 573K 673K E1: Cr 2O3(c)+ 6HCl(g)=2CrCl 2(g) + 3H 2O(g)+Cl 2(g) Figure 7(a). Possibly Achieved Maximum CrCl 2 Pressure in HCl Environment -10 -5 05-20 -15 -10 -5 0 llogP CrCl 3(Unit: bar) logP HCl (Unit: bar) 373K 473K 573K 673K E2: Cr 2O3(c)+6HCl(g)=2CrCl 3(g)+3H 2O(g) Figure 7(b). Possibly Achieved Maximum CrCl 3 Pressure in HCl Environment 50 -10 -5 0 5-25 -20 -15 -10 -5 0 logP CrCl 4(Unit: bar) logP HCl (Unit: bar) 373K 473K 573K 673K E3: Cr 2O3(c) + 8HCl(g) =2CrCl 4(g) + 3H 2O(g)+H 2(g) Figure 7(c). Possibly Achieved Maximum CrCl 4 in HCl Environment -10 -5 0 5-60 -58 -56 -54 -52 -50 -48 -46 -44 -42 -40 -38 -36 -34 -32 -30 -28 -26 logP CrO 2 Cl 2(Unit: bar) logP HCl (Unit: bar) 373K 473K 573K 673K E4: 2Cr 2O3(c)+2HCl(g)=CrO 2Cl 2(g)+3CrO(g)+ H 2O(g) Figure 7(d). Possibly Achieved Maximum CrO 2 Cl 2 Pressure in HCl Environment 51 -10 -5 0 5-48 -46 -44 -42 -40 -38 -36 -34 -32 -30 -28 -26 -24 -22 -20 -18 logP CrO 2Cl 2(Unit: bar) logP HCl (Unit: bar) 373K 473K 573K 673K E5: Cr 2O3(c) + 2HCl (g)= CrO 2Cl 2(g) +Cr(g)+H 2O(g) Figure 7(e). Possibly Achieved Maximum CrO 2 Cl 2 Pressure in HCl Environment -10 -5 0 5-20 -15 -10 -5 0 logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP HCl (Unit: bar) logP CrCl 2 logP CrCl 3 logP CrCl 4Cr 2O3 etched with HCl at 373K Figure 8(a). Cr 2 O3 (c) etched by HCl at 373K 52 -10 -5 0 5-20 -15 -10 -5 0 logP CrCl 2 logP CrCl 3 logP CrCl 4Cr 2O3 etched with HCl at 473K logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP HCl (Unit: bar) Figure 8(b). Cr 2 O3 (c) etched by HCl at 473K -10 -5 0 5-20 -15 -10 -5 0 logP CrCl 2 logP CrCl 3 logP CrCl 4Cr 2O3 etched with HCl at 573K logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP HCl (Unit: bar) Figure 8(c). Cr 2 O3 (c) etched by HCl at 573K 53 -10 -5 0 5-20 -15 -10 -5 0 logP CrCl 2 logP CrCl 3 logP CrCl 4Cr 2O3 etched with HCl at 673K logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP HCl (Unit: bar) Figure 8(d). Cr 2 O3 (c) etched by HCl at 673K Table IV. The chlorine partial pressure from HCl dissociation (unit:bar) 373K 473K 573K 673K 1 -12.911 -10.178 -8.397 -7.144 0 -13.911 -11.172 -9.397 -8.144 -1 -14.911 -12.178 -10.397 -9.144 -2 -15.911 -13.178 -11.397 -10.144 -3 -16.911 -14.178 -12.397 -11.144 54 Table V. Thermodynamic data utilized for construction of volitility diagrams in the Cr-Cl system (T= 298.15K, P=0.1MPa) C0 =a+b10 -3 T+c10 6 T-2 +d10 -6 T2 No. Species s o h0 a b c d (JK -1 mol -1 ) (Jmol -1 ) (JK -1 mol -1 ) Ref. 1 Cr(c) 23.640 0 24.514 2.050 -0.180 5.950 2 Cl 2 (g) 223.078 0 36.610 1.079 -0.272 0 3 Cr (g) 174.310 397480 20.983 -1.992 0.038 1.828 4 CrCl 2 (c) 115.269 -395388 71.362 12.996 -0.527 0 5 CrCl 3 (c) 122.962 -556472 98.830 13.975 -0.996 0 6 CrCl 4 (g) 364.401 -426768 106.433 1.314 -0.950 0 7 CrCl 2 (g) 307.947 -136247 61.116 1.356 -0.343 0 8 CrCl 3 (g) 317.645 -325180 83.345 3.155 -0.736 0 9 Cl(g) 165.184 121294 23.736 -1.284 -0.126 0 10 H2 (g) 130.679 0 26.882 3.586 0.105 0 11 HCl(g) 186.908 -92307 26.527 4.602 0.109 0 12 O2 (g) 205.146 0 29.154 6.477 -0.184 -1.017 13 H2 O(g) 188.824 -241856 34.376 7.841 -0.423 0 55 Table VI. Reactions utilized for plotting volatility diagrams in the Cr-Cl system. No. Reaction log(K)= -ΔG/(2.303RT) ΔH (Jmol -1 ) 373K 473K 573K 673K 373K 473K 573K 673K 1 Cr (c) + Cl 2(g) =CrCl 2(c) 48.461 36.812 29.255 23.962 -394239 -392948 -391599 -390211 2 CrCl 2(c) +1/2 Cl 2(g) =CrCl 3(c) 17.117 12.378 9.307 7.161 -160505 -159759 -158942 -158085 3 Cr (c) =Cr (g) -47.765 -36.018 -28.382 -23.023 397177.5 396702.9 396105.7 395397 4 CrCl 2(c) = Cr (g) +Cl 2(g) -96.226 -72.830 -57.637 -46.986 791416.8 789650.5 787704.4 785608.1 5 CrCl 3(c) = Cr (g) + 3/2 Cl 2(g) -113.345 -85.209 -66.945 -54.147 951921.8 949409.3 946646.8 943693.2 6 Cr (c) + Cl 2(g) =CrCl 2(g) 22.265 18.232 15.603 13.752 -136263 -136384 -136601 -136915 7 CrCl 2(c) =CrCl 2 (g) -26.195 -18.580 -13.651 -10.210 257976.3 256563.6 254998.1 253296 8 CrCl 3(c) =CrCl 2(g) + ½ Cl 2(g) -43.313 -30.958 -22.959 -17.371 418481.3 416322.4 413940.5 411381.2 9 Cr (c) + 3/2 Cl 2(g) =CrCl 3(g) 43.392 33.777 27.521 23.125 -325020 -324841 -324694 -324604 10 CrCl 2(c) +1/2 Cl 2(g) =CrCl 3(g) -5.068 -3.0352 -1.734 -0.836 69219.54 68106.87 66904.83 65607 11 CrCl 3(c) = CrCl 3(g) -22.186 -15.413 -11.041 -7.997 229724.5 227865.7 225847.3 223692.1 12 CrCl 4(g) = Cr (C) + 2 Cl 2(g) -54.224 -41.616 -33.419 -27.663 426298.4 425786 425314.1 424914.5 13 CrCl 2(c) + Cl 2(g) =CrCl 4(g) 5.763 4.804 4.164 3.701 -32059.1 -32838.4 -33715.4 -34703.5 14 CrCl 3(c) +1/2 Cl 2(g) = CrCl 4(g) -11.354 -7.573 -5.143 -3.460 128445.9 126920.4 125227 123381.6 56 57 Table VII. Possible etching reactions between Cr 2 O 3 (c) and HCl gases No. Reaction Log(K)= -ΔG/(2.303RT) ΔH (Jmol -1 ) 373K 473K 573K 673K 373K 473K 573K 673K E1 Cr 2 O 3 (c)+ 6HCl(g)=2CrCl 2(g) + 3H 2O(g)+Cl 2(g) -86.796 -66.344 -53.084 -43.799 692561.408 689577.862 687006.742 684593.977 E2 Cr 2 O 3 (c)+6HCl(g)=2CrCl 3(g)+3H 2O(g) -44.543 -35.254 -29.248 -25.051 315047.957 312664.430 310820.206 309215.886 E3 Cr 2 O 3 (c) + 8HCl(g) =2CrCl 4(g) + 3H 2O(g)+H 2(g) -49.772 -40.982 -35.276 -31.271 297603.117 296519.961 295990.137 295666.536 E4 2Cr 2 O 3(c)+2HCl(g)=CrO 2Cl 2(g)+3CrO(g)+ H 2O(g) -278.283 -211.992 -168.952 -138.776 2243040.382 2236970.619 2231435.969 2226078.740 E5 Cr 2 O 3 (c) + 2HCl (g)= CrO 2Cl 2(g) +Cr(g)+H 2O(g) -119.422 -91.674 -73.656 -61.020 938881.386 936377.389 934272.149 932321.338 E6 Cr 2 O 3 (c) +H 2O(g)+4HCl(g)= 2CrO 2Cl 2(g) +3H 2(g) -127.614 -102.474 -86.221 -74.873 852409.106 846361.774 840806.795 835427.661 Chapter 5 Etching of Chromium(VI) oxide 5.1 Etching of CrO 3 due to formation of Cr-based gas species in HCl environment. From the Cr-O volatility diagrams (in chapter 3), we can see another Cr-O condensed phase: Chromium(VI) oxide. CrO 3 (c) can be formed in the extremely oxidation environments, such as ozone or atomic oxygen. We list the possible etching reactions between CrO 3 (c) and HCl gases in Table VIII. From the equilibrium constant in Table VIII, we expect the etching reactions of CrO 3 (c) would be far more effective than the etching reactions of Cr 2 O3 (c) in HCl environments. In Figure 9(a)-(d), the partial pressure of all the gaseous products in these reactions satisfy the effective etching criterion at 373-673K. By the way, it is noticed that the temperature effect on these reactions (in Table VIII) are different: the partial pressures of CrCl 2and CrCl 3 increase with increasing temperature (in Figure 9(a)-(b)), while the partial pressures of CrCl 4 and CrO 2 Cl 2 decrease with increasing temperature (in Figure 9(c)-(d)). When the partial pressure of HCl is about 1 bar, the dominating gaseous product is always CrCl 4 in Figure 10 (a)-(d). CrO 3 (c) can be etched much more effectively than Cr 2O3 (c) in the same HCl environment. However, the passive film on the stainless steel is usually made of Cr 2 O3 (c). Cr 2 O3 (c) is etched effectively only when the temperature is high enough. If we want to use the more effective etching reactions of CrO 3 (c), we need to oxidize Cr 2 O3 (C) into CrO 3 (c) first and then etch CrO 3 (c) in the HCl environment. From the Cr-O volatility diagram in chapter 3, we can see that the oxygen gas pressure have to be larger than 10 5 bar to oxidize Cr 2 O3 (c) into CrO 3 (c). It is 58 technically difficult to make the oxygen pressure as higher as 10 5 bar, so we can not oxidize Cr 2O3 (c) into CrO 3 (c) by oxygen gas. A stronger oxidizing reactant is needed to oxidize Cr 2 O3 (c) into CrO 3 (c). Thus, we will consider ozone and atomic oxygen in section 5.2. 5.2 Construction of volatility diagram for Cr-O system in ozone and atomic oxygen environments Atomic oxygen or ozone can be generated in the discharge region of an oxygen plasma device. The two atoms in oxygen molecule can be separated when the high-voltage electric field is applied. The atomic oxygen may also react with oxygen molecule to form an ozone molecule. The partial atomic oxygen pressure can be as high as several mbar. We take the atmospheric pressure plasma jet (APPJ) for example to show how the atomic oxygen is generated in Figure 11. The temperature map in the APPJ is shown in Figure 12, since we need to know if the temperature range is acceptable for the low temperature stainless steel processing. The method of constructing volatility diagram is the same as in chapter 3. All the reactions used to construct volatility diagram of Cr-O system in atomic oxygen gas are listed in Table IX. The more detailed construction of volatility diagram in the chromium - atomic oxygen system can be seen in Appendix C. In Figure 13, the volatility diagram of Cr-O system in atomic oxygen gas shows that Cr 2 O3 (c) can be oxidized into CrO 3 (c) in the presence of atomic oxygen partial pressure higher than about 10 -13 bar at 673K. If the temperature is lower, then the required partial pressure of atomic oxygen is even lower. Atomic oxygen partial pressure of 10 -13 bar is achieved easily in the high energy oxygen plasma, so atomic oxygen can be chosen as an oxidizing reagent in this case. 59 Volatility diagram of Cr-O system in ozone can be constructed by similar method. The more detailed construction of volatility diagram in the chromium - atomic oxygen system can be seen in Appendix D. All the involved reactions are listed in Table X. In Figure 14, the volatility diagram of Cr-O system in ozone gas shows that Cr 2 O3 (c) can be oxidized into CrO 3(c) in the presence of ozone partial pressure higher than about 10 -5 bar at 673K. If the temperature is lower, then the required partial pressure of ozone is even lower. Ozone partial pressure of 10 -5 bar is achievable in the oxygen plasma, and therefore ozone can also be a candidate for the oxidizing reagent in this case. -10 -5 05-15 -10 -5 0 logP CrCl 2(Unit: bar) logP HCl (Unit: bar) 373K 473K 573K 673K E4: CrO 3(C) +6HCl (g)=CrCl 2(g) + 3H 2O(g)+2Cl 2(g) Figure 9(a). Possibly Achieved Maximum CrCl 2 in HCl Environment 60 -10 -5 05-10 -5 0 E5: CrO 3(C)+6HCl(g)=CrCl 3(g)+3H 2O(g)+(3/2)Cl 2(g) logP CrCl 3(Unit: bar) A 373K 473K 573K 673K Figure 9(b). Possibly Achieved Maximum CrCl 3 Pressure in HCl Environment -10 -5 05-10 -5 0 E6: CrO 3(C) + 6HCl(g)=CrCl 4(g) + 3H 2O(g) +Cl 2(g) logP CrCl 4(Unit: bar) logP HCl (Unit: bar) 373K 473K 573K 673K Figure 9(c). Possibly Achieved Maximum CrCl 4 Pressure in HCl Environment 61 -10 -5 0 5-10 -5 05 logP CrO 2 Cl 2(Unit: bar) logP HCl (Unit: bar) 373K 473K 573K 673K E10: CrO 3(C) + 2HCl (g)= CrO 2Cl 2(g) + H 2O(g) Figure 9(d). Possibly Achieved Maximum CrO 2 Cl 2 Pressure in HCl Environment -10 -5 0 5-15 -10 -5 05 CrO 3 etched with HCl at 373K logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP CrCl 2 logP CrCl 3 logP CrCl 4logP HCl (Unit: bar) Figure 10(a). CrO 3 (c) etched by HCl at 373K 62 -10 -5 0 5-15 -10 -5 05 CrO 3 etched with HCl at 473K logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP CrCl 2 logP CrCl 3 logP CrCl 4logP HCl (Unit: bar) Figure 10(b). CrO 3 (c) etched by HCl at 473K -10 -5 0 5-15 -10 -5 05 CrO 3 etched with HCl at 573K logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP CrCl 2 logP CrCl 3 logP CrCl 4logP HCl (Unit: bar) Figure 10(c). CrO 3 (c) etched by HCl at 573K 63 -10 -5 0 5-15 -10 -5 05 CrO 3 etched with HCl at 673K logP CrCl 2or logP CrCl 3or logP CrCl 4(Unit: bar) logP CrCl 2 logP CrCl 3 logP CrCl 4logP HCl (Unit: bar) Figure 10(d). CrO 3 (c) etched by HCl at 673K Figure 11. Side view (a) and top view (b) schematics of the planar APPJ. 19 64 Figure 12. Temperature map of the APPJ’s effluent in the centre position of the jet’s discharge gap. The jet is operated at 150W RF power with a helium gas flux of 2m 3 h -1 and 0.5 vol% O 2 admixture 19 . -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O (Unit: bar) T=573K logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O (Unit: bar) T=373K Cr (g) Cr (g) CrO 2(g) CrO 3(g) CrO 3(g) logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O (Unit: bar) T=473K logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O (Unit: bar) T=673K Figure 13. Volatility diagram for Cr-O system in atomic oxygen environment 65 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O3(Unit: bar) T=473K logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O3(Unit: bar) T=373K logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O3(Unit: bar) T=673K logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O3(Unit: bar) T=573K Cr (g) Cr (g) CrO 2(g) CrO 3(g) CrO 3(g) Figure 14. Volatility diagram for Cr-O system in ozone environment 66 Table VIII. Possible etching reactions between CrO 3 (c) and HCl gases No. Reaction Log(K)= -ΔG/(2.303RT) ΔH (Jmol -1 ) 373K 473K 573K 673K 373K 473K 573K 673K O1 CrO 3 (C) +6HCl (g)=CrCl 2(g) + 3H 2O(g)+2Cl 2(g) -32.433 -24.318 -19.114 -15.521 275888.360 272158.106 267768.203 262618.469 O2 CrO 3 (C)+6HCl(g)=CrCl 3(g)+3H 2O(g)+(3/2)Cl 2(g) -11.306 -8.773 -7.196 -6.147 87131.635 83701.389 79674.934 74929.424 O3 CrO 3 (C) + 6HCl(g)=CrCl 4(g) + 3H 2O(g) +Cl 2(g) -0.473 -0.933 -1.298 -1.60912 -14147.050 -17243.900 -20945.300 -25381.100 O4 CrO 3 (C) + 2HCl (g)= CrO 2Cl 2(g) + H 2O(g) 4.973 4.603 4.300 4.034 -11232.204 -14129.274 -17672.698 -21966.187 67 Table IX. Reactions utilized for ploting volitility diagrams in the Cr-O system (in atomic oxygen environment). No. Reaction log(K)= -ΔG/(2.303RT) ΔH (Jmol -1 ) 373K 473K 573K 673K 373K 473K 573K 673K 1 2Cr (c) + 3 O (g) =Cr 2 O3(c) 240.811 184.945 148.580 123.023 -1887954 -1887846 -1887750 -1887683 2 Cr 2 O3(c) + 3 O (g) =2CrO 3(c) 87.553 64.428 49.433 38.963 -782081.8 -780238 -776072 -769902 3 Cr (c) =Cr (g) -47.765 -36.018 -28.382 -23.023 397177.5 396702.9 396105.7 395397 4 Cr 2 O3(c) =2 Cr (g) +3 O (g) -336.342 -256.982 -205.345 -169.071 2682309 2681252 2679962 2678477 5 CrO 3(c) = Cr (g) + 3 O (g) -211.948 -160.705 -127.389 -104.018 1732195 1730745 1728017 1724190 6 Cr (c) + O (g) =CrO (g) 11.394 9.536 8.300 7.414 -62205.82 -63516.6 -64826.9 -66176.2 7 Cr 2 O3(c) =2CrO(g) + O (g) -218.021 -165.873 -131.979 -108.194 1763542 1760813 1758096 1755331 8 CrO 3(c) = CrO (g) + 2O (g) -152.787 -115.151 -90.705 -73.579 1272812 1270525 1267084 1262616 9 Cr (c) + 2O (g) =CrO 2(g) 76.339 59.273 48.128 40.270 -575806.8 -577749 -579557 -581324 10 Cr 2 O3(c) + O (g) =2CrO 2(g) -88.131 -66.397 -52.324 -42.482 736340.4 732348 728636.1 725036 11 CrO 3(c) = CrO 2 (g) + O (g) -87.842 -65.413 -50.878 -40.723 759211.1 756293.1 752354 747469 12 Cr ( c) + 3 O (g) = CrO 3(g) 133.119 102.215 82.057 67.864 -1043270 -1045555 -1047490 -1049258 13 Cr 2 O3( C) +3 O (g) =2CrO 3(g) 25.428 19.485 15.533 12.704 -198586.9 -203265 -207230 -210832 14 CrO 3(c) =CrO 3(g) -31.062 -22.471 -16.949 -13.129 291747.4 288486.5 284421 279535 68 69 Table X. Reactions utilized for ploting volitility diagrams in the Cr-O system (in ozone environment). No. Reaction log(K)= -ΔG/(2.303RT) ΔH (Jmol -1 ) 373K 473K 573K 673K 373K 473K 573K 673K 1 2Cr (c) + O 3(g) =Cr 2 O3(c) 172.626 132.557 106.460 88.103 -1353936 -1354346 -1355319 -1356639 2 Cr 2 O3(c) + O 3(g) =2CrO 3(c) 19.368 12.041 7.313 4.043 -248063.8 -246739 -243640 -238858 3 Cr (c) =Cr (g) -47.765 -36.018 -28.382 -23.023 397177.5 396702.9 396105.7 395397 4 Cr 2 O3(c) =2 Cr (g) + O 3(g) -268.158 -204.595 -163.225 -134.151 2148291 2147752 2147530 2147433 5 CrO 3(c) = Cr (g) + O 3(g) -143.763 -108.318 -85.269 -69.097 1198177 1197245 1195585 1193146 6 Cr (c) + 1/3 O 3(g) =CrO (g) -11.333 -7.926 -5.738 -4.225 115800.2 114316.6 112650.3 110838.4 7 Cr 2 O3(c) =2CrO(g) + 1/3 O 3 (g) -195.293 -148.410 -117.939 -96.554 1585536 1582979 1580619 1578316 8 CrO 3(c) = CrO (g) + 2/3 O 3(g) -107.331 -80.225 -62.626 -50.299 916800.1 914859.1 912129.8 908587 9 Cr (c) + (2/3) O 3(g) =CrO 2(g) 30.883 24.3487 20.048 16.990 -219794.8 -222083 -224603 -227294 10 Cr 2 O3(c) + 1/3 O 3(g) =2CrO 2(g) -110.860 -83.860 -66.364 -54.122 914346.4 910181.1 906113.3 902050.7 11 CrO 3(c) = CrO 2 (g) + 1/3 O 3(g) -65.114 -47.950 -36.838 -29.083 581205.1 578459.9 574876.8 570454.4 12 Cr ( c) + O 3(g) = CrO 3(g) 64.935 49.828 39.937 32.944 -509252.5 -512056 -515058 -518214 13 Cr 2 O3(C) + O 3(g) =2CrO 3(g) -42.755 -32.901 -26.585 -22.214 335431.1 330234.4 325201.7 320211.9 14 CrO 3(c) =CrO 3(g) -31.062 -22.471 -16.949 -13.129 291747.4 288486.5 284421 279535 Chapter 6 Conclusions Volatility diagrams are used to investigate the etching reactions in the activation step of LTCSS. When HCl pressure is around 1 atm (LTCSS conditions), the Cr 2 O3(c) of the passive film on the stainless steel can be etched effectively only when the temperature is high enough (at least 473K). Higher temperature or higher HCl partial pressure can result in more effective etching in HCl environment. Monoatomic hydrogen can reduce Cr 2 O3(c) to Cr (g) and the isobaric points in the volatility diagram indicate the maximum vapor pressure of Cr (g) over Cr 2 O3(c) is far higher than the effective etching criterion, even when the atomic hydrogen partial pressure is very low (10 -4 bar). Thus, etching Cr 2O3(c) on the stainless steel is expected to be an effective etching method. Another alternative way to etch the passive film on the stainless steel is to oxidize Cr 2 O3 (c) into CrO 3 (c) first and then etch CrO 3 (c) in the HCl environment. That is because CrO 3 (c) can be etched more effectively than Cr 2 O3 (c) in HCl environment. However, based on the volatility diagram in Cr-O system, the oxygen gas pressure has to be larger than 10 5 bar to oxidize Cr 2 O3 (c) into CrO 3 (c). It is very difficult to make the oxygen pressure as high as 10 5 bar. Thus, ozone or atomic oxygen, instead of oxygen gas, should be used to oxidize Cr 2 O3(c) into CrO 3 (c). Then, we can etch CrO 3 (c) in HCl environment; the dominating Cr-based gas species generated here is expected to be CrO 2 Cl 2 (g). 70 Chapter 7 Future Work If we plan to use HCl to etch away Cr 2 O3 (c), the only way to increase the etching effectiveness is to increase the temperature or to increase HCl partial pressure. Although increasing temperature can help remove passive film effectively, there is an upper limit of the applicable temperature range in LTCSS processing because of microstructural changes in steel. 673K is probably an acceptable upper limit for HCl etching. Increasing HCl partial pressure can also improve the etching effectiveness, but there is also an upper limit of applicable pressure range due to technical problem. Generally, increasing temperature is technically easier than increasing HCl partial pressure. Since CrO 3 (c) can be etched away more effectively than Cr 2O3 (c) in HCl environment, we consider oxidizing Cr 2 O3 (c) into CrO 3 (c) first by O or O 3 and then etching CrO 3 (c) away by HCl. Actually, other oxidizing reagent may also work well to oxidize Cr 2 O3 (c) into CrO 3 (c). Available choices of strong oxidizing reagents could include hydrogen peroxide, perchloric acid, etc. Although HCl can effectively etch Cr 2 O3 (c) away at enough high temperature, other gases may also etch away the passive film effectively. For example, CO gas may react with Cr 2 O3 (c) to form chromium hexacarbonyl, Cr(CO) 6 . At room temperature , the Cr(CO) 6 solid is stable to air, but it does have a high vapor pressure and sublimes readily. Actually, the Cr(CO) 6 has a melting point of 425K and a boiling point of 483K, which is acceptable temperature for LTCSS processing. Cr(CO) 6 gas is decomposed readily and therefore we have no thermochemical data to calculate its gas pressure. Although the thermochemical data of Cr(CO) 6 gas is not available for calculation, the formation of Cr(CO) 6 may still assistant the etching reaction of Cr 2 O3 (c) significantly. Thus, etching with CO gas may experimentally be a nice try. 71 Appendix A Cr 2 O 3 (c) reduction into CrO(g) in the hydrogen environment CrO(g) is never a dominating gas species in the volatility diagram of Cr-O system. That is, the partial pressure of CrO is too low to be considered. Even though the Cr 2 O3(c) is put in the constant flowing hydrogen environment, the partial pressure of product CrO, which is the possibly achieved maximum CrO partial pressure, is still too low to be useful. Table XI. Cr 2 O3 (c) reduction into CrO( g) in the hydro gen environment: Cr 2 O 3 (c) + H 2 (g) = 2CrO(g) + H 2O(g) H2 Partial pressure Temperature 10 bar 373K -51.110 473K -38.595 573K -30.474 673K -24.785 1 bar 373K -51.444 473K -38.928 573K -30.808 673K -25.119 0.1 bar 373K -51.777 473K -39.261 573K -31.141 673K -25.452 72 Appendix B. Cr 2O3(c) reduction into CrO (g) in the atomic hydrogen environment Atomic hydrogen is stronger reducing reagent than hydrogen gas. However, even if the Cr 2 O3 (c) is put in the constant flowing atomic hydrogen environment, the partial pressure of product CrO, which is the possibly achieved maximum CrO partial pressure, is still too low to be considered in the effective dry etching process. Table XII Cr 2 O 3 (c) reduction into CrO(g) in the monoatomic hydrogen environment: Cr 2 O3(c) + 2H(g) = 2CrO(g) + H 2 O(g) H Partial Pressure Temperature 0.1 bar 373K -33.484 473K -25.285 573K -19.984 673K -16.281 0.01 bar 373K -34.150 473K -25.952 573K -20.650 673K -16.948 0.001 bar 373K -34.817 473K -26.619 573K -21.317 673K -17.615 0.0001 bar 373K -35.484 473K -27.285 573K -21.984 673K -18.281 73 Appendix C. Volatility diagrams for chromium-atomic oxygen system The volatility diagram of Cr-O system in the atomic oxygen environment at a certain temperature is constructed in Figure 14(a)-(d): -140 -120 -100 -80 -60 -40 -20 020 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O (Unit: bar) T=373K Cr (g) Cr (g) CrO 2(g) CrO 3(g) CrO 3(g) Cr (c) Cr 2O3 (c) CrO 3 (c) Figure 15(a). Complete volatility diagram for Cr-O system in the atomic oxygen environment at 373K -140 -120 -100 -80 -60 -40 -20 020 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O (Unit: bar) T=473K Cr (c) Cr (g) Cr (g) CrO 2(g) CrO 3(g) CrO 3(g) CrO 3 (c) Cr 2O3 (c) Figure 15(b). Complete volatility diagram for Cr-O system in the atomic oxygen environment at 473K 74 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O (Unit: bar) T=573K Cr (c) Cr (g) Cr (g) CrO 2(g) CrO 3(g) CrO 3(g) CrO 3 (c) Cr 2O3 (c) Figure 15(c). Complete volatility diagram for Cr-O system in the atomic oxygen environment at 573K -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O (Unit: bar) T=673K Cr (c) Cr (g) Cr (g) CrO 2(g) CrO 3(g) CrO 3(g) CrO 3 (c) Cr 2O3 (c) Figure 15(d). Complete volatility diagram for Cr-O system in the atomic oxygen environment at 673K 75 Appendix D. Volatility diagrams for chromium-ozone system The volatility diagram of Cr-O system in the ozone environment at a certain temperature is constructed in Figure 15(a)-(d): -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O3(Unit: bar) T=373K Cr (c) Cr (g) Cr (g) CrO 2(g) CrO 3(g) CrO 3(g) CrO 3 (c) Cr 2O3 (c) Figure 16(a). Complete volatility diagram for Cr-O system in the ozone environment at 373K -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O3(Unit: bar) T=473K Cr (c) Cr (g) Cr (g) CrO 2(g) CrO 3(g) CrO 3(g) CrO 3 (c) Cr 2O3 (c) Figure 16(b). Complete volatility diagram for Cr-O system in the ozone environment at 473K 76 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 CrO 2(g) logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O3(Unit: bar) T=573K Cr (c) Cr (g) Cr (g) CrO 3(g) CrO 3(g) CrO 3 (c) Cr 2O3 (c) Figure 16(c). Complete volatility diagram for Cr-O system in the ozone environment at 573K -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 -80 -60 -40 -20 0 logP Cr or logP CrO or logP CrO 2or logP CrO 3(Unit: bar) logP O3(Unit: bar) T=673K Cr (c) Cr 2O3 (c) CrO 3 (c) CrO 3(g) CrO 3(g) CrO 2(g) Cr (g) Cr (g) Figure 16(d). Complete volatility diagram for Cr-O system in the ozone environment at 673K 77 Appendix E Possibly achieved maximum partial pressure of Cr-Cl gaseous species in the etching environment of HCl (g) Possibly achieved maximum partial pressures of CrCl 2 (g) or CrCl 3 (g) or CrCl 4 (g) for Cr 2 O3 (c) in the constant flowing HCl environment are shown in the following tables according to different conditions: Table XIII . Possibly achieved maximum partial pressure of CrCl 2 (g) for Cr 2 O 3 (c) in the constant flowing HCl environment: Cr 2 O 3 (c) + 6HCl(g) = 2CrCl 2 (g) + 3H 2 O(g)+Cl 2 (g) (Unit: bar) 373K 473K 573K 673K 2 -12.504 -9.095 -6.885 -5.338 1 -13.504 -10.095 -7.885 -6.338 0 -14.504 -11.095 -8.885 -7.338 -1 -15.504 -12.095 -9.885 -8.338 -2 -16.504 -13.095 -10.885 -9.338 -3 -17.504 -14.095 -11.885 -10.338 -4 -18.504 -15.095 -12.885 -11.338 -5 -19.504 -16.095 -13.885 -12.338 -6 -20.504 -17.095 -14.885 -13.338 -7 -21.504 -18.095 -15.885 -14.338 -8 -22.504 -19.095 -16.885 -15.338 -9 -23.504 -20.095 -17.885 -16.338 -10 -24.504 -21.095 -18.885 -17.338 78 Table XIV . Possibly achieved maximum partial pressure of CrCl 3 (g) for Cr 2 O 3 (c) in the constant flowing HCl environment: Cr 2 O 3 (c) + 6HCl(g) = 2CrCl 3 (g) + 3H 2 O(g) (Unit: bar) 373K 473K 573K 673K 2 -6.614 -4.756 -3.555 -2.716 1 -7.814 -5.956 -4.755 -3.916 0 -9.014 -7.156 -5.955 -5.116 -1 -10.214 -8.356 -7.155 -6.316 -2 -11.414 -9.556 -8.355 -7.516 -3 -12.614 -10.756 -9.555 -8.716 -4 -13.814 -11.956 -10.755 -9.916 -5 -15.014 -13.156 -11.955 -11.116 -6 -16.214 -14.356 -13.155 -12.316 -7 -17.414 -15.556 -14.355 -13.516 -8 -18.614 -16.756 -15.555 -14.716 -9 -19.814 -17.956 -16.755 -15.916 -10 -21.014 -19.156 -17.955 -17.116 79 Table XV . Possibly achieved maximum partial pressure of CrCl 4 (g) for Cr 2 O 3 (c) in the constant flowing HCl environment: Cr 2 O 3 (c) + 8HCl (g)= 2CrCl 4 (g) + 3H 2 O(g)+H2 (g) (Unit: bar) 373K 473K 573K 673K 2 -5.667 -4.202 -3.250 -2.583 1 -7.000 -5.535 -4.584 -3.916 0 -8.333 -6.868 -5.917 -5.250 -1 -9.667 -8.202 -7.250 -6.583 -2 -11.000 -9.535 -8.584 -7.916 -3 -12.333 -10.868 -9.917 -9.250 -4 -13.667 -12.202 -11.250 -10.583 -5 -15.000 -13.535 -12.584 -11.916 -6 -16.333 -14.868 -13.917 -13.250 -7 -17.667 -16.202 -15.250 -14.583 -8 -19.000 -17.535 -16.584 -15.916 -9 -20.333 -18.868 -17.917 -17.250 -10 -21.667 -20.202 -19.250 -18.583 80 Appendix F Possibly achieved maximum partial pressure of Cr-Cl gaseous species in the etching environment of HCl Possibly achieved maximum partial pressures of CrCl 2 (g) or CrCl 3 (g) or CrCl 4 (g) for CrO 3 (c) in the constant flowing HCl environment are shown in the following tables according to different conditions: Table XVI . Possibly achieved maximum partial pressure of CrCl 2 (g) for CrO 3 (c) in the constant flowing HCl environment: CrO 3 (C) + 6HCl(g) =CrCl 2 (g) + 3H 2 O(g)+2Cl 2 (g) (Unit: bar) 373K 473K 573K 673K 2 -3.744 -2.392 -1.525 -0.926 1 -4.744 -3.392 -2.525 -1.926 0 -5.744 -4.392 -3.525 -2.926 -1 -6.744 -5.392 -4.525 -3.926 -2 -7.744 -6.392 -5.525 -4.926 -3 -8.744 -7.392 -6.525 -5.926 -4 -9.744 -8.392 -7.525 -6.926 -5 -10.744 -9.392 -8.525 -7.926 -6 -11.744 -10.392 -9.525 -8.926 -7 -12.744 -11.392 -10.525 -9.926 -8 -13.744 -12.392 -11.525 -10.926 -9 -14.744 -13.392 -12.525 -11.926 -10 -15.744 -14.392 -13.525 -12.926 81 Table XVII . Possibly achieved maximum partial pressure of CrCl 3 (g) for CrO 3 (c) in the constant flowing HCl environment: CrO 3 (C) + 6HCl(g) = CrCl 3 (g) + 3H 2 O(g)+(3/2) Cl 2(g) (Unit: bar) 373K 473K 573K 673K 2 -1.273 -0.812 -0.526 -0.335 1 -1.818 -1.358 -1.071 -0.880 0 -2.364 -1.903 -1.617 -1.426 -1 -2.909 -2.449 -2.162 -1.971 -2 -3.455 -2.994 -2.708 -2.517 -3 -4.000 -3.540 -3.253 -3.062 -4 -4.546 -4.085 -3.798 -3.608 -5 -5.091 -4.631 -4.344 -4.153 -6 -5.637 -5.176 -4.889 -4.699 -7 -6.182 -5.722 -5.435 -5.244 -8 -6.728 -6.267 -5.980 -5.790 -9 -7.273 -6.812 -6.526 -6.335 -10 -7.818 -7.358 -7.071 -6.880 82 Table XVIII . Possibly achieved maximum partial pressure of CrCl 4 (g) for CrO 3 (c) in the constant flowing HCl environment: CrO 3 (C) + 6HCl(g) = CrCl 4 (g) + 3H 2 O(g) +Cl 2 (g) ((Unit: bar) 373K 473K 573K 673K 2 2.019 1.927 1.854 1.792 1 0.819 0.727 0.654 0.592 0 -0.381 -0.473 -0.546 -0.608 -1 -1.581 -1.673 -1.746 -1.808 -2 -2.781 -2.873 -2.946 -3.008 -3 -3.981 -4.073 -4.146 -4.208 -4 -5.181 -5.273 -5.346 -5.408 -5 -6.381 -6.473 -6.546 -6.608 -6 -7.581 -7.673 -7.746 -7.808 -7 -8.781 -8.873 -8.946 -9.008 -8 -9.981 -10.073 -10.146 -10.208 -9 -11.181 -11.273 -11.346 -11.408 -10 -12.381 -12.473 -12.546 -12.608 83 Reference Michal GM, Ernst F, Heuer AH. Carbon paraequilibrium in austenitic stainless steel. Metallurgical and Materials Transactions a-Physical Metallurgy and Materials Science 2006;37A(6):1819-1824. Michal GM, Ernst F, Kahn H, Cao Y, Oba F, Agarwal N, Heuer AH. Carbon supersaturation due to paraequilibrium carburization: Stainless steels with greatly improved mechanical properties. Acta Materialia 2006;54(6):1597-1606. Agarwal N, Kahn H, Avishai A, Michal G, Ernst F, Heuer AH. Enhanced fatigue resistance in 316L austenitic stainless steel due to low-temperature paraequilibrium carburization. Acta Materialia 2007;55(16):5572-5580. O'Donnell LJ, Michal GM, Ernst F, Kahn H, Heuer AH. Wear maps for low temperature carburised 316L austenitic stainless steel sliding against alumina. Surface Engineering 2010;26(4):284-292. Qu J, Blau PJ, Zhang LG, Xu HB. Effects of multiple treatments of low-temperature colossal supersaturation on tribological characteristics of austenitic stainless steels. Wear 2008;265(11-12):1909-1913. Ernst F, Cao Y, Michal GM, Heuer AH. Carbide precipitation in austenitic stainless steel carburized at low temperature. Acta Materialia 2007;55(6):1895-1906. Heuer AH, Ernst F, Kahn H, Avishai A, Michal GM, Pitchure DJ, Ricker RE. Interstitial defects in 316L austenitic stainless steel containing "colossal" carbon concentrations: An internal friction study. Scripta Materialia 2007;56(12):1067-1070. Martin FJ, Lemieux EJ, Newbauer TM, Bayles RA, Natishan PM, Kahn H, 84 Michal GM, Ernst F, Heuer AH. Carburization-induced passivity of 316 L austenitic stainless steel. Electrochemical and Solid State Letters 2007;10(12):C76-C78. JANAF Thermochemical Tables, 2nd ed. National Bureau of standards 1971. Metals Handbook. American Society for Metals 1948. Kubaschewski OaA, C. B. Metallurgical Thermochemistry, 5th ed. . Pergamon Press 1977. Barin I. Thermochemical Properties of Inorganic Substances. Springer-Verlag, Berlin and New York 1973:-. Lou VLK, Mitchell TE, Heuer AH. Graphical Displays of the Thermodynamics of High-Temperature Gas-Solid Reactions and Their Application to Oxidation of Metals and Evaporation of Oxides. Journal of the American Ceramic Society 1985;68(2):49-58. Yokoyama K, Wada S. Solid-gas reaction during sintering of Si3N4 ceramics (Part 1) - Stability of SiO2, Y2O3 and Al2O3 at high temperature. Journal of the Ceramic Society of Japan 2000;108(1):6-9. Goto T. High-temperature oxidation behavior of chemical vapor deposited silicon carbide. Journal of the Ceramic Society of Japan 2002;110(10):884-889. Fahrenholtz WG. The ZrB2 volatility diagram. Journal of the American Ceramic Society 2005;88(12):3509-3512. Kulkarni NS, DeHoff RT. Application of volatility diagrams for low temperature, dry etching, and planarization of copper. Journal of the Electrochemical Society 2002;149(11):G620-G632. Barin I. Thermochemical Data of Pure Substances 85 86 VCH Verlagsgesellschaft mbH. D-69451 Weinheim (Federal Republic of Germany) 1995. Reuter S, Niemi K, Schulz-von der Gathen V, Dobele HF. Generation of atomic oxygen in the effluent of an atmospheric pressure plasma jet. Plasma Sources Science & Technology 2009;18(1):-.
4671	https://godinton.kent.sch.uk/media/2538/year-3-adding-fractions-help-sheet.pdf	A dding and S ubtracting Fractions Fractions are equal parts of a whole. We add and subtract fractions in a similar way to how we add and subtract whole numbers. When adding and subtracting fractions within 1, with the same denominator we add or subtract the numerator. numerator denominator Key Vocabulary Mixed fractions – fraction that shows a whole number and fraction eg: Improper fraction – when the numerator is greater than the denominator eg: Equivalent fraction – Two fraction that are worth the same (equal) eg: 1/2 and 2/4 Simplifying fractions How do you simplify fractions step by step? Here are the steps to follow: 1. Write down the factors for the numerator and the denominator. 2. Determine the largest factor that is common between the two. 3. Divide the numerator and denominator by the greatest common factor. 4. Write down the reduced fraction. Adding and subtracting fractions with different denominators. When children are confident adding and subtracting within 1 we can challenge the children further by asking them to add and subtract fractions with unlike denominators. The children will use their times tables to find a common multiple of each denominator. Subtracting follows the same rules. When the common denominator has been found subtract the numerators. Refer to above BBC Bitesize clip. pics/zhdwxnb/articles/zcdgxfr Simplifying fractions – BBC Bitesize S TEM S entences: Use these sentences in your Unexpected Adventure Trail book to help you explain adding or subtracting fractions. C ompare fractions: I know that ½ is > greater than ¼ because the smaller the denominator the bigger each of the parts. ¾ > ½ I know this because it is more than half. Half of 4 is 2 and the numerator is greater than 2. ½ is = equal to 2/4. Challenge: Can you write your own instructions for adding and subtracting fractions with unlike denominators? L earn to say and spell this maths vocabulary: 1 whole, numerator, denominator, fraction, improper, equal parts
4672	https://www.themathpage.com/aTrig/30-60-90-triangle.htm	\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- \| \| The Topics \| Home THE 30°-60°-90° TRIANGLE THERE ARE TWO special triangles in trigonometry. One is the 30°-60°-90° triangle. The other is the isosceles right triangle. They are special because with simple geometry we can know the ratios of their sides, and therefore solve any such triangle. Theorem. In a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : . We will prove that below. Note that the smallest side, 1, is opposite the smallest angle, 30°; while the largest side, 2, is opposite the largest angle, 90°. (Theorem 6). (For, 2 is larger than . Also, while 1 : : 2 correctly corresponds to the sides opposite 30°-60°-90°, many find the sequence 1 : 2 : easier to remember.) The cited theorems are from the Appendix, Some theorems of plane geometry. Here are examples of how we take advantage of knowing those ratios. First, we can evaluate the functions of 60° and 30°. Example 1. Evaluate cos 60°. Answer. For any problem involving a 30°-60°-90° triangle, the student should not use a table. The student should sketch the triangle and place the ratio numbers. Since the cosine is the ratio of the adjacent side to the hypotenuse, we can see that cos 60° = ½. Example 2. Evaluate sin 30°. Answer. According to the property of cofunctions, sin 30° is equal to cos 60°. sin 30° = ½. On the other hand, you can see that directly in the figure above. Problem 1. Evaluate sin 60° and tan 60°. To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload"). The sine is the ratio of the opposite side to the hypotenuse. \| \| \| \| --- \| sin 60° = \| 2 \| = ½. \| Lesson 5 of Algebra The tangent is ratio of the opposite side to the adjacent. \| \| \| \| --- \| tan 60° = \| 1 \| = . \| Problem 2. Evaluate cot 30° and cos 30°. The cotangent is the ratio of the adjacent side to the opposite. \| \| \| --- \| \| Therefore, on inspecting the figure above, cot 30° = \| 1 \| = . Or, more simply, cot 30° = tan 60°. Problem 1 As for the cosine, it is the ratio of the adjacent side to the hypotenuse. Therefore, \| \| \| \| --- \| cos 30° = \| 2 \| = ½. \| Before we come to the next Example, here is how we relate the sides and angles of a triangle: If an angle is labeled capital A, then the side opposite will be labeled small a. Similarly for angle B and side b, angle C and side c. Example 3. Solve the right triangle ABC if angle A is 60°, and side AB is 10 cm. Solution. To solve a triangle means to know all three sides and all three angles. Since this is a right triangle and angle A is 60°, then the remaining angle B is its complement, 30°. Again, in every 30°-60°-90° triangle, the sides are in the ratio 1 : 2 : , as shown on the left. When we know the ratios of the sides, then to solve a triangle we do not require the trigonometric functions or the Pythagorean theorem. We can solve it by the method of similar figures. Now, the sides that make the equal angles are in the same ratio. Proportionally, 2 : 1 = 10 : AC. 2 is two times 1. Therefore 10 is two times AC. AC is 5 cm. The side adjacent to 60°, we see, is always half the hypotenuse. As for BC—proportionally, 2 : = 10 : BC. To produce 10, 2 has been multiplied by 5. Therefore, will also be multiplied by 5. BC is 5 cm. In other words, since one side of the standard triangle has been multiplied by 5, then every side will be multiplied by 5. 1 : 2 : = 5 : 10 : 5. Compare Example 11 here. Again: When we know the ratio numbers, then to solve the triangle the student should use this method of similar figures, not the trigonometric functions. (In Topic 10, we will solve right triangles whose ratios of sides we do not know.) Problem 3. In the right triangle DFE, angle D is 30° and side DF is 3 inches. How long are sides d and f ? The student should draw a similar triangle in the same orientation. Then see that the side corresponding to was multiplied by . Lesson 26 of Algebra Therefore, each side will be multiplied by . Side d will be 1 = . Side f will be 2. Problem 4. In the right triangle PQR, angle P is 30°, and side r is 1 cm. How long are sides p and q ? The side corresponding to 2 has been divided by 2. Therefore, each side must be divided by 2. Side p will be ½, and side q will be ½. Problem 5. Solve the right triangle ABC if angle A is 60°, and the hypotenuse is 18.6 cm. The side adjacent to 60° is always half of the hypotenuse -- therefore, side b is 9.3 cm. But this is the side that corresponds to 1. And it has been multiplied by 9.3. Therefore, side a will be multiplied by 9.3. It will be 9.3 cm. Problem 6. Prove: The area A of an equilateral triangle whose side is s, is A = ¼s2. The area A of any triangle is equal to one-half the sine of any angle times the product of the two sides that make the angle. (Topic 2, Problem 6.) In an equilateral triangle each side is s , and each angle is 60°. Therefore, A = ½ sin 60°s2. Since sin 60° = ½, Problem 1 A = ½· ½ s2 = ¼s2. Problem 7. Prove: The area A of an equilateral triangle inscribed in a circle of radius r, is \| \| \| \| \| --- --- \| \| A = \| 3 4 \| \| r2. \| The three radii divide the triangle into three congruent triangles. Side-side-side Hence each radius bisects each vertex into two 30° angles. If we extend the radius AO, then AD is the perpendicular bisector of the side CB. Theorem 2 Triangle OBD is therefore a 30-60-90 triangle. If we call each side of the equilateral triangle s, then in the right triangle OBD, \| \| \| \| --- \| ½s r \| = \| cos 30° = ½. \| Therefore, s = r so that s2 = 3r2. Now, the area A of an equilateral triangle is A = ¼s2. Problem 6 Therefore, \| \| \| \| \| --- --- \| \| A = ¼s2 = ¼· 3r2 \| = \| 3 4 \| r2. \| That is what we wanted to prove. Problem 8. Prove: The angle bisectors of an equilateral triangle meet at a point that is two thirds of the distance from the vertex of the triangle to the base. Let ABC be an equilateral triangle, let AD, BF, CE be the angle bisectors of angles A, B, C respectively; then those angle bisectors meet at the point P such that AP is two thirds of AD. First, triangles BPD, APE are congruent. For, since the triangle is equilateral and BF, AD are the angle bisectors, then angles PBD, PAE are equal and each 30°; and the side BD is equal to the side AE, because in an equilateral triangle the angle bisector is the perpendicular bisector of the base. Theorem 2 Angles PDB, AEP then are right angles and equal. Therefore, Angle-side-angle triangles BPD, APE are congruent. \| \| \| \| \| --- --- \| \| \| Now, \| BP PD \| = csc 30° = 2. \| Problem 2 Therefore, BP = 2PD. But AP = BP, because triangles APE, BPD are conguent, and those are the sides opposite the equal angles. Therefore, AP = 2PD. Therefore AP is two thirds of the whole AD. Which is what we wanted to prove. The proof Here is the proof that in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : . It is based on the fact that a 30°-60°-90° triangle is half of an equilateral triangle. Draw the equilateral triangle ABC. Then each of its equal angles is 60°. (Theorems 3 and 9) Draw the straight line AD bisecting the angle at A into two 30° angles. Then AD is the perpendicular bisector of BC (Theorem 2). Triangle ABD therefore is a 30°-60°-90° triangle. Now, since BD is equal to DC, then BD is half of BC. This implies that BD is also half of AB, because AB is equal to BC. That is, BD : AB = 1 : 2 From the Pythagorean theorem, we can find the third side AD: \| \| \| \| --- \| AD2 + 12 \| = \| 22 \| \| AD2 \| = \| 4 − 1 = 3 \| \| AD \| = \| . \| Therefore in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : ; which is what we set out to prove. Corollary. The square drawn on the height of an equalateral triangle is three fourths of the square drawn on the side. Next Topic: The Isosceles Right Triangle The Topics \| Home --- Please make a donation to keep TheMathPage online. Even $1 will help. --- Copyright © 2022 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com \|
4673	https://www.ck12.org/flexi/cbse-math/divisibility-rules/lesspgreaterprove-that-one-of-any-three-consecutive-positive-integers-must-be-divisible-by-3-less-by-pgreater/	Flexi answers - Prove that one of any three consecutive positive integers must be divisible by 3 . \| CK-12 Foundation Subjects Explore Donate Sign InSign Up All Subjects CBSE Math Divisibility Rules Question Prove that one of any three consecutive positive integers must be divisible by 3 . Flexi Says: To prove that one of any three consecutive positive integers is divisible by 3, let's consider three consecutive integers: n, n+1, and n+2. Step-by-step: Remainders when divided by 3: Any integer n can be expressed in one of the forms: 3 k, 3 k+1, or 3 k+2, where k is an integer. Check each case: Case 1: If n=3 k, then n is divisible by 3. Case 2: If n=3 k+1, then n+1=3 k+2 and n+2=3 k+3. Here, n+2 is divisible by 3. Case 3: If n=3 k+2, then n+1=3 k+3. Here, n+1 is divisible by 3. In each case, at least one of the numbers n, n+1, or n+2 is divisible by 3. Conclusion: Therefore, one of any three consecutive positive integers must be divisible by 3. Q.E.D. Analogy / Example Try Asking: By what numbers is 107 divisible?What is 69 divisible by?What is 144 divisible by? How can Flexi help? By messaging Flexi, you agree to our Terms and Privacy Policy
4674	https://pdfcoffee.com/inorganic-chemistry-atkins-shriverpdf-pdf-free.html	Inorganic Chemistry (Atkins, Shriver).PDF Author / Uploaded Luedu Jkdhask Categories Documents The elements Name Symbol Atomic number Molar mass (g mol⫺1) Name Symbol Atomic number Molar mass (g mol⫺1) Actin Views 16,253 Downloads 5,114 File size 31MB Report DMCA / Copyright DOWNLOAD FILE Recommend Stories ###### Inorganic Chemistry c c c p gp gp gp gp gp gp gp \| p p p pp p p p p pp 1,518 89 21KB Read more ###### Chemistry Conceptual Inorganic Chemistry INORGANIC ASSIGNMENT (d) Covalent and co-ordinate CHEMICAL BONDING 12. Which contains a co-ordinate and covalent bond 2,337 130 3MB Read more ###### Inorganic Chemistry INTRODUCTION TO CHEMISTRY CHEMISTRY – Study of the structure, composition, properties and reactions of matter and the en 1 0 2MB Read more ###### Inorganic Chemistry I-N-O-R-G-A-N-I-C C-H-E-M-I-S-T-R-Y Single Choice Questions Chemical Bonding Q. 1. The bond order of individual carbon 1 0 325KB Read more ###### Inorganic Chemistry 1 Coordination Chemistry 2 Coordination Chemistry 1. Ni(CN4)2- and Ni(Cl4)2- complex ions are : (a) (b) (c) (d) 2 0 6MB Read more ###### Inorganic Chemistry The elements Name Symbol Atomic number Molar mass (g mol⫺1) Name Symbol Atomic number Molar mass (g mol⫺1) Actin 8,073 7,704 31MB Read more ###### Inorganic Chemistry 5e Atkins, Overton, Rourke, Weller, Armstrong and Hagerman The elements Name Symbol Atomic number Molar mass (g mol⫺1) Name Symbol Atomic number Molar mass (g mol⫺1) Actin 6,460 1,194 31MB Read more ###### Inorganic Chemistry shriver and atkins 5th edition New York The elements Name Symbol Atomic number Molar mass (g mol⫺1) Name Symbol Atomic number Molar mass (g mol⫺1) Actin 6,211 2,268 31MB Read more ###### Atkins - Physical Chemistry 9ed This page intentionally left blank General data and fundamental constants Quantity Symbol Value Power of ten Units 1 0 129MB Read more Citation preview The elements Name Symbol Atomic number Molar mass (g mol⫺1) Name Symbol Atomic number Molar mass (g mol⫺1) Actinium Aluminium (aluminum) Americium Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Bohrium Boron Bromine Cadmium Caesium (cesium) Calcium Californium Carbon Cerium Chlorine Chromium Cobalt Copernicum Copper Curium Darmstadtium Dubnium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold Hafnium Hassium Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Ac Al Am Sb Ar As At Ba Bk Be Bi Bh B Br Cd Cs Ca Cf C Ce Cl Cr Co ? Cu Cm Ds Db Dy Es Er Eu Fm F Fr Gd Ga Ge Au Hf Hs He Ho H In I Ir Fe Kr La Lr Pb Li Lu Mg Mn 89 13 95 51 18 33 85 56 97 4 83 107 5 35 48 55 20 98 6 58 17 24 27 112 29 96 110 105 66 99 68 63 100 9 87 64 31 32 79 72 108 2 67 1 49 53 77 26 36 57 103 82 3 71 12 25 227 26.98 243 121.76 39.95 74.92 210 137.33 247 9.01 208.98 264 10.81 79.90 112.41 132.91 40.08 251 12.01 140.12 35.45 52.00 58.93 ? 63.55 247 271 262 162.50 252 167.27 151.96 257 19.00 223 157.25 69.72 72.64 196.97 178.49 269 4.00 164.93 1.008 114.82 126.90 192.22 55.84 83.80 138.91 262 207.2 6.94 174.97 24.31 54.94 Meitnerium Mendelevium Mercury Molybdenun Neodymium Neon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Roentgenium Rubidium Ruthenium Rutherfordium Samarium Scandium Seaborgium Selenium Silicon Silver Sodium Strontium Sulfur Tantalum Technetium Tellurium Terbium Thallium Thorium Thulium Tin Titanium Tungsten Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium Mt Md Hg Mo Nd Ne Np Ni Nb N No Os O Pd P Pt Pu Po K Pr Pm Pa Ra Rn Re Rh Rg Rb Ru Rf Sm Sc Sg Se Si Ag Na Sr S Ta Tc Te Tb TI Th Tm Sn Ti W U V Xe Yb Y Zn Zr 109 101 80 42 60 10 93 28 41 7 102 76 8 46 15 78 94 84 19 59 61 91 88 86 75 45 111 37 44 104 62 21 106 34 14 47 11 38 16 73 43 52 65 81 90 69 50 22 74 92 23 54 70 39 30 40 268 258 200.59 95.94 144.24 20.18 237 58.69 92.91 14.01 259 190.23 16.00 106.42 30.97 195.08 244 209 39.10 140.91 145 231.04 226 222 186.21 102.91 272 85.47 101.07 261 150.36 44.96 266 78.96 28.09 107.87 22.99 87.62 32.06 180.95 98 127.60 158.93 204.38 232.04 168.93 118.71 47.87 183.84 238.03 50.94 131.29 173.04 88.91 65.41 91.22 This page intentionally left blank Shriver & Atkins’ This page intentionally left blank Shriver & Atkins’ W. H. Freeman and Company New York Shriver and Atkins' Inorganic Chemistry, Fifth Edition © 2010 P.W. Atkins, T.L. Overton, J.P. Rourke, M.T. Weller, and F.A. Armstrong All rights reserved. ISBN 978–1–42–921820–7 Published in Great Britain by Oxford University Press This edition has been authorized by Oxford University Press for sale in the United States and Canada only and not for export therefrom. First printing W. H. Freeman and Company, 41 Madison Avenue, New York, NY 10010 www.whfreeman.com Preface Our aim in the fifth edition of Shriver and Atkins’ Inorganic Chemistry is to provide a comprehensive and contemporary introduction to the diverse and fascinating discipline of inorganic chemistry. Inorganic chemistry deals with the properties of all of the elements in the periodic table. These elements range from highly reactive metals, such as sodium, to noble metals, such as gold. The nonmetals include solids, liquids, and gases, and range from the aggressive oxidizing agent fluorine to unreactive gases such as helium. Although this variety and diversity are features of any study of inorganic chemistry, there are underlying patterns and trends which enrich and enhance our understanding of the discipline. These trends in reactivity, structure, and properties of the elements and their compounds provide an insight into the landscape of the periodic table and provide a foundation on which to build understanding. Inorganic compounds vary from ionic solids, which can be described by simple applications of classical electrostatics, to covalent compounds and metals, which are best described by models that have their origin in quantum mechanics. We can rationalize and interpret the properties of most inorganic compounds by using qualitative models that are based on quantum mechanics, such as atomic orbitals and their use to form molecular orbitals. The text builds on similar qualitative bonding models that should already be familiar from introductory chemistry courses. Although qualitative models of bonding and reactivity clarify and systematize the subject, inorganic chemistry is essentially an experimental subject. New areas of inorganic chemistry are constantly being explored and new and often unusual inorganic compounds are constantly being synthesized and identified. These new inorganic syntheses continue to enrich the field with compounds that give us new perspectives on structure, bonding, and reactivity. Inorganic chemistry has considerable impact on our everyday lives and on other scientific disciplines. The chemical industry is strongly dependent on it. Inorganic chemistry is essential to the formulation and improvement of modern materials such as catalysts, semiconductors, optical devices, superconductors, and advanced ceramic materials. The environmental and biological impact of inorganic chemistry is also huge. Current topics in industrial, biological, and environmental chemistry are mentioned throughout the book and are developed more thoroughly in later chapters. In this new edition we have refined the presentation, organization, and visual representation. All of the book has been revised, much has been rewritten and there is some completely new material. We have written with the student in mind, and we have added new pedagogical features and have enhanced others. The topics in Part 1, Foundations, have been revised to make them more accessible to the reader with more qualitative explanation accompanying the more mathematical treatments. Part 2, The elements and their compounds, has been reorganized. The section starts with a new chapter which draws together periodic trends and cross references forward to the descriptive chapters. The remaining chapters start with hydrogen and proceed across the periodic table from the s-block metals, across the p block, and finishing with the d- and f-block elements. Most of these chapters have been reorganized into two sections: Essentials describes the essential chemistry of the elements and the Detail provides a more thorough account. The chemical properties of each group of elements and their compounds are enriched with descriptions of current applications. The patterns and trends that emerge are rationalized by drawing on the principles introduced in Part 1. Part 3, Frontiers, takes the reader to the edge of knowledge in several areas of current research. These chapters explore specialized subjects that are of importance to industry, materials, and biology, and include catalysis, nanomaterials, and bioinorganic chemistry. All the illustrations and the marginal structures—nearly 1500 in all—have been redrawn and are presented in full colour. We have used colour systematically rather than just for decoration, and have ensured that it serves a pedagogical purpose. viii Preface We are confident that this text will serve the undergraduate chemist well. It provides the theoretical building blocks with which to build knowledge and understanding of inorganic chemistry. It should help to rationalize the sometimes bewildering diversity of descriptive chemistry. It also takes the student to the forefront of the discipline and should therefore complement many courses taken in the later stages of a programme. Peter Atkins Tina Overton Jonathan Rourke Mark Weller Fraser Armstrong Mike Hagerman March 2009 Acknowledgements We have taken care to ensure that the text is free of errors. This is difficult in a rapidly changing field, where today’s knowledge is soon replaced by tomorrow’s. We would particularly like to thank Jennifer Armstrong, University of Southampton; Sandra Dann, University of Loughborough; Rob Deeth, University of Warwick; Martin Jones, Jennifer Creen, and Russ Egdell, University of Oxford, for their guidance and advice. Many of the figures in Chapter 27 were produced using PyMOL software; for more information see DeLano, W.L. The PyMOL Molecular Graphics System (2002), De Lano Scientific, San Carlos, CA, USA. We acknowledge and thank all those colleagues who so willingly gave their time and expertise to a careful reading of a variety of draft chapters. Rolf Berger, University of Uppsala, Sweden Richard Henderson, University of Newcastle Harry Bitter, University of Utrecht, The Netherlands Eva Hervia, University of Strathclyde Richard Blair, University of Central Florida Brendan Howlin, University of Surrey Andrew Bond, University of Southern Denmark, Denmark Songping Huang, Kent State University Darren Bradshaw, University of Liverpool Carl Hultman, Gannon University Paul Brandt, North Central College Stephanie Hurst, Northern Arizona University Karen Brewer, Hamilton College Jon Iggo, University of Liverpool George Britovsek, Imperial College, London S. Jackson, University of Glasgow Scott Bunge, Kent State University Michael Jensen, Ohio University David Cardin, University of Reading Pavel Karen, University of Oslo, Norway Claire Carmalt, University College London Terry Kee, University of Leeds Carl Carrano, San Diego State University Paul King, Birbeck, University of London Neil Champness, University of Nottingham Rachael Kipp, Suffolk University Ferman Chavez, Oakland University Caroline Kirk, University of Loughborough Ann Chippindale, University of Reading Lars Kloo, KTH Royal Institute of Technology, Sweden Karl Coleman, University of Durham Randolph Kohn, University of Bath Simon Collison, University of Nottingham Simon Lancaster, University of East Anglia Bill Connick, University of Cincinnati Paul Lickiss, Imperial College, London Stephen Daff, University of Edinburgh Sven Lindin, University of Stockholm, Sweden Sandra Dann, University of Loughborough Paul Loeffler, Sam Houston State University Nancy Dervisi, University of Cardiff Paul Low, University of Durham Richard Douthwaite, University of York Astrid Lund Ramstrad, University of Bergen, Norway Simon Duckett, University of York Jason Lynam, University of York A.W. Ehlers, Free University of Amsterdam, The Netherlands Joel Mague, Tulane University Anders Eriksson, University of Uppsala, Sweden Francis Mair, University of Manchester Andrew Fogg, University of Liverpool Mikhail Maliarik, University of Uppsala, Sweden Margaret Geselbracht, Reed College David E. Marx, University of Scranton Gregory Grant, University of Tennessee Katrina Miranda, University of Arizona Yurii Gun’ko, Trinity College Dublin Grace Morgan, University College Dublin Simon Hall, University of Bristol Ebbe Nordlander, University of Lund, Sweden Justin Hargreaves, University of Glasgow Lars Öhrström, Chalmers (Goteborg), Sweden x Acknowledgements Ivan Parkin, University College London Martin B. Smith, University of Loughborough Dan Price, University of Glasgow Sheila Smith, University of Michigan T. B. Rauchfuss, University of Illinois Jake Soper, Georgia Institute of Technology Jan Reedijk, University of Leiden, The Netherlands Jonathan Steed, University of Durham David Richens, St Andrews University Gunnar Svensson, University of Stockholm, Sweden Denise Rooney, National University of Ireland, Maynooth Andrei Verdernikov, University of Maryland Graham Saunders, Queens University Belfast Ramon Vilar, Imperial College, London Ian Shannon, University of Birmingham Keith Walters, Northern Kentucky University P. Shiv Halasyamani, University of Houston Robert Wang, Salem State College Stephen Skinner, Imperial College, London David Weatherburn, University of Victoria, Wellington Bob Slade, University of Surrey Paul Wilson, University of Bath Peter Slater, University of Surrey Jingdong Zhang, Denmark Technical University LeGrande Slaughter, Oklahoma State University About the book Inorganic chemistry is an extensive subject that at first sight can seem daunting. We have made every effort to help by organizing the information in this textbook systematically, and by including numerous features that are designed to make learning inorganic chemistry more effective and more enjoyable. Whether you work through the book chronologically or dip in at an appropriate point in your studies, this text will engage you and help you to develop a deeper understanding of the subject. We have also provided further electronic resources in the accompanying Book Companion Site. The following paragraphs explain the features of the text and website in more detail. Organizing the information Key points The key points act as a summary of the main take-home message(s) of the section that follows. They will alert you to the principal ideas being introduced. 2.1 The octet rule Key point: Atoms share electron pairs until they have acquired an octet of valence electrons. Lewis found that he could account for the existence of a wide range of molecules by proposing the octet rule: Context boxes The numerous context boxes illustrate the diversity of inorganic chemistry and its applications to advanced materials, industrial processes, environmental chemistry, and everyday life, and are set out distinctly from the text itself. B OX 11.1 Lithium batteries The very negative standard potential and low molar mass of lithium make it an ideal anode material for batteries. These batteries have high specific energy (energy production divided by the mass of the battery) because lithium metal and compounds containing lithium are relatively light in comparison with some other materials used in batteries, such as lead and zinc. Lithium batteries are common, but there are many types based on different lithium compounds and reactions. The lithium rechargeable battery, used in portable computers and phones, mainly uses Li1⫺xCoO2 (x ⬍ 1) as the cathode with a lithium/graphite anode, the redox reaction in a similar way to the cobalt. The latest generation of electric cars uses lithium battery technology rather than lead-acid cells. Another popular lithium battery uses thionyl chloride, SOCl2. This system produces a light, high-voltage cell with a stable energy output. The overall reaction in the battery is 2 Li(s) ⫹ 3 SOCl2(l) q LiCl(s) ⫹ S(s) ⫹ SO2(l) The battery requires no additional solvent as both SOCl2 and SO2 are liquids at the internal battery pressure. This battery is not rechargeable as Further reading Each chapter lists sources where more information can be found. We have tried to ensure that these sources are easily available and have indicated the type of information each one provided. Resource section At the back of the book is a collection of resources, including an extensive data section and information relating to group theory and spectroscopy. FURTHER READING P. Atkins and J. de Paula, Physical chemistry. Oxford University Press and W.H. Freeman & Co (2010). An account of the generation and use of character tables without too much mathematical background. For more rigorous introductions, see: J.S. Ogden, Introduction to molecular symmetry. Oxford University Press (2001). P. Atkins and R. Friedman, Molecular quantum mechanics. Oxford University Press (2005). xii About the book Problem solving Examples and Self-tests E X A M PL E 6 .1 Identifying symmetry elements Identify the symmetry elements in the eclipsed and staggered conformations of an ethane molecule. Answer We need to identify the rotations, reflections, and inversions that leave the molecule apparently unchanged. Don’t forget that the identity is a symmetry operation. By inspection of the molecular models, we see that the eclipsed conformation of a CH3CH3 molecule (1) has the elements E, C3, C2, ␴h, ␴v, and S3. The staggered conformation (2) has the elements E, C3, ␴d, i, and S6. Self-test 6.1 Sketch the S4 axis of an NH⫹4 ion. How many of these axes does the ion possess? We have provided numerous Worked examples throughout the text. Each one illustrates an important aspect of the topic under discussion or provides practice with calculations and problems. Each Example is followed by a Self-test, where the answer is provided as a check that the method has been mastered. Think of Self-tests as in-chapter exercises designed to help you monitor your progress. Exercises EXERCISES 6.1 Draw sketches to identify the following symmetry elements: (a) a C3 axis and a ␴v plane in the NH3 molecule, (b) a C4 axis and a ␴h plane in the square-planar [PtCl4]2– ion. 220, 213, and 83 cm–1. Detailed analysis of the 369 and 295 cm–1 bands show them to arise from totally symmetric modes. Show that the Raman spectrum is consistent with a trigonal-bipyamidal geometry. 6.2 Which of the following molecules and ions has (a) a centre of inversion, (b) an S4 axis: (i) CO2, (ii) C2H2, (iii) BF3, (iv) SO42–? 6.9 How many vibrational modes does an SO3 molecule have (a) in the plane of the nuclei, (b) perpendicular to the molecular plane? 6.3 Determine the symmetry elements and assign the point group of (a) NH2Cl, (b) CO32–, (c) SiF4, (d) HCN, (e) SiFClBrI, (f) BF4–. 6.10 What are the symmetry species of the vibrations of (a) SF6, (b) BF3 that are both IR and Raman active? 6.4 How many planes of symmetry does a benzene molecule possess? What chloro-substituted benzene of formula C6HnCl6–n has exactly four planes of symmetry? 6.11 What are the symmetry species of the vibrational modes of a C6v molecule that are neither IR nor Raman active? 6.5 Determine the symmetry elements of objects with the same shape as the boundary surface of (a) an s orbital, (b) a p orbital, (c) a dxy orbital, (d) a dz^2 orbital. 6.6 (a) Determine the symmetry group of an SO32– ion. (b) What is the maximum degeneracy of a molecular orbital in this ion? (c) If the sulfur orbitals are 3s and 3p, which of them can contribute to molecular orbitals of this maximum degeneracy? 6.7 (a) Determine the point group of the PF5 molecule. (Use VSEPR, if necessary, to assign geometry.) (b) What is the maximum degeneracy of its molecular orbitals? (c) Which P3p orbitals contribute to a molecular orbital of this degeneracy? 6.12 The [AuCl4]– ion has D4h symmetry. Determine the representations ⌫ of all 3N displacements and reduce it to obtain the symmetry species of the irreducible representations. There are many brief Exercises at the end of each chapter. Answers are found in the Answers section and fully worked answers are available in the separate Solutions manual. The Exercises can be used to check your understanding and gain experience and practice in tasks such as balancing equations, predicting and drawing structures, and manipulating data. 6.13 How could IR and Raman spectroscopy be used to distinguish between: (a) planar and pyramidal forms of PF3, (b) planar and 90º-twisted forms of B2F4 (D2h and D2d, respectively). 6.14 (a) Take the four hydrogen 1s orbitals of CH4 and determine how they transform under Td. (b) Confirm that it is possible to reduce this representation to A1 + T2. (c) With which atomic orbitals on C would it be possible to form MOs with H1s SALCs of symmetry A1 + T2? 6.15 Consider CH4. Use the projection operator method to construct the SALCs of A1 + T2 symmetry that derive from the four H1s orbitals. Problems PROBLEMS 6.1 Consider a molecule IF3O2 (with I as the central atom). How many isomers are possible? Assign point group designations to each isomer. 6.2 (a) Determine the point group of the most symmetric planar conformation of B(OH)3 and the most symmetric nonplanar conformation of B(OH)3. Assume that the B⫺O⫺H bond angles are 109.5º in all conformations. (b) Sketch a conformation of B(OH)3 that is chiral, once again keeping all three B⫺O⫺H bond angles equal to 109.5º. The Problems are more demanding in content and style than the Exercises and are often based on a research paper or other additional source of information. Problems generally require a discursive response and there may not be a single correct answer. They may be used as essay type questions or for classroom discussion. New Molecular Modelling Problems Over the past two decades computational chemistry has evolved from a highly specialized tool, available to relatively few researchers, into a powerful and practical alternative to experimentation, accessible to all chemists. The driving force behind this evolution is the remarkable progress in computer technology. Calculations that previously required hours or days on giant mainframe computers may now be completed in a fraction of time on a personal computer. It is natural and necessary that computational chemistry finds its way into the undergraduate chemistry curriculum. This requires a hands-on approach, just as teaching experimental chemistry requires a laboratory. With this edition we have the addition of new molecular modelling problems for almost every chapter, which can be found on the text’s companion web site. The problems were written to be performed using the popular Spartan StudentTM software. With purchase of this text, students can purchase Wavefunction’s Spartan StudentTM at a significant discount from www.wavefun.com/cart/spartaned.html using the code WHFICHEM. While the problems are written to be performed using Spartan StudentTM they can be completed using any electronic structure program that allows Hartree-Fock, density functional, and MP2 calculations. About the Book Companion Site The Book Companion Site which accompanies this book provides teaching and learning resources to augment the printed book. It is free of charge, and provides additional material for download, much of which can be incorporated into a virtual learning environment. You can access the Book Companion Site by visiting www.whfreeman.com/ichem5e Please note that instructor resources are available only to registered adopters of the textbook. To register, simply visit www.whfreeman.com/ichem5e and follow the appropriate links. You will be given the opportunity to select your own username and password, which will be activated once your adoption has been verified. Student resources are openly available to all, without registration. Instructor resources Artwork An instructor may wish to use the figures from this text in a lecture. Almost all the figures are available in PowerPoint® format and can be used for lectures without charge (but not for commercial purposes without specific permission). Tables of data All the tables of data that appear in the chapter text are available and may be used under the same conditions as the figures. New Molecular Modelling Problems With this edition we have the addition of new molecular modelling problems for almost every chapter, which can be found on the text’s companion web site. The problems were written to be performed using the popular Spartan StudentTM software. With purchase of this text, students can purchase Wavefunction’s Spartan StudentTM at a significant discount from www.wavefun.com/cart/spartaned.html using the code WHFICHEM. While the problems are written to be performed using Spartan StudentTM they can be completed using any electronic structure program that allows Hartree-Fock, density functional, and MP2 calculations. Student resources 3D rotatable molecular structures Nearly all the numbered molecular structures featured in the book are available in a three-dimensional, viewable, rotatable form along with many of the crystal structures and bioinorganic molecules. These have been produced in collaboration with Dr Karl Harrison, University of Oxford. Group theory tables Comprehensive group theory tables are available for downloading. Videos of chemical reactions Video clips showing demonstrations of inorganic chemistry reactions are available for viewing. Solutions manual As with the previous edition, Michael Hagerman, Christopher Schnabel, and Kandalam Ramanujachary have produced the solutions manual to accompany this book. A Solution Manual (978-142-925255-3) provides completed solutions to most end of chapter Exercises and Self-tests. Spartan Student discount With purchase of this text, students can purchase Wavefunction’s Spartan StudentTM at a significant discount at www.wavefun.com/cart/spartaned.html using the code WHFICHEM. Answers to Self-tests and Exercises Please visit the Book Companion Site at www.whfreeman.com/ichem5e/ to download a PDF document containing answers to the end-of-chapter exercises in this book. Summary of contents Part 1 Foundations 1 1 Atomic structure 2 Molecular structure and bonding 34 3 The structures of simple solids 65 4 Acids and bases 111 5 Oxidation and reduction 147 6 Molecular symmetry 179 7 An introduction to coordination compounds 199 8 Physical techniques in inorganic chemistry 223 Part 2 The elements and their compounds 9 3 255 Periodic trends 257 10 Hydrogen 274 11 The Group 1 elements 293 12 The Group 2 elements 309 13 The Group 13 elements 325 14 The Group 14 elements 350 15 The Group 15 elements 375 16 The Group 16 elements 398 17 The Group 17 elements 419 18 The Group 18 elements 440 19 The d-block elements 449 20 d-Metal complexes: electronic structure and properties 473 21 Coordination chemistry: reactions of complexes 507 22 d-Metal organometallic chemistry 534 23 The f-block metals 579 Part 3 Frontiers 599 24 Solid-state and materials chemistry 601 25 Nanomaterials, nanoscience, and nanotechnology 653 26 Catalysis 690 27 Biological inorganic chemistry 722 Resource section 1: Resource section 2: Resource section 3: Resource section 4: Resource section 5: Resource section 6: Index Selected ionic radii Electronic properties of the elements Standard potentials Character tables Symmetry-adapted orbitals Tanabe–Sugano diagrams 783 785 787 800 805 809 813 This page intentionally left blank Contents Part 1 Foundations 1 3 The structures of simple solids The description of the structures of solids 65 66 Atomic structure 3 3.1 Unit cells and the description of crystal structures 66 The origin of the elements 4 3.2 The close packing of spheres 68 3.3 Holes in close-packed structures 70 1 1.1 The nucleosynthesis of light elements 1.2 The nucleosynthesis of heavy elements 5 6 The structures of metals and alloys 71 The structures of hydrogenic atoms 8 3.4 Polytypism 72 1.3 Spectroscopic information 8 3.5 Non-close-packed structures 72 1.4 Some principles of quantum mechanics 9 3.6 Polymorphism of metals 73 10 3.7 Atomic radii of metals 74 15 3.8 Alloys 75 1.5 Atomic orbitals Many-electron atoms 1.6 Penetration and shielding 16 1.7 The building-up principle 18 1.8 The classification of the elements 20 1.9 Atomic parameters 22 83 86 32 3.12 The calculation of lattice enthalpies 88 33 3.13 Comparison of experimental and theoretical values 90 3.14 The Kapustinskii equation 91 3.15 Consequences of lattice enthalpies 91 EXERCISES PROBLEMS Lewis structures The energetics of ionic bonding 77 87 32 Molecular structure and bonding 3.9 Characteristic structures of ionic solids 3.10 The rationalization of structures 77 3.11 Lattice enthalpy and the Born–Haber cycle FURTHER READING 2 Ionic solids 34 34 Defects and nonstoichiometry 95 2.1 The octet rule 34 3.16 The origins and types of defects 96 2.2 Resonance 35 3.17 Nonstoichiometric compounds and solid solutions 99 2.3 The VSEPR model 36 The electronic structures of solids 101 39 3.18 The conductivities of inorganic solids 101 2.4 The hydrogen molecule 39 3.19 Bands formed from overlapping atomic orbitals 101 2.5 Homonuclear diatomic molecules 40 3.20 Semiconduction 104 2.6 Polyatomic molecules 40 FURTHER INFORMATION 3.1 The Born–Mayer equation 42 FURTHER READING 107 2.7 An introduction to the theory 43 EXERCISES 108 2.8 Homonuclear diatomic molecules 45 PROBLEMS 108 2.9 Heteronuclear diatomic molecules 48 Valence bond theory Molecular orbital theory 2.10 Bond properties 50 2.11 Polyatomic molecules 52 2.12 Molecular shape in terms of molecular orbitals 56 Structure and bond properties 58 2.13 Bond length 58 2.14 Bond strength 59 2.15 Electronegativity and bond enthalpy 59 2.16 Oxidation states 61 FURTHER READING 62 EXERCISES 62 PROBLEMS 63 4 Acids and bases Brønsted acidity 106 111 111 4.1 Proton transfer equilibria in water 112 4.2 Solvent levelling 119 4.3 The solvent system definition of acids and bases 121 Characteristics of Brønsted acids 122 4.4 Periodic trends in aqua acid strength 122 4.5 Simple oxoacids 123 4.6 Anhydrous oxides 126 4.7 Polyoxo compound formation 127 4.8 Nonaqueous solvents 129 xviii Contents Lewis acidity 131 Applications of symmetry 186 4.9 Examples of Lewis acids and bases 132 6.3 Polar molecules 186 4.10 Group characteristics of Lewis acids 133 6.4 Chiral molecules 187 136 6.5 Molecular vibrations 188 Reactions and properties of lewis acids and bases 4.11 The fundamental types of reaction 137 4.12 Hard and soft acids and bases 138 6.6 Symmetry-adapted linear combinations 191 4.13 Thermodynamic acidity parameters 140 6.7 The construction of molecular orbitals 192 4.14 Solvents as acids and bases 141 6.8 The vibrational analogy 194 Applications of acid–base chemistry 142 4.15 Superacids and superbases 142 4.16 Heterogeneous acid–base reactions 143 The symmetries of molecular orbitals Representations 6.9 The reduction of a representation 6.10 Projection operators 191 194 194 196 FURTHER READING 144 FURTHER READING 197 EXERCISES 144 EXERCISES 197 PROBLEMS 145 PROBLEMS 197 5 Oxidation and reduction Reduction potentials 147 7 An introduction to coordination compounds 148 The language of coordination chemistry 199 5.1 Redox half-reactions 148 7.1 Representative ligands 200 5.2 Standard potentials and spontaneity 149 7.2 Nomenclature 202 5.3 Trends in standard potentials 151 Constitution and geometry 203 5.4 The electrochemical series 153 7.3 Low coordination numbers 204 5.5 The Nernst equation 154 7.4 Intermediate coordination numbers 204 156 7.5 Higher coordination numbers 206 156 7.6 Polymetallic complexes 208 Redox stability 5.6 The influence of pH 5.7 Reactions with water 157 5.8 Oxidation by atmospheric oxygen 159 7.7 Square-planar complexes 209 5.9 Disproportionation and comproportionation 160 7.8 Tetrahedral complexes 210 161 7.9 Trigonal-bipyramidal and square-pyramidal complexes 210 5.10 The influence of complexation 5.11 The relation between solubility and standard potentials The diagrammatic presentation of potential data Isomerism and chirality 208 162 7.10 Octahedral complexes 211 162 7.11 Ligand chirality 214 5.12 Latimer diagrams 162 5.13 Frost diagrams 164 7.12 Formation constants 215 5.14 Pourbaix diagrams 168 7.13 Trends in successive formation constants 216 5.15 Natural waters 169 7.14 The chelate and macrocyclic effects 218 169 7.15 Steric effects and electron delocalization 219 Chemical extraction of the elements The thermodynamics of complex formation 215 5.16 Chemical reduction 170 FURTHER READING 220 5.17 Chemical oxidation 174 EXERCISES 221 174 PROBLEMS 221 5.18 Electrochemical extraction FURTHER READING 175 EXERCISES 176 8 PROBLEMS 177 Diffraction methods Physical techniques in inorganic chemistry 8.1 X-ray diffraction 6 199 Molecular symmetry An introduction to symmetry analysis 179 179 8.2 Neutron diffraction Absorption spectroscopy 223 223 223 226 227 6.1 Symmetry operations, elements and point groups 179 8.3 Ultraviolet–visible spectroscopy 228 6.2 Character tables 183 8.4 Infrared and Raman spectroscopy 230 Contents Resonance techniques xix 233 10.5 Reactions of dihydrogen 281 8.5 Nuclear magnetic resonance 233 10.6 Compounds of hydrogen 283 8.6 Electron paramagnetic resonance 238 10.7 General methods for synthesis 291 8.7 Mössbauer spectroscopy 240 FURTHER READING 291 292 292 241 EXERCISES 8.8 Photoelectron spectroscopy 241 PROBLEMS 8.9 X-ray absorption spectroscopy 242 Ionization-based techniques 8.10 Mass spectrometry Chemical analysis 243 11 The Group 1 elements 245 Part A: The essentials 293 293 8.11 Atomic absorption spectroscopy 245 11.1 The elements 8.12 CHN analysis 246 11.2 Simple compounds 295 8.13 X-ray fluorescence elemental analysis 247 11.3 The atypical properties of lithium 296 8.14 Thermal analysis 247 Part B: The detail 293 296 Magnetometry 249 11.4 Occurrence and extraction Electrochemical techniques 249 11.5 Uses of the elements and their compounds 297 Computational techniques 250 11.6 Hydrides 298 FURTHER READING 251 11.7 Halides 299 300 301 EXERCISES 252 11.8 Oxides and related compounds PROBLEMS 253 11.9 Sulfides, selenides, and tellurides 11.10 Hydroxides Part 2 The elements and their compounds 9 Periodic trends Periodic properties of the elements 255 296 301 11.11 Compounds of oxoacids 302 11.12 Nitrides and carbides 304 257 11.13 Solubility and hydration 304 257 11.14 Solutions in liquid ammonia 305 257 11.15 Zintl phases containing alkali metals 305 9.2 Atomic parameters 257 11.16 Coordination compounds 305 9.3 Occurrence 261 11.17 Organometallic compounds 307 9.4 Metallic character 263 9.1 Valence electron configurations 9.5 Oxidation states 264 Periodic characteristics of compounds 265 9.6 Coordination numbers 265 9.7 Bond enthalpy trends 265 9.8 Anomalies 266 9.9 Binary compounds 268 9.10 Wider aspects of periodicity 270 FURTHER READING 272 EXERCISES 272 PROBLEMS 273 FURTHER READING 308 EXERCISES 308 PROBLEMS 308 12 The Group 2 elements Part A: The essentials 309 309 12.1 The elements 309 12.2 Simple compounds 310 12.3 The anomalous properties of beryllium 311 Part B: The detail 312 12.4 Occurrence and extraction 312 12.5 Uses of the elements and their compounds 313 314 10 Hydrogen 274 12.6 Hydrides Part A: The essentials 274 12.7 Halides 315 10.1 The element 274 12.8 Oxides, sulfides, and hydroxides 316 10.2 Simple compounds 276 12.9 Nitrides and carbides 317 Part B: The detail 10.3 Nuclear properties 10.4 Production of dihydrogen 12.10 Salts of oxoacids 318 279 12.11 Solubility, hydration, and beryllates 320 280 12.12 Coordination compounds 321 279 xx Contents 12.13 Organometallic compounds 322 14.10 Simple compounds of silicon with oxygen 364 FURTHER READING 323 14.11 Oxides of germanium, tin, and lead 365 EXERCISES 323 14.12 Compounds with nitrogen 365 PROBLEMS 324 14.13 Carbides 366 14.14 Silicides 368 14.15 Extended silicon–oxygen compounds 368 325 14.16 Organosilicon compounds 371 13.1 The elements 325 14.17 Organometallic compounds 371 13.2 Compounds 327 13.3 Boron clusters 329 EXERCISES 373 330 PROBLEMS 373 13 The Group 13 elements Part A: The essentials Part B: The detail 325 FURTHER READING 373 13.4 Occurrence and recovery 330 13.5 Uses of the elements and their compounds 330 15 The Group 15 elements 375 13.6 Simple hydrides of boron 330 Part A: The essentials 375 13.7 Boron trihalides 333 15.1 The elements 13.8 Boron–oxygen compounds 334 15.2 Simple compounds 376 13.9 Compounds of boron with nitrogen 335 15.3 Oxides and oxanions of nitrogen 377 Part B: The detail 375 378 13.10 Metal borides 337 13.11 Higher boranes and borohydrides 338 15.4 Occurrence and recovery 378 13.12 Metallaboranes and carboranes 342 15.5 Uses 379 13.13 The hydrides of aluminium and gallium 344 15.6 Nitrogen activation 381 13.14 Trihalides of aluminium, gallium, indium, and thallium 15.7 Nitrides and azides 382 344 15.8 Phosphides 382 13.15 Low-oxidation-state halides of aluminium, gallium, indium, and thallium 345 15.9 Arsenides, antimonides, and bismuthides 383 15.10 Hydrides 383 13.16 Oxo compounds of aluminium, gallium, indium, and thallium 346 15.11 Halides 385 13.17 Sulfides of gallium, indium, and thallium 346 15.12 Oxohalides 386 13.18 Compounds with Group 15 elements 346 15.13 Oxides and oxoanions of nitrogen 387 13.19 Zintl phases 347 15.14 Oxides of phosphorus, arsenic, antimony, and bismuth 390 13.20 Organometallic compounds 347 15.15 Oxoanions of phosphorus, arsenic, antimony, and bismuth 391 FURTHER READING 348 15.16 Condensed phosphates 391 EXERCISES 348 15.17 Phosphazenes 393 PROBLEMS 348 15.18 Organometallic compounds of arsenic, antimony, and bismuth 394 14 The Group 14 elements Part A: The essentials 350 350 14.1 The elements 350 14.2 Simple compounds 352 14.3 Extended silicon–oxygen compounds 353 Part B: The detail 354 14.4 Occurrence and recovery 354 14.5 Diamond and graphite 354 14.6 Other forms of carbon 356 14.7 Hydrides 358 14.8 Compounds with halogens 359 14.9 Compounds of carbon with oxygen and sulfur 361 FURTHER READING 396 EXERCISES 396 PROBLEMS 397 16 The Group 16 elements Part A: The essentials 398 398 16.1 The elements 398 16.2 Simple compounds 400 16.3 Ring and cluster compounds 402 Part B: The detail 403 16.4 Oxygen 403 16.5 Reactivity of oxygen 404 Contents xxi 16.6 Sulfur 404 18.6 Reactions of xenon fluorides 443 16.7 Selenium, tellurium, and polonium 405 18.7 Xenon–oxygen compounds 444 16.8 Hydrides 406 18.8 Xenon insertion compounds 445 16.9 Halides 407 18.9 Organoxenon compounds 445 16.10 Metal oxides 409 18.10 Coordination compounds 446 16.11 Metal sulfides, selenides, tellurides, and polonides 409 18.11 Other compounds of noble gases 446 16.12 Oxides 410 FURTHER READING 447 16.13 Oxoacids of sulfur 412 EXERCISES 447 16.14 Polyanions of sulfur, selenium, and tellurium 415 PROBLEMS 447 16.15 Polycations of sulfur, selenium, and tellurium 416 16.16 Sulfur–nitrogen compounds 416 FURTHER READING 417 EXERCISES 417 PROBLEMS 418 19 The d-Block elements The elements 19.1 Occurrence and recovery 19.2 Physical properties Trends in chemical properties 449 449 449 450 453 17 The Group 17 elements 419 19.3 Oxidation states across a series 453 Part A: The essentials 419 19.4 Oxidation states down a group 456 419 19.5 Structural trends 458 17.2 Simple compounds 421 19.6 Noble character 459 17.3 The interhalogens 422 Representative compounds 460 424 19.7 Metal halides 460 424 19.8 Metal oxides and oxido complexes 460 17.5 Molecular structure and properties 425 19.9 Metal sulfides and sulfide complexes 464 17.6 Reactivity trends 427 19.10 Nitrido and alkylidyne complexes 466 17.7 Pseudohalogens 427 19.11 Metal–metal bonded compounds and clusters 466 17.8 Special properties of fluorine compounds 428 FURTHER READING 471 17.9 Structural features 429 EXERCISES 472 17.10 The interhalogens 429 PROBLEMS 472 17.11 Halogen oxides 432 17.12 Oxoacids and oxoanions 433 17.13 Thermodynamic aspects of oxoanion redox reactions 434 17.14 Trends in rates of oxoanion redox reactions 435 17.15 Redox properties of individual oxidation states 435 17.16 Fluorocarbons 437 17.1 The elements Part B: The detail 17.4 Occurrence, recovery, and uses 20 d-Metal complexes: electronic structure and properties 473 Electronic structure 473 20.1 Crystal-field theory 473 20.2 Ligand-field theory 483 Electronic spectra 487 FURTHER READING 438 20.3 Electronic spectra of atoms 487 EXERCISES 438 20.4 Electronic spectra of complexes 493 PROBLEMS 439 20.5 Charge-transfer bands 497 20.6 Selection rules and intensities 499 20.7 Luminescence 501 18 The Group 18 elements Part A: The essentials 440 440 18.1 The elements 440 18.2 Simple compounds 441 Part B: The detail Magnetism 502 20.8 Cooperative magnetism 502 20.9 Spin crossover complexes 504 442 18.3 Occurrence and recovery 442 18.4 Uses 442 18.5 Synthesis and structure of xenon fluorides 442 FURTHER READING 504 EXERCISES 505 PROBLEMS 505 xxii Contents 21 Coordination chemistry: reactions of complexes Compounds 507 553 22.18 d-Block carbonyls 553 507 22.19 Metallocenes 560 21.1 Rates of ligand substitution 507 22.20 Metal–metal bonding and metal clusters 564 21.2 The classification of mechanisms 509 Ligand substitution reactions Ligand substitution in square-planar complexes Reactions 568 512 22.21 Ligand substitution 568 21.3 The nucleophilicity of the entering group 513 22.22 Oxidative addition and reductive elimination 571 21.4 The shape of the transition state 514 22.23 σ-Bond metathesis 572 Ligand substitution in octahedral complexes 517 22.24 1,1-Migratory insertion reactions 573 21.5 Rate laws and their interpretation 517 22.25 1,2-Insertions and β-hydride elimination 574 21.6 The activation of octahedral complexes 519 21.7 Base hydrolysis 522 22.26 α-, β-, and δ-Hydride eliminations and cyclometallations 575 21.8 Stereochemistry 522 FURTHER READING 21.9 Isomerization reactions 523 EXERCISES 576 524 PROBLEMS 577 Redox reactions 576 21.10 The classification of redox reactions 524 21.11 The inner-sphere mechanism 524 23 The f-Block elements 579 21.12 The outer-sphere mechanism 527 The elements 579 Photochemical reactions 530 23.1 Occurrence and recovery 579 23.2 Physical properties and applications 580 21.13 Prompt and delayed reactions 530 21.14 d–d and charge-transfer reactions 530 21.15 Transitions in metal–metal bonded systems 531 FURTHER READING 532 Lanthanoid chemistry 23.3 General trends 581 581 23.4 Electronic, optical, and magnetic properties 583 586 EXERCISES 532 23.5 Binary ionic compounds PROBLEMS 533 23.6 Ternary and complex oxides 588 23.7 Coordination compounds 589 22 d-Metal organometallic chemistry Bonding 534 535 22.1 Stable electron configurations 535 22.2 Electron count preference 536 22.3 Electron counting and oxidation states 22.4 Nomenclature Ligands 23.8 Organometallic compounds Actinoid chemistry 23.9 General trends 590 592 593 23.10 Electronic spectra 594 537 23.11 Thorium and uranium 595 539 23.12 Neptunium, plutonium, and americium 596 540 FURTHER READING 597 22.5 Carbon monoxide 540 EXERCISES 597 22.6 Phosphines 542 PROBLEMS 598 22.7 Hydrides and dihydrogen complexes 543 22.8 η1-Alkyl, -alkenyl, -alkynyl, and -aryl ligands 544 22.9 η -Alkene and -alkyne ligands 545 2 PART 3 FRONTIERS 599 22.10 Nonconjugated diene and polyene ligands 545 24 Solid-state and materials chemistry 601 22.11 Butadiene, cyclobutadiene, and cyclooctatetraene 546 Synthesis of materials 602 22.12 Benzene and other arenes 548 24.1 The formation of bulk material 602 22.13 The allyl ligand 549 24.2 Chemical deposition 604 22.14 Cyclopentadiene and cycloheptatriene 550 Defects and ion transport 605 22.15 Carbenes 551 24.3 Extended defects 605 22.16 Alkanes, agostic hydrogens, and noble gases 552 24.4 Atom and ion diffusion 606 22.17 Dinitrogen and nitrogen monoxide 552 24.5 Solid electrolytes 607 Contents xxiii Metal oxides, nitrides, and fluorides 611 24.6 Monoxides of the 3d metals 611 FURTHER READING 24.7 Higher oxides and complex oxides 613 EXERCISES 687 24.8 Oxide glasses 623 PROBLEMS 688 24.9 Nitrides and fluorides 625 Chalcogenides, intercalation compounds, and metal-rich phases 627 24.10 Layered MS2 compounds and intercalation 627 24.11 Chevrel phases and chalcogenide thermoelectrics 630 Framework structures 24.12 Structures based on tetrahedral oxoanions 24.13 Structures based on octahedra and tetrahedra 631 25.12 Bionanocomposites 682 687 26 Catalysis 690 General principles 690 26.1 The language of catalysis 26.2 Homogeneous and heterogeneous catalysts Homogeneous catalysis 691 694 694 631 26.3 Alkene metathesis 695 636 26.4 Hydrogenation of alkenes 696 639 26.5 Hydroformylation 698 24.14 Metal hydrides 639 26.6 Wacker oxidation of alkenes 700 24.15 Other inorganic hydrogen storage materials 641 26.7 Asymmetric oxidations 701 642 26.8 Palladium-catalysed C–C bond-forming reactions 701 Hydrides and hydrogen-storage materials Inorganic pigments 24.16 Coloured solids 642 24.17 White and black pigments 643 Semiconductor chemistry 26.9 Methanol carbonylation: ethanoic acid synthesis Heterogeneous catalysis 703 704 644 26.10 The nature of heterogeneous catalysts 704 24.18 Group 14 semiconductors 645 26.11 Hydrogenation catalysts 709 24.19 Semiconductor systems isoelectronic with silicon 645 26.12 Ammonia synthesis 709 647 26.13 Sulfur dioxide oxidation 710 24.20 Fullerides 647 24.21 Molecular materials chemistry 648 26.14 Catalytic cracking and the interconversion of aromatics by zeolites 710 26.15 Fischer–Tropsch synthesis 713 Molecular materials and fullerides FURTHER READING 650 EXERCISES 651 26.16 Alkene polymerization 713 PROBLEMS 651 26.17 Electrocatalysis 717 26.18 New directions in heterogeneous catalysis 25 Nanomaterials, nanoscience, and nanotechnology Fundamentals Hybrid catalysis 653 653 718 718 26.19 Tethered catalysts 719 26.20 Biphasic systems 719 25.1 Terminology and history 653 FURTHER READING 720 25.2 Novel optical properties of nanomaterials 654 EXERCISES 720 657 PROBLEMS 721 Characterization and fabrication 25.3 Characterization methods 657 25.4 Top-down and bottom-up fabrication 658 25.5 Templated synthesis using frameworks, supports, and substrates Self-assembled nanostructures 25.6 Control of nanoarchitecture 25.7 One-dimensonal control: carbon nanotubes and inorganic nanowires 662 666 666 669 27 Biological inorganic chemistry 722 The organization of cells 722 27.1 The physical structure of cells 722 27.2 The inorganic composition of cells 723 Transport, transfer, and transcription 731 27.3 Sodium and potassium transport 731 27.4 Calcium signalling proteins 733 734 25.8 Two-dimensional control: quantum wells and solid-state superlattices 672 27.5 Zinc in transcription 25.9 Three-dimensional control 675 27.6 Selective transport and storage of iron 735 681 27.7 Oxygen transport and storage 738 25.10 DNA and nanomaterials 681 27.8 Electron transfer 741 25.11 Natural and artificial nanomaterials: biomimetics 682 Bioinorganic nanomaterials xxiv Contents Catalytic processes 27.9 Acid–base catalysis 745 Perspectives 776 746 27.21 The contributions of individual elements 777 27.10 Enzymes dealing with H2O2 and O2 751 27.22 Future directions 778 27.11 The reactions of cobalt-containing enzymes 759 27.12 Oxygen atom transfer by molybdenum and tungsten enzymes Biological cycles 763 27.13 The nitrogen cycle 765 768 27.15 Iron proteins as sensors 27.16 Proteins that sense Cu and Zn levels 779 EXERCISES 780 PROBLEMS 781 Resource section 783 765 27.14 The hydrogen cycle Sensors FURTHER READING 768 768 771 Biomineralization 771 The chemistry of elements in medicine 772 27.17 Chelation therapy 773 27.18 Cancer treatment 773 27.19 Anti-arthritis drugs 775 27.20 Imaging agents 775 Resource section 1: Resource section 2: Resource section 3: Resource section 4: Resource section 5: Resource section 6: Index Selected ionic radii Electronic properties of the elements Standard potentials Character tables Symmetry-adapted orbitals Tanabe–Sugano diagrams 783 785 787 800 805 809 813 Glossary of chemical abbreviations Ac acetyl, CH3CO acac acetylacetonato aq aqueous solution species bpy 2,2'-bipyridine cod 1,5-cyclooctadiene cot cyclooctatetraene Cy cyclohexyl Cp cyclopentadienyl Cp pentamethylcyclopentadienyl cyclam tetraazacyclotetradecane dien diethylenetriamine DMSO dimethylsulfoxide DMF dimethylformamide ␩ hapticity edta ethylenediaminetetraacetato en ethylenediamine (1,2-diaminoethane) Et ethyl gly glycinato Hal halide i isopropyl Pr KCP K2Pt(CN)4Br0.3·3H2O L a ligand µ signifies a bridging ligand M a metal Me methyl mes mesityl, 2,4,6-trimethylphenyl Ox an oxidized species ox oxalato Ph phenyl phen phenanthroline py pyridine Sol solvent, or a solvent molecule soln nonaqueous solution species t Bu tertiary butyl THF tetrahydrofuran TMEDA N, N,N′,N′-tetramethylethylenediamine trien 2,2′,2′′-triaminotriethylene X generally halogen, also a leaving group or an anion Y an entering group This page intentionally left blank PART 1 Foundations The eight chapters in this part of the book lay the foundations of inorganic chemistry. The first three chapters develop an understanding of the structures of atoms, molecules, and solids. Chapter 1 introduces the structure of atoms in terms of quantum theory and describes important periodic trends in their properties. Chapter 2 develops molecular structure in terms of increasingly sophisticated models of covalent bonding. Chapter 3 describes ionic bonding and the structures and properties of a range of typical ionic solids. The next two chapters focus on two major types of reactions. Chapter 4 introduces the definitions of acids and bases, and uses their properties to systematize many inorganic reactions. Chapter 5 describes oxidation and reduction, and demonstrates how electrochemical data can be used to predict and explain the outcomes of redox reactions. Chapter 6 shows how a systematic consideration of the symmetry of molecules can be used to discuss the bonding and structure of molecules and help interpret the techniques described in Chapter 8. Chapter 7 describes the coordination compounds of the elements. We discuss bonding, structure, and reactions of complexes, and see how symmetry considerations can provide useful insight into this important class of compounds. Chapter 8 provides a toolbox for inorganic chemistry: it describes a wide range of the instrumental techniques that are used to identify and determine the structures of compounds. This page intentionally left blank 1 Atomic structure This chapter lays the foundations for the explanation of the trends in the physical and chemical properties of all inorganic compounds. To understand the behaviour of molecules and solids we need to understand atoms: our study of inorganic chemistry must therefore begin with a review of their structures and properties. We begin with discussion of the origin of matter in the solar system and then consider the development of our understanding of atomic structure and the behaviour of electrons in atoms. We introduce quantum theory qualitatively and use the results to rationalize properties such as atomic radii, ionization energy, electron affinity, and electronegativity. An understanding of these properties allows us to begin to rationalize the diverse chemical properties of the more than 110 elements known today. The observation that the universe is expanding has led to the current view that about 15 billion years ago the currently visible universe was concentrated into a point-like region that exploded in an event called the Big Bang. With initial temperatures immediately after the Big Bang of about 109 K, the fundamental particles produced in the explosion had too much kinetic energy to bind together in the forms we know today. However, the universe cooled as it expanded, the particles moved more slowly, and they soon began to adhere together under the influence of a variety of forces. In particular, the strong force, a short-range but powerful attractive force between nucleons (protons and neutrons), bound these particles together into nuclei. As the temperature fell still further, the electromagnetic force, a relatively weak but long-range force between electric charges, bound electrons to nuclei to form atoms, and the universe acquired the potential for complex chemistry and the existence of life. Table 1.1 summarizes the properties of the only subatomic particles that we need to consider in chemistry. All the known elements—by 2008, 112 had been confirmed and several more are candidates for confirmation—that are formed from these subatomic particles are distinguished by their atomic number, Z, the number of protons in the nucleus of an atom of the element. Many elements have a number of isotopes, which are atoms with the same atomic number but different atomic masses. These isotopes are distinguished by the mass Table 1.1 Subatomic particles of relevance to chemistry Practice Symbol Mass/mu Mass number Charge/e† Electron e 5.486 104 0 1 1 2 Proton p 1.0073 1 1 1 2 Neutron n 1.0087 1 0 1 2 Photon 0 0 0 1 Neutrino v c. 0 0 0 1 2 0 1 1 2 Spin Positron e 5.486 10 particle [ 24 He2 nucleus ] 4 2 0 particle [e ejected from nucleus] 0 1 1 2 photon [electromagnetic radiation from nucleus] 0 0 1 4 Masses are expressed relative to the atomic mass constant, mu 1.6605 10 † The elementary charge is e 1.602 10–19 C. 27 kg. The origin of the elements 1.1 The nucleosynthesis of light elements 1.2 The nucleosynthesis of heavy elements The structures of hydrogenic atoms 1.3 Spectroscopic information 1.4 Some principles of quantum mechanics 1.5 Atomic orbitals Many-electron atoms 1.6 Penetration and shielding 1.7 The building-up principle 1.8 The classification of the elements 1.9 Atomic properties FURTHER READING EXERCISES PROBLEMS 1 Atomic structure number, A, which is the total number of protons and neutrons in the nucleus. The mass number is also sometimes termed more appropriately the nucleon number. Hydrogen, for instance, has three isotopes. In each case Z 1, indicating that the nucleus contains one proton. The most abundant isotope has A 1, denoted 1H, its nucleus consisting of a single proton. Far less abundant (only 1 atom in 6000) is deuterium, with A 2. This mass number indicates that, in addition to a proton, the nucleus contains one neutron. The formal designation of deuterium is 2H, but it is commonly denoted D. The third, short-lived, radioactive isotope of hydrogen is tritium, 3H or T. Its nucleus consists of one proton and two neutrons. In certain cases it is helpful to display the atomic number of the element as a left suffix; so the three isotopes of hydrogen would then be denoted 11 H, 21 H,and 31 H. The origin of the elements About two hours after the start of the universe, the temperature had fallen so much that most of the matter was in the form of H atoms (89 per cent) and He atoms (11 per cent). In one sense, not much has happened since then for, as Fig. 1.1 shows, hydrogen and helium log (mass fraction, ppb) 10 O Si Ca Fe Earth's crust H Sr 6 Ba Pb 2 Ar Ne He Kr –2 Xe Rn –6 10 30 50 70 Atomic number, Z 90 H 11 log (atoms per 1012 H) 4 Sun O Fe 7 F 3 Sc Li As –1 10 30 50 70 Atomic number, Z 90 Figure 1.1 The abundances of the elements in the Earth’s crust and the Sun. Elements with odd Z are less stable than their neighbours with even Z. The origin of the elements remain overwhelmingly the most abundant elements in the universe. However, nuclear reactions have formed a wide assortment of other elements and have immeasurably enriched the variety of matter in the universe, and thus given rise to the whole area of chemistry. 1.1 The nucleosynthesis of light elements Key points: The light elements were formed by nuclear reactions in stars formed from primeval hydrogen and helium; total mass number and overall charge are conserved in nuclear reactions; a large binding energy signifies a stable nucleus. The earliest stars resulted from the gravitational condensation of clouds of H and He atoms. The compression of these clouds under the influence of gravity gave rise to high temperatures and densities within them, and fusion reactions began as nuclei merged together. The earliest nuclear reactions are closely related to those now being studied in connection with the development of controlled nuclear fusion. Energy is released when light nuclei fuse together to give elements of higher atomic number. For example, the nuclear reaction in which an particle (a 4He nucleus with two protons and two neutrons) fuses with a carbon-12 nucleus to give an oxygen-16 nucleus and a -ray photon () is 12 6 C 42 α ➝ 16 8 O This reaction releases 7.2 MeV of energy.1 Nuclear reactions are very much more energetic than normal chemical reactions because the strong force is much stronger than the electromagnetic force that binds electrons to nuclei. Whereas a typical chemical reaction might release about 103 kJ mol1, a nuclear reaction typically releases a million times more energy, about 109 kJ mol1. In this nuclear equation, the nuclide, a nucleus of specific atomic number Z and mass number A, is designated ZA E , where E is the chemical symbol of the element. Note that, in a balanced nuclear equation, the sum of the mass numbers of the reactants is equal to the sum of the mass numbers of the products (12 4 16). The atomic numbers sum similarly (6 2 8) provided an electron, e, when it appears as a particle, is denoted 10 e and a positron, e, is denoted 01 e. A positron is a positively charged version of an electron: it has zero mass number (but not zero mass) and a single positive charge. When it is emitted, the mass number of the nuclide is unchanged but the atomic number decreases by 1 because the nucleus has lost one positive charge. Its emission is equivalent to the conversion of a proton in the nucleus into a neutron: 11 p ➝ 01 n e ν. A neutrino, (nu), is electrically neutral and has a very small (possibly zero) mass. Elements up to Z 26 were formed inside stars. Such elements are the products of the nuclear fusion reactions referred to as ‘nuclear burning’. The burning reactions, which should not be confused with chemical combustion, involved H and He nuclei and a complicated fusion cycle catalysed by C nuclei. (The stars that formed in the earliest stages of the evolution of the cosmos lacked C nuclei and used noncatalysed H-burning reactions.) Some of the most important nuclear reactions in the cycle are Proton (p) capture by carbon-12: 12 6 C 11 p ➝ Positron decay accompanied by neutrino () emission: 13 7 N➝ Proton capture by carbon-13: 13 6 C 11 p ➝ 14 7 Nγ Proton capture by nitrogen-14: 14 7 N 11p ➝ 15 8 O γ Positron decay, accompanied by neutrino emission: 15 8 O ➝ Proton capture by nitrogen-15: 15 7 N 11p ➝ 13 6 13 7 Nγ C e ν 15 7 N e+ ν 12 6 C 42 α The net result of this sequence of nuclear reactions is the conversion of four protons (four 1H nuclei) into an particle (a 4He nucleus): 4 11p ➝ 42 2e+ 2 ␯ 3 1 An electronvolt (1 eV) is the energy required to move an electron through a potential difference of 1 V. It follows that 1 eV 1.602 1019 J, which is equivalent to 96.48 kJ mol1; 1 MeV 106 eV. 5 6 1 Atomic structure The reactions in the sequence are rapid at temperatures between 5 and 10 MK (where 1 MK 106 K). Here we have another contrast between chemical and nuclear reactions, because chemical reactions take place at temperatures a hundred thousand times lower. Moderately energetic collisions between species can result in chemical change, but only highly vigorous collisions can provide the energy required to bring about most nuclear processes. Heavier elements are produced in significant quantities when hydrogen burning is complete and the collapse of the star’s core raises the density there to 108 kg m3 (about 105 times the density of water) and the temperature to 100 MK. Under these extreme conditions, helium burning becomes viable. The low abundance of beryllium in the present-day universe is consistent with the observation that 48 Be formed by collisions between particles goes on to react with more particles to produce the more stable carbon nuclide 216 C: 8 4 Be 42 ➝ 12 6 C Thus, the helium-burning stage of stellar evolution does not result in the formation of Be as a stable end product; for similar reasons, low concentrations of Li and B are also formed. The nuclear reactions leading to these three elements are still uncertain, but they may result from the fragmentation of C, N, and O nuclei by collisions with high-energy particles. Elements can also be produced by nuclear reactions such as neutron (n) capture accompanied by proton emission: Binding energy per nucleon/MeV 14 7 8 56 4 55 57 58 60 59 H C 11 p 1.2 The nucleosynthesis of heavy elements 2 Key point: Heavier nuclides are formed by processes that include neutron capture and subsequent decay. He 0 14 6 This reaction still continues in our atmosphere as a result of the impact of cosmic rays and contributes to the steady-state concentration of radioactive carbon-14 on Earth. The high abundance of iron and nickel in the universe is consistent with these elements having the most stable of all nuclei. This stability is expressed in terms of the binding energy, which represents the difference in energy between the nucleus itself and the same numbers of individual protons and neutrons. This binding energy is often presented in terms of a difference in mass between the nucleus and its individual protons and neutrons because, according to Einstein’s theory of relativity, mass and energy are related by E mc2, where c is the speed of light. Therefore, if the mass of a nucleus differs from the total mass of its components by ∆m mnucleons mnucleus, then its binding energy is Ebind (∆m)c2. The binding energy of 56Fe, for example, is the difference in energy between the 56Fe nucleus and 26 protons and 30 neutrons. A positive binding energy corresponds to a nucleus that has a lower, more favourable, energy (and lower mass) than its constituent nucleons (Box 1.1). Figure 1.2 shows the binding energy per nucleon, Ebind/A (obtained by dividing the total binding energy by the number of nucleons), for all the elements. Iron and nickel occur at the maximum of the curve, showing that their nucleons are bound more strongly than in any other nuclide. Harder to see from the graph is an alternation of binding energies as the atomic number varies from even to odd, with even-Z nuclides slightly more stable than their odd-Z neighbours. There is a corresponding alternation in cosmic abundances, with nuclides of even atomic number being marginally more abundant than those of odd atomic number. This stability of even-Z nuclides is attributed to the lowering of energy by pairing nucleons in the nucleus. Fe 6 N 01 n ➝ 10 30 50 70 90 Atomic number, Z Figure 1.2 Nuclear binding energies. The greater the binding energy, the more stable is the nucleus. Note the alternation in stability shown in the inset. Nuclei close to iron are the most stable and heavier elements are produced by a variety of processes that require energy. These processes include the capture of free neutrons, which are not present in the earliest stages of stellar evolution but are produced later in reactions such as 23 10 Na 42 ➝ 26 12 Mg 01 n The origin of the elements 7 B OX 1.1 Nuclear fusion and nuclear fission If two nuclei with mass numbers lower than 56 merge to produce a new nucleus with a larger nuclear binding energy, the excess energy is released. This process is called fusion. For example, two neon-20 nuclei may fuse to give a calcium-40 nucleus: 20 2 10 Ne ➝ 40 20 Ca The value of Ebind /A for Ne is approximately 8.0 MeV. Therefore, the total binding energy of the species on the left-hand side of the equation is 2 20 8.0 MeV 320 MeV. The value of Ebind /A for Ca is close to 8.6 MeV and so the total energy of the species on the right-hand side is 40 8.6 MeV 344 MeV. The difference in the binding energies of the products and reactants is therefore 24 MeV. For nuclei with A 56, binding energy can be released when they split into lighter products with higher values of Ebind /A. This process is called fission. For example, uranium-236 can undergo fission into (among many other modes) xenon-140 and strontium-93 nuclei: U ➝ 236 92 140 54 Xe 93 Sr 01 n 38 The values of Ebind /A for 236U, 140Xe, and 93Sr nuclei are 7.6, 8.4, and 8.7 MeV, respectively. Therefore, the energy released in this reaction is (140 8.4) (93 8.7) (236 7.6) MeV 191.5 MeV for the fission of each 236U nucleus. Fission can also be induced by bombarding heavy elements with neutrons: 235 92 U 01n ➝ fissionproducts neutrons The kinetic energy of fission products from 235U is about 165 MeV, that of the neutrons is about 5 MeV, and the -rays produced have an energy of about 7 MeV. The fission products are themselves radioactive and decay by -, -, and X-radiation, releasing about 23 MeV. In a nuclear fission reactor the neutrons that are not consumed by fission are captured with the release of about 10 MeV. The energy produced is reduced by about 10 MeV, which escapes from the reactor as radiation, and about 1 MeV which remains as undecayed fission products in the spent fuel. Therefore, the total energy produced for one fission event is about 200 MeV, or 32 pJ. It follows that about 1 W of reactor heat (where 1 W 1 J s1) corresponds to about 3.1 1010 fission events per second. A nuclear reactor producing 3 GW has an electrical output of approximately 1 GW and corresponds to the fission of 3 kg of 235U per day. The use of nuclear power is controversial in large part on account of the risks associated with the highly radioactive, long-lived spent fuel. The declining stocks of fossil fuels, however, make nuclear power very attractive as it is estimated that stocks of uranium could last for about 100 years. The cost of uranium ores is currently very low and one small pellet of uranium oxide generates as much energy as three barrels of oil or 1 tonne of coal. The use of nuclear power would also drastically reduce the rate of emission of greenhouse gases. The environmental drawback with nuclear power is the storage and disposal of radioactive waste and the public’s continued nervousness about possible nuclear accidents and misuse in pursuit of political ambitions. Under conditions of intense neutron flux, as in a supernova (one type of stellar explosion), a given nucleus may capture a succession of neutrons and become a progressively heavier isotope. However, there comes a point at which the nucleus will eject an electron from the nucleus as a particle (a high-velocity electron, e). Because decay leaves the mass number of the nuclide unchanged but increases its atomic number by 1 (the nuclear charge increases by 1 unit when an electron is ejected), a new element is formed. An example is Neutron capture: 98 42 Mo 01 n ➝ Followed by decay accompanied by neutrino emission: Mo ➝ 99 42 99 42 Mo + Tc + e + ␯ 99 43 99 The daughter nuclide, the product of a nuclear reaction ( 33 Tc , an isotope of technetium, in this example), can absorb another neutron, and the process can continue, gradually building up the heavier elements (Box 1.2). B OX 1. 2 Technetium—the first synthetic element A synthetic element is one that does not occur naturally on Earth but that can be artificially generated by nuclear reactions. The first synthetic element was technetium (Tc, Z 43), named from the Greek word for ‘artificial’. Its discovery—more precisely, its preparation—filled a gap in the periodic table and its properties matched those predicted by Mendeleev. The longest-lived isotope of technetium (98Tc) has a half-life of 4.2 million years so any produced when the Earth was formed has long since decayed. Technetium is produced in red giant stars. The most widely used isotope of technetium is 99mTc, where the ‘m’ indicates a metastable isotope. Technetium-99m emits high-energy -rays but has a relatively short half-life of 6.01 hours. These properties make the isotope particularly attractive for use in vivo as the -ray energy is sufficient for it to be detected outside the body and its half-life means that most of it will have decayed within 24 hours. Consequently, 99mTc is widely used in nuclear medicine, for example in radiopharmaceuticals for imaging and functional studies of the brain, bones, blood, lungs, liver, heart, thyroid gland, and kidneys. Technetium-99m is generated through nuclear fission in nuclear power plants but a more useful laboratory source of the isotope is a technetium generator, which uses the decay of 99Mo to 99mTc. The half-life of 99Mo is 66 hours, which makes it more convenient for transport and storage than 99mTc itself. Most commercial generators are based on 99Mo in the form of the molybdate ion, MoO 42, adsorbed on Al2O3. The 99MoO42 ion decays to the pertechnetate ion, 99m TcO4, which is less tightly bound to the alumina. Sterile saline solution is washed through a column of the immobilized 99Mo and the 99mTc solution is collected. 8 1 Atomic structure E X A M PL E 1.1 Balancing equations for nuclear reactions Synthesis of heavy elements occurs in the neutron-capture reactions believed to take place in the interior of 69 cool ‘red giant’ stars. One such reaction is the conversion of 68 Zn to 69 Ga by neutron capture to form 30 Zn, 30 31 which then undergoes decay. Write balanced nuclear equations for this process. Answer We use the fact that the sum of the mass numbers and the sum of the atomic numbers on each side of the equation must be the same. Neutron capture increases the mass number of a nuclide by 1 but leaves the atomic number (and hence the identity of the element) unchanged: 68 30 Zn 01n ➝ 69 30 Zn The excess energy is carried away as a photon. The loss of an electron from the nucleus by decay leaves the mass number unchanged but increases the atomic number by 1. Because zinc has atomic number 30, the daughter nuclide has Z 31, corresponding to gallium. Therefore, the nuclear reaction is 69 30 Zn ➝ 69 31 Ga e − In fact, a neutrino is also emitted, but this cannot be inferred from the data as a neutrino is effectively massless and electrically neutral. Self-test 1.1 Write the balanced nuclear equation for neutron capture by 80 35 Br. The structures of hydrogenic atoms The organization of the periodic table is a direct consequence of periodic variations in the electronic structure of atoms. Initially, we consider hydrogen-like or hydrogenic atoms, which have only one electron and so are free of the complicating effects of electron– electron repulsions. Hydrogenic atoms include ions such as He and C5 (found in stellar interiors) as well as the hydrogen atom itself. Then we use the concepts these atoms introduce to build up an approximate description of the structures of many-electron atoms (or polyelectron atoms), which are atoms with more than one electron. 1.3 Spectroscopic information Key points: Spectroscopic observations on hydrogen atoms suggest that an electron can occupy only certain energy levels and that the emission of discrete frequencies of electromagnetic radiation occurs when an electron makes a transition between these levels. Electromagnetic radiation is emitted when an electric discharge is passed through hydrogen gas. When passed through a prism or diffraction grating, this radiation is found to consist of a series of components: one in the ultraviolet region, one in the visible region, and several in the infrared region of the electromagnetic spectrum (Fig. 1.3; Box 1.3). Total Balmer Paschen Figure 1.3 The spectrum of atomic hydrogen and its analysis into series. Brackett Lyman 100 120 150 200 300 /nm 400 500 1000 800 600 2000 Visible The structures of hydrogenic atoms 9 B OX 1. 3 Sodium street lamps The emission of light when atoms are excited is put to good use in lighting streets in many parts of the world. The widely used yellow street lamps are based on the emission of light from excited sodium atoms. Low pressure sodium (LPS) lamps consist of a glass tube coated with indium tin oxide (ITO), a solid solution of In2O3 with typically 10 per cent by mass SnO2. The indium tin oxide reflects the infrared radiation and transmits the visible light. Two inner glass tubes hold solid sodium and a small amount of neon and argon, the same mixture as found in neon lights. When the lamp is turned on, the neon and argon emit a red glow and heat the sodium metal. The sodium rapidly starts to vaporize and the electrical discharge excites the atoms and they re-emit the energy as yellow light from the transition 3p → 3s. One advantage of sodium lamps over other types of street lighting is that their light output does not diminish with age. They do, however, use more energy towards the end of their life, which may make them less attractive from environmental and economic perspectives. The nineteenth-century spectroscopist Johann Rydberg found that all the wavelengths (, lambda) can be described by the expression ⎛ 1 1 1⎞ R⎜ 2 − 2 ⎟ (1.1) ⎝ n1 n2 ⎠ where R is the Rydberg constant, an empirical constant with the value 1.097 107 m1. The n are integers, with n1 1, 2, . . . and n2 n1 1, n1 2, . . . . The series with n1 1 is called the Lyman series and lies in the ultraviolet. The series with n1 2 lies in the visible region and is called the Balmer series. The infrared series include the Paschen series (n1 3) and the Brackett series (n1 4). The structure of the spectrum is explained if it is supposed that the emission of radiation takes place when an electron makes a transition from a state of energy hcR/n22 to a state of energy hcR/n12 and that the difference, which is equal to hcR(1/n12 1/n22), is carried away as a photon of energy hc/. By equating these two energies, and cancelling hc, we obtain eqn 1.1. The question these observations raise is why the energy of the electron in the atom is limited to the values hcR/n2 and why R has the value observed. An initial attempt to explain these features was made by Niels Bohr in 1913 using an early form of quantum theory in which he supposed that the electron could exist in only certain circular orbits. Although he obtained the correct value of R, his model was later shown to be untenable as it conflicted with the version of quantum theory developed by Erwin Schrödinger and Werner Heisenberg in 1926. 1.4 Some principles of quantum mechanics Key points: Electrons can behave as particles or as waves; solution of the Schrödinger equation gives wavefunctions, which describe the location and properties of electrons in atoms. The probability of finding an electron at a given location is proportional to the square of the wavefunction. Wavefunctions generally have regions of positive and negative amplitude, and may undergo constructive or destructive interference with one another. In 1924, Louis de Broglie suggested that because electromagnetic radiation could be considered to consist of particles called photons yet at the same time exhibit wave-like properties, such as interference and diffraction, then the same might be true of electrons. This dual nature is called wave–particle duality. An immediate consequence of duality is that it is impossible to know the linear momentum (the product of mass and velocity) and the location of an electron (and any particle) simultaneously. This restriction is the content of Heisenberg’s uncertainty principle, that the product of the uncertainty in momentum and the uncertainty in position cannot be less than a quantity of the order of Planck’s constant –, where h – h/2π). (specifically, 12 h Schrödinger formulated an equation that took account of wave–particle duality and accounted for the motion of electrons in atoms. To do so, he introduced the wavefunction, (psi), a mathematical function of the position coordinates x, y, and z which describes the behaviour of an electron. The Schrödinger equation, of which the wavefunction is a solution, for an electron free to move in one dimension is Kinetic energy Potential energy contribution Total energy contribution contribution –h 2 d2 V (x) (x) E (x) 2 2me dx (1.2) 10 1 Atomic structure Wavefunction, Figure 1.4 The Born interpretation of the wavefunction is that its square is a probability density. There is zero probability density at a node. Resultant Wave 1 Wave 2 (a) Wave 1 Wave 2 Resultant (b) Figure 1.5 Wavefunctions interfere where they spread into the same region of space. (a) If they have the same sign in a region, they interfere constructively and the total wavefunction has an enhanced amplitude in the region. (b) If the wavefunctions have opposite signs, then they interfere destructively, and the resulting superposition has a reduced amplitude. where me is the mass of an electron, V is the potential energy of the electron, and E is its total energy. The Schrödinger equation is a second-order differential equation that can be solved exactly for a number of simple systems (such as a hydrogen atom) and can be solved numerically for many more complex systems (such as many-electron atoms and molecules). However, we shall need only qualitative aspects of its solutions. The generalization of eqn 1.2 to three dimensions is straightforward, but we do not need its explicit form. One crucial feature of eqn 1.2 and its analogues in three dimensions is that physically acceptable solutions exist only for certain values of E. Therefore, the quantization of energy, the fact that an electron can possess only certain discrete energies in an atom, follows naturally from the Schrödinger equation, in addition to the imposition of certain requirements (‘boundary conditions’) that restrict the number of acceptable solutions. A wavefunction contains all the dynamical information possible about the electron, including where it is and what it is doing. Specifically, the probability of finding an electron at a given location is proportional to the square of the wavefunction at that point, 2. According to this interpretation, there is a high probability of finding the electron where 2 is large, and the electron will not be found where 2 is zero (Fig. 1.4). The quantity 2 is called the probability density of the electron. It is a ‘density’ in the sense that the product of 2 and the infinitesimal volume element d dxdydz (where is tau) is proportional to the probability of finding the electron in that volume. The probability is equal to 2d if the wavefunction is ‘normalized’. A normalized wavefunction is one that is scaled so that the total probability of finding the electron somewhere is 1. Like other waves, wavefunctions in general have regions of positive and negative amplitude, or sign. The sign of the wavefunction is of crucial importance when two wavefunctions spread into the same region of space and interact. Then a positive region of one wavefunction may add to a positive region of the other wavefunction to give a region of enhanced amplitude. This enhancement is called constructive interference (Fig. 1.5a). It means that, where the two wavefunctions spread into the same region of space, such as occurs when two atoms are close together, there may be a significantly enhanced probability of finding the electrons in that region. Conversely, a positive region of one wavefunction may be cancelled by a negative region of the second wavefunction (Fig. 1.5b). This destructive interference between wavefunctions reduces the probability that an electron will be found in that region. As we shall see, the interference of wavefunctions is of great importance in the explanation of chemical bonding. To help keep track of the relative signs of different regions of a wavefunction in illustrations, we label regions of opposite sign with dark and light shading (sometimes white in the place of light shading). 1.5 Atomic orbitals The wavefunction of an electron in an atom is called an atomic orbital. Chemists use hydrogenic atomic orbitals to develop models that are central to the interpretation of inorganic chemistry, and we shall spend some time describing their shapes and significance. (a) Hydrogenic energy levels Key points: The energy of the bound electron is determined by n, the principal quantum number; in addition, l specifies the magnitude of the orbital angular momentum and ml specifies the orientation of that angular momentum. Each of the wavefunctions obtained by solving the Schrödinger equation for a hydrogenic atom is uniquely labelled by a set of three integers called quantum numbers. These quantum numbers are designated n, l, and ml: n is called the principal quantum number, l is the orbital angular momentum quantum number (formerly the ‘azimuthal quantum number’), and ml is called the magnetic quantum number. Each quantum number specifies a physical property of the electron: n specifies the energy, l labels the magnitude of the orbital angular momentum, and ml labels the orientation of that angular momentum. The value of n also indicates the size of the orbital, with high n, high-energy orbitals more diffuse than low n compact, tightly bound, low-energy orbitals. The value of l also indicates the angular shape of the orbital, with the number of lobes increasing as l increases. The value of ml also indicates the orientation of these lobes. The structures of hydrogenic atoms n The allowed energies are specified by the principal quantum number, n. For a hydrogenic atom of atomic number Z, they are given by En = Z=1 Z=2 ∞ –R/9 hcRZ 2 n2 (1.3) 11 4 –R/4 2 –R 1 3 with n 1, 2, 3, . . . and R= mee4 8h3c 02 (1.4) 2 Energy (The fundamental constants in this expression are given inside the back cover.) The calculated numerical value of R is 1.097 107 m1, in excellent agreement with the empirical value determined spectroscopically. For future reference, the value of hcR corresponds to 13.6 eV. The zero of energy (at n ∞) corresponds to the electron and nucleus being widely separated and stationary. Positive values of the energy correspond to unbound states of the electron in which it may travel with any velocity and hence possess any energy. The energies given by eqn 1.3 are all negative, signifying that the energy of the electron in a bound state is lower than a widely separated stationary electron and nucleus. Finally, because the energy is proportional to 1/n2, the energy levels converge as the energy increases (becomes less negative, Fig. 1.6). The value of l specifies the magnitude of the orbital angular momentum through –, with l 0, 1, 2, . . . . We can think of l as indicating the rate at which {l(l 1)}1/2h the electron circulates around the nucleus. As we shall see shortly, the third quantum number ml specifies the orientation of this momentum, for instance whether the circulation is clockwise or anticlockwise. 1 (b) Shells, subshells, and orbitals Key points: All orbitals with a given value of n belong to the same shell, all orbitals of a given shell with the same value of l belong to the same subshell, and individual orbitals are distinguished by the value of ml. Figure 1.6 The quantized energy levels of an H atom (Z 1) and an He ion (Z 2). The energy levels of a hydrogenic atom are proportional to Z 2. In a hydrogenic atom, all orbitals with the same value of n have the same energy and are said to be degenerate. The principal quantum number therefore defines a series of shells of the atom, or sets of orbitals with the same value of n and hence with the same energy and approximately the same radial extent. Shells with n 1, 2, 3 . . . are commonly referred to as K, L, M, . . . shells. The orbitals belonging to each shell are classified into subshells distinguished by a quantum number l. For a given value of n, the quantum number l can have the values l 0, 1,..., n 1, giving n different values in all. For example, the shell with n 1 consists of just one subshell with l 0, the shell with n 2 consists of two subshells, one with l 0 and the other with l 1, the shell with n 3 consists of three subshells, with values of l of 0, 1, and 2. It is common practice to refer to each subshell by a letter: Value of l 0 1 2 3 4 ... Subshell designation s p d f g ... For most purposes in chemistry we need consider only s, p, d, and f subshells. A subshell with quantum number l consists of 2l 1 individual orbitals. These orbitals are distinguished by the magnetic quantum number, ml, which can have the 2l 1 integer values from l down to l. This quantum number specifies the component of orbital angular momentum around an arbitrary axis (commonly designated z) passing through the nucleus. So, for example, a d subshell of an atom (l 2) consists of five individual atomic orbitals that are distinguished by the values ml 2, 1, 0, 1, 2. A note on good practice Write the sign of ml, even when it is positive. Thus, we write ml 2, not ml 2. The practical conclusion for chemistry from these remarks is that there is only one orbital in an s subshell (l 0), the one with ml 0: this orbital is called an s orbital. There are three orbitals in a p subshell (l 1), with quantum numbers ml 1, 0, 1; they are called p orbitals. The five orbitals of a d subshell (l 2) are called d orbitals, and so on (Fig. 1.7). Subshells s p d f 4 3 2 Shell 1 Figure 1.7 The classification of orbitals into subshells (same value of l) and shells (same value of n). 12 1 Atomic structure E X A M PL E 1. 2 Identifying orbitals from quantum numbers Which set of orbitals is defined by n 4 and l 1? How many orbitals are there in this set? Answer We need to remember that the principal quantum number n identifies the shell and that the orbital quantum number l identifies the subshell. The subshell with l 1 consists of p orbitals. The allowed values of ml l, l 1, . . . , l give the number of orbitals of that type. In this case, ml 1, 0, and 1. There are therefore three 4p orbitals. Self-test 1.2 Which set of orbitals is defined by the quantum numbers n 3 and l 2? How many orbitals are there in this set? (c) Electron spin Key points: The intrinsic spin angular momentum of an electron is defined by the two quantum numbers s and ms. Four quantum numbers are needed to define the state of an electron in a hydrogenic atom. In addition to the three quantum numbers required to specify the spatial distribution of an electron in a hydrogenic atom, two more quantum numbers are needed to define the state of an electron. These additional quantum numbers relate to the intrinsic angular momentum of an electron, its spin. This evocative name suggests that an electron can be regarded as having an angular momentum arising from a spinning motion, rather like the daily rotation of a planet as it travels in its annual orbit around the sun. However, spin is a quantum mechanical property and this analogy must be viewed with great caution. Spin is described by two quantum numbers, s and ms. The former is the analogue of l for 1 orbital motion but it is restricted to the single, unchangeable value s 2 . The magnitude 1/2– of the spin angular momentum is given by the expression {s(s 1)} h, so for an electron – for any electron. The second quantum number, the spin this magnitude is fixed at 12 3 h magnetic quantum number, ms, may take only two values, 12 (anticlockwise spin, imagined from above) and 12 (clockwise spin). The two states are often represented by the two arrows ↑ (‘spin-up’, ms 12 ) and ↓ (‘spin-down’, ms 12 ) or by the Greek letters and , respectively. Because the spin state of an electron must be specified if the state of the atom is to be specified fully, it is common to say that the state of an electron in a hydrogenic atom is characterized by four quantum numbers, namely n, l, ml, and ms (the fifth quantum number, s, is fixed at 12 ). (d) Nodes Key point: Regions where wavefunctions pass through zero are called nodes. Inorganic chemists generally find it adequate to use visual representations of atomic orbitals rather than mathematical expressions. However, we need to be aware of the mathematical expressions that underlie these representations. Because the potential energy of an electron in the field of a nucleus is spherically symmetric (it is proportional to Z/r and independent of orientation relative to the nucleus), the orbitals are best expressed in terms of the spherical polar coordinates defined in Fig. 1.8. In these coordinates, the orbitals all have the form z r y x Figure 1.8 Spherical polar coordinates: r is the radius, (theta) the colatitude, and (phi) the azimuth. nlm = l Variation with radius Rnl ( r ) Variation with angle Ylm ( , ) l (1.5) This expression expresses the simple idea that a hydrogenic orbital can be written as the product of a function R(r) of the radius and a function Y(,) of the angular coordinates. The positions where either component of the wavefunction passes through zero are called nodes. Consequently, there are two types of nodes. Radial nodes occur where the radial component of the wavefunction passes through zero and angular nodes occur where the angular component of the wavefunction passes through zero. The numbers of both types of node increase with increasing energy and are related to the quantum numbers n and l. The structures of hydrogenic atoms 13 Key point: An s orbital has nonzero amplitude at the nucleus; all other orbitals (those with l > 0) vanish at the nucleus. Figures 1.9 and 1.10 show the radial variation of some atomic orbitals. A 1s orbital, the wavefunction with n 1, l 0, and ml 0, decays exponentially with distance from the nucleus and never passes through zero. All orbitals decay exponentially at sufficiently great distances from the nucleus and this distance increases as n increases. Some orbitals oscillate through zero close to the nucleus and thus have one or more radial nodes before beginning their final exponential decay. As the principal quantum number of an electron increases, it is found further away from the nucleus and its energy increases. An orbital with quantum numbers n and l in general has n l 1 radial nodes. This oscillation is evident in the 2s orbital, the orbital with n 2, l 0, and ml 0, which passes through zero once and hence has one radial node. A 3s orbital passes through zero twice and so has two radial nodes. A 2p orbital (one of the three orbitals with n 2 and l 1) has no radial nodes because its radial wavefunction does not pass through zero anywhere. However, a 2p orbital, like all orbitals other than s orbitals, is zero at the nucleus. For any series of the same type of orbital, the first occurrence has no radial nodes, the second has one radial node, and so on. Although an electron in an s orbital may be found at the nucleus, an electron in any other type of orbital will not be found there. We shall soon see that this apparently minor detail, which is a consequence of the absence of orbital angular momentum when l 0, is one of the key concepts for understanding chemistry. Radial wavefunction, R/(Z/a0)3/2 (e) The radial variation of atomic orbitals 1.8 1.2 0.6 1s 3s 0 2s 10 20 Radius, Zr/a0 30 Figure 1.9 The radial wavefunctions of the 1s, 2s, and 3s hydrogenic orbitals. Note that the number of radial nodes is 0, 1, and 2, respectively. Each orbital has a nonzero amplitude at the nucleus (at r 0). 1 How many radial nodes do 3p, 3d, and 4f orbitals have? Answer We need to make use of the fact that the number of radial nodes is given by the expression n l 1 and use it to find the values of n and l. The 3p orbitals have n 3 and l 1 and the number of radial nodes will be n l 1 1. The 3d orbitals have n 3 and l 2. Therefore, the number of radial nodes will n l 1 0. The 4f orbitals have n 4 and l 3 and the number of radial nodes will be n l 1 0. The 3d and 4f orbitals are the first occurrence of the d and f orbitals so this also indicates that they will have no radial nodes. Self-test 1.3 How many radial nodes does a 5s orbital have? Radial wavefunction, R/(Z/a0)3/2 E X A MPL E 1. 3 Predicting numbers of radial nodes 0.8 0.6 0.4 2p 0.2 0 3p (f) The radial distribution function –0.2 Key point: A radial distribution function gives the probability that an electron will be found at a given distance from the nucleus, regardless of the direction. The Coulombic (electrostatic) force that binds the electron is centred on the nucleus, so it is often of interest to know the probability of finding an electron at a given distance from the nucleus, regardless of its direction. This information enables us to judge how tightly the electron is bound. The total probability of finding the electron in a spherical shell of radius r and thickness dr is the integral of 2d over all angles. This result is written P(r) dr, where P(r) is called the radial distribution function. In general, P(r) r2 R(r)2 (1.6) (For s orbitals, this expression is the same as P 4πr .) If we know the value of P at some radius r, then we can state the probability of finding the electron somewhere in a shell of thickness dr at that radius simply by multiplying P by dr. In general, a radial distribution function for an orbital in a shell of principal quantum number n has n 1 peaks, the outermost peak being the highest. Because the wavefunction of a 1s orbital decreases exponentially with distance from the nucleus and the factor r2 in eqn 1.6 increases, the radial distribution function of a 1s orbital goes through a maximum (Fig. 1.11). Therefore, there is a distance at which the electron is most likely to be found. In general, this most probable distance decreases as the 2 2 0 10 20 Radius, Zr/a0 30 Figure 1.10 The radial wavefunctions of the 2p and 3p hydrogenic orbitals. Note that the number of radial nodes is 0 and 1, respectively. Each orbital has zero amplitude at the nucleus (at r 0). 14 1 Atomic structure R nuclear charge increases (because the electron is attracted more strongly to the nucleus), and specifically 2 rmax = r 2R 2 a0 Z (1.7) where a0 is the Bohr radius, a0 0 h 2/πmee2, a quantity that appeared in Bohr’s formulation of his model of the atom; its numerical value is 52.9 pm. The most probable distance increases as n increases because the higher the energy, the more likely it is that the electron will be found far from the nucleus. r2 E X A M PL E 1. 4 Interpreting radial distribution functions Figure 1.12 shows the radial distribution functions for 2s and 2p hydrogenic orbitals. Which orbital gives the electron a greater probability of close approach to the nucleus? 0 1 2 3 4 Radius, Zr/a0 5 Radial distribution function, r 2R 2 Figure 1.11 The radial distribution function of a hydrogenic 1s orbital. The product of 4πr2 (which increases as r increases) and 2 (which decreases exponentially) passes through a maximum at r a0 /Z. 0 Answer By examining Fig. 1.12 we can see that the radial distribution function of a 2p orbital approaches zero near the nucleus faster than a 2s electron does. This difference is a consequence of the fact that a 2p orbital has zero amplitude at the nucleus on account of its orbital angular momentum. Thus, the 2s electron has a greater probability of close approach to the nucleus. Self-test 1.4 Which orbital, 3p or 3d, gives an electron a greater probability of being found close to the nucleus? (g) The angular variation of atomic orbitals Key points: The boundary surface of an orbital indicates the region of space within which the electron is most likely to be found; orbitals with the quantum number l have l nodal planes. 2p 2s 15 Radius, Zr/a0 Figure 1.12 The radial distribution functions of hydrogenic orbitals. Although the 2p orbital is on average closer to the nucleus (note where its maximum lies), the 2s orbital has a high probability of being close to the nucleus on account of the inner maximum. The angular wavefunction expresses the variation of angle around the nucleus and this describes the orbital’s angular shape. An s orbital has the same amplitude at a given distance from the nucleus whatever the angular coordinates of the point of interest: that is, an s orbital is spherically symmetrical. The orbital is normally represented by a spherical surface with the nucleus at its centre. The surface is called the boundary surface of the orbital, and defines the region of space within which there is a high (typically 90 per cent) probability of finding the electron. The planes on which the angular wavefunction passes through zero are called angular nodes or nodal planes. An electron will not be found anywhere on a nodal plane. A nodal plane cuts through the nucleus and separates the regions of positive and negative sign of the wavefunction. In general, an orbital with the quantum number l has l nodal planes. An s orbital, with l 0, has no nodal planes and the boundary surface of the orbital is spherical (Fig. 1.13). All orbitals with l 0 have amplitudes that vary with angle. In the most common graphical representation, the boundary surfaces of the three p orbitals of a given shell are identical apart from the fact that their axes lie parallel to each of the three different Cartesian axes centred on the nucleus, and each one possesses a nodal plane passing through the nucleus (Fig. 1.14). This representation is the origin of the labels px, py, and pz, which are z z z z y y y x Figure 1.13 The spherical boundary surface of an s orbital. x px y x py x pz Figure 1.14 The boundary surfaces of p orbitals. Each orbital has one nodal plane running through the nucleus. For example, the nodal plane of the p orbital is the xy-plane. The lightly shaded lobe has a positive amplitude, the more darkly shaded one is negative. Many-electron atoms 15 z y x dx2–y2 dz 2 dzx dyz Figure 1.15 One representation of the boundary surfaces of the d orbitals. Four of the orbitals have two perpendicular nodal planes that intersect in a line passing through the nucleus. In the dz2 orbital, the nodal surface forms two cones that meet at the nucleus. dxy z x y f5z3–3zr2 fzx2–zy2 f5xz2–3xr2 fxyz fy3–3yx2 f5yz2–yr2 fx3–3xy2 alternatives to the use of ml to label the individual orbitals. Each p orbital, with l 1, has a single nodal plane. The boundary surfaces and labels we use for the d and f orbitals are shown in Figs 1.15 and 1.16, respectively. The dz2 orbital looks different from the remaining d orbitals. There are in fact six possible combinations of double dumb-bell shaped orbitals around three axes: three with lobes between the axes, as in dxy, dyz, and dzx, and three with lobes along the axis. One of these orbitals is dx2y2. The dz2 orbital can be thought of as the superposition of two contributions, one with lobes along the z- and x-axes and the other with lobes along the z- and y-axes. Note that a d orbital with (l 2) has two nodal planes that intersect at the nucleus; a typical f orbital (l 3) has three nodal planes. Many-electron atoms As we have remarked, a ‘many-electron atom’ is an atom with more than one electron, so even He, with two electrons, is technically a many-electron atom. The exact solution of the Schrödinger equation for an atom with N electrons would be a function of the 3N coordinates of all the electrons. There is no hope of finding exact formulas for such complicated functions; however, it is straightforward to perform numerical computations by using widely available software to obtain precise energies and probability densities. This software can also generate graphical representations of the resulting orbitals that can assist in the interpretation of the properties of the atom. For most of inorganic chemistry we rely on the orbital approximation, in which each electron occupies an atomic orbital that resembles those found in hydrogenic atoms. When we say that an electron ‘occupies’ an atomic orbital, we mean that it is described by the corresponding wavefunction. Figure 1.16 One representation of the boundary surfaces of the f orbitals. Other representations (with different shapes) are also sometimes encountered. 16 1 Atomic structure 1.6 Penetration and shielding Key points: The ground-state electron configuration is a specification of the orbital occupation of an atom in its lowest energy state. The exclusion principle forbids more than two electrons to occupy a single orbital. The nuclear charge experienced by an electron is reduced by shielding by other electrons. Trends in effective nuclear charge can be used to rationalize the trends in many properties. As a result of the combined effects of penetration and shielding, the order of energy levels in a shell of a manyelectron atom is s p d f. It is quite easy to account for the electronic structure of the helium atom in its ground state, its state of lowest energy. According to the orbital approximation, we suppose that both electrons occupy an atomic orbital that has the same spherical shape as a hydrogenic 1s orbital. However, the orbital will be more compact because, as the nuclear charge of helium is greater than that of hydrogen, the electrons are drawn in towards the nucleus more closely than is the one electron of an H atom. The ground-state configuration of an atom is a statement of the orbitals its electrons occupy in the ground state. For helium, with two electrons in the 1s orbital, the ground-state configuration is denoted 1s2 (read ‘one s two’). As soon as we come to the next atom in the periodic table, lithium (Z 3), we encounter several major new features. The configuration 1s3 is forbidden by a fundamental feature of nature known as the Pauli exclusion principle: Charge does not contribute r Charge contributes Radial distribution function, r 2R 2 Figure 1.17 The electron at the r radius experiences a repulsion from the total charge within the sphere of radius r; charge outside that radius has no net effect. 2p 2s 0 Radius, Zr/a0 15 Figure 1.18 The penetration of a 2s electron through the inner core is greater than that of a 2p electron because the latter vanishes at the nucleus. Therefore, the 2s electrons are less shielded than the 2p electrons. No more than two electrons may occupy a single orbital and, if two do occupy a single orbital, then their spins must be paired. By ‘paired’ we mean that one electron spin must be ↑ and the other ↓; the pair is denoted ↑↓. Another way of expressing the principle is to note that, because an electron in an atom is described by four variable quantum numbers, n, l, ml, and ms, no two electrons can have the same four quantum numbers. The Pauli principle was introduced originally to account for the absence of certain transitions in the spectrum of atomic helium. Because the configuration 1s3 is forbidden by the Pauli exclusion principle, the third electron must occupy an orbital of the next higher shell, the shell with n 2. The question that now arises is whether the third electron occupies a 2s orbital or one of the three 2p orbitals. To answer this question, we need to examine the energies of the two subshells and the effect of the other electrons in the atom. Although 2s and 2p orbitals have the same energy in a hydrogenic atom, spectroscopic data and calculations show that this is not the case in a many-electron atom. In the orbital approximation we treat the repulsion between electrons in an approximate manner by supposing that the electronic charge is distributed spherically around the nucleus. Then each electron moves in the attractive field of the nucleus and experiences an average repulsive charge from the other electrons. According to classical electrostatics, the field that arises from a spherical distribution of charge is equivalent to the field generated by a single point charge at the centre of the distribution (Fig. 1.17). This negative charge reduces the actual charge of the nucleus, Ze, to Zeffe, where Zeff (more precisely, Zeffe) is called the effective nuclear charge. This effective nuclear charge depends on the values of n and l of the electron of interest because electrons in different shells and subshells approach the nucleus to different extents. The reduction of the true nuclear charge to the effective nuclear charge by the other electrons is called shielding. The effective nuclear charge is sometimes expressed in terms of the true nuclear charge and an empirical shielding constant, , by writing Zeff Z . The shielding constant can be determined by fitting hydrogenic orbitals to those computed numerically. The closer to the nucleus that an electron can approach, the closer is the value of Zeff to Z itself because the electron is repelled less by the other electrons present in the atom. With this point in mind, consider a 2s electron in the Li atom. There is a nonzero probability that the 2s electron can be found inside the 1s shell and experience the full nuclear charge (Fig. 1.18). The presence of an electron inside shells of other electrons is called penetration. A 2p electron does not penetrate so effectively through the core, the filled inner shells of electrons, because its wavefunction goes to zero at the nucleus. As a consequence, it is more fully shielded from the nucleus by the core electrons. We can conclude that a 2s electron has a lower energy (is bound more tightly) than a 2p electron, and therefore that the 2s orbital will be occupied before the 2p orbitals, giving a ground-state electron configuration for Li of 1s22s1. This configuration is commonly denoted [He]2s1, where [He] denotes the atom’s helium-like 1s2 core. 17 Many-electron atoms Table 1.2 Effective nuclear charge, Zeff H He Z 1 2 1s 1.00 1.69 Li Be B C N O F Ne Z 3 4 5 6 7 8 9 10 1s 2.69 3.68 4.68 5.67 6.66 7.66 8.65 2s 1.28 1.91 2.58 3.22 3.85 4.49 5.13 5.76 2.42 3.14 3.83 4.45 5.10 5.76 Na Mg Al Si P S Cl Ar Z 11 12 13 14 15 16 17 18 1s 10.63 11.61 12.59 13.57 14.56 15.54 16.52 17.51 2s 6.57 7.39 8.21 9.02 9.82 10.63 11.43 12.23 2p 6.80 7.83 8.96 9.94 10.96 11.98 12.99 14.01 3s 2.51 3.31 4.12 4.90 5.64 6.37 7.07 7.76 4.07 4.29 4.89 5.48 6.12 6.76 3p The pattern of energies in lithium, with 2s lower than 2p, and in general ns lower than np, is a general feature of many-electron atoms. This pattern can be seen from Table 1.2, which gives the values of Zeff for a number of valence-shell atomic orbitals in the groundstate electron configuration of atoms. The typical trend in effective nuclear charge is an increase across a period, for in most cases the increase in nuclear charge in successive groups is not cancelled by the additional electron. The values in the table also confirm that an s electron in the outermost shell of the atom is generally less shielded than a p electron of that shell. So, for example, Zeff 5.13 for a 2s electron in an F atom, whereas for a 2p electron Zeff 5.10, a lower value. Similarly, the effective nuclear charge is larger for an electron in an np orbital than for one in an nd orbital. As a result of penetration and shielding, the order of energies in many-electron atoms is typically ns np nd nf because, in a given shell, s orbitals are the most penetrating and f orbitals are the least penetrating. The overall effect of penetration and shielding is depicted in the energy-level diagram for a neutral atom shown in Fig. 1.19. Figure 1.20 summarizes the energies of the orbitals through the periodic table. The effects are quite subtle, and the order of the orbitals depends strongly on the numbers of K 3d n 5 4p Ca Sc Ti 4 4p 4s 3s Energy 2p 9.64 4d 3p 3d 4f Z < 21 Z ≥ 21 2p 2s 1s Figure 1.19 A schematic diagram of the energy levels of a many-electron atom with Z 21 (as far as calcium). There is a change in order for Z 21 (from scandium onwards). This is the diagram that justifies the building-up principle, with up to two electrons being allowed to occupy each orbital. 4s V Energy 3 2 1 1 25 50 Atomic number, Z 75 100 Figure 1.20 A more detailed portrayal of the energy levels of many-electron atoms in the periodic table. The inset shows a magnified view of the order near Z 20, where the 3d series of elements begins. 18 1 Atomic structure electrons present in the atom and may change on ionization. For example, the effects of penetration are very pronounced for 4s electrons in K and Ca, and in these atoms the 4s orbitals lie lower in energy than the 3d orbitals. However, from Sc through Zn, the 3d orbitals in the neutral atoms lie close to but lower than the 4s orbitals. In atoms from Ga (Z 31) onwards, the 3d orbitals lie well below the 4s orbital in energy, and the outermost electrons are unambiguously those of the 4s and 4p subshells. 1.7 The building-up principle The ground-state electron configurations of many-electron atoms are determined experimentally by spectroscopy and are summarized in Resource section 2. To account for them, we need to consider both the effects of penetration and shielding on the energies of the orbitals and the role of the Pauli exclusion principle. The building-up principle (which is also known as the Aufbau principle and is described below) is a procedure that leads to plausible ground-state configurations. It is not infallible, but it is an excellent starting point for the discussion. Moreover, as we shall see, it provides a theoretical framework for understanding the structure and implications of the periodic table. (a) Ground-state electron configurations Key points: The order of occupation of atomic orbitals follows the order 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, . . . . Degenerate orbitals are occupied singly before being doubly occupied; certain modifications of the order of occupation occur for d and f orbitals. According to the building-up principle, orbitals of neutral atoms are treated as being occupied in the order determined in part by the principal quantum number and in part by penetration and shielding: Order of occupation: 1s 2s 3s 3p 4s 3d 4p ... Each orbital can accommodate up to two electrons. Thus, the three orbitals in a p subshell can accommodate a total of six electrons and the five orbitals in a d subshell can accommodate up to ten electrons. The ground-state configurations of the first five elements are therefore expected to be H 1s He 1 2 1s Li Be 2 1 1s 2s 2 B 2 1s 2s 1s22s22p1 This order agrees with experiment. When more than one orbital of the same energy is available for occupation, such as when the 2p orbitals begin to be filled in B and C, we adopt Hund’s rule: When more than one orbital has the same energy, electrons occupy separate orbitals and do so with parallel spins (↑↑). The occupation of separate orbitals of the same value of l (such as a px orbital and a py orbital) can be understood in terms of the weaker repulsive interactions that exist between electrons occupying different regions of space (electrons in different orbitals) than between those occupying the same region of space (electrons in the same orbital). The requirement of parallel spins for electrons that do occupy different orbitals is a consequence of a quantum mechanical effect called spin correlation, the tendency for two electrons with parallel spins to stay apart from one another and hence to repel each other less. One consequence of this effect is that half-filled shells of electrons with parallel spins are particularly stable. For example, the ground state of the chromium atom is 4s13d5 rather than 4s23d4. Further examples of the effect of spin correlation will be seen later in this chapter. It is arbitrary which of the p orbitals of a subshell is occupied first because they are degenerate, but it is common to adopt the alphabetical order px, py, pz. It then follows from the building-up principle that the ground-state configuration of C is 1s22s22p1x2py1 or, more simply, 1s22s22p2. If we recognize the helium-like core (1s2), an even briefer notation is [He]2s22p2, and we can think of the electronic valence structure of the atom as consisting of two paired 2s electrons and two parallel 2p electrons surrounding a closed helium-like core. The electron configurations of the remaining elements in the period are similarly Many-electron atoms C N 2 [He]2s 2p 2 O 2 3 [He]2s 2p F 2 4 [He]2s 2p Ne 2 [He]2s 2p 5 [He]2s22p6 The 2s22p6 configuration of neon is another example of a closed shell, a shell with its full complement of electrons. The configuration 1s22s22p6 is denoted [Ne] when it occurs as a core. E X A MPL E 1. 5 Accounting for trends in effective nuclear charge The increase in Zeff between C and N is 0.69 whereas the increase between N and O is only 0.62. Suggest a reason why the increase in Zeff for a 2p electron is smaller between N and O than between C and N given the configurations of the atoms listed above. Answer We need to identify the general trend and then think about an additional effect that might modify it. In this case, we expect to see an increase in effective nuclear charge across a period. However, on going from C to N, the additional electron occupies an empty 2p orbital whereas on going from N to O, the additional electron must occupy a 2p orbital that is already occupied by one electron. It therefore experiences stronger electron–electron repulsion, and the increase in Zeff is not as great. Self-test 1.5 Account for the larger increase in effective nuclear charge for a 2p electron on going from B to C compared with a 2s electron on going from Li to Be. The ground-state configuration of Na is obtained by adding one more electron to a neon-like core, and is [Ne]3s1, showing that it consists of a single electron outside a completely filled 1s22s22p6 core. Now a similar sequence of filling subshells begins again, with the 3s and 3p orbitals complete at argon, with configuration [Ne]3s23p6, which can be denoted [Ar]. Because the 3d orbitals are so much higher in energy, this configuration is effectively closed. Moreover, the 4s orbital is next in line for occupation, so the configuration of K is analogous to that of Na, with a single electron outside a noble-gas core: specifically, it is [Ar]4s1. The next electron, for Ca, also enters the 4s orbital, giving [Ar]4s2, which is the analogue of Mg. However, in the next element, Sc, the added electron occupies a 3d orbital, and filling of the d orbitals begins. (b) Exceptions The energy levels in Figs 1.19 and 1.20 are for individual atomic orbitals and do not fully take into account repulsion between electrons. For elements with an incompletely filled d subshell, the determination of actual ground states by spectroscopy and calculation shows that it is advantageous to occupy orbitals predicted to be higher in energy (the 4s orbitals). The explanation for this order is that the occupation of orbitals of higher energy can result in a reduction in the repulsions between electrons that would occur if the lower-energy 3d orbitals were occupied. It is essential when assessing the total energy of the electrons to consider all contributions to the energy of a configuration, not merely the one-electron orbital energies. Spectroscopic data show that the ground-state configurations of these atoms are mostly of the form 3dn4s2, with the 4s orbitals fully occupied despite individual 3d orbitals being lower in energy. An additional feature, another consequence of spin correlation, is that in some cases a lower total energy may be obtained by forming a half-filled or filled d subshell, even though that may mean moving an s electron into the d subshell. Therefore, as a half-filled d shell is approached the ground-state configuration is likely to be d5s1 and not d4s2 (as for Cr). As a full d subshell is approached the configuration is likely to be d10s1 rather than d9s2 (as for Cu) or d10s0 rather than d8s2 (as for Pd). A similar effect occurs where f orbitals are being occupied, and a d electron may be moved into the f subshell so as to achieve an f7 or an f14 configuration, with a net lowering of energy. For instance, the ground-state electron configuration of Gd is [Xe]4f 75d16s2 and not [Xe]4f 86s2. For cations and complexes of the d-block elements the removal of electrons reduces the complicating effects of electron–electron repulsions and the 3d orbital energies fall well below that of the 4s orbitals. Consequently, all d-block cations and complexes have dn configurations and no electrons in the outermost s orbitals. For example, the configuration of Fe is [Ar]3d64s2 whereas that of [Fe(CO)5] is [Ar]3d8 and Fe2 is [Ar]3d6. For the 19 20 1 Atomic structure purposes of chemistry, the electron configurations of the d-block ions are more important than those of the neutral atoms. In later chapters (starting in Chapter 19), we shall see the great significance of the configurations of the d-metal ions, for the subtle modulations of their energies provide the basis for the explanations of important properties of their compounds. E X A M PL E 1.6 Deriving an electron configuration Predict the ground-state electron configurations of (a) Ti and (b) Ti3. Answer We need to use the building-up principle and Hund’s rule to populate atomic orbitals with electrons. (a) For the neutral atom, for which Z 22, we must add 22 electrons in the order specified above, with no more than two electrons in any one orbital. This procedure results in the configuration [Ar]4s23d2, with the two 3d electrons in different orbitals with parallel spins. However, because the 3d orbitals lie below the 4s orbitals for elements beyond Ca, it is appropriate to reverse the order in which they are written. The configuration is therefore reported as [Ar]3d24s2. (b) The cation has 19 electrons. We should fill the orbitals in the order specified above remembering, however, that the cation will have a dn configuration and no electrons in the s orbital. The configuration of Ti3 is therefore [Ar]3d1. Self-test 1.6 Predict the ground-state electron configurations of Ni and Ni2. 1.8 The classification of the elements Key points: The elements are broadly divided into metals, nonmetals, and metalloids according to their physical and chemical properties; the organization of elements into the form resembling the modern periodic table is accredited to Mendeleev. A useful broad division of elements is into metals and nonmetals. Metallic elements (such as iron and copper) are typically lustrous, malleable, ductile, electrically conducting solids at about room temperature. Nonmetals are often gases (oxygen), liquids (bromine), or solids that do not conduct electricity appreciably (sulfur). The chemical implications of this classification should already be clear from introductory chemistry: 1. Metallic elements combine with nonmetallic elements to give compounds that are typically hard, nonvolatile solids (for example sodium chloride). 2. When combined with each other, the nonmetals often form volatile molecular compounds (for example phosphorus trichloride). 3. When metals combine (or simply mix together) they produce alloys that have most of the physical characteristics of metals (for example brass from copper and zinc). Some elements have properties that make it difficult to classify them as metals or nonmetals. These elements are called metalloids. Examples of metalloids are silicon, germanium, arsenic, and tellurium. A note on good practice You will sometimes see metalloids referred to as ‘semi-metals’. This name is best avoided because a semi-metal has a well defined and quite distinct meaning in physics (see Section 3.19). (a) The periodic table A more detailed classification of the elements is the one devised by Dmitri Mendeleev in 1869; this scheme is familiar to every chemist as the periodic table. Mendeleev arranged the known elements in order of increasing atomic weight (molar mass). This arrangement resulted in families of elements with similar chemical properties, which he arranged into the groups of the periodic table. For example, the fact that C, Si, Ge, and Sn all form hydrides of the general formula EH4 suggests that they belong to the same group. That N, P, As, and Sb all form hydrides with the general formula EH3 suggests that they belong to a different group. Other compounds of these elements show family similarities, as in the formulas CF4 and SiF4 in the first group, and NF3 and PF3 in the second. Many-electron atoms 21 Cs Molar atomic volume/(cm3 mol−1) 70 60 Rb 50 K Xe 40 He Kr Eu 30 Yb Na Ar Po Cm 20 U 10 0 B 10 30 50 Atomic number, Z 70 90 Mendeleev concentrated on the chemical properties of the elements. At about the same time Lothar Meyer in Germany was investigating their physical properties, and found that similar values repeated periodically with increasing molar mass. Figure 1.21 shows a classic example, where the molar volume of the element (its volume per mole of atoms) at 1 bar and 298 K is plotted against atomic number. Mendeleev provided a spectacular demonstration of the usefulness of the periodic table by predicting the general chemical properties, such as the numbers of bonds they form, of unknown elements corresponding to gaps in his original periodic table. (He also predicted elements that we now know cannot exist and denied the presence of elements that we now know do exist, but that is overshadowed by his positive achievement and has been quietly forgotten.) The same process of inference from periodic trends is still used by inorganic chemists to rationalize trends in the physical and chemical properties of compounds and to suggest the synthesis of previously unknown compounds. For instance, by recognizing that carbon and silicon are in the same family, the existence of alkenes R2CCR2 suggests that R2SiSiR2 ought to exist too. Compounds with silicon–silicon double bonds (disilaethenes) do indeed exist, but it was not until 1981 that chemists succeeded in isolating one. The periodic trends in the properties of the elements are explored further in Chapter 9. (b) The format of the periodic table Key points: The blocks of the periodic table reflect the identity of the orbitals that are occupied last in the building-up process. The period number is the principal quantum number of the valence shell. The group number is related to the number of valence electrons. The layout of the periodic table reflects the electronic structure of the atoms of the elements (Fig. 1.22). We can now see, for instance, that a block of the table indicates the type of subshell currently being occupied according to the building-up principle. Each period, or row, of the table corresponds to the completion of the s and p subshells of a given shell. The period number is the value of the principal quantum number n of the shell which according to the building-up principle is currently being occupied in the main groups of the table. For example, Period 2 corresponds to the n 2 shell and the filling of the 2s and 2p subshells. The group numbers, G, are closely related to the number of electrons in the valence shell, the outermost shell of the atom. In the ‘1–18’ numbering system recommended by IUPAC: Block: s p d Number of electrons in valence shell: G G – 10 G Figure 1.21 The periodic variation of molar volume with atomic number. 22 1 Atomic structure s p u ro g in a M I V 18 I 1 I 1 2 I IV V H 13 14 IV V I 15 16 17 iv ta rsn p e R lm 2 lT e m sito n ra 3 3 4 5 6 7 8 9 11 10 12 4 se a lg b o N s n e g lo a H s n e g lco a h C lka A ts e im 5 lsA a e m th kin r 6 7 ksB c lo kd c lo B kp c lo B s id o th n a L Figure 1.22 The general structure of the periodic table. Compare this template with the complete table inside the front cover for the identities of the elements that belong to each block. s d o ctin A kfB c lo For the purpose of this expression, the ‘valence shell’ of a d-block element consists of the ns and (n 1)d orbitals, so a Sc atom has three valence electrons (two 4s and one 3d electron). The number of valence electrons for the p-block element Se (Group 16) is 16 10 6, which corresponds to the configuration s2p4. E X A M PL E 1.7 Placing elements within the periodic table. State to which period, group, and block of the periodic table the element with the electron configuration 1s22s22p63s23p4 belongs. Identify the element. Answer We need to remember that the period number is given by the principal quantum number, n, that the group number can be found from the number of valence electrons, and that the identity of the block is given by the type of orbital last occupied according to the building-up principle. The valence electrons have n 3, therefore the element is in Period 3 of the periodic table. The six valence electrons identify the element as a member of Group 16. The electron added last is a p electron, so the element is in the p block. The element is sulfur. Self-test 1.7 State to which period, group, and block of the periodic table the element with the electron configuration 1s22s22p63s23p64s2 belongs. Identify the element. 1.9 Atomic properties Certain characteristic properties of atoms, particularly their radii and the energies associated with the removal and addition of electrons, show regular periodic variations with atomic number. These atomic properties are of considerable importance for understanding the chemical properties of the elements and are discussed further in Chapter 9. A knowledge of these trends enables chemists to rationalize observations and predict likely chemical and structural behaviour without having to refer to tabulated data for each element. (a) Atomic and ionic radii Key points: Atomic radii increase down a group and, within the s and p blocks, decrease from left to right across a period. The lanthanide contraction results in a decrease in atomic radius for elements following the f block. All monatomic anions are larger than their parent atoms and all monatomic cations are smaller. Many-electron atoms One of the most useful atomic characteristics of an element is the size of its atoms and ions. As we shall see in later chapters, geometrical considerations are central to explaining the structures of many solids and individual molecules. In addition, the average distance of electrons from the nucleus of an atom correlates with the energy needed to remove it in the process of forming a cation. An atom does not have a precise radius because far from the nucleus the electron density falls off only exponentially (but sharply). However, we can expect atoms with numerous electrons to be larger, in some sense, than atoms that have only a few electrons. Such considerations have led chemists to propose a variety of definitions of atomic radius on the basis of empirical considerations. The metallic radius of a metallic element is defined as half the experimentally determined distance between the centres of nearest-neighbour atoms in the solid (Fig. 1.23a, but see Section 3.7 for a refinement of this definition). The covalent radius of a nonmetallic element is similarly defined as half the internuclear distance between neighbouring atoms of the same element in a molecule (Fig. 1.23b). We shall refer to metallic and covalent radii jointly as atomic radii (Table 1.3). The periodic trends in metallic and covalent radii can be seen from the data in the table and are illustrated in Fig. 1.24. As will be familiar from introductory chemistry, atoms may be linked by single, double, and triple bonds, with multiple bonds shorter than single bonds between the same two elements. The ionic radius (Fig. 1.23c) of an element is related to the distance between the centres of neighbouring cations and anions in an ionic compound. An arbitrary decision has to be taken on how to apportion the cation– anion distance between the two ions. There have been many suggestions: in one common scheme, the radius of the O2 ion is taken to be 140 pm (Table 1.4; see Section 3.7 for a refinement of this definition). For example, the ionic radius of Mg2 is obtained by subtracting 140 pm from the internuclear distance between adjacent Mg2 and O2 ions in solid MgO. The data in Table 1.3 show that atomic radii increase down a group, and that they decrease from left to right across a period. These trends are readily interpreted in terms of the electronic structure of the atoms. On descending a group, the valence electrons are found in orbitals of successively higher principal quantum number. The atoms within the group have a greater number of completed shells of electrons in successive periods and hence their radii increase down the group. Across a period, the valence electrons enter orbitals of the same shell; however, the increase in effective nuclear charge across the period draws in the electrons and results in progressively more compact atoms. The general increase in radius down a group and decrease across a period should be remembered as they correlate well with trends in many chemical properties. Period 6 shows an interesting and important modification to these otherwise general trends. We see from Fig. 1.24 that the metallic radii in the third row of the d block are very similar to those in the second row, and not significantly larger as might be expected given their considerably greater numbers of electrons. For example, the atomic radii of Mo (Z 42) and W (Z 74) are 140 and 141 pm, respectively, despite the latter having many more electrons. The reduction of radius below that expected on the basis of a simple extrapolation down the group is called the lanthanide contraction. The name points to the Table 1.3 Atomic radii, r/pm Li Be B C N O F 157 112 88 77 74 73 71 Na Mg Al Si P S Cl 191 160 143 118 110 104 99 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se 235 197 164 147 135 129 137 126 125 125 128 137 140 122 122 117 114 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I 250 215 182 160 147 140 135 134 134 137 144 152 150 140 141 135 133 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi 272 224 188 159 147 141 137 135 136 139 144 155 155 154 152 The values refer to coordination number 12 for metallic radii (see Section 3.2). Br 2rM (a) 2rcov (b) r+ + r– (c) Figure 1.23 A representation of (a) metallic radius, (b) covalent radius, and (c) ionic radius. 23 24 1 Atomic structure 300 Figure 1.24 The variation of atomic radii through the periodic table. Note the contraction of radii following the lanthanoids in Period 6. Metallic radii have been used for the metallic elements and covalent radii have been used for the nonmetallic elements. Atomic radius, r/pm Cs Rb K 200 Na Pb Ac Li 100 I Cl 0 1 Po Br Am F 20 40 60 Atomic number, Z 80 100 origin of the effect. The elements in the third row of the d block (Period 6) are preceded by the elements of the first row of the f block, the lanthanoids, in which the 4f orbitals are being occupied. These orbitals have poor shielding properties and so the valence electrons experience more attraction from the nuclear charge than might be expected. The repulsions between electrons being added on crossing the f block fail to compensate for the increasing nuclear charge, so Zeff increases from left to right across a period. The dominating effect of the latter is to draw in all the electrons and hence to result in a more compact atom. A similar contraction is found in the elements that follow the d block for the same reasons. For example, although there is a substantial increase in atomic radius between C and Si (77 and 118 pm, respectively), the atomic radius of Ge (122 pm) is only slightly greater than that of Al. Relativistic effects, especially the increase in mass as particles approach the speed of light, have an important role to play on the elements in and following Period 6 but are rather subtle. Electrons in s and p orbitals, which approach closely to the highly charged nucleus and experience strong accelerations, contract whereas electrons in the less penetrating d and f orbitals expand. One consequence of the latter expansion is that d and f electrons become less effective at shielding other electrons, and the outermost s electrons contract further. For light elements, relativistic effects can be neglected but for the heavier elements with high atomic numbers they become significant and can result in an approximately 20 per cent reduction in the size of the atom. Another general feature apparent from Table 1.4 is that all monatomic anions are larger than their parent atoms and all monatomic cations are smaller than their parent atoms (in some cases markedly so). The increase in radius of an atom on anion formation is a result of the greater electron–electron repulsions that occur when an additional electron is added to form an anion. There is also an associated decrease in the value of Zeff. The smaller radius of a cation compared with its parent atom is a consequence not only of the reduction in electron–electron repulsions that follow electron loss but also of the fact that cation formation typically results in the loss of the valence electrons and an increase in Zeff. That loss often leaves behind only the much more compact closed shells of electrons. Once these gross differences are taken into account, the variation in ionic radii through the periodic table mirrors that of the atoms. Although small variations in atomic radii may seem of little importance, in fact atomic radius plays a central role in the chemical properties of the elements. Small changes can have profound consequences, as we shall see in Chapter 9. (b) Ionization energy Key points: First ionization energies are lowest at the lower left of the periodic table (near caesium) and greatest near the upper right (near helium). Successive ionizations of a species require higher energies. The ease with which an electron can be removed from an atom is measured by its ionization energy, I, the minimum energy needed to remove an electron from a gas-phase atom: A ( g ) ➝ A ( g ) e– ( g ) I E ( A ,g ) E ( A,g ) (1.8) Many-electron atoms Table 1.4 Ionic radii, r/pm Li Be2 B3 N3 O2 F 59(4) 27(4) 11(4) 146 135(2) 128(2) 138(4) 131(4) 140(6) 133(6) 76(6) 142(8) Na Mg2 Al3 P3 S2 Cl 99(4) 49(4) 39(4) 212 184(6) 181(6) 102(6) 72(6) 53(6) 132(8) 103(8) K Ca2 Ga3 As3 Se2 Br 138(6) 100(6) 62(6) 222 198(6) 196(6) 151(8) 112(8) 159(10) 123(10) 160(12) 134(12) Rb Sr2 148(6) 118(6) 160(8) 125(8) Sn2 Sn4 Te2 I 80(6) 83(6) 69(6) 221(6) 220(6) 92(8) 93(8) In3 173(12) 144(12) Cs Ba2 Tl3 167(6) 135(6) 89(6) 174(8) 142(8) Tl 188(12) 175(12) 150(6) Numbers in parentheses are the coordination number of the ion. For more values, see Resource section 1. The first ionization energy, I1, is the energy required to remove the least tightly bound electron from the neutral atom, the second ionization energy, I2, is the energy required to remove the least tightly bound electron from the resulting cation, and so on. Ionization energies are conveniently expressed in electronvolts (eV), but are easily converted into kilojoules per mole by using 1 eV 96.485 kJ mol1. The ionization energy of the H atom is 13.6 eV, so to remove an electron from an H atom is equivalent to dragging the electron through a potential difference of 13.6 V. In thermodynamic calculations it is often more appropriate to use the ionization enthalpy, the standard enthalpy of the process in eqn 1.8, typically at 298 K. The molar ionization enthalpy is larger by 25 RT than the ionization energy. This difference stems from the change from T 0 (assumed implicitly for I) to the temperature T (typically 298 K) to which the enthalpy value refers, and the replacement of 1 mol of gas particles by 2 mol of gaseous ions plus electrons. However, because RT is only 2.5 kJ mol1 (corresponding to 0.026 eV) at room temperature and ionization energies are of the order of 102103 kJ mol1 (110 eV), the difference between ionization energy and enthalpy can often be ignored. To a large extent, the first ionization energy of an element is determined by the energy of the highest occupied orbital of its ground-state atom. First ionization energies vary systematically through the periodic table (Table 1.5), being smallest at the lower left (near Cs) and greatest near the upper right (near He). The variation follows the pattern of effective nuclear charge, and (as Zeff itself shows) there are some subtle modulations arising from the effect of electron–electron repulsions within the same subshell. A useful approximation is that for an electron from a shell with principal quantum number n 2 Zeff n2 Ionization energies also correlate strongly with atomic radii, and elements that have small atomic radii generally have high ionization energies. The explanation of the correlation is 25 26 1 Atomic structure Table 1.5 First, second, and third (and some fourth) ionization energies of the elements, I/(kJ mol1) H He 1312 2373 5259 Li Be B C N O F Ne 513 899 801 1086 1402 1314 1681 2080 7297 1757 2426 2352 2855 3386 3375 3952 11809 14844 3660 4619 4577 5300 6050 6122 25018 Na Mg Al Si P S Cl Ar 495 737 577 786 1011 1000 1251 1520 4562 1476 1816 1577 1903 2251 2296 2665 6911 7732 2744 3231 2911 3361 3826 3928 K Ca Ga Ge As Se Br Kr 419 589 579 762 947 941 1139 1351 3051 1145 1979 1537 1798 2044 2103 3314 4410 4910 2963 3302 2734 2974 3500 3565 Rb Sr In Sn Sb Te I Xe 403 549 558 708 834 869 1008 1170 2632 1064 1821 1412 1794 1795 1846 2045 3900 4210 2704 2943 2443 2698 3197 3097 Cs Ba Tl Pb Bi Po At Rn 375 502 590 716 704 812 926 1036 2420 965 1971 1450 1610 1800 1600 3400 3619 2878 3080 2466 2700 2900 11574 that in a small atom an electron is close to the nucleus and experiences a strong Coulombic attraction, making it difficult to remove. Therefore, as the atomic radius increases down a group, the ionization energy decreases and the decrease in radius across a period is accompanied by a gradual increase in ionization energy. Some deviation from this general trend in ionization energy can be explained quite readily. An example is the observation that the first ionization energy of boron is smaller than that of beryllium, despite the former’s higher nuclear charge. This anomaly is readily explained by noting that, on going to boron, the outermost electron occupies a 2p orbital and hence is less strongly bound than if it had occupied a 2s orbital. As a result, the value of I1 decreases from Be to B. The decrease between N and O has a slightly different explanation. The configurations of the two atoms are N [He]2s2 2p1x 2p1y 2p1z O [He]2s2 2p2x 2p1y 2p1z We see that, in an O atom, two electrons are present in a single 2p orbital. They repel each other strongly, and this strong repulsion offsets the greater nuclear charge. Another contribution to the difference is the lower energy of the O ion on account of its having a 2s22p3 configuration: as we have seen, a half-filled subshell has a relatively low energy (Fig. 1.25). Additionally, the half-filled shell of p orbitals of nitrogen is a particularly stable configuration. When considering F and Ne on the right of Period 2, the last electrons enter orbitals that are already half full, and continue the trend from O towards higher ionization energy. The higher values of the ionization energies of these two elements reflect the high value of Zeff. The value of I1 falls back sharply from Ne to Na as the outermost electron occupies the next shell with an increased principal quantum number and is therefore further from the nucleus. Many-electron atoms 27 30 I/eV e H e N Ioniza tion energ ,y 20 Ar r K e X H g H n R 10 i L a N 0 1 K 20 b R s C 40 60 Atomic number, Z l T 80 100 Figure 1.25 The periodic variation of first ionization energies. E X A MPL E 1. 8 Accounting for a variation in ionization energy Account for the decrease in first ionization energy between phosphorus and sulfur. Answer We approach this question by considering the ground-state configurations of the two atoms: P [Ne]3s2 3p1x 3p1y 3p1z S [Ne]3s2 3p2x 3p1y 3p1z As in the analogous case of N and O, in the ground state of S, two electrons are present in a single 3p orbital. They are so close together that they repel each other strongly, and this increased repulsion offsets the effect of the greater nuclear charge of S compared with P. As in the difference between N and O, the half-filled subshell of S also contributes to the lowering of energy of the ion and hence to the smaller ionization energy. Self-test 1.8 Account for the decrease in first ionization energy between fluorine and chlorine. (c) Electron affinity Key point: Electron affinities are highest for elements near fluorine in the periodic table. The electron-gain enthalpy, eg H ° , is the change in standard molar enthalpy when a gaseous atom gains an electron: A(g) + e− (g) ➝ A − (g) 40 Ionization energy, I/eV Another important pattern is that successive ionizations of an element require increasingly higher energies (Fig. 1.26). Thus, the second ionization energy of an element E (the energy needed to remove an electron from the cation E) is higher than its first ionization energy, and its third ionization energy (the energy needed to remove an electron from E2) is higher still. The explanation is that the higher the positive charge of a species, the greater the electrostatic attraction experienced by the electron being removed. Moreover, when an electron is removed, Zeff increases and the atom contracts. It is then even more difficult to remove an electron from this smaller, more compact, cation. The difference in ionization energy is greatly magnified when the electron is removed from a closed shell of the atom (as is the case for the second ionization energy of Li and any of its congeners) because the electron must then be extracted from a compact orbital in which it interacts strongly with the nucleus. The first ionization energy of Li, for instance, is 513 kJ mol1, but its second ionization energy is 7297 kJ mol1, more than ten times greater. The pattern of successive ionization energies down a group is far from simple. Figure 1.26 shows the first, second, and third ionization energies of the members of Group 13. Although they lie in the expected order I1 I2 I3, there is no simple trend. The lesson to be drawn is that whenever an argument hangs on trends in small differences in ionization energies, it is always best to refer to actual numerical values rather than to guess a likely outcome. 30 20 10 B Al 2 3 In Tl 4 5 Period 6 Ga Figure 1.26 The first, second, and third ionization energies of the elements of Group 13. Successive ionization energies increase, but there is no clear pattern of ionization energies down the group. 28 1 Atomic structure E X A M PL E 1. 9 Accounting for values of successive energies of ionization Rationalize the following values for successive ionization energies of boron, where ∆ionH(N) is the Nth enthalpy of ionization: 1 2 3 4 5 N 807 2433 3666 25033 32834 ∆ionH(N)/(kJ mol1) Answer When considering trends in ionization energy, a sensible starting point is the electron configurations of the atoms. The electron configuration of B is 1s22s22p1. The first ionization energy corresponds to removal of the electron in the 2p orbital. This electron is shielded from nuclear charge by the core and the full 2s orbital. The second value corresponds to removal of a 2s electron from the B cation. This electron is more difficult to remove on account of the increased effective nuclear charge. The effective nuclear charge increases further on removal of this electron, resulting in an increase between ∆ionH(2) and ∆ionH(3). There is a large increase between ∆ionH(3) and ∆ionH(4) because the 1s shell lies at very low energy as it experiences almost the full nuclear charge and also has n 1. The final electron to be removed experiences no shielding of nuclear charge so ∆ionH(5) is very high, and is given by hcRZ 2 with Z 5, corresponding to (13.6 eV) 25 340 eV (32.8 MJ mol1). Self-test 1.9 Study the values listed below of the first five ionization energies of an element and deduce to which group of the periodic table the element belongs. Give your reasoning. N ∆ionH(N)/(kJ mol1) 1 1093 2 2359 3 4627 4 6229 5 37838 Electron gain may be either exothermic or endothermic. Although the electron-gain enthalpy is the thermodynamically appropriate term, much of inorganic chemistry is discussed in terms of a closely related property, the electron affinity, Ea, of an element (Table 1.6), which is the difference in energy between the gaseous atoms and the gaseous ions at T 0. Ea E(A, g) E(A , g) (1.9) Although the precise relation is ∆ eg H ° = Ea 25 RT, the contribution 25 RT is commonly ignored. A positive electron affinity indicates that the ion A has a lower, more negative energy than the neutral atom, A. The second electron-gain enthalpy, the enthalpy change for the attachment of a second electron to an initially neutral atom, is invariably positive because the electron repulsion outweighs the nuclear attraction. The electron affinity of an element is largely determined by the energy of the lowest unfilled (or half-filled) orbital of the ground-state atom. This orbital is one of the two frontier orbitals of an atom, the other one being the highest filled atomic orbital. The frontier orbitals are the sites of many of the changes in electron distributions when bonds form, Table 1.6 First electron affinities of the main-group elements, Ea /(kJ mol1) H He 72 48 Li Be B C N O F Ne 60 ≤0 27 122 8 141 328 116 Na Mg Al Si P S Cl Ar 53 ≤0 43 134 72 200 349 96 780 492 K Ca Ga Ge As Se Br Kr 48 2 29 116 78 195 325 96 Rb Sr In Sn Sb Te I Xe 47 5 29 116 103 190 295 77 The first values refer to the formation of the ion X from the neutral atom; the second value to the formation of X2 from X. Many-electron atoms E X A MPL E 1.10 Accounting for the variation in electron affinity Account for the large decrease in electron affinity between Li and Be despite the increase in nuclear charge. Answer When considering trends in electron affinities, as in the case of ionization energies, a sensible starting point is the electron configurations of the atoms. The electron configurations of Li and Be are [He]2s1 and [He]2s2, respectively. The additional electron enters the 2s orbital of Li but it enters the 2p orbital of Be, and hence is much less tightly bound. In fact, the nuclear charge is so well shielded in Be that electron gain is endothermic. Self-test 1.10 Account for the decrease in electron affinity between C and N. and we shall see more of their importance as the text progresses. An element has a high electron affinity if the additional electron can enter a shell where it experiences a strong effective nuclear charge. This is the case for elements towards the top right of the periodic table, as we have already explained. Therefore, elements close to fluorine (specifically O and Cl, but not the noble gases) can be expected to have the highest electron affinities as their Zeff is large and it is possible to add electrons to the valence shell. Nitrogen has very low electron affinity because there is a high electron repulsion when the incoming electron enters an orbital that is already half full. A note on good practice Be alert to the fact that some people use the terms ‘electron affinity’ and ‘electron-gain enthalpy’ interchangeably. In such cases, a positive electron affinity could indicate that A has a higher energy than A. (d) Electronegativity Key points: The electronegativity of an element is the power of an atom of the element to attract electrons when it is part of a compound; there is a general increase in electronegativity across a period and a general decrease down a group. The electronegativity, (chi), of an element is the power of an atom of the element to attract electrons to itself when it is part of a compound. If an atom has a strong tendency to acquire electrons, it is said to be highly electronegative (like the elements close to fluorine). Electronegativity is a very useful concept in chemistry and has numerous applications, which include a rationalization of bond energies and the types of reactions that substances undergo and the prediction of the polarities of bonds and molecules (Chapter 2). Periodic trends in electronegativity can be related to the size of the atoms and electron configuration. If an atom is small and has an almost closed shell of electrons, then it is more likely to attract an electron to itself than a large atom with few valence electrons. Consequently, the electronegativities of the elements typically increase left to right across a period and decrease down a group. Quantitative measures of electronegativity have been defined in many different ways. Linus Pauling’s original formulation (which results in the values denoted P in Table 1.7) draws on concepts relating to the energetics of bond formation, which will be dealt with in Chapter 2.2 A definition more in the spirit of this chapter, in the sense that it is based on the properties of individual atoms, was proposed by Robert Mulliken. He observed that, if an atom has a high ionization energy, I, and a high electron affinity, Ea, then it will be likely to acquire rather than lose electrons when it is part of a compound, and hence be classified as highly electronegative. Conversely, if its ionization energy and electron affinity are both low, then the atom will tend to lose electrons rather than gain them, and hence be classified as electropositive. These observations motivate the definition of the Mulliken electronegativity, M, as the average value of the ionization energy and the electron affinity of the element (both expressed in electronvolts): M = 12 (I + Ea ) 2 Pauling values of electronegativity are used throughout the following chapters. (1.10) 29 30 1 Atomic structure Table 1.7 Pauling P , Mulliken, M, and AllredRochow, AR, electronegativities H He 2.20 5.5 3.06 2.20 Li Be B C N O F 0.98 1.57 2.04 2.55 3.04 3.44 3.98 Ne 1.28 1.99 1.83 2.67 3.08 3.22 4.43 4.60 0.97 1.47 2.01 2.50 3.07 3.50 4.10 5.10 Na Mg Al Si P S Cl Ar 0.93 1.31 1.61 1.90 2.19 2.58 3.16 1.21 1.63 1.37 2.03 2.39 2.65 3.54 3.36 1.01 1.23 1.47 1.74 2.06 2.44 2.83 3.30 K Ca Ga Ge As Se Br Kr 0.82 1.00 1.81 2.01 2.18 2.55 2.96 3.0 1.03 1.30 1.34 1.95 2.26 2.51 3.24 2.98 0.91 1.04 1.82 2.02 2.20 2.48 2.74 3.10 Rb Sr In Sn Sb Te I Xe 0.82 0.95 1.78 1.96 2.05 2.10 2.66 2.6 0.99 1.21 1.30 1.83 2.06 2.34 2.88 2.59 0.89 0.99 1.49 1.72 1.82 2.01 2.21 2.40 Cs Ba Tl Pb Bi 0.79 0.89 2.04 2.33 2.02 0.70 0.90 1.80 1.90 1.90 0.86 0.97 1.44 1.55 1.67 The hidden complication in the apparently simple definition of the Mulliken electronegativity is that the ionization energy and electron affinity in the definition relate to the valence state, the electron configuration the atom is supposed to have when it is part of a molecule. Hence, some calculation is required because the ionization energy and electron affinity to be used in calculating M are mixtures of values for various actual spectroscopically observable states of the atom. We need not go into the calculation, but the resulting values given in Table 1.7 may be compared with the Pauling values (Fig. 1.27). The two Pauling electronegativity, χ 5 F 4 Cl Br 3 I Pb H Tl 2 Bi 1 Li Na K Rb Cs 0 Figure 1.27 The periodic variation of Pauling electronegativities. 10 30 50 Atomic number, Z 70 90 Many-electron atoms 31 scales give similar values and show the same trends. One reasonably reliable conversion between the two is P = 1.35 1M/ 2 1.37 (1.11) Because the elements near F (other than the noble gases) have high ionization energies and appreciable electron affinities, these elements have the highest Mulliken electronegativities. Because M depends on atomic energy levels—and in particular on the location of the highest filled and lowest empty orbitals—the electronegativity of an element is high if the two frontier orbitals of its atoms are low in energy. Various alternative ‘atomic’ definitions of electronegativity have been proposed. A widely used scale, suggested by A.L. Allred and E. Rochow, is based on the view that electronegativity is determined by the electric field at the surface of an atom. As we have seen, an electron in an atom experiences an effective nuclear charge Zeff. The Coulombic potential at the surface of such an atom is proportional to Zeff /r, and the electric field there is proportional to Zeff/r2. In the Allred–Rochow definition of electronegativity, AR is assumed to be proportional to this field, with r taken to be the covalent radius of the atom: AR 0.744 35.90Zeff (r / pm)2 (1.12) The numerical constants have been chosen to give values comparable to Pauling electronegativities. According to the Allred–Rochow definition, elements with high electronegativity are those with high effective nuclear charge and the small covalent radius: such elements lie close to F. The Allred–Rochow values parallel closely those of the Pauling electronegativities and are useful for discussing the electron distributions in compounds. (e) Polarizability Key points: A polarizable atom or ion is one with orbitals that lie close in energy; large, heavy atoms and ions tend to be highly polarizable. r Small, highly charged cations have polarizing ability. r Large, highly charged anions are easily polarized. r Cations that do not have a noble-gas electron configuration are easily polarized. The last rule is particularly important for the d-block elements. E X A MPL E 1.11 Identifying polarizable species Which would be the more polarizable, an F ion or an I ion? Answer We can make use of the fact that polarizable anions are typically large and highly charged. An F ion is small and singly charged. An I ion has the same charge but is large. Therefore, an I ion is likely to be the more polarizable. Self-test 1.11 Which would be more polarizing, Na or Cs? I Ea I cM Energy The polarizability, , of an atom is its ability to be distorted by an electric field (such as that of a neighbouring ion). An atom or ion (most commonly, an anion) is highly polarizable if its electron distribution can be distorted readily, which is the case if unfilled atomic orbitals lie close to the highest-energy filled orbitals. That is, the polarizability is likely to be high if the separation of the frontier orbitals is small and the polarizability will be low if the separation of the frontier orbitals is large (Fig. 1.28). Closely separated frontier orbitals are typically found for large, heavy atoms and ions, such as the atoms and ions of the heavier alkali metals and the heavier halogens, so these atoms and ions are the most polarizable. Small, light atoms, such as the atoms and ions near fluorine, typically have widely spaced energy levels, so these atoms and ions are least polarizable. Species that effectively distort the electron distribution of a neighbouring atom or anion are described as having polarizing ability. We shall see the consequences of polarizability when considering the nature of bonding in Section 2.2, but it is appropriate to anticipate here that extensive polarization leads to covalency. Fajan’s rules summarize the factors that affect polarization: Ionization limit cM α α (a) Ea (b) Figure 1.28 The interpretation of the electronegativity and polarizability of an element in terms of the energies of the frontier orbitals (the highest filled and lowest unfilled atomic orbitals). (a) Low electronegativity and polarizability; (b) high electronegativity and polarizability. 32 1 Atomic structure FURTHER READING M. Laing, The different periodic tables of Dmitrii Mendeleev. J. Chem. Educ., 2008, 85, 63. M.W. Cronyn, The proper place for hydrogen in the periodic table. J. Chem. Educ. 2003, 80, 947. P.A. Cox, Introduction to quantum theory and atomic structure. Oxford University Press (1996). An introduction to the subject. P. Atkins and J. de Paula, Physical chemistry. Oxford University Press and W.H. Freeman & Co. (2010). Chapters 7 and 8 give an account of quantum theory and atomic structure. J. Emsley, Nature’s building blocks. Oxford University Press (2003). An interesting guide to the elements. D.M.P. Mingos, Essential trends in inorganic chemistry. Oxford University Press (1998). Includes a detailed discussion of the important horizontal, vertical, and diagonal trends in the properties of the atoms. P.A. Cox, The elements: their origin, abundance, and distribution. Oxford University Press (1989). Examines the origin of the elements, the factors controlling their widely differing abundances, and their distributions in the Earth, the solar system, and the universe. N.G. Connelly, T. Danhus, R.M. Hartshoin, and A.T Hutton, Nomenclature of inorganic chemistry. Recommendations 2005. Royal Society of Chemistry (2005). This book outlines the conventions for the periodic table and inorganic substances. It is known colloquially as the ‘Red Book’ on account of its distinctive red cover. EXERCISES 1.1 Write balanced equations for the following nuclear reactions (show emission of excess energy as a photon of electromagnetic radiation, ): (a) 14N 4He to produce 17O, (b) 12C p to produce 13N, (c) 14N n to produce 3H and 12C. (The last reaction produces a steady-state concentration of radioactive 3H in the upper atmosphere.) 1.2 Balance the following nuclear reaction: 246 96 1.14 Complete the following table: n l ml Orbital designation 2 3 Number of orbitals 2p 2 4s Cm 126 C ➝ ? 01 n 3, 2, . . . . . . , 3 4 1.3 In general, ionization energies increase across a period from left to right. Explain why the second ionization energy of Cr is higher, not lower, than that of Mn. 1.4 One possible source of neutrons for the neutron-capture processes mentioned in the text is the reaction of 22Ne with particles to produce 25Mg and neutrons. Write the balanced equation for the nuclear reaction. 1.15 What are the values of the n, l, and ml quantum numbers that describe the 5f orbitals? 1.16 Use the data in Table 1.2 to calculate the screening constants for the outermost electron in the elements Li to F. Comment on the values you obtain. 1.5 9Be undergoes decay to produce 12C and neutrons. Write a balanced equation for this reaction. 1.17 Consider the process of shielding in atoms, using Be as an example. What is being shielded? What is it shielded from? What is doing the shielding? 1.6 The natural abundances of adjacent elements in the periodic table usually differ by a factor of 10 or more. Explain this phenomenon. 1.18 Use sketches of 2s and 2p orbitals to distinguish between (a) the radial wavefunction and (b) the radial distribution function. 1.7 Explain how you would determine, using data you would look up in tables, whether or not the nuclear reaction in Exercise 1.2 corresponds to a release of energy. 1.19 Compare the first ionization energy of Ca with that of Zn. Explain the difference in terms of the balance between shielding with increasing numbers of d electrons and the effect of increasing nuclear charge. 1.8 What is the ratio of the energy of a ground-state He ion to that of a Be3 ion? 1.20 Compare the first ionization energies of Sr, Ba, and Ra. Relate the irregularity to the lanthanide contraction. 1.9 The ionization energy of H is 13.6 eV. What is the difference in energy between the n 1 and n 6 levels? 1.21 The second ionization energies of some Period 4 elements are Ca Sc Ti V Cr Mn 1145 1235 1310 1365 1592 1509 kJ mol1 ~ 1.10 Calculate the wavenumber ( 1/) and wavelength of the first transition in the visible region of the atomic spectrum of hydrogen. 1.11 Show that the following four lines in the Lyman series can be predicted from equation 1.1: 91.127, 97.202, 102.52, and 121.57 nm. 1.12 What is the relation of the possible angular momentum quantum numbers to the principal quantum number? 1.13 How many orbitals are there in a shell of principal quantum number n? (Hint: begin with n 1, 2, and 3 and see if you can recognize the pattern.) Identify the orbital from which ionization occurs and account for the trend in values. 1.22 Give the ground-state electron configurations of (a) C, (b) F, (c) Ca, (d) Ga3, (e) Bi, (f) Pb2. 1.23 Give the ground-state electron configurations of (a) Sc, (b) V3, (c) Mn2, (d) Cr2, (e) Co3, (f) Cr6, (g) Cu, (h) Gd3. 1.24 Give the ground-state electron configurations of (a) W, (b) Rh3, (c) Eu3, (d) Eu2, (e) V5, (f) Mo4. Problems 1.25 Identify the elements that have the ground-state electron configurations: (a) [Ne]3s23p4, (b) [Kr]5s2, (c) [Ar]4s23d3, (d) [Kr]5s24d5, (e) [Kr]5s24d105p1, (f) [Xe]6s24f6. 1.26 Without consulting reference material, draw the form of the periodic table with the numbers of the groups and the periods and identify the s, p, and d blocks. Identify as many elements as you can. (As you progress through your study of inorganic chemistry, you should learn the positions of all the s-, p-, and d-block elements and associate their positions in the periodic table with their chemical properties.) 33 1.27 Account for the trends across Period 3 in (a) ionization energy, (b) electron affinity, (c) electronegativity. 1.28 Account for the fact that the two Group 5 elements niobium (Period 5) and tantalum (Period 6) have the same atomic radii. 1.29 Identify the frontier orbitals of a Be atom in its ground state. 1.30 Use the data in Tables 1.6 and 1.7 to test Mulliken’s proposition that electronegativity values are proportional to I Ea. PROBLEMS 1.1 Show that an atom with the configuration ns2np6 is spherically symmetrical. Is the same true of an atom with the configuration ns2np3? 1.2 According to the Born interpretation, the probability of finding an electron in a volume element d is proportional to 2d. (a) What is the most probable location of an electron in an H atom in its ground state? (b) What is its most probable distance from the nucleus, and why is this different? (c) What is the most probable distance of a 2s electron from the nucleus? 1.3 The ionization energies of rubidium and silver are 4.18 and 7.57 eV, respectively. Calculate the ionization energies of an H atom with its electron in the same orbitals as in these two atoms and account for the differences in values. 1.4 When 58.4 nm radiation from a helium discharge lamp is directed on a sample of krypton, electrons are ejected with a velocity of 1.59 106 m s1. The same radiation ejects electrons from Rb atoms with a velocity of 2.45 106 m s1. What are the ionization energies (in electronvolts, eV) of the two elements? 1.5 Survey the early and modern proposals for the construction of the periodic table. You should consider attempts to arrange the elements on helices and cones as well as the more practical twodimensional surfaces. What, in your judgement, are the advantages and disadvantages of the various arrangements? 1.6 The decision about which elements should be identified as belonging to the f block has been a matter of some controversy. A view has been expressed by W.B. Jensen (J. Chem. Educ., 1982, 59, 635). Summarize the controversy and Jensen’s arguments. An alternative view has been expressed by L. Lavalle (J. Chem. Educ., 2008, 85, 1482). Summarize the controversy and the arguments. 1.7 Draw pictures of the two d orbitals in the xy-plane as flat projections in the plane of the paper. Label each drawing with the appropriate mathematical function, and include a labelled pair of Cartesian coordinate axes. Label the orbital lobes correctly with and signs. 1.8 During 1999 several papers appeared in the scientific literature claiming that d orbitals of Cu2O had been observed experimentally. In his paper ‘Have orbitals really been observed?’ (J. Chem. Educ., 2000, 77, 1494), Eric Scerri reviews these claims and discusses whether orbitals can be observed physically. Summarize his arguments briefly. 1.9 At various times the following two sequences have been proposed for the elements to be included in Group 3: (a) Sc, Y, La, Ac, (b) Sc, Y, Lu, Lr. Because ionic radii strongly influence the chemical properties of the metallic elements, it might be thought that ionic radii could be used as one criterion for the periodic arrangement of the elements. Use this criterion to describe which of these sequences is preferred. 1.10 In the paper ‘Ionization energies of atoms and atomic ions’ (P.F. Lang and B.C. Smith, J. Chem. Educ., 2003, 80, 938) the authors discuss the apparent irregularities in the first and second ionization energies of d- and f-block elements. Describe how these inconsistencies are rationalized. 2 Lewis structures 2.1 The octet rule 2.2 Resonance 2.3 The VSEPR model Valence bond theory 2.4 The hydrogen molecule Molecular structure and bonding The interpretation of structures and reactions in inorganic chemistry is often based on semiquantitative models. In this chapter we examine the development of models of molecular structure in terms of the concepts of valence bond and molecular orbital theory. In addition, we review methods for predicting the shapes of molecules. This chapter introduces concepts that will be used throughout the text to explain the structures and reactions of a wide variety of species. The chapter also illustrates the importance of the interplay between qualitative models, experiment, and calculation. 2.5 Homonuclear diatomic molecules 2.6 Polyatomic molecules Molecular orbital theory 2.7 An introduction to the theory 2.8 Homonuclear diatomic molecules 2.9 Heteronuclear diatomic molecules 2.10 Bond properties 2.11 Polyatomic molecules 2.12 Molecular shape in terms of molecular orbitals Structure and bond properties 2.13 Bond length Lewis structures Lewis proposed that a covalent bond is formed when two neighbouring atoms share an electron pair. A single bond, a shared electron pair (A:B), is denoted AB; likewise, a double bond, two shared electron pairs (A::B), is denoted AB, and a triple bond, three shared pairs of electrons (A:::B), is denoted A⬅B. An unshared pair of valence electrons on an atom (A:) is called a lone pair. Although lone pairs do not contribute directly to the bonding, they do influence the shape of the molecule and play an important role in its properties. 2.1 The octet rule Key point: Atoms share electron pairs until they have acquired an octet of valence electrons. 2.14 Bond strength 2.15 Electronegativity and bond enthalpy 2.16 Oxidation states FURTHER READING EXERCISES PROBLEMS Lewis found that he could account for the existence of a wide range of molecules by proposing the octet rule: Each atom shares electrons with neighbouring atoms to achieve a total of eight valence electrons (an ‘octet’). As we saw in Section 1.8, a closed-shell, noble-gas configuration is achieved when eight electrons occupy the s and p subshells of the valence shell. One exception is the hydrogen atom, which fills its valence shell, the 1s orbital, with two electrons (a ‘duplet’). The octet rule provides a simple way of constructing a Lewis structure, a diagram that shows the pattern of bonds and lone pairs in a molecule. In most cases we can construct a Lewis structure in three steps. 1. Decide on the number of electrons that are to be included in the structure by adding together the numbers of all the valence electrons provided by the atoms. Each atom provides all its valence electrons (thus, H provides one electron and O, with the configuration [He]2s22p4, provides six). Each negative charge on an ion corresponds to an additional electron; each positive charge corresponds to one electron less. 2. Write the chemical symbols of the atoms in the arrangement that shows which atoms are bonded together. In most cases we know the arrangement or can make an informed guess. The less electronegative element is usually the central atom of a molecule, as in CO2 and SO42, but there are many well-known exceptions (H2O and NH3 among them). 35 Lewis structures Distribute the electrons in pairs so that there is one pair of electrons forming a single bond between each pair of atoms bonded together, and then supply electron pairs (to form lone pairs or multiple bonds) until each atom has an octet. Each bonding pair (:) is then represented by a single line (). The net charge of a polyatomic ion is supposed to be possessed by the ion as a whole, not by a particular individual atom. E X A M PL E 2 .1 Writing a Lewis structure Write a Lewis structure for the BF4 ion. Answer We need to consider the total number of electrons supplied and how they are shared to complete an octet around each atom. The atoms supply 3 (4 7) 31 valence electrons; the single negative charge of the ion reflects the presence of an additional electron. We must therefore accommodate 32 electrons in 16 pairs around the five atoms. One solution is (1). The negative charge is ascribed to the ion as a whole, not to a particular individual atom. Self-test 2.1 Write a Lewis structure for the PCl3 molecule. Table 2.1 gives examples of Lewis structures of some common molecules and ions. Except in simple cases, a Lewis structure does not portray the shape of the species, but only the pattern of bonds and lone pairs: it shows the number of the links, not the geometry of the molecule. For example the BF4 ion is actually tetrahedral (2), not planar, and PF3 is trigonal pyramidal (3). – F F B F F 1 BF4– 2.2 Resonance Key points: Resonance between Lewis structures lowers the calculated energy of the molecule and distributes the bonding character of electrons over the molecule; Lewis structures with similar energies provide the greatest resonance stabilization. – F A single Lewis structure is often an inadequate description of the molecule. The Lewis structure of O3 (4), for instance, suggests incorrectly that one OO bond is different from the other, whereas in fact they have identical lengths (128 pm) intermediate between those of typical single OO and double OO bonds (148 pm and 121 pm, respectively). This deficiency of the Lewis description is overcome by introducing the concept of resonance, B 2 BF4− Table 2.1 Lewis structures of some simple molecules O O C O N N H H O O S O O N – P O F O H H N 3– O O P O O H O S 3 PF3 O O 2– O O S O O Cl O O O – Only representative resonance structures are given. Shapes are indicated only for diatomic and triatomic molecules. O O O 4 O3 36 2 Molecular structure and bonding in which the actual structure of the molecule is taken to be a superposition, or average, of all the feasible Lewis structures corresponding to a given atomic arrangement. Resonance is indicated by a double-headed arrow, as in O O O O O O At this stage we are not indicating the shape of the molecule. Resonance should be pictured as a blending of structures, not a flickering alternation between them. In quantum mechanical terms, the electron distribution of each structure is represented by a wavefunction, and the actual wavefunction, , of the molecule is the superposition of the individual wavefunctions for each contributing structure:1 (OOO) (OOO) The overall wavefunction is written as a superposition with equal contributions from both structures because the two structures have identical energies. The blended structure of two or more Lewis structures is called a resonance hybrid. Note that resonance occurs between structures that differ only in the allocation of electrons; resonance does not occur between structures in which the atoms themselves lie in different positions. For instance, there is no resonance between the structures SOO and OSO. Resonance has two main effects: 1. Resonance averages the bond characteristics over the molecule. 2. The energy of a resonance hybrid structure is lower than that of any single contributing structure. The energy of the O3 resonance hybrid, for instance, is lower than that of either individual structure alone. Resonance is most important when there are several structures of identical energy that can be written to describe the molecule, as for O3. In such cases, all the structures of the same energy contribute equally to the overall structure. Structures with different energies may also contribute to an overall resonance hybrid but, in general, the greater the energy difference between two Lewis structures, the smaller the contribution of the higher energy structure. The BF3 molecule, for instance, could be regarded as a resonance hybrid of the structures shown in (5), but the first structure dominates even though the octet is incomplete. Consequently, BF3 is regarded primarily as having that structure with a small admixture of double-bond character. In contrast, for the NO3 ion (6), the last three structures dominate, and we treat the ion as having partial double-bond character. F B F F F F F B F F B F F F B F 5 BF3 – O O N O – O O N O – O O N O – O O N O 6 NO3– 2.3 The VSEPR model There is no simple method for predicting the numerical value of bond angles even in simple molecules, except where the shape is governed by symmetry. However, the valence 1 This wavefunction is not normalized (Section 1.5). We shall often omit normalization constants from linear combinations in order to clarify their structure. The wavefunctions themselves are formulated in the valence bond theory, which is described later. 37 Lewis structures shell electron pair repulsion (VSEPR) model of molecular shape, which is based on some simple ideas about electrostatic repulsion and the presence or absence of lone pairs, is surprisingly useful. Table 2.2 The basic arrangement of regions of electron density according to the VSEPR model (a) The basic shapes Number of electron regions Arrangement Key points: In the VSEPR model, regions of enhanced electron density take up positions as far apart as possible, and the shape of the molecule is identified by referring to the locations of the atoms in the resulting structure. 2 Linear 3 Trigonal planar 4 Tetrahedral 5 Trigonal bipyramidal 6 Octahedral The primary assumption of the VSEPR model is that regions of enhanced electron density, by which we mean bonding pairs, lone pairs, or the concentrations of electrons associated with multiple bonds, take up positions as far apart as possible so that the repulsions between them are minimized. For instance, four such regions of electron density will lie at the corners of a regular tetrahedron, five will lie at the corners of a trigonal bipyramid, and so on (Table 2.2). Although the arrangement of regions of electron density, both bonding regions and regions associated with lone pairs, governs the shape of the molecule, the name of the shape is determined by the arrangement of atoms, not the arrangement of the regions of electron density (Table 2.3). For instance, the NH3 molecule has four electron pairs that are disposed tetrahedrally, but as one of them is a lone pair the molecule itself is classified as trigonal pyramidal. One apex of the pyramid is occupied by the lone pair. Similarly, H2O has a tetrahedral arrangement of its electron pairs but, as two of the pairs are lone pairs, the molecule is classified as angular (or ‘bent’). To apply the VSEPR model systematically, we first write down the Lewis structure for the molecule or ion and identify the central atom. Next, we count the number of atoms and lone pairs carried by that atom because each atom (whether it is singly or multiply bonded to the central atom) and each lone pair counts as one region of high electron density. To achieve lowest energy, these regions take up positions as far apart as possible, so we identify the basic shape they adopt by referring to Table 2.2. Finally, we note which locations correspond to atoms and identify the shape of the molecule from Table 2.3. Thus, a PCl5 molecule, with five single bonds and therefore five regions of electron density around the central atom, is predicted (and found) to be trigonal bipyramidal (7). Cl P 7 PCl5 F B E X A M PL E 2 . 2 Using the VSEPR model to predict shapes 8 BF3 Predict the shape of (a) a BF3 molecule, (b) an SO32 ion, and (c) a PCl4 ion. Answer We begin by drawing the Lewis structure of each species and then consider the number of bonding and lone pairs of electrons and how they are arranged around the central atom. (a) The Lewis structure of BF3 is shown in (5).To the central B atom there are attached three F atoms but no lone pairs. The basic arrangement of three regions of electron density is trigonal planar. Because each location carries an F atom, the shape of the molecule is also trigonal planar (8). (b) Two Lewis structures for SO32 are shown in (9): they are representative of a variety of structures that contribute to the overall resonance structure. In each case there are three atoms attached to the central S atom and one lone pair, corresponding to four regions of electron density. The basic arrangement of these regions is tetrahedral. Three of the locations correspond to atoms, so the shape of the ion is trigonal pyramidal (10). Note that the shape deduced in this way is independent of which resonance structure is being considered. (c) Phosphorus has five valence electrons. Four of these electrons are used to form bonds to the four Cl atoms. One electron is removed to give the 1 charge on the ion, so all the electrons supplied by the P atom are used in bonding and there is no lone pair. Four regions adopt a tetrahedral arrangement and, as each one is associated with a Cl atom, the ion is tetrahedral (11). Self-test 2.2 Predict the shape of (a) an H2S molecule, (b) an XeO4 molecule. The VSEPR model is highly successful, but sometimes runs into difficulty when there is more than one basic shape of similar energy. For example, with five electron-dense regions around the central atom, a square-pyramidal arrangement is only slightly higher in energy than a trigonal-bipyramidal arrangement, and there are several examples of the former (12). Similarly, the basic shapes for seven electron-dense regions are less readily predicted than others, partly because so many different conformations correspond to similar energies. However, in the p block, seven-coordination is dominated by pentagonal-bipyramidal structures. For example, IF7 is pentagonal bipyramidal and XeF5, with five bonds and two O S O 2– O O S O O 9 SO32– – 2 O S 10 SO2– 3 Cl P 11 PCl4+ 2– 38 2 Molecular structure and bonding Table 2.3 The description of molecular shapes Shape Examples Linear HCN, CO2 Angular (bent) H2O, O3, NO2− Trigonal planar BF3, SO3, NO3− , CO2− 3 Trigonal pyramidal NH3, SO2− 3 Tetrahedral CH4, SO2− 4 Square planar XeF4 Square pyramidal Sb(Ph)5 Trigonal bipyramidal PCl5(g), SOF4 Octahedral SF6, lC P −, 6 lO(OH)5 Approximate shape. Cl 2– In lone pairs, is pentagonal planar. Lone pairs are stereochemically less influential when they belong to heavy p-block elements. The SeF62 and TeCl62 ions, for instance, are octahedral despite the presence of a lone pair on the Se and Te atoms. Lone pairs that do not influence the molecular geometry are said to be stereochemically inert and are usually in the non-directional s orbitals. (b) Modifications of the basic shapes 12 [InCl5]2– Key point: Lone pairs repel other pairs more strongly than bonding pairs do. Once the basic shape of a molecule has been identified, adjustments are made by taking into account the differences in electrostatic repulsion between bonding regions and lone pairs. These repulsions are assumed to lie in the order lone pair/lone pair lone pair/bonding region bonding region/bonding region In elementary accounts, the greater repelling effect of a lone pair is explained by supposing that the lone pair is on average closer to the nucleus than a bonding pair and therefore repels other electron pairs more strongly. However, the true origin of the difference is obscure. An additional detail about this order of repulsions is that, given the choice between an axial and an equatorial site for a lone pair in a trigonal-bipyramidal array, the lone pair occupies the equatorial site. Whereas in the equatorial site the lone pair is repelled by the two bonding pairs at 90° (Fig. 2.1), in the axial position the lone pair is repelled by three bonding pairs at 90°. In an octahedral basic shape, a single lone pair can occupy any position but a second lone pair will occupy the position directly trans (opposite) to the first, which results in a square-planar structure. In a molecule with two adjacent bonding pairs and one or more lone pairs, the bond angle is decreased relative to that expected when all pairs are bonding. Thus, the HNH angle in NH3 is reduced from the tetrahedral angle (109.5°) of the underlying basic shape to a smaller value. This decrease is consistent with the observed HNH angle of 107°. Similarly, the HOH angle in H2O is decreased from the tetrahedral value as the two lone pairs move apart. This decrease is in agreement with the observed HOH bond angle of 104.5°. A deficiency of the VSEPR model, however, is that it cannot be used to predict the actual bond angle adopted by the molecule.2 ) (a (b ) Fig. 2.1 In the VSEPR model a lone pair in (a) the equatorial position of a trigonalbipyramidal arrangement interacts strongly with two bonding pairs, but in (b) an axial position it interacts strongly with three bonding pairs. The former arrangement is generally lower in energy. E X A M PL E 2 . 3 Accounting for the effect of lone pairs on molecular shape Predict the shape of an SF4 molecule. Answer We begin by drawing the Lewis structure of the molecule and identify the number of bonding and lone pairs of electrons; then we identify the shape of the molecule and finally consider any modifications 2 There are also problems with hydrides and fluorides. See Further reading. Valence bond theory 39 F due to the presence of lone pairs. The Lewis structure of SF4 is shown in (13). The central S atom has four F atoms attached to it and one lone pair. The basic shape adopted by these five regions is trigonal bipyramidal. The potential energy is least if the lone pair occupies an equatorial site to give a molecular shape that resembles a see-saw, with the axial bonds forming the ‘plank’ of the see-saw and the equatorial bonds the ‘pivot’. The SF bonds then bend away from the lone pair (14). F S F F 13 SF4 Self-test 2.3 Predict the shape of an XeF2 molecule. F S Valence bond theory The valence bond theory (VB theory) of bonding was the first quantum mechanical theory of bonding to be developed. Valence bond theory considers the interaction of atomic orbitals on separate atoms as they are brought together to form a molecule. Although the computational techniques involved have been largely superseded by molecular orbital theory, much of the language and some of the concepts of VB theory still remain and are used throughout chemistry. 14 SF4 2.4 The hydrogen molecule Key points: In valence bond theory, the wavefunction of an electron pair is formed by superimposing the wavefunctions for the separated fragments of the molecule; a molecular potential energy curve shows the variation of the molecular energy with internuclear separation. The two-electron wavefunction for two widely separated H atoms is A(1)B(2), where A and B are H1s orbitals on atoms A and B. (Although , chi, is also used for electronegativity, the context makes it unlikely that the two usages will be confused: is commonly used to denote an atomic orbital in computational chemistry.) When the atoms are close, it is not possible to know whether it is electron 1 that is on A or electron 2. An equally valid description is therefore A(2)B(1), in which electron 2 is on A and electron 1 is on B. When two outcomes are equally probable, quantum mechanics instructs us to describe the true state of the system as a superposition of the wavefunctions for each possibility, so a better description of the molecule than either wavefunction alone is the linear combination of the two possibilities. A(1)B(2) A(2)B(1) (2.1) This function is the (unnormalized) VB wavefunction for an HH bond. The formation of the bond can be pictured as being due to the high probability that the two electrons will be found between the two nuclei and hence will bind them together (Fig. 2.2). More formally, the wave pattern represented by the term A(1)B(2) interferes constructively with the wave pattern represented by the contribution A(2)B(1) and there is an enhancement in the amplitude of the wavefunction in the internuclear region. For technical reasons stemming from the Pauli principle, only electrons with paired spins can be described by a wavefunction of the type written in eqn 2.1, so only paired electrons can contribute to a bond in VB theory. We say, therefore, that a VB wavefunction is formed by spin pairing of the electrons in the two contributing atomic orbitals. The electron distribution described by the wavefunction in eqn 2.1 is called a bond. As shown in Fig. 2.2, a bond has cylindrical symmetry around the internuclear axis, and the electrons in it have zero orbital angular momentum about that axis. The molecular potential energy curve for H2, a graph showing the variation of the energy of the molecule with internuclear separation, is calculated by changing the internuclear separation R and evaluating the energy at each selected separation (Fig. 2.3). The energy is found to fall below that of two separated H atoms as the two atoms are brought within bonding distance and each electron becomes free to migrate to the other atom. However, the resulting lowering of energy is counteracted by an increase in energy from the Coulombic (electrostatic) repulsion between the two positively charged nuclei. This positive contribution to the energy becomes large as R becomes small. Consequently, the total potential energy curve passes through a minimum and then climbs (a) (b) Fig. 2.2 The formation of a bond from (a) s orbital overlap, (b) p orbital overlap. A bond has cylindrical symmetry around the internuclear axis. 40 Internuclear Re separation Energy 0 2 Molecular structure and bonding to a strongly positive value at small internuclear separations. The depth of the minimum of the curve is denoted De. The deeper the minimum, the more strongly the atoms are bonded together. The steepness of the well shows how rapidly the energy of the molecule rises as the bond is stretched or compressed. The steepness of the curve, an indication of the stiffness of the bond, therefore governs the vibrational frequency of the molecule (Section 8.4). 2.5 Homonuclear diatomic molecules Key point: Electrons in atomic orbitals of the same symmetry but on neighbouring atoms are paired to form and bonds. De Fig. 2.3 A molecular potential energy curve showing how the total energy of a molecule varies as the internuclear separation is changed. Fig. 2.4 The formation of a bond. A similar description can be applied to more complex molecules, and we begin by considering homonuclear diatomic molecules, diatomic molecules in which both atoms belong to the same element (dinitrogen, N2, is an example). To construct the VB description of N2, we consider the valence electron configuration of each atom, which from Section 1.8 we know to be 2s22p1z 2p1y 2px1. It is conventional to take the z-axis to be the internuclear axis, so we can imagine each atom as having a 2pz orbital pointing towards a 2pz orbital on the other atom, with the 2px and 2py orbitals perpendicular to the axis. A bond is then formed by spin pairing between the two electrons in the opposing 2pz orbitals. Its spatial wavefunction is still given by eqn 2.1, but now A and B stand for the two 2pz orbitals. A simple way of identifying a bond is to envisage rotation of the bond around the internuclear axis: if the wavefunction remains unchanged, the bond is classified as . The remaining 2p orbitals cannot merge to give bonds as they do not have cylindrical symmetry around the internuclear axis. Instead, the orbitals merge to form two bonds. A bond arises from the spin pairing of electrons in two p orbitals that approach side by side (Fig. 2.4). The bond is so-called because, viewed along the internuclear axis, it resembles a pair of electrons in a p orbital. More precisely, an electron in a bond has one unit of orbital angular momentum about the internuclear axis. A simple way of identifying a bond is to envisage rotation of the bond through 180° around the internuclear axis. If the signs (as indicated by the shading) of the lobes of the orbital are interchanged, then the bond is classified as . There are two bonds in N2, one formed by spin pairing in two neighbouring 2px orbitals and the other by spin pairing in two neighbouring 2py orbitals. The overall bonding pattern in N2 is therefore a bond plus two bonds (Fig. 2.5), which is consistent with the structure N⬅N. Analysis of the total electron density in a triple bond shows that it has cylindrical symmetry around the internuclear axis, with the four electrons in the two bonds forming a ring of electron density around the central bond. 2.6 Polyatomic molecules Fig. 2.5 The VB description of N2. Two electrons form a bond and another two pairs form two bonds. In linear molecules, where the x- and y-axes are not specified, the electron density of bonds is cylindrically symmetrical around the internuclear axis. Key points: Each bond in a polyatomic molecule is formed by the spin pairing of electrons in any neighbouring atomic orbitals with cylindrical symmetry about the relevant internuclear axis; bonds are formed by pairing electrons that occupy neighbouring atomic orbitals of the appropriate symmetry. To introduce polyatomic molecules we consider the VB description of H2O. The valence electron configuration of a hydrogen atom is 1s1 and that of an O atom is 2s22p2z 2p1y 2px1. The two unpaired electrons in the O2p orbitals can each pair with an electron in an H1s orbital, and each combination results in the formation of a bond (each bond has cylindrical symmetry about the respective OH internuclear axis). Because the 2py and 2pz orbitals lie at 90° to each other, the two bonds also lie at 90° to each other (Fig. 2.6). We can predict, therefore, that H2O should be an angular molecule, which it is. However, the theory predicts a bond angle of 90º whereas the actual bond angle is 104.5°. Similarly, to predict the structure of an ammonia molecule, NH3, we start by noting that the valence electron configuration of an N atom given previously suggests that three H atoms can form bonds by spin pairing with the electrons in the three half-filled 2p orbitals. The latter are perpendicular to each other, so we predict a trigonal-pyramidal molecule with a bond angle of 90°. An NH3 molecule is indeed trigonal pyramidal, but the experimental bond angle is 107º. Another deficiency of the VB theory presented so far is its inability to account for the tetravalence of carbon, its ability to form four bonds. The ground-state configuration of Valence bond theory C is 2s22p1z 2p1y , which suggests that a C atom should be capable of forming only two bonds, not four. Clearly, something is missing from the VB approach. These two deficiencies—the failure to account for bond angles and the valence of carbon—are overcome by introducing two new features, promotion and hybridization. 41 O (a) Promotion Key point: Promotion of electrons may occur if the outcome is to achieve more or stronger bonds and a lower overall energy. Promotion is the excitation of an electron to an orbital of higher energy in the course of bond formation. Although electron promotion requires an investment of energy, that investment is worthwhile if the energy can be more than recovered from the greater strength or number of bonds that it allows to be formed. Promotion is not a ‘real’ process in which an atom somehow becomes excited and then forms bonds: it is a contribution to the overall energy change that occurs when bonds form. In carbon, for example, the promotion of a 2s electron to a 2p orbital can be thought of as leading to the configuration 2s12p1z 2p1y 2px1, with four unpaired electrons in separate orbitals. These electrons may pair with four electrons in orbitals provided by four other atoms, such as four H1s orbitals if the molecule is CH4, and hence form four bonds. Although energy was required to promote the electron, it is more than recovered by the atom’s ability to form four bonds in place of the two bonds of the unpromoted atom. Promotion, and the formation of four bonds, is a characteristic feature of carbon and of its congeners in Group 14 (Chapter 14) because the promotion energy is quite small: the promoted electron leaves a doubly occupied ns orbital and enters a vacant np orbital, hence significantly relieving the electron–electron repulsion it experiences in the ground state. This promotion on an electron becomes energetically less favourable as the group is descended and divalent compounds are common for tin and lead (Section 9.5). H H Fig. 2.6 The VB description of H2O. There are two bonds formed by pairing electrons in O2p and H1s orbitals. This model predicts a bond angle of 90°. (b) Hypervalence Key point: Hypervalence and octet expansion occur for elements following Period 2. The elements of Period 2, Li through Ne, obey the octet rule quite well, but elements of later periods show deviations from it. For example, the bonding in PCl5 requires the P atom to have 10 electrons in its valence shell, one pair for each PCl bond (15). Similarly, in SF6 the S atom must have 12 electrons if each F atom is to be bound to the central S atom by an electron pair (16). Species of this kind, which in terms of Lewis structures demand the presence of more than an octet of electrons around at least one atom, are called hypervalent. The traditional explanation of hypervalence invokes the availability of low-lying unfilled d orbitals, which can accommodate the additional electrons. According to this explanation, a P atom can accommodate more than eight electrons if it uses its vacant 3d orbitals. In PCl5, with its five pairs of bonding electrons, at least one 3d orbital must be used in addition to the four 3s and 3p orbitals of the valence shell. The rarity of hypervalence in Period 2 is then ascribed to the absence of 2d orbitals. However, the real reason for the rarity of hypervalence in Period 2 may be the geometrical difficulty of packing more than four atoms around a small central atom and may in fact have little to do with the availability of d orbitals. The molecular orbital theory of bonding, which is described later in this chapter, describes the bonding in hypervalent compounds without invoking participation of d orbitals. (c) Hybridization Key points: Hybrid orbitals are formed when atomic orbitals on the same atom interfere; specific hybridization schemes correspond to each local molecular geometry. The description of the bonding in AB4 molecules of Group 14 is still incomplete because it appears to imply the presence of three bonds of one type (formed from B and A2p orbitals) and a fourth bond of a distinctly different character (formed from B and A2s), whereas all the experimental evidence (bond lengths and strengths) points to the equivalence of all four AB bonds, as in CH4, for example. This problem is overcome by realizing that the electron density distribution in the promoted atom is equivalent to the electron density in which each electron occupies a hybrid Cl Cl Cl P Cl Cl 15 PCl5 F F S F F F F 16 SF6 42 2 Molecular structure and bonding orbital formed by interference, or ‘mixing’, between the A2s and the A2p orbitals. The origin of the hybridization can be appreciated by thinking of the four atomic orbitals, which are waves centred on a nucleus, as being like ripples spreading from a single point on the surface of a lake: the waves interfere destructively and constructively in different regions, and give rise to four new shapes. The specific linear combinations that give rise to four equivalent hybrid orbitals are h1 s px py pz h2 s px py pz (2.2) h3 s px py pz h4 s px py pz Fig. 2.7 One of the four equivalent sp3 hybrid orbitals. Each one points towards a different vertex of a regular tetrahedron. As a result of the interference between the component orbitals, each hybrid orbital consists of a large lobe pointing in the direction of one corner of a regular tetrahedron and a smaller lobe pointing in the opposite direction (Fig. 2.7). The angle between the axes of the hybrid orbitals is the tetrahedral angle, 109.47°. Because each hybrid is built from one s orbital and three p orbitals, it is called an sp3 hybrid orbital. It is now easy to see how the VB description of a CH4 molecule is consistent with a tetrahedral shape with four equivalent CH bonds. Each hybrid orbital of the promoted carbon atom contains a single unpaired electron; an electron in H1s can pair with each one, giving rise to a bond pointing in a tetrahedral direction. Because each sp3 hybrid orbital has the same composition, all four bonds are identical apart from their orientation in space. A further feature of hybridization is that a hybrid orbital has pronounced directional character, in the sense that it has enhanced amplitude in the internuclear region. This directional character arises from the constructive interference between the s orbital and the positive lobes of the p orbitals. As a result of the enhanced amplitude in the internuclear region, the bond strength is greater than for an s or p orbital alone. This increased bond strength is another factor that helps to repay the promotion energy. Hybrid orbitals of different compositions are used to match different molecular geometries and to provide a basis for their VB description. For example, sp2 hybridization is used to reproduce the electron distribution needed for trigonal-planar species, such as on B in BF3 and N in NO3, and sp hybridization reproduces a linear distribution. Table 2.4 gives the hybrids needed to match the geometries of a variety of electron distributions. Table 2.4 Some hybridization schemes Coordination number Arrangement 2 Linear sp, pd, sd Angular sd Trigonal planar sp2, p2d Unsymmetrical planar spd Trigonal pyramidal pd2 Tetrahedral sp3, sd3 Irregular tetrahedral spd2, p3d, pd3 Square planar p2d2, sp2d Trigonal bipyramidal sp3d, spd3 Tetragonal pyramidal sp2d2, sd4, pd4, p3d2 Pentagonal planar p2d3 Octahedral sp3d2 Trigonal prismatic spd4, pd5 Trigonal antiprismatic p3d3 3 4 5 6 Composition Molecular orbital theory We have seen that VB theory provides a reasonable description of bonding in simple molecules. However, it does not handle polyatomic molecules very elegantly. Molecular orbital theory (MO theory) is a more sophisticated model of bonding that can be applied equally successfully to simple and complex molecules. In MO theory, we generalize the atomic Molecular orbital theory orbital description of atoms in a very natural way to a molecular orbital description of molecules in which electrons spread over all the atoms in a molecule and bind them all together. In the spirit of this chapter, we continue to treat the concepts qualitatively and to give a sense of how inorganic chemists discuss the electronic structures of molecules by using MO theory. Almost all qualitative discussions and calculations on inorganic molecules and ions are now carried out within the framework of MO theory. 2.7 An introduction to the theory We begin by considering homonuclear diatomic molecules and diatomic ions formed by two atoms of the same element. The concepts these species introduce are readily extended to heteronuclear diatomic molecules formed between two atoms or ions of different elements. They are also easily extended to polyatomic molecules and solids composed of huge numbers of atoms and ions. In parts of this section we shall include molecular fragments in the discussion, such as the SF diatomic group in the SF6 molecule or the OO diatomic group in H2O2 as similar concepts also apply to pairs of atoms bound together as parts of larger molecules. (a) The approximations of the theory Key points: Molecular orbitals are constructed as linear combinations of atomic orbitals; there is a high probability of finding electrons in atomic orbitals that have large coefficients in the linear combination; each molecular orbital can be occupied by up to two electrons. As in the description of the electronic structures of atoms, we set out by making the orbital approximation, in which we assume that the wavefunction, , of the Ne electrons in the molecule can be written as a product of one-electron wavefunctions: (1)(2) ... (Ne). The interpretation of this expression is that electron 1 is described by the wavefunction (1), electron 2 by the wavefunction (2), and so on. These one-electron wavefunctions are the molecular orbitals of the theory. As for atoms, the square of a one-electron wavefunction gives the probability distribution for that electron in the molecule: an electron in a molecular orbital is likely to be found where the orbital has a large amplitude, and will not be found at all at any of its nodes. The next approximation is motivated by noting that, when an electron is close to the nucleus of one atom, its wavefunction closely resembles an atomic orbital of that atom. For instance, when an electron is close to the nucleus of an H atom in a molecule, its wavefunction is like a 1s orbital of that atom. Therefore, we may suspect that we can construct a reasonable first approximation to the molecular orbital by superimposing atomic orbitals contributed by each atom. This modelling of a molecular orbital in terms of contributing atomic orbitals is called the linear combination of atomic orbitals (LCAO) approximation. A ‘linear combination’ is a sum with various weighting coefficients. In simple terms, we combine the atomic orbitals of contributing atoms to give molecular orbitals that extend over the entire molecule. In the most elementary form of MO theory, only the valence shell atomic orbitals are used to form molecular orbitals. Thus, the molecular orbitals of H2 are approximated by using two hydrogen 1s orbitals, one from each atom: cAA cBB (2.3) In this case the basis set, the atomic orbitals from which the molecular orbital is built, consists of two H1s orbitals, one on atom A and the other on atom B. The principle is exactly the same for more complex molecules. For example, the basis set for the methane molecule consists of the 2s and 2p orbitals on carbon and four 1s orbitals on the hydrogen atoms. The coefficients c in the linear combination show the extent to which each atomic orbital contributes to the molecular orbital: the greater the value of c, the greater the contribution of that atomic orbital to the molecular orbital. To interpret the coefficients in eqn 2.3 we note that cA2 is the probability that the electron will be found in the orbital A and cB2 is the probability that the electron will be found in the orbital B. The fact that both atomic orbitals contribute to the molecular orbital implies that there is interference between them where their amplitudes are nonzero, with the probability distribution being given by 2 cA2A2 2cAcBAB cB2B2 (2.4) 43 44 2 Molecular structure and bonding The term 2cAcBAB represents the contribution to the probability density arising from this interference. Because H2 is a homonuclear diatomic molecule, its electrons are equally likely to be found near each nucleus, so the linear combination that gives the lowest energy will have equal contributions from each 1s orbital (cA2 cB2), leaving open the possibility that cA cB or cA cB. Thus, ignoring normalization, the two molecular orbitals are Enhanced density ± A B Fig. 2.8 The enhancement of electron density in the internuclear region arising from the constructive interference between the atomic orbitals on neighbouring atoms. Node (2.5) The relative signs of coefficients in LCAOs play a very important role in determining the energies of the orbitals. As we shall see, they determine whether atomic orbitals interfere constructively or destructively where they spread into the same region and hence lead to an accumulation or a reduction of electron density in those regions. Two more preliminary points should be noted. We see from this discussion that two molecular orbitals may be constructed from two atomic orbitals. In due course, we shall see the importance of the general point that N molecular orbitals can be constructed from a basis set of N atomic orbitals. For example, if we use all four valence orbitals on each O atom in O2, then from the total of eight atomic orbitals we can construct eight molecular orbitals. In addition, as in atoms, the Pauli exclusion principle implies that each molecular orbital may be occupied by up to two electrons; if two electrons are present, then their spins must be paired. Thus, in a diatomic molecule constructed from two Period 2 atoms and in which there are eight molecular orbitals available for occupation, up to 16 electrons may be accommodated before all the molecular orbitals are full. The same rules that are used for filling atomic orbitals with electrons (the building-up principle and Hund’s rule, Section 1.8) apply to filling molecular orbitals with electrons. The general pattern of the energies of molecular orbitals formed from N atomic orbitals is that one molecular orbital lies below that of the parent atomic energy levels, one lies higher in energy than they do, and the remainder are distributed between these two extremes. (b) Bonding and antibonding orbitals Key points: A bonding orbital arises from the constructive interference of neighbouring atomic orbitals; an antibonding orbital arises from their destructive interference, as indicated by a node between the atoms. Fig. 2.9 The destructive interference that arises if the overlapping orbitals have opposite signs. This interference leads to a nodal surface in an antibonding molecular orbital. Energy – A B Fig. 2.10 The molecular orbital energy level diagram for H2 and analogous molecules. The orbital is an example of a bonding orbital. It is so-called because the energy of the molecule is lowered relative to that of the separated atoms if this orbital is occupied by electrons. The bonding character of is ascribed to the constructive interference between the two atomic orbitals and the resulting enhanced amplitude between the two nuclei (Fig. 2.8). An electron that occupies has an enhanced probability of being found in the internuclear region and can interact strongly with both nuclei. Hence orbital overlap, the spreading of one orbital into the region occupied by another, leading to enhanced probability of electrons being found in the internuclear region, is taken to be the origin of the strength of bonds. The orbital is an example of an antibonding orbital. It is so-called because, if it is occupied, the energy of the molecule is higher than for the two separated atoms. The greater energy of an electron in this orbital arises from the destructive interference between the two atomic orbitals, which cancels their amplitudes and gives rise to a nodal plane between the two nuclei (Fig. 2.9). Electrons that occupy are largely excluded from the internuclear region and are forced to occupy energetically less favourable locations. It is generally true that the energy of a molecular orbital in a polyatomic molecule is higher the more internuclear nodes it has. The increase in energy reflects an increasingly complete exclusion of electrons from the regions between nuclei. Note that an antibonding orbital is slightly more antibonding than its partner bonding orbital is bonding: the asymmetry arises partly from the details of the electron distribution and partly from the fact that internuclear repulsion pushes the entire energy level diagram upwards. The energies of the two molecular orbitals in H2 are depicted in Fig. 2.10, which is an example of a molecular orbital energy level diagram, a diagram depicting the relative energies of molecular orbitals. The two electrons occupy the lower energy molecular orbital. An indication of the size of the energy gap between the two molecular orbitals is the observation of a spectroscopic absorption in H2 at 11.4 eV (in the ultraviolet at 109 nm), which can be ascribed to the transition of an electron from the bonding orbital 45 Molecular orbital theory N2+ Energy to the antibonding orbital. The dissociation energy of H2 is 4.5 eV (434 kJ mol1), which gives an indication of the location of the bonding orbital relative to the separated atoms. The Pauli exclusion principle limits to two the number of electrons that can occupy any molecular orbital and requires that those two electrons be paired (↑↓). The exclusion principle is the origin of the importance of the pairing of the electrons in bond formation in MO theory just as it is in VB theory: in the context of MO theory, two is the maximum number of electrons that can occupy an orbital that contributes to the stability of the molecule. The H2 molecule, for example, has a lower energy than that of the separated atoms because two electrons can occupy the orbital and both can contribute to the lowering of its energy (as shown in Fig. 2.10). A weaker bond can be expected if only one electron is present in a bonding orbital, but nevertheless H2 is known as a transient gas-phase ion; its dissociation energy is 2.6 eV (250.9 kJ mol1). Three electrons (as in H2) are less effective than two electrons because the third electron must occupy the antibonding orbital and hence destabilize the molecule. With four electrons, the antibonding effect of two electrons in overcomes the bonding effect of two electrons in . There is then no net bonding. It follows that a four-electron molecule with only 1s orbitals available for bond formation, such as He2, is not expected to be stable relative to dissociation into its atoms. So far, we have discussed interactions of atomic orbitals that give rise to molecular orbitals that are lower in energy (bonding) and higher in energy (antibonding) than the separated atoms. In addition, it is possible to generate a molecular orbital that has the same energy as the initial atomic orbitals. In this case, occupation of this orbital neither stabilizes nor destabilizes the molecule and so it is described as a nonbonding orbital. Typically, a nonbonding orbital is a molecular orbital that consists of a single orbital on one atom, perhaps because there is no atomic orbital of the correct symmetry for it to overlap on a neighbouring atom. N2 2σg I = 15.6 eV 1πu I = 16.7 eV 1σu I = 18.8 eV Fig. 2.11 The UV photoelectron spectrum of N2. The fine structure in the spectrum arises from excitation of vibrations in the cation formed by photoejection of an electron. 2σu 1πg 2.8 Homonuclear diatomic molecules 2p Energy Although the structures of diatomic molecules can be calculated effortlessly by using commercial software packages, the validity of any such calculations must, at some point, be confirmed by experimental data. Moreover, elucidation of molecular structure can often be achieved by drawing on experimental information. One of the most direct portrayals of electronic structure is obtained from ultraviolet photoelectron spectroscopy (UPS, Section 8.8) in which electrons are ejected from the orbitals they occupy in molecules and their energies determined. Because the peaks in a photoelectron spectrum correspond to the various kinetic energies of photoelectrons ejected from different orbitals of the molecule, the spectrum gives a vivid portrayal of the molecular orbital energy levels of a molecule (Fig. 2.11). 2p 1πu 2σg 1σu 2s 2s 1σg (a) The orbitals Key points: Molecular orbitals are classified as , , or according to their rotational symmetry about the internuclear axis, and (in centrosymmetric species) as g or u according to their symmetry with respect to inversion. Our task is to see how MO theory can account for the features revealed by photoelectron spectroscopy and the other techniques, principally absorption spectroscopy, that are used to study diatomic molecules. We are concerned predominantly with outer-shell valence orbitals, rather than core orbitals. As with H2, the starting point in the theoretical discussion is the minimal basis set, the smallest set of atomic orbitals from which useful molecular orbitals can be built. In Period 2 diatomic molecules, the minimal basis set consists of the one valence s orbital and three valence p orbitals on each atom, giving eight atomic orbitals in all. We shall now see how the minimal basis set of eight valence shell atomic orbitals (four from each atom, one s and three p) is used to construct eight molecular orbitals. Then we shall use the Pauli principle to predict the ground-state electron configurations of the molecules. The energies of the atomic orbitals that form the basis set are shown on either side of the molecular orbital diagram in Fig. 2.12. We form orbitals by allowing overlap between atomic orbitals that have cylindrical symmetry around the internuclear axis, which (as remarked earlier) is conventionally labelled z. The notation signifies that the orbital has cylindrical symmetry; atomic orbitals that can form orbitals include the 2s and 2pz orbitals on the two atoms (Fig. 2.13). From these four orbitals (the 2s and the 2pz orbitals Fig. 2.12 The molecular orbital energy level diagram for the later Period 2 homonuclear diatomic molecules. This diagram should be used for O2 and F2. s s p s p p Fig. 2.13 A orbital can be formed in several ways, including s,s overlap, s,p overlap, and p,p overlap, with the p orbitals directed along the internuclear axis. 46 2 Molecular structure and bonding Nodal plane Fig. 2.14 Two p orbitals can overlap to form a orbital. The orbital has a nodal plane passing through the internuclear axis, shown here from the side. σg (a) From a basis set of four atomic orbitals on each atom, eight molecular orbitals are constructed. – Four of these eight molecular orbitals are orbitals and four are orbitals. σu (b) Fig 2.15 (a) Bonding and (b) antibonding interactions with the arrow indicating the inversion. – – πu (a) + on atom A and the corresponding orbitals on atom B) with cylindrical symmetry we can construct four molecular orbitals, two of which arise predominantly from interaction of the 2s orbitals, and two from interaction of the 2pz orbitals. These molecular orbitals are labelled 1g, 1u, 2g, and 2u, respectively. Their energies resemble those shown in Fig. 2.12 but it is difficult to predict the precise locations of the central two orbitals. Interaction between a 2s on one atom and a 2pz orbital on the other atom is possible if their relative energies are similar. The remaining two 2p orbitals on each atom, which have a nodal plane containing the z-axis, overlap to give orbitals (Fig. 2.14). Bonding and antibonding orbitals can be formed from the mutual overlap of the two 2px orbitals, and also from the mutual overlap of the two 2py orbitals. This pattern of overlap gives rise to the two pairs of doubly degenerate energy levels (two energy levels of the same energy) shown in Fig. 2.12 and labelled 1u and 1g. For homonuclear diatomics, it is sometimes convenient (particularly for spectroscopic discussions) to signify the symmetry of the molecular orbitals with respect to their behaviour under inversion through the centre of the molecule. The operation of inversion consists of starting at an arbitrary point in the molecule, travelling in a straight line to the centre of the molecule, and then continuing an equal distance out on the other side of the centre. This procedure is indicated by the arrows in Figs 2.15 and 2.16. The orbital is designated g (for gerade, even) if it is identical under inversion, and u (for ungerade, odd) if it changes sign. Thus, a bonding orbital is g and an antibonding orbital is u (Fig. 2.15). On the other hand, a bonding orbital is u and an antibonding orbital is g (Fig. 2.16). Note that the g orbitals are numbered separately from the u orbitals, and similarly for the orbitals. The procedure can be summarized as follows: – The four orbitals span a range of energies, one being strongly bonding and another strongly antibonding; the remaining two lie between these extremes. 4. The four orbitals form one doubly degenerate pair of bonding orbitals and one doubly degenerate pair of antibonding orbitals. To establish the actual location of the energy levels, it is necessary to use electronic absorption spectroscopy, photoelectron spectroscopy, or detailed computation. Photoelectron spectroscopy and detailed computation (the numerical solution of the Schrödinger equation for the molecules) enable us to build the orbital energy schemes shown in Fig. 2.17. As we see there, from Li2 to N2 the arrangement of orbitals is that shown in Fig. 2.18, whereas for O2 and F2 the order of the and orbitals is reversed and the array is that shown in Fig. 2.12. The reversal of order can be traced to the increasing separation of the 2s and 2p orbitals that occurs on going to the right across Period 2. A general principle of quantum mechanics is that the mixing of wavefunctions is strongest if their energies are similar; mixing is not important if their energies differ by more than about 1 eV. When the s,p energy separation is small, each molecular orbital is a mixture of s and p character on each atom. As the s and p energy separation increases, the molecular orbitals become more purely s-like and p-like. When considering species containing two neighbouring d-block atoms, as in Hg22 and [Cl4ReReCl4]2, we should also allow for the possibility of forming bonds from d orbitals. A dz orbital has cylindrical symmetry with respect to the internuclear (z) axis, and hence can contribute to the orbitals that are formed from s and pz orbitals. The dyz and dzx orbitals both look like p orbitals when viewed along the internuclear axis, and hence can contribute to the orbitals formed from px and py. The new feature is the role of dx y and dxy, which have no counterpart in the orbitals discussed up to now. These two orbitals can overlap with matching orbitals on the other atom to give rise to doubly degenerate pairs of bonding and antibonding orbitals (Fig. 2.19). As we shall see in Chapter 19, orbitals are important for the discussion of bonds between d-metal atoms, in d-metal complexes, and in organometallic compounds. 2 2 (b) – πg Fig 2.16 (a) Bonding and (b) antibonding interactions with the arrow indicating the inversions. 2 47 Molecular orbital theory Li2 Be2 B2 C2 N2 O2 F2 2σu 2σu 1πg 2σu 1πg 2p 2σg 1πu 1σu 2σg Energy Energy 1πg 1πu 1πu 2σg 2p 1σu 2s 2s 1σg 1σu 1σg 1σg Fig. 2.18 The molecular orbital energy level diagram for Period 2 homonuclear diatomic molecules from Li2 to N2. Fig. 2.17 The variation of orbital energies for Period 2 homonuclear diatomic molecules from Li2 to F2. (b) The building-up principle for molecules Key points: The building-up principle is used to predict the ground-state electron configurations by accommodating electrons in the array of molecular orbitals summarized in Fig. 2.12 or Fig. 2.18 and recognizing the constraints of the Pauli principle. We use the building-up principle in conjunction with the molecular orbital energy level diagram in the same way as for atoms. The order of occupation of the orbitals is the order of increasing energy as depicted in Fig. 2.12 or Fig. 2.18. Each orbital can accommodate up to two electrons. If more than one orbital is available for occupation (because they happen to have identical energies, as in the case of pairs of orbitals), then the orbitals are occupied separately. In that case, the electrons in the half-filled orbitals adopt parallel spins (↑↑), just as is required by Hund’s rule for atoms (Section 1.7a). With very few exceptions, these rules lead to the actual ground-state configuration of the Period 2 diatomic molecules. For example, the electron configuration of N2, with 10 valence electrons, is N2: 12g 1u2 1u4 22g Molecular orbital configurations are written like those for atoms: the orbitals are listed in order of increasing energy, and the number of electrons in each one is indicated by a superscript. Note that 4 is shorthand for the occupation of two different orbitals. E X A M PL E 2 . 4 Predicting the electron configurations of diatomic molecules Predict the ground-state electron configurations of the oxygen molecule, O2, the superoxide ion, O2, and the peroxide ion, O22. Answer We need to determine the number of valence electrons and then populate the molecular orbitals with them in accord with the building-up principle. An O2 molecule has 12 valence electrons. The first ten electrons recreate the N2 configuration except for the reversal of the order of the 1u and 2g orbitals (see Fig. 2.17). Next in line for occupation are the doubly degenerate 1g orbitals. The last two electrons enter these orbitals separately and have parallel spins. The configuration is therefore O2: 12g 12u 22g 14u 12g Fig. 2.19 The formation of orbitals by d-orbital overlap. The orbital has two mutually perpendicular nodal planes that intersect along the internuclear axis. 48 2 Molecular structure and bonding The O2 molecule is interesting because the lowest energy configuration has two unpaired electrons in different orbitals. Hence, O2 is paramagnetic (tends to be attracted into a magnetic field). The next two electrons can be accommodated in the 1g orbitals, giving O2: 12g12u22g14u13g O22: 12g12u22g14u14g We are assuming that the orbital order remains that shown in Fig. 2.17; this might not be the case. Self-test 2.4 Write the valence electron configuration for S22 and Cl22. The highest occupied molecular orbital (HOMO) is the molecular orbital that, according to the building-up principle, is occupied last. The lowest unoccupied molecular orbital (LUMO) is the next higher molecular orbital. In Fig. 2.17, the HOMO of F2 is 1g and its LUMO is 2u; for N2 the HOMO is 2g and the LUMO is 1g. We shall increasingly see that these frontier orbitals, the LUMO and the HOMO, play special roles in the interpretation of structural and kinetic studies. The term SOMO, denoting a singly occupied molecular orbital, is sometimes encountered and is of crucial importance for the properties of radical species. 2.9 Heteronuclear diatomic molecules The molecular orbitals of heteronuclear diatomic molecules differ from those of homonuclear diatomic molecules in having unequal contributions from each atomic orbital. Each molecular orbital has the form cAA cBB ... (2.6) The unwritten orbitals include all the other orbitals of the correct symmetry for forming or bonds but which typically make a smaller contribution than the two valence shell orbitals we are considering. In contrast to orbitals for homonuclear species, the coefficients cA and cB are not necessarily equal in magnitude. If cA2 cB2, the orbital is composed principally of A and an electron that occupies the molecular orbital is more likely to be found near atom A than atom B. The opposite is true for a molecular orbital in which cA2 cB2. In heteronuclear diatomic molecules, the more electronegative element makes the larger contribution to bonding orbitals and the less electronegative element makes the greater contribution to the antibonding orbitals. (a) Heteronuclear molecular orbitals = cA’ A cB’ Key points: Heteronuclear diatomic molecules are polar; bonding electrons tend to be found on the more electronegative atom and antibonding electrons on the less electronegative atom. B Energy B A = cA A cB B Fig. 2.20 The molecular orbital energy level diagram arising from interaction of two atomic orbitals with different energies. The lower molecular orbital is primarily composed of the lower energy atomic orbital, and vice versa. The shift in energies of the two levels is less than if the atomic orbitals had the same energy. The greater contribution to a bonding molecular orbital normally comes from the more electronegative atom: the bonding electrons are then likely to be found close to that atom and hence be in an energetically favourable location. The extreme case of a polar covalent bond, a covalent bond formed by an electron pair that is unequally shared by the two atoms, is an ionic bond. In an ionic bond, one atom gains complete control over the electron pair. The less electronegative atom normally contributes more to an antibonding orbital (Fig. 2.20), that is antibonding electrons are more likely to be found in an energetically unfavourable location, close to the less electronegative atom. A second difference between homonuclear and heteronuclear diatomic molecules stems from the energy mismatch in the latter between the two sets of atomic orbitals. We have already remarked that two wavefunctions interact less strongly as their energies diverge. This dependence on energy separation implies that the lowering of energy as a result of the overlap of atomic orbitals on different atoms in a heteronuclear molecule is less pronounced than in a homonuclear molecule, in which the orbitals have the same energies. However, we cannot necessarily conclude that AB bonds are weaker than AA bonds because other factors (including orbital size and closeness of approach) are also important. The heteronuclear CO molecule, for example, which is isoelectronic with its homonuclear counterpart N2, has an even higher bond enthalpy (1070 kJ mol1) than N2 (946 kJ mol1). Molecular orbital theory (b) Hydrogen fluoride 3σ Key points: In hydrogen fluoride the bonding orbital is more concentrated on the F atom and the antibonding orbital is more concentrated on the H atom. Energy 1π Mainly H F2p 2σ Mainly F 1σ F2s Fig. 2.21 The molecular orbital energy level diagram for HF. The relative positions of the atomic orbitals reflect the ionization energies of the atoms. 4σ C2p 2π Energy As an illustration of these general points, consider a simple heteronuclear diatomic molecule, HF. The five valence orbitals available for molecular orbital formation are the 1s orbital of H and the 2s and 2p orbitals of F; there are 1 7 8 valence electrons to accommodate in the five molecular orbitals that can be constructed from the five basis orbitals. The orbitals of HF can be constructed by allowing an H1s orbital to overlap the F2s and F2pz orbitals (z being the internuclear axis). These three atomic orbitals combine to give three molecular orbitals of the form c1H1s c2F2s c3F2p. This procedure leaves the F2px and F2py orbitals unaffected as they have symmetry and there is no valence H orbital of that symmetry. These orbitals are therefore examples of the nonbonding orbitals mentioned earlier, and are molecular orbitals confined to a single atom. Note that, because there is no centre of inversion in a heteronuclear diatomic molecule, we do not use the g,u classification for its molecular orbitals. Figure 2.21 shows the resulting energy level diagram. The 1 bonding orbital is predominantly F2s in character as the energy difference between it and the H1s orbital is large. It is, therefore, confined mainly to the F atom and essentially nonbonding. The 2 orbital is more bonding than the 1 orbital and has both H1s and F2p character. The 3 orbital is antibonding, and principally H1s in character: the 1s orbital has a relatively high energy (compared with the fluorine orbitals) and hence contributes predominantly to the highenergy antibonding molecular orbital. Two of the eight valence electrons enter the 2 orbital, forming a bond between the two atoms. Six more enter the 1 and 1 orbitals; these two orbitals are largely nonbonding and confined mainly to the F atom. This is consistent with the conventional model of three lone pairs on the fluorine atom. All the electrons are now accommodated, so the configuration of the molecule is 122214. One important feature to note is that all the electrons occupy orbitals that are predominantly on the F atom. It follows that we can expect the HF molecule to be polar, with a partial negative charge on the F atom, which is found experimentally. H1s 49 3σ 1π C2s O2p 2σ (c) Carbon monoxide O2s Key points: The HOMO of a carbon monoxide molecule is an almost nonbonding orbital largely localized on C; the LUMO is an antibonding orbital. 1σ The molecular orbital energy level diagram for carbon monoxide is a somewhat more complicated example than HF because both atoms have 2s and 2p orbitals that can participate in the formation of and orbitals. The energy level diagram is shown in Fig. 2.22. The ground-state configuration is Fig. 2.22 The molecular orbital energy level diagram for CO. CO: 12221432 4σ 2π Energy The 1 orbital is localized mostly on the O atom and essentially nonbonding. The 2 orbital is bonding. The 1 orbitals constitute the doubly degenerate pair of bonding orbitals, with mainly C2p orbital character. The HOMO in CO is 3, which is predominantly C2pz in character, largely nonbonding, and located on the C atom. The LUMO is the doubly degenerate pair of antibonding orbitals, with mainly C2p orbital character (Fig. 2.23). This combination of frontier orbitals—a full orbital largely localized on C and a pair of empty orbitals—is one reason why metal carbonyls are such a characteristic feature of the d metals: in d-metal carbonyls, the HOMO lone pair orbital of CO participates in the formation of a bond and the LUMO antibonding orbital participates in the formation of bonds to the metal atom (Chapter 22). Although the difference in electronegativity between C and O is large, the experimental value of the electric dipole moment of the CO molecule (0.1 D) is small. Moreover, the negative end of the dipole is on the C atom despite that being the less electronegative atom. This odd situation stems from the fact that the lone pairs and bonding pairs have a complex distribution. It is wrong to conclude that, because the bonding electrons are mainly on the O atom, O is the negative end of the dipole, as this ignores the balancing effect of the lone pair on the C atom. The inference of polarity from electronegativity is particularly unreliable when antibonding orbitals are occupied. 3σ 1π 2σ 1σ Fig. 2.23 A schematic illustration of the molecular orbitals of CO, with the size of the atomic orbital indicating the magnitude of its contribution to the molecular orbital. 50 2 Molecular structure and bonding ICl Energy I5p E X A M PL E 2 . 5 Accounting for the structure of a heteronuclear diatomic molecule 4σ 2π The halogens form compounds among themselves. One of these ‘interhalogen’ compounds is iodine monochloride, ICl, in which the order of orbitals is 1, 2, 3, 1, 2, 4 (from calculation). What is the ground-state electron configuration of ICl? 3σ I5s Cl3p 1π 2σ Answer First, we identify the atomic orbitals that are to be used to construct molecular orbitals: these are the Cl3s and Cl3p valence shell orbitals of Cl and the I5s and I5p valence shell orbitals of I. As for Period 2 elements, an array of and orbitals can be constructed, and is shown in Fig. 2.24. The bonding orbitals are predominantly Cl in character (because that is the more electronegative element) and the antibonding orbitals are predominantly I in character. There are 7 7 14 valence electrons to accommodate, which results in the ground-state electron configuration 1222143224. Self-test 2.5 Predict the ground-state electron configuration of the hypochlorite ion, ClO. 1σ Cl3s Fig. 2.24 A schematic illustration of the energies of the molecular orbitals of ICl. 2.10 Bond properties We have already seen the origin of the importance of the electron pair: two electrons is the maximum number that can occupy a bonding orbital and hence contribute to a chemical bond. We now extend this concept by introducing the concept of ‘bond order’. (a) Bond order Key points: The bond order assesses the net number of bonds between two atoms in the molecular orbital formalism; the greater the bond order between a given pair of atoms, the greater the bond strength. The bond order, b, identifies a shared electron pair as counting as a ‘bond’ and an electron pair in an antibonding orbital as an ‘antibond’ between two atoms. More precisely, the bond order is defined as b = 12 (n −n) (2.7) where n is the number of electrons in bonding orbitals and n is the number in antibonding orbitals. Nonbonding electrons are ignored when calculating bond order. ■ A brief illustration. Difluorine, F2, has the configuration 12g12u22g14u14g and, because 1g, 1u, and 2g orbitals are bonding but 1u and 1g are antibonding, b 21 (2 2 4 2 4) 1. The bond order of F2 is 1, which is consistent with the structure FF and the conventional description of the molecule as having a single bond. Dinitrogen, N2, has the configuration 12g12u14u22g and b 21 (2 4 2 2) 3. A bond order of 3 corresponds to a triply bonded molecule, which is in line with the structure N⬅N. The high bond order is reflected in the high bond enthalpy of the molecule (946 kJ mol1), one of the highest for any molecule. ■ Isoelectronic molecules and ions have the same bond order, so F2 and O22 both have bond order 1. The bond order of the CO molecule, like that of the isoelectronic molecule N2, is 3, in accord with the analogous structure C⬅O. However, this method of assessing bonding is primitive, especially for heteronuclear species. For instance, inspection of the computed molecular orbitals suggests that 1 and 3 are best regarded as nonbonding orbitals largely localized on O and C, and hence should really be disregarded in the calculation of b. The resulting bond order is unchanged by this modification. The lesson is that the definition of bond order provides a useful indication of the multiplicity of the bond, but any interpretation of contributions to b needs to be made in the light of guidance from the composition of computed orbitals. The definition of bond order allows for the possibility that an orbital is only singly occupied. The bond order in O2, for example, is 1.5 because three electrons occupy the 1g antibonding orbitals. Electron loss from N2 leads to the formation of the transient species N2 in which the bond order is reduced from 3 to 2.5. This reduction in bond order is accompanied by a corresponding decrease in bond strength (from 946 to 855 kJ mol1) and increase in the bond length from 109 pm for N2 to 112 pm for N2. 51 Molecular orbital theory E X A M PL E 2 .6 Determining bond order Determine the bond order of the oxygen molecule, O2, the superoxide ion, O2, and the peroxide ion, O22. Answer We must determine the number of valence electrons, use them to populate the molecular orbitals, and then use eqn 2.7 to calculate b. The species O2, O2 and O22 have 12, 13, and 14 valence electrons, respectively. Their configurations are O2: 12g12u22g14u12g 1000 NN CN CC O2: 12g12u22g14u13g 800 The 1g, 1u, and 2g orbitals are bonding and the 1u and 1g orbitals are antibonding. Therefore, the bond orders are O2: b 1 2 (2 2 2 4 2) 2 O :b 1 2 (2 2 2 4 3) 1.5 2 O :b 2 2 1 2 (2 B/(kJ mol–1) O22: 12g12u22g14u14g CO CC CN 600 CC 200 NN OO 2 2 4 4) 1 0 Self-test 2.6 Predict the bond order of the carbide anion, C22. NN OO CC CO CN 400 0 1 2 Bond order, b 3 Fig. 2.25 The correlation between bond strength and bond order. (b) Bond correlations 160 Key point: For a given pair of elements, bond strength increases and bond length decreases as bond order increases. These trends are illustrated in Figs 2.25 and 2.26. The strength of the dependence varies with the elements. In Period 2 the correlation is relatively weak for CC bonds, with the result that a CC double bond is less than twice as strong as a CC single bond. This difference has profound consequences in organic chemistry, particularly for the reactions of unsaturated compounds. It implies, for example, that it is energetically favourable (but slow in the absence of a catalyst) for ethene and ethyne to polymerize: in this process, CC single bonds form at the expense of the appropriate numbers of multiple bonds. Familiarity with carbon’s properties, however, must not be extrapolated without caution to the bonds between other elements. An NN double bond (409 kJ mol1) is more than twice as strong as an NN single bond (163 kJ mol1), and an N⬅N triple bond (946 kJ mol1) is more than five times as strong. It is on account of this trend that NN multiply bonded compounds are stable relative to polymers or three-dimensional compounds having only single bonds. The same is not true of phosphorus, where the PP, PP, and P⬅P bond enthalpies are 200, 310, and 490 kJ mol1, respectively. For phosphorus, single bonds are stable relative to the matching number of multiple bonds. Thus, phosphorus exists in a variety of solid forms in which PP single bonds are present, including the tetrahedral P4 molecules of white phosphorus. Diphosphorus molecules, P2, are transient species generated at high temperatures and low pressures. The two correlations with bond order taken together imply that, for a given pair of elements: Bond enthalpy increases as bond length decreases. This correlation is illustrated in Fig. 2.27: it is a useful feature to bear in mind when considering the stabilities of molecules because bond lengths may be readily available from independent sources. 140 Re/pm Bond enthalpy increases as bond order increases. Bond length decreases as bond order increases. 150 130 120 110 0 OO NN CN CO CC CC CN CC CO OO NN CN NN 1 2 3 Bond order, b Fig. 2.26 The correlation between bond length and bond order. 1000 800 B/(kJ mol–1) The strengths and lengths of bonds correlate quite well with each other and with the bond order. For a given pair of atoms: CC 600 OO 400 CC NN CO 120 140 Re/pm CN 200 0 160 Fig. 2.27 The correlation between bond length and bond strength. 52 2 Molecular structure and bonding E X A M PL E 2 .7 Predicting correlations between bond order, bond length, and bond strength Use the bond orders of the oxygen molecule, O2, the superoxide ion, O2, and the peroxide ion, O22, calculated in Example 2.6 to predict the relative bond lengths and strengths of the species. Answer We need to remember that bond enthalpy increases as bond order increases. The bond orders of O2, O2, and O22 are 2, 1.5, and 1, respectively. Therefore, we expect the bond enthalpies to increase in the order O22 O2 O2. Bond length decreases as the bond enthalpy increases, so bond length should follow the opposite trend: O22 O2 O2. These predictions are supported by the gas phase bond enthalpies of OO bonds (146 kJ mol1) and OO bonds (496 kJ mol1) and the associated bond lengths of 132 and 121 pm, respectively. Self-test 2.7 Predict the order of bond enthalpies and bond lengths for CN, CN, and C⬅N bonds. 2.11 Polyatomic molecules Molecular orbital theory can be used to discuss in a uniform manner the electronic structures of triatomic molecules, finite groups of atoms, and the almost infinite arrays of atoms in solids. In each case the molecular orbitals resemble those of diatomic molecules, the only important difference being that the orbitals are built from a more extensive basis set of atomic orbitals. As remarked earlier, a key point to bear in mind is that from N atomic orbitals it is possible to construct N molecular orbitals. We saw in Section 2.8 that the general structure of molecular orbital energy level diagrams can be derived by grouping the orbitals into different sets, the and orbitals, according to their shapes. The same procedure is used in the discussion of the molecular orbitals of polyatomic molecules. However, because their shapes are more complex than diatomic molecules, we need a more powerful approach. The discussion of polyatomic molecules will therefore be carried out in two stages. In this chapter we use intuitive ideas about molecular shape to construct molecular orbitals. In Chapter 6 we discuss the shapes of molecules and the use of their symmetry characteristics to construct molecular orbitals and account for other properties. That chapter rationalizes the procedures presented here. The photoelectron spectrum of NH3 (Fig. 2.28) indicates some of the features that a theory of the structure of polyatomic molecules must elucidate. The spectrum shows two bands. The one with the lower ionization energy (in the region of 11 eV) has considerable vibrational structure. This structure indicates that the orbital from which the electron is ejected plays a considerable role in the determination of the molecule’s shape. The broad band in the region of 16 eV arises from electrons that are bound more tightly. (a) Polyatomic molecular orbitals Photoelectron flux Key points: Molecular orbitals are formed from linear combinations of atomic orbitals of the same symmetry; their energies can be determined experimentally from gas-phase photoelectron spectra and interpreted in terms of the pattern of orbital overlap. The features that have been introduced in connection with diatomic molecules are present in all polyatomic molecules. In each case, we write the molecular orbital of a given symmetry (such as the orbitals of a linear molecule) as a sum of all the atomic orbitals that can overlap to form orbitals of that symmetry: = ∑c i i i (2.8) In this linear combination, the i are atomic orbitals (usually the valence orbitals of each atom in the molecule) and the index i runs over all the atomic orbitals that have the appropriate symmetry. From N atomic orbitals we can construct N molecular orbitals. Then: 1. The greater the number of nodes in a molecular orbital, the greater the antibonding 11 15 19 character and the higher the orbital energy. Ionization energy, I/eV 2. Orbitals constructed from lower energy atomic orbitals lie lower in energy (so atomic Fig. 2.28 The UV photoelectron spectrum of NH3, obtained using He 21 eV radiation. s orbitals typically produce lower energy molecular orbitals than atomic p orbitals of the same shell). Molecular orbital theory 53 Interactions between nonnearest-neighbour atoms are weakly bonding (lower the energy slightly) if the orbital lobes on these atoms have the same sign (and interfere constructively). They are weakly antibonding if the signs are opposite (and interfere destructively). 2e to build molecular orbitals that will accommodate the eight valence electrons in the molecule. Each molecular orbital is the combination of seven atomic orbitals: the three H1s orbitals, the N2s orbital, and the three N2p orbitals. It is possible to construct seven molecular orbitals from these seven atomic orbitals (Fig. 2.29). ■ It is not always strictly appropriate to use the notation and in polyatomic molecules because these labels apply to a linear molecule. However, it is often convenient to continue to use the notation when concentrating on the local form of an orbital, its shape relative to the internuclear axis between two neighbouring atoms (this is an example of how the language of valence bond theory survives in MO theory). The correct procedure for labelling orbitals in polyatomic molecules according to their symmetry is described in Chapter 6. For our present purposes all we need know of this more appropriate procedure is the following: a, b denote a nondegenerate orbital e denotes a doubly degenerate orbital (two orbitals of the same energy) t denotes a triply degenerate orbital (three orbitals of the same energy). Subscripts and superscripts are sometimes added to these letters, as in a1, b, eg, and t2 because it is sometimes necessary to distinguish different a, b, e, and t orbitals according to a more detailed analysis of their symmetries. The formal rules for the construction of the orbitals are described in Chapter 6, but it is possible to obtain a sense of their origin by imagining viewing the NH3 molecule along its threefold axis (designated z). The N2pz and N2s orbitals both have cylindrical symmetry about that axis. If the three H1s orbitals are superimposed with the same sign relative to each other (that is, so that all have the same size and tint in the diagram, Fig. 2.29), then they match this cylindrical symmetry. It follows that we can form molecular orbitals of the form c1N2s c2Np c3{H1sA H1sB H1sC} z (2.9) From these three basis orbitals (the specific combination of H1s orbitals counts as a single ‘symmetry adapted’ basis orbital), it is possible to construct three molecular orbitals (with different values of the coefficients c). The orbital with no nodes between the N and H atoms is the lowest in energy, that with a node between all the NH neighbours is the highest in energy, and the third orbital lies between the two. The three orbitals are nondegenerate and are labelled 1a1, 2a1, and 3a1 in order of increasing energy. The N2px and N2py orbitals have symmetry with respect to the z-axis, and can be used to form orbitals with combinations of the H1s orbitals that have a matching symmetry. For example, one such superposition will have the form c1N2p c2{H1sA H1sB} x (2.10) As can be seen from Fig. 2.29, the signs of the H1s orbital combination match those of the N2px orbital. The N2s orbital cannot contribute to this superposition, so only two combinations can be formed, one without a node between the N and H orbitals and the other with a node. The two orbitals differ in energy, the former being lower. A similar combination of orbitals can be formed with the N2py orbital, and it turns out (by the symmetry arguments that we use in Chapter 6) that the two orbitals are degenerate with the two we have just described. The combinations are examples of e orbitals (because they form doubly degenerate pairs), and are labelled 1e and 2e in order of increasing energy. The general form of the molecular orbital energy level diagram is shown in Fig. 2.30. The actual location of the orbitals (particularly the relative positions of the a and the e sets), can be found only by detailed computation or by identifying the orbitals responsible for the photoelectron spectrum. We have indicated the probable assignment of the 11 eV and 16 eV peaks, which fixes the locations of two of the occupied orbitals. The third occupied orbital is out of range of the 21 eV radiation used to obtain the spectrum. Energy ■ A brief illustration. To account for the features in the photoelectron spectrum of NH3, we need 3a1 2a1 1e 1a1 Fig. 2.29 A schematic illustration of the molecular orbitals of NH3 with the size of the atomic orbital indicating the magnitude of its contribution to the molecular orbital. The view is along the z-axis. 54 2 Molecular structure and bonding N 16 eV Energy H3 NH3 N2p 11 eV 2e H3 3a1 e N2s 2a1 a 1e 1a1 Fig. 2.30 The molecular orbital energy level diagram for NH3 when the molecule has the observed bond angle (107°) and bond length. The photoelectron spectrum is consistent with the need to accommodate eight electrons in the orbitals. The electrons enter the molecular orbitals in increasing order of energy, starting with the orbital of lowest energy, and taking note of the requirement of the exclusion principle that no more than two electrons can occupy any one orbital. The first two electrons enter 1a1 and fill it. The next four enter the doubly degenerate 1e orbitals and fill them. The last two enter the 2a1 orbital, which calculations show is almost nonbonding and localized on the N atom. The resulting overall ground-state electron configuration is therefore 1a121e142a12. No antibonding orbitals are occupied, so the molecule has a lower energy than the separated atoms. The conventional description of NH3 as a molecule with a lone pair is also mirrored in the configuration: the HOMO is 2a1, which is largely confined to the N atom and makes only a small contribution to the bonding. We saw in Section 2.3 that lone pair electrons play a considerable role in determining the shapes of molecules. The extensive vibrational structure in the 11 eV band of the photoelectron spectrum is consistent with this observation, as photoejection of a 2a1 electron removes the effectiveness of the lone pair and the shape of the ionized molecule is considerably different from that of NH3 itself. Photoionization therefore results in extensive vibrational structure in the spectrum. (b) Hypervalence in the context of molecular orbitals Key point: The delocalization of molecular orbitals means that an electron pair can contribute to the bonding of more than two atoms. S F6 SF6 2a1 2t1 Energy S3p S3s t 1e e a 1t1 1a1 Fig. 2.31 A schematic molecular orbital energy level diagram for SF6. In Section 2.3 we used valence bond theory to explain hypervalence by using d orbitals to allow the valence shell of an atom to accommodate more than eight electrons. Molecular orbital theory explains it rather more elegantly. We consider SF6, which has six SF bonds and hence 12 electrons involved in forming bonds and is therefore hypervalent. The simple basis set of atomic orbitals that are used to construct the molecular orbitals consists of the valence shell s and p orbitals of the S atom and one p orbital of each of the six F atoms and pointing towards the S atom. We use the F2p orbitals rather than the F2s orbitals because they match the S orbitals more closely in energy. From these ten atomic orbitals it is possible to construct ten molecular orbitals. Calculations indicate that four of the orbitals are bonding and four are antibonding; the two remaining orbitals are nonbonding (Fig. 2.31). There are 12 electrons to accommodate. The first two enter 1a1 and the next six enter 1t1. The remaining four fill the nonbonding pair of orbitals, resulting in the configuration 1a121t161e4. As we see, none of the antibonding orbitals (2a1 and 2t1) is occupied. Molecular orbital theory, therefore, accounts for the formation of SF6, with four bonding orbitals and two nonbonding orbitals occupied and does not need to invoke S3d orbitals and octet expansion. This does not mean that d orbitals cannot participate in the bonding, but it does show that they are not necessary for bonding six F atoms to the central S atom. The limitation of valence bond theory is the assumption that each atomic orbital on the central atom can participate in the formation of only one bond. Molecular orbital theory takes hypervalence into its stride by having available plenty of orbitals, not all of which are antibonding. Therefore, the question of when hypervalence can occur appears to depend on factors other than d-orbital availability, such as the ability of small atoms to pack around a large atom. (c) Localization Key points: Localized and delocalized descriptions of bonds are mathematically equivalent, but one description may be more suitable for a particular property, as summarized in Table 2.5. A striking feature of the VB approach to chemical bonding is its accord with chemical instinct, as it identifies something that can be called ‘an AB bond’. Both OH bonds in H2O, for instance, are treated as localized, equivalent structures because each one consists of an electron pair shared between O and H. This feature appears to be absent from MO theory because molecular orbitals are delocalized and the electrons that occupy them bind all the atoms together, not just a specific pair of neighbouring atoms. The concept of an AB bond as existing independently of other bonds in the molecule, and of being transferable from one molecule to another, seems to have been lost. However, we shall now show that the molecular orbital description is mathematically almost equivalent to a localized Molecular orbital theory 55 Table 2.5 A general indication of the properties for which localized and delocalized descriptions are appropriate Localized appropriate Delocalized appropriate Bond strengths Electronic spectra Force constants Photoionization Bond lengths Electron attachment Brønsted acidity Magnetism VSEPR description of molecular geometry Walsh description of molecular geometry Standard potentials† Chapter 4. † Chapter 5. description of the overall electron distribution. The demonstration hinges on the fact that linear combinations of molecular orbitals can be formed that result in the same overall electron distribution, but the individual orbitals are distinctly different. Consider the H2O molecule. The two occupied bonding orbitals of the delocalized description, 1a1 and 1b2, are shown in Fig. 2.32. If we form the sum 1a1 1b2, the negative half of 1b2 cancels half the 1a1 orbital almost completely, leaving a localized orbital between O and the other H. Likewise, when we form the difference 1a1 1b2, the other half of the 1a1 orbital is cancelled almost completely, so leaving a localized orbital between the other pair of atoms. Therefore, by taking sums and differences of delocalized orbitals, localized orbitals are created (and vice versa). Because these are two equivalent ways of describing the same overall electron population, one description cannot be said to be better than the other. Table 2.5 suggests when it is appropriate to select a delocalized description or a localized description. In general, a delocalized description is needed for dealing with global properties of the entire molecule. Such properties include electronic spectra (UV and visible transitions, Section 8.3), photoionization spectra, ionization and electron attachment energies (Section 1.9), and reduction potentials (Section 5.1). In contrast, a localized description is most appropriate for dealing with properties of a fragment of a total molecule. Such properties include bond strength, bond length, bond force constant, and some aspects of reactions (such as acidbase character): in these aspects the localized description is more appropriate because it focuses attention on the distribution of electrons in and around a particular bond. (d) Localized bonds and hybridization Key point: Hybrid atomic orbitals are sometimes used in the discussion of localized molecular orbitals. The localized molecular orbital description of bonding can be taken a stage further by invoking the concept of hybridization. Strictly speaking, hybridization belongs to VB theory, but it is commonly invoked in simple qualitative descriptions of molecular orbitals. We have seen that in general a molecular orbital is constructed from all atomic orbitals of the appropriate symmetry. However, it is sometimes convenient to form a mixture of orbitals on one atom (the O atom in H2O, for instance), and then to use these hybrid orbitals to construct localized molecular orbitals. In H2O, for instance, each OH bond can be regarded as formed by the overlap of an H1s orbital and a hybrid orbital composed of O2s and O2p orbitals (Fig. 2.33). We have already seen that the mixing of s and p orbitals on a given atom results in hybrid orbitals that have a definite direction in space, as in the formation of tetrahedral hybrids. Once the hybrid orbitals have been selected, a localized molecular orbital description can be constructed. For example, four bonds in CF4 can be formed by building bonding and antibonding localized orbitals by overlap of each hybrid and one F2p orbital directed towards it. Similarly, to describe the electron distribution of BF3, we could consider each localized BF orbital as formed by the overlap of an sp2 hybrid with an F2p orbital. A localized orbital description of a PCl5 molecule would be in terms of five PCl bonds formed by overlap of each of the five trigonal-bipyramidal sp3d hybrid orbitals with a 2p b1 a1 + b2 a1 a1 – b2 Fig. 2.32 The two occupied 1a1 and 1b2 orbitals of the H2O molecule and their sum 1a1 1b2 and difference 1a1 1b2. In each case we form an almost fully delocalized orbital between a pair of atoms. – Hybrid + H1s + Fig. 2.33 The formation of localized OH orbitals in H2O by the overlap of hybrid orbitals on the O atom and H1s orbitals. The hybrid orbitals are a close approximation to the sp3 hybrids shown in Fig. 2.6. 56 2 Molecular structure and bonding orbital of a Cl atom. Similarly, where we wanted to form six localized orbitals in a regular octahedral arrangement (for example, in SF6), we would need two d orbitals: the resulting six sp3d2 hybrids point in the required directions. (e) Electron deficiency Key point: The existence of electron-deficient species is explained by the delocalization of the bonding influence of electrons over several atoms. H B 17 Diborane, B2H6 The VB model of bonding fails to account for the existence of electron-deficient compounds, which are compounds for which, according to Lewis’s approach, there are not enough electrons to form the required number of bonds. This point can be illustrated most easily with diborane, B2H6 (17). There are only 12 valence electrons but, according to Lewis’s approach, at least eight electron pairs are needed to bind eight atoms together. The formation of molecular orbitals by combining several atomic orbitals accounts effortlessly for the existence of these compounds. The eight atoms of this molecule contribute a total of 14 valence orbitals (three p and one s orbital from each B atom, making eight, and one s orbital each from the six H atoms). These 14 atomic orbitals can be used to construct 14 molecular orbitals. About seven of these molecular orbitals will be bonding or nonbonding, which is more than enough to accommodate the 12 valence electrons provided by the atoms. The bonding can be best understood if we consider that the MOs produced are associated with either the terminal BH fragments or with the bridging BHB fragments. The localized MOs associated with the terminal BH bonds are constructed simply from atomic orbitals on two atoms (the H1s and a B2s2pn hybrid). The molecular orbitals associated with the two BHB fragments are linear combinations of the B2s2pn hybrids on each of the two B atoms and an H1s orbital of the H atom lying between them (Fig. 2.34). Three molecular orbitals are formed from these three atomic orbitals: one is bonding, one nonbonding, and the third is antibonding. The bonding orbital can accommodate two electrons and hold the BHB fragment together. The same remark applies to the second BHB fragment, and the two occupied ‘bridging’ bonding molecular orbitals hold the molecule together. Thus, overall, 12 electrons account for the stability of the molecule because their influence is spread over more than six pairs of atoms. Electron deficiency is well developed not only in boron (where it was first clearly recognized) but also in carbocations and a variety of other classes of compounds that we encounter later in the text. 2.12 Molecular shape in terms of molecular orbitals Key point: In the Walsh model, the shape of a molecule is predicted on the basis of the occupation of molecular orbitals that, in a correlation diagram, show a strong dependence on bond angle. Fig. 2.34 The molecular orbital formed between two B atoms and one H atom lying between them, as in B2H6. Two electrons occupy the bonding combination and hold all three atoms together. In MO theory, the electrons responsible for bonding are delocalized over the entire molecule. Current ab initio and semiempirical molecular orbital calculations, which are easily carried out with software, are able to predict the shapes of even quite complicated molecules with high reliability. Nevertheless, there is still a need to understand the qualitative factors that contribute to the shape of a molecule within the framework of MO theory. Figure 2.35 shows the Walsh diagram for an XH2 molecule. A Walsh diagram is a special case of a correlation diagram, a diagram that shows how one set of orbitals evolves into another as a parameter (such as a bond angle) is changed; we shall meet other examples later. A Walsh diagram adopts a simple pictorial approach to the task of analysing molecular shape in terms of delocalized molecular orbitals and was devised by A.D. Walsh in a classic series of papers published in 1953. Such diagrams play an important role in understanding the shapes, spectra, and reactions of polyatomic molecules. The XH2 diagram has been constructed by considering how the composition and energy of each molecular orbital changes as the bond angle is varied from 90° to 180°. The molecular orbitals are constructed from the 2s, 2pz, 2py, and 2px atomic orbitals on X and the two H1s atomic orbitals. It is convenient to consider the possible combinations of H1s orbitals before forming the molecular orbitals with X. The linear combinations of Molecular orbital theory H1s orbitals and are illustrated in Fig. 2.36. ‘Symmetry adapted’ combinations such as these will figure extensively in later discussions. The molecular orbitals to consider in the angular molecule are3 1b1 57 1πu a1 c12s c22p c3 (2.11) x b2 c42p c5 y There are three a1 orbitals and two b2 orbitals; the lowest energy orbitals of each type are shown on the left of Fig. 2.37. In the linear molecule, the molecular orbitals are g c12s c2 1b2 2a1 1σu 1a1 1σg u 2p and 2p y Energy z b1 2p (2.12) z u c32p c4 y These orbitals are shown on the right in Fig. 2.37. The lowest energy molecular orbital in 90° H2X is the one labelled 1a1, which is built from the overlap of the X2pz orbital with the combination of H1s orbitals. As the bond angle changes to 180° the energy of this orbital decreases (Fig. 2.35) in part because the HH overlap decreases and in part because the reduced involvement of the X2pz orbital decreases the overlap with (Fig. 2.37). The energy of the 1b2 orbital is lowered because the H1s orbitals move into a better position for overlap with the X2py orbital. The weakly antibonding HH contribution is also decreased. The biggest change occurs for the 2a1 orbital. It has considerable X2s character in the 90° molecule, but the corresponding molecular orbital in the 180° molecule has pure X2pz orbital character. Hence, it shows a steep rise in energy as the bond angle increases. The 1b1 orbital is a nonbonding X2p orbital perpendicular to the molecular plane in the 90° molecule and remains nonbonding in the linear molecule. Hence, its energy barely changes with angle. Each of the curves plotted on Fig. 2.35 will have a maximum or minimum on the 180° axis. The two lowest energy curves have a minimum on the 180° line. Therefore, we would expect XH2 molecules having four valence electrons to be linear. XH2 molecules with more than four valence electrons are expected to be angular because at least one electron is in the nonbonding 2a1 orbital. If the molecule is close to linear then the order of filling the molecular orbitals will be in the order of their increasing energy, which is 1a1 2a1 1b2 1b1. The simplest XH2 molecule in Period 2 is the transient gas-phase BeH2 molecule (BeH2 normally exists as a polymeric solid, with four-coordinate Be atoms). There are four valence electrons in a BeH2 molecule, which occupy the lowest two molecular orbitals. If the lowest energy is achieved with the molecule angular, then that will be its shape. We can decide whether or not the molecule is likely to be angular by accommodating the electrons in the two lowest-energy orbitals corresponding to an arbitrary bond angle in Fig. 2.35. We then note that the HOMO decreases in energy on going to the right of the diagram (in the direction of increasing bond angle) and that the lowest total energy is obtained when the molecule is linear. Hence, BeH2 is predicted to be linear and to have the configuration 1222. In CH2, which has two more electrons than BeH2, three of the molecular orbitals must be occupied. In this case, the lowest energy is achieved if the molecule is angular and has configuration 1a122a121b22. In general, any XH2 molecule with from five to eight valence electrons is predicted to be angular. The observed bond angles are BeH2 BH2 CH2 NH2 OH2 180° 131° 136° 103° 105° These experimental observations are qualitatively in line with Walsh’s approach, but for quantitative predictions we have to turn to detailed molecular orbital calculations. 3 We continue to use the letters a and b to label nondegenerate orbitals, and will explain their full significance in Chapter 6. 90° Bond angle 180° Fig. 2.35 The Walsh diagram for XH2 molecules. Only the bonding and nonbonding orbitals are shown. – (a) – (b) Fig. 2.36 The combination of H1s orbitals that are used to construct molecular orbitals in (a) angular and (b) linear XH2. 58 2 Molecular structure and bonding 90° 180° E X A M PL E 2 . 8 Using a Walsh diagram to predict a shape Predict the shape of an H2O molecule on the basis of a Walsh diagram for an XH2 molecule. 1b1 1πu 1σu 1b2 Self-test 2.8 Is any XH2 molecule, in which X denotes an atom of a Period 3 element, expected to be linear? If so, which? 1πu 2a1 1a1 Answer We need to choose an angle between 90 and 180°, refer to the appropriate Walsh diagram, fill the molecular orbitals with the valence electrons, and assess whether the resulting configuration implies a linear or an angular shape. In this case we choose an intermediate bond angle along the horizontal axis of the XH2 diagram in Fig. 2.37 and accommodate eight electrons. The resulting configuration is 1a122a121b221b12. The 2a1 orbital is occupied, so we expect the nonlinear molecule to have a lower energy than the linear molecule. 1σg Fig. 2.37 The composition of the molecular orbitals of an XH2 molecule at the two extremes of the correlation diagram shown in Fig. 2.35. Walsh applied his approach to molecules other than compounds of hydrogen, but the correlation diagrams soon become very complicated. His approach represents a valuable model because it traces the influences on molecular shapes of the occupation of orbitals spreading over the entire molecule. Correlation diagrams like those introduced by Walsh are frequently encountered in contemporary discussions of the shapes of complex molecules, and we shall see a number of examples in later chapters. They illustrate how inorganic chemists can sometimes identify and weigh competing influences by considering two extreme cases (such as linear and 90° XH2 molecules), and then rationalize the fact that the state of a molecule is a compromise intermediate between the two extremes. Structure and bond properties Certain properties of bonds are approximately the same in different compounds of the elements. Thus, if we know the strength of an OH bond in H2O, then with some confidence we can use the same value for the OH bond in CH3OH. At this stage we confine our attention to two of the most important characteristics of a bond: its length and its strength. We also extend our understanding of bonds to predict the shapes of simple inorganic molecules. 2.13 Bond length Key points: The equilibrium bond length in a molecule is the separation of the centres of the two bonded atoms; covalent radii vary through the periodic table in much the same way as metallic and ionic radii. Table 2.6 Equilibrium bond lengths, Re /pm H2+ 106 H2 74 HF 92 HCl 127 HBr 141 HI 160 N2 109 O2 121 F2 144 Cl2 199 I2 267 The equilibrium bond length in a molecule is the distance between the centres of the two bonded atoms. A wealth of useful and accurate information about bond lengths is available in the literature, most of it obtained by X-ray diffraction on solids (Section 8.1). Equilibrium bond lengths of molecules in the gas phase are usually determined by infrared or microwave spectroscopy, or more directly by electron diffraction. Some typical values are given in Table 2.6. To a reasonable first approximation, equilibrium bond lengths can be partitioned into contributions from each atom of the bonded pair. The contribution of an atom to a covalent bond is called the covalent radius of the element (18). We can use the covalent radii in Table 2.7 to predict, for example, that the length of a PN bond is 110 pm 74 pm 184 pm; experimentally, this bond length is close to 180 pm in a number of compounds. Experimental bond lengths should be used whenever possible, but covalent radii are useful for making cautious estimates when experimental data are not available. Covalent radii vary through the periodic table in much the same way as metallic and ionic radii (Section 1.9a), for the same reasons, and are smallest close to F. Covalent radii are approximately equal to the separation of nuclei when the cores of the two atoms are in contact: the valence electrons draw the two atoms together until the repulsion between the cores starts to dominate. A covalent radius expresses the closeness of approach of bonded atoms; the closeness of approach of nonbonded atoms in neighbouring molecules that are in contact is expressed in terms of the van der Waals radius of the element, which is the internuclear separation when the valence shells of the two atoms are in nonbonding contact (19). van der Waals radii are of paramount importance for understanding the packing of molecular compounds in crystals, the conformations adopted by small but flexible molecules, and the shapes of biological macromolecules (Chapter 27). 59 Structure and bond properties Table 2.7 Covalent radii, rcov /pm 2.14 Bond strength Key points: The strength of a bond is measured by its dissociation enthalpy; mean bond enthalpies are used to make estimates of reaction enthalpies. H A convenient thermodynamic measure of the strength of an AB bond is the bond dissociation enthalpy, H O (AB), the standard reaction enthalpy for the process C N O F 77 (1) 74 (1) 66 (1) 64 67 (2) 65 (2) 57 (2) 60 (3) 54 (3) AB(g) → A(g) B(g) The mean bond enthalpy, B, is the average bond dissociation enthalpy taken over a series of AB bonds in different molecules (Table 2.8). Mean bond enthalpies can be used to estimate reaction enthalpies. However, thermodynamic data on actual species should be used whenever possible in preference to mean values because the latter can be misleading. For instance, the SiSi bond enthalpy ranges from 226 kJ mol1 in Si2H6 to 322 kJ mol1 in Si2(CH3)6. The values in Table 2.8 are best considered as data of last resort: they may be used to make rough estimates of reaction enthalpies when enthalpies of formation or actual bond enthalpies are unavailable. E X A M PL E 2 . 9 Making estimates using mean bond enthalpies Estimate the reaction enthalpy for the production of SF6(g) from SF4(g) given that the mean bond enthalpies of F2, SF4, and SF6 are 158, 343, and 327 kJ mol1, respectively, at 25C. 37 70 (a) Si P 118 110 S Cl 104 (1) 99 95 (2) Ge As Se Br 122 121 117 114 Sb Te I 141 137 133 Values are for single bonds except where otherwise stated (in parentheses); (a) denotes aromatic. Answer We make use of the fact that the enthalpy of a reaction is equal to the difference between the sum of the bond enthalpies for broken bonds and the sum of the enthalpies of the bonds that are formed. The reaction is SF4(g) F2(g) → SF6(g) In this reaction, 1 mol FF bonds and 4 mol SF bonds (in SF4) must be broken, corresponding to an enthalpy change of 158 kJ (4 343 kJ) 1530 kJ. This enthalpy change is positive because energy will be used in breaking bonds. Then 6 mol SF bonds (in SF6) must be formed, corresponding to an enthalpy change of 6 (327 kJ) 1962 kJ. This enthalpy change is negative because energy is released when the bonds are formed. The net enthalpy change is therefore R H O 1530 kJ 1962 kJ 432 kJ A Hence, the reaction is strongly exothermic. The experimental value for the reaction is 434 kJ, which is in excellent agreement with the estimated value. Self-test 2.9 Estimate the enthalpy of formation of H2S from S8 (a cyclic molecule) and H2. B r A rB 18 Covalent radius 2.15 Electronegativity and bond enthalpy Key points: The Pauling scale of electronegativity is useful for estimating bond enthalpies and for assessing the polarities of bonds. The concept of electronegativity was introduced in Section 1.9d, where it was defined as the power of an atom of the element to attract electrons to itself when it is part of a compound. The greater the difference in electronegativity between two elements A and B, the greater the ionic character of the AB bond. Linus Pauling’s original formulation of electronegativity drew on concepts relating to the energetics of bond formation. For example, in the formation of AB from the diatomic A2 and B2 molecules, A2(g) B2(g) → 2 AB(g) He argued that the excess energy, ∆E, of the AB bond over the average energy of AA and BB bonds can be attributed to the presence of ionic contributions to the covalent bonding. He defined the difference in electronegativity as \|P(A) P(B)\| 0.102(∆E/kJ mol1)1/2 (2.13a) where ∆E B(AB) 12{B(AA) B(BB)} (2.13b) A B rA rB 19 van der Waals radius 60 2 Molecular structure and bonding Table 2.8 Mean bond enthalpies, B/(kJ mol1) H H 436 C 412 C N O F Cl Br I S P Si 348 (1) 612 (2) 837 (3) 518 (a) N 388 305 (1) 163 (1) 613 (2) 409 (2) 890 (3) 946 (3) O 463 360 (1) 157 F 565 484 270 185 155 Cl 431 338 200 203 254 Br 366 276 I 299 238 S 338 259 P 322 (1) 743 (2) 146 (1) 497 (2) 464 523 343 242 219 193 210 178 250 212 151 264 201 480 (3) Si 318 466 226 Values are for single bonds except where otherwise stated (in parentheses); (a) denotes aromatic. CsF 3 MgO 2 Δ Ionic 1 SiO2 Metallic 0 1 Covalent F2 Cs 2 3 4 mean Fig. 2.38 A Ketelaar triangle, showing how a plot of average electronegativity against electronegativity difference can be used to classify the bond type for binary compounds. with B(AB) the mean AB bond enthalpy. Thus, if the AB bond enthalpy is significantly greater than the average of the nonpolar AA and BB bonds, then it is presumed that there is a substantial ionic contribution to the wavefunction and hence a large difference in electronegativity between the two atoms. Pauling electronegativities increase with increasing oxidation number of the element and the values in Table 1.7 are for the most common oxidation state. Pauling electronegativities are useful for estimating the enthalpies of bonds between elements of different electronegativity and to make qualitative assessments of the polarities of bonds. Binary compounds in which the difference in electronegativity between the two elements is greater than about 1.7 can generally be regarded as being predominantly ionic. However, this crude distinction was refined by Anton van Arkel and Jan Ketelaar in the 1940s, when they drew a triangle with vertices representing ionic, covalent, and metallic bonding. The Ketelaar triangle (more appropriately, the van ArkelKetelaar triangle) has been elaborated by Gordon Sproul, who constructed a triangle based on the difference in electronegativities (∆) of the elements in a binary compound and their average electronegativity (mean) (Fig. 2.38). The Ketelaar triangle is used extensively in Chapter 3, where we shall see how this basic concept can be used to classify a wide range of compounds of different kinds. Ionic bonding is characterized by a large difference in electronegativity. Because a large difference indicates that the electronegativity of one element is high and that of the other is low, the average electronegativity must be intermediate in value. The compound CsF, for instance, with ∆ 3.19 and mean 2.38, lies at the ‘ionic’ apex of the triangle. Covalent bonding is characterized by a small difference in electronegativities. Such compounds lie at the base of the triangle. Binary compounds that are predominantly covalently bonded are typically formed between nonmetals, which commonly have high electronegativities. It follows that the covalent region of the triangle is the lower, right-hand corner. This corner of the triangle is occupied by F2, which has ∆ 0 and mean 3.98 (the maximum value of any Pauling electronegativity). Metallic bonding is also characterized by a small electronegativity difference, and also lies towards the base of the triangle. In metallic bonding, Structure and bond properties however, electronegativities are low, the average values are therefore also low, and consequently metallic bonding occupies the lower, left-hand corner of the triangle. The outer corner is occupied by Cs, which has ∆ 0 and mean 0.79 (the lowest value of Pauling electronegativity). The advantage of using a Ketelaar triangle over simple electronegativity difference is that it allows us to distinguish between covalent and metallic bonding, which are both indicated by a small electronegativity difference. ■ A brief illustration. For MgO, ∆ 3.44 1.31 2.13 and mean 2.38. These values place MgO in the ionic region of the triangle. By contrast, for SiO2, ∆ 2.58 1.90 0.68 and mean 2.24. These values place SiO2 lower on the triangle compared to MgO and in the covalent bonding region. ■ 2.16 Oxidation states Key point: Oxidation numbers are assigned by applying the rules in Table 2.9. The oxidation number, Nox,4 is a parameter obtained by exaggerating the ionic character of a bond. It can be regarded as the charge that an atom would have if the more electronegative atom in a bond acquired the two electrons of the bond completely. The oxidation state is the physical state of the element corresponding to its oxidation number. Thus, an atom may be assigned an oxidation number and be in the corresponding oxidation state.5 The alkali metals are the most electropositive elements in the periodic table, so we can assume they will always be present as M and are assigned an oxidation number of 1. Because oxygen’s electronegativity is exceeded only by that of F, we can regard it as O2 in combination with any element other than F, and hence it is ascribed an oxidation number of 2. Likewise, the exaggerated ionic structure of NO3 is N5(O2)3, so the oxidation number of nitrogen in this compound is 5, which is denoted either N(V) or N(5). These conventions may be used even if the oxidation number is negative, so oxygen has oxidation number 2, denoted O(2) or more rarely O(II), in most of its compounds. Table 2.9 The determination of oxidation number Oxidation number 1. The sum of the oxidation numbers of all the atoms in the species is equal to its total charge 2. For atoms in their elemental form 0 For atoms of Group 1 For atoms of Group 2 For atoms of Group 13 (except B) For atoms of Group 14 (except C, Si) 1 2 3(EX3), 1(EX) 4(EX4), 2(EX2) For hydrogen 1 in combination with nonmetals 1 in combination with metals For fluorine 1 in all its compounds For oxygen 2 unless combined with F 1 in peroxides ( O2− 2 ) −21 in superoxides ( O2− ) −31 in ozonides ( O3− ) Halogens 1 in most compounds, unless the other elements include oxygen or more electronegative halogens To determine an oxidation number, work through the rules in the order given. Stop as soon as the oxidation number has been assigned. These rules are not exhaustive, but they are applicable to a wide range of common compounds. 4 There is no formally agreed symbol for oxidation number. In practice, inorganic chemists use the terms ‘oxidation number’ and ‘oxidation state’ interchangeably, but in this text we shall preserve the distinction. 5 61 62 2 Molecular structure and bonding In practice, oxidation numbers are assigned by applying a set of simple rules (Table 2.9). These rules reflect the consequences of electronegativity for the ‘exaggerated ionic’ structures of compounds and match the increase in the degree of oxidation that we would expect as the number of oxygen atoms in a compound increases (as in going from NO to NO3). This aspect of oxidation number is taken further in Chapter 5. Many elements, for example nitrogen, the halogens, and the d-block elements, can exist in a variety of oxidation states (Table 2.9). E X A M PL E 2 .10 Assigning an oxidation number to an element What is the oxidation number of (a) S in hydrogen sulfide, H2S, (b) Mn in the permanganate ion, MnO4? Answer We need to work through the steps set out in Table 2.9 in the order given. (a) The overall charge of the species is 0; so 2Nox(H) Nox(S) 0. Because Nox(H) 1 in combination with a nonmetal, it follows that Nox(S) 2. (b) The sum of the oxidation numbers of all the atoms is 1, so Nox(Mn) 4Nox(O) 1. Because Nox(O) 2, it follows that Nox(Mn) 1 4(2) 7. That is, MnO4 is a compound of Mn(VII). Its formal name is tetraoxomanganese(VII) ion. Self-test 2.10 What is the oxidation number of (a) O in O2, (b) P in PO43? FURTHER READING R.J. Gillespie and I. Hargittai, The VSEPR model of molecular geometry. Prentice Hall (1992). An excellent introduction to modern attitudes to VSEPR theory. D.M.P. Mingos, Essential trends in inorganic chemistry. Oxford University Press (1998). An overview of inorganic chemistry from the perspective of structure and bonding. R.J. Gillespie and P.L.A. Popelier, Chemical bonding and molecular geometry: from Lewis to electron densities. Oxford University Press (2001). A comprehensive survey of modern theories of chemical bonding and geometry. I.D. Brown, The chemical bond in inorganic chemistry. Oxford University Press (2006). M.J. Winter, Chemical bonding. Oxford University Press (1994). This short text introduces some concepts of chemical bonding in a descriptive and non-mathematical way. J.N. Murrell, S.F.A. Kettle, and J.M. Tedder, The chemical bond. Wiley, New York (1985). T. Albright, Orbital interactions in chemistry. Wiley, New York (2005). This text covers the application of molecular orbital theory to organic, organometallic, inorganic, and solid-state chemistry. K. Bansal, Molecular structure and orbital theory. Campus Books International (2000). T. Albright and J.K. Burdett, Problems in molecular orbital theory. Oxford University Press (1993). EXERCISES 2.1 What shapes would you expect for the species (a) H2S, (b) BF4, (c) NH4? 2.8 The common forms of nitrogen and phosphorus are N2(g) and P4(s), respectively. Account for the difference in terms of the single and multiple bond enthalpies. 2.2 What shapes would you expect for the species (a) SO3, (b) SO32, (c) IF5? 2.9 Use the data in Table 2.8 to calculate the standard enthalpy of the reaction 2 H2(g) O2(g) → 2H2O(g). The experimental value is 484 kJ mol1. Account for the difference between the estimated and experimental values. 2.3 What shapes would you expect for the species (a) ClF3, (b) ICl4, (c) I3? 2.4 In which of the species ICl6 and SF4 is the bond angle closest to that predicted by the VSEPR model? 2.10 Predict the standard enthalpies of the reactions 4 2.5 Solid phosphorus pentachoride is an ionic solid composed of PCl cations and PCl6 anions, but the vapour is molecular. What are the shapes of the ions in the solid? 2.6 Use the covalent radii in Table 2.7 to calculate the bond lengths in (a) CCl4 (177 pm), (b) SiCl4 (201 pm), (c) GeCl4 (210 pm). (The values in parentheses are experimental bond lengths and are included for comparison.) 2.7 Given that B(SiO) 640 kJ mol1, show that bond enthalpy considerations predict that siliconoxygen compounds are likely to contain networks of tetrahedra with SiO single bonds and not discrete molecules with SiO double bonds. (a) S22(g) 1–4 S8(g) → S42(g) (b) O22(g) O2(g) → O42(g) by using mean bond enthalpy data. Assume that the unknown species O42 is a singly bonded chain analogue of S42. 2.11 Four elements arbitrarily labelled A, B, C, and D have electronegativities 3.8, 3.3, 2.8, and 1.3, respectively. Place the compounds AB, AD, BD, and AC in order of increasing covalent character. 2.12 Use the Ketelaar triangle in Fig. 2.38 and the electronegativity values in Table 1.7 to predict what type of bonding is likely to dominate in (a) BCl3, (b) KCl, (c) BeO. Problems 2.13 Predict the hybridization of orbitals required in (a) BCl3, (b) NH4, (c) SF4, (d) XeF4. 2.14 Use molecular orbital diagrams to determine the number of unpaired electrons in (a) O2, (b) O2, (c) BN, (d) NO2. 2.15 Use Fig. 2.17 to write the electron configurations of (a) Be2, (b) B2, (c) C2, (d) F2 and sketch the form of the HOMO in each case. 2.16 When acetylene (ethyne) is passed through a solution of copper(I) chloride a red precipitate of copper acetylide, CuC2, is formed. This is a common test for the presence of acetylene. Describe the bonding in the C22 ion in terms of molecular orbital theory and compare the bond order to that of C2. 2.17 Assume that the MO diagram of IBr is analogous to that of ICl (Fig. 2.24). (a) What basis set of atomic orbital would be used to generate the IBr molecular orbitals? (b) Calculate the bond order of IBr. (c) Comment on the relative stabilities and bond orders of IBr and IBr2. 2.18 Determine the bond orders of (a) S2, (b) Cl2, and (c) NO2 from their molecular orbital configurations and compare the values with the bond orders determined from Lewis structures. (NO has orbitals like those of O2.) 2.19 What are the expected changes in bond order and bond distance that accompany the following ionization processes? 63 2.20 (a) How many independent linear combinations are possible for four 1s orbitals? (b) Draw pictures of the linear combinations of H1s orbitals for a hypothetical linear H4 molecule. (c) From a consideration of the number of nonbonding and antibonding interactions, arrange these molecular orbitals in order of increasing energy. 2.21 (a) Construct the form of each molecular orbital in linear [HHeH]2 using 1s basis atomic orbitals on each atom and considering successive nodal surfaces. (b) Arrange the MOs in increasing energy. (c) Indicate the electron population of the MOs. (d) Should [HHeH]2 be stable in isolation or in solution? Explain your reasoning. 2.22 (a) Based on the MO discussion of NH3 in the text, find the average NH bond order in NH3 by calculating the net number of bonds and dividing by the number of NH groups. 2.23 From the relative atomic orbital and molecular orbital energies depicted in Fig. 2.31, describe the character as mainly F or mainly S for the frontier orbitals e (the HOMO) and 2t (the LUMO) in SF6. Explain your reasoning. 2.24 Classify the hypothetical species (a) square H42, (b) angular O32 as electron precise or electron deficient. Explain your answer and decide whether either of them is likely to exist. (a) O2 → O2 e (b) N2 e → N2 (c) NO → NO e PROBLEMS 2.1 Use the concepts from Chapter 1, particularly the effects of penetration and shielding on the radial wavefunction, to account for the variation of single bond covalent radii with position in the periodic table. 2.2 In valence bond theory, hypervalence is usually explained in terms of d-orbital participation in bonding. In the paper ‘On the role of orbital hybridisation’ (J. Chem. Educ. 2007, 84, 783) the author argues that this is not the case. Give a concise summary of the method used and the author’s reasoning. 2.3 Develop an argument based on bond enthalpies for the importance of SiO bonds in substances common in the Earth’s crust in preference to SiSi or SiH bonds. How and why does the behaviour of silicon differ from that of carbon? 2.4 The van ArkelKetelaar triangle has been in use since the 1940s. A quantitative treatment of the triangle was carried out by Gordon Sproul in 1994 (J. Phys. Chem., 1994, 98, 6699). How many scales of electronegativity and how many compounds did Sproul investigate? What criteria were used to select compounds for the study? Which two electronegativity scales were found to give the best separation between areas of the triangle? What were the theoretical bases of these two scales? 2.5 When an He atom absorbs a photon to form the excited configuration 1s12s1 (here called He) a weak bond forms with another He atom to give the diatomic molecule HeHe. Construct a molecular orbital description of the bonding in this species. 2.6 In their article ‘Some observation on molecular orbital theory’ (J.F. Harrison and D. Lawson, J. Chem. Educ., 2005, 82, 1205) the authors discuss several limitations of the theory. What are these limitations? Sketch the MO diagram for Li2 given in the paper. Why do you think this version does not appear in textbooks? Use the data given in the paper to construct MO diagrams for B2 and C2. Do these versions differ from that in Fig. 2.17 in this textbook? Discuss any variations. 2.7 Construct an approximate molecular orbital energy diagram for a hypothetical planar form of NH3. You may refer to Resource section 4 to determine the form of the appropriate orbitals on the central N atom and on the triangle of H3 atoms. From a consideration of the atomic energy levels, place the N and H3 orbitals on either side of a molecular orbital energy-level diagram. Then use your judgement about the effect of bonding and antibonding interactions and energies of the parent orbitals to construct the molecular orbital energy levels in the centre of your diagram and draw lines indicating the contributions of the atomic orbitals to each molecular orbital. Ionization energies are I(H1s) 13.6 eV, I(N2s) 26.0 eV, and I(N2p) 13.4 eV. 2.8 (a) Use a molecular orbital program or input and output from software supplied by your instructor to construct a molecular orbital energy level diagram to correlate the MO (from the output) and AO (from the input) energies and indicate the occupancy of the MOs (in the manner of Fig. 2.17) for one of the following molecules: HF (bond length 92 pm), HCl (127 pm), or CS (153 pm). (b) Use the output to sketch the form of the occupied orbitals, showing signs of the AO lobes by shading and their amplitudes by means of size of the orbital. 2.9 Use software to perform an MO calculation on H3 by using the H energy given in Problem 2.7 and HH distances from NH3 (NH length 102 pm, HNH bond angle 107°) and then carry out the same 64 2 Molecular structure and bonding type of calculation for NH3. Use energy data for N2s and N2p orbitals from Problem 2.7. From the output plot the molecular orbital energy levels with proper symmetry labels and correlate them with the N orbitals and H3 orbitals of the appropriate symmetries. Compare the results of this calculation with the qualitative description in Problem 2.7. 2.10 Assign the lines in the UV photoelectron spectrum of CO shown in Fig. 2.39 and predict the appearance of the UV photoelectron spectrum of the SO molecule (see Section 8.3). 3σ 1π 2σ 11 13 15 I/eV 17 19 Fig. 2.39 The UV photoelectron spectrum of CO obtained using 21 eV radiation. The structures of simple solids An understanding of the chemistry of compounds in the solid state is central to the study of many important inorganic materials, such as alloys, simple metal salts, inorganic pigments, nanomaterials, zeolites, and high-temperature superconductors. This chapter surveys the structures adopted by atoms and ions in simple solids and explores why one arrangement may be preferred to another. We begin with the simplest model, in which atoms are represented by hard spheres and the structure of the solid is the outcome of stacking these spheres densely together. This ‘close-packed’ arrangement provides a good description of many metals and alloys and is a useful starting point for the discussion of numerous ionic solids. These simple solid structures can then be considered as building blocks for the construction of more complex inorganic materials. Introduction of partial covalent character into the bonding influences the choice of structure and thus trends in the adopted structural type correlate with the electronegativities of the constituent atoms. The chapter also describes some of the energy considerations that can be used to rationalize the trends in structure and reactivity. These arguments also systematize the discussion of the thermal stabilities and solubilities of ionic solids formed by the elements of Groups 1 and 2. Finally the electronic structures of materials are discussed in terms of an extension of molecular orbital theory to the almost infinite arrays of atoms found in solids. The classification of inorganic solids as conductors, semiconductors, and insulators is described in terms of this theory. The majority of inorganic compounds exist as solids and comprise ordered arrays of atoms, ions, or molecules. Some of the simplest solids are the metals, the structures of which can be described in terms of regular, space-filling arrangements of the metal atoms. These metal centres interact through metallic bonding, a type of bonding that can be described in two ways. One view is that bonding occurs in metals when each atom loses one or more electrons to a common ‘sea’. The strength of the bonding results from the combined attractions between all these freely moving electrons and the resulting cations. An alternative view is that metals are effectively enormous molecules with a multitude of atomic orbitals that overlap to produce molecular orbitals extending throughout the sample. Metallic bonding is characteristic of elements with low ionization energies, such as those on the left of the periodic table, through the d block, and into part of the p block close to the d block. Most of the elements are metals, but metallic bonding also occurs in many other solids, especially compounds of the d-metals such as their oxides and sulfides. Compounds such as the lustrous-red rhenium oxide ReO3 and ‘fool’s gold’ (iron pyrites, FeS2), illustrate the occurrence of metallic bonding in compounds. The familiar properties of a metal stem from the characteristics of its bonding and in particular the delocalization of electrons throughout the solid. Thus, metals are malleable (easily deformed by the application of pressure) and ductile (able to be drawn into a wire) because the electrons can adjust rapidly to relocation of the metal atom nuclei and there is no directionality in the bonding. They are lustrous because the electrons can respond almost freely to an incident wave of electromagnetic radiation and reflect it. In ionic bonding ions of different elements are held together in rigid, symmetrical arrays as a result of the attraction between their opposite charges. Ionic bonding also depends on electron loss and gain, so it is found typically in compounds of metals with electronegative elements. However, there are plenty of exceptions: not all compounds of metals are ionic and some compounds of nonmetals (such as ammonium nitrate) contain features of ionic 3 The description of the structures of solids 3.1 Unit cells and the description of crystal structures 3.2 The close packing of spheres 3.3 Holes in close-packed structures The structures of metals and alloys 3.4 Polytypism 3.5 Nonclose-packed structures 3.6 Polymorphism of metals 3.7 Atomic radii of metals 3.8 Alloys Ionic solids 3.9 Characteristic structures of ionic solids 3.10 The rationalization of structures The energetics of ionic bonding 3.11 Lattice enthalpy and the Born–Haber cycle 3.12 The calculation of lattice enthalpies 3.13 Comparison of experimental and theoretical values 3.14 The Kapustinskii equation 3.15 Consequences of lattice enthalpies Defects and nonstoichiometry 3.16 The origins and types of defects 3.17 Nonstoichiometric compounds and solid solutions The electronic structures of solids 3.18 The conductivities of inorganic solids 3.19 Bands formed from overlapping atomic orbitals 3.20 Semiconduction Further information 3.1 The Born–Mayer equation FURTHER READING EXERCISES PROBLEMS 66 3 The structures of simple solids bonding as well as covalent interactions. There are also materials that exhibit features of both ionic and metallic bonding. Ionic and metallic bonding are nondirectional, so structures where these types of bonding occur are most easily understood in terms of space-filling models that maximize, for example, the number and strength of the electrostatic interactions between the ions. The regular arrays of atoms, ions, or molecules in solids that produce these structures are best represented in terms of the repeating units that are produced as a result of the efficient methods of filling space. The description of the structures of solids The arrangement of atoms or ions in simple solid structures can often be represented by different arrangements of hard spheres. The spheres used to describe metallic solids represent neutral atoms because each cation is still surrounded by its full complement of electrons. The spheres used to describe ionic solids represent the cations and anions because there has been a substantial transfer of electrons from one type of atom to the other. 3.1 Unit cells and the description of crystal structures A crystal of an element or compound can be regarded as constructed from regularly repeating structural elements, which may be atoms, molecules, or ions. The ‘crystal lattice’ is the pattern formed by the points and used to represent the positions of these repeating structural elements. (a) Lattices and unit cells Key points: The lattice defines a network of identical points that has the translational symmetry of a structure. A unit cell is a subdivision of a crystal that, when stacked together without rotation or reflection, reproduces the crystal. (a) (b) Fig. 3.1 A two-dimensional solid and two choices of a unit cell. The entire crystal is reproduced by translational displacements of either unit cell, but (b) is generally preferred to (a) because it is smaller. A lattice is a three-dimensional, infinite array of points, the lattice points, each of which is surrounded in an identical way by neighbouring points, and which defines the basic repeating structure of the crystal. In some cases the structural unit may be centred on the lattice point, but that is not necessary. The crystal structure itself is obtained by associating one or more identical structural units (such as molecules or ions) with each lattice point. A unit cell of the crystal is an imaginary parallel-sided region (a ‘parallelepiped’) from which the entire crystal can be built up by purely translational displacements;1 unit cells so generated fit perfectly together with no space excluded. Unit cells may be chosen in a variety of ways but it is generally preferable to choose the smallest cell that exhibits the greatest symmetry. Thus, in the two-dimensional pattern in Fig. 3.1, a variety of unit cells may be chosen, each of which repeats the contents of the box under translational displacements. Two possible choices of repeating unit are shown but (b) would be preferred to (a) because it is smaller. The relationship between the lattice parameters in three dimensions as a result of the symmetry of the structure gives rise to the seven crystal systems (Table 3.1 and Fig. 3.2). All ordered structures adopted by compounds belong to one of these crystal systems; most of those described in this chapter, which deals with simple compositions and stoichiometries, belong to the higher symmetry cubic and hexagonal systems. The angles (, , ) and lengths (a, b, c) used to define the size and shape of a unit cell are the unit cell parameters (the ‘lattice parameters’); the angle between a and b is denoted , that between b and c is , and that between a and c is ; see the triclinic unit cell in Fig. 3.2. A primitive unit cell (denoted by the symbol P) has just one lattice point in the unit cell (Fig. 3.3) and the translational symmetry present is just that on the repeating unit cell. More complex lattice types are body-centred (I, from the German word innenzentriet, referring to the lattice point at the unit cell centre) and face-centred (F) with two and four lattice points in each unit cell, respectively, and additional translational symmetry beyond that of 1 A translation exists where it is possible to move an original figure or motif in a defined direction by a certain distance to produce an exact image. In this case a unit cell reproduces itself exactly by translation parallel to a unit cell edge by a distance equal to the unit cell parameter. The description of the structures of solids 67 Table 3.1 The seven crystal systems System Relationships between lattice parameters Unit cell defined by Essential symmetries Triclinic a≠b≠c ≠ ≠ ≠ 90° abc None Monoclinic a≠b≠c ≠ ≠ 90° 90° abc One twofold rotation axis and/or a mirror plane Orthorhombic a≠b≠c 90° abc Three perpendicular twofold axes and/or mirror planes Rhombohedral abc ≠ 90° One threefold rotation axis Tetragonal ab≠c 90° ac One fourfold rotation axis Hexagonal ab≠c 90° 120° ac One sixfold rotation axis Cubic a b c 90° a Four threefold rotation axes tetrahedrally arranged b a a c c a a Cubic a Tetragonal a Orthorhombic a a c c c a a b a Monoclinic b Triclinic 120 a a Rhombohedral (trigonal) Hexagonal Fig. 3.3 Lattice points describing the translational symmetry of a primitive cubic unit cell. Fig. 3.2 The seven crystal systems. the unit cell (Figs 3.4 and 3.5). The additional translational symmetry in the body-centred cubic (bcc) lattice, equivalent to the displacement ( + 12 , + 12 , + 12 ) from the unit cell origin at (0,0,0), produces a lattice point at the unit cell centre; note that the surroundings of each lattice point are identical, consisting of eight other lattice points at the corners of a cube. Centred lattices are sometimes preferred to primitive (although it is always possible to use a primitive lattice for any structure) for with them the essential structural symmetry of the cell is more apparent. We use the following rules to work out the number of lattice points in a three-dimensional unit cell. The same process can be used to count the number of atoms, ions, or molecules that the unit cell contains (Section 3.9). Fig. 3.4 Lattice points describing the translational symmetry of a body-centred cubic unit cell. A lattice point in the body of, that is fully inside, a cell belongs entirely to that cell and counts as 1. 2. A lattice point on a face is shared by two cells and contributes 1 2 to the cell. A lattice point on an edge is shared by four cells and hence contributes 41 . 4. A lattice point at a corner is shared by eight cells that share the corner, and so contributes 81 . Thus, for the face-centred cubic lattice depicted in Fig. 3.5 the total number of lattice points in the unit cell is (8 81 ) (6 12 ) 4. For the body-centred cubic lattice depicted in Fig. 3.4, the number of lattice points is (1 1) (8 81 ) 2. Fig. 3.5 Lattice points describing the translational symmetry of a face-centred cubic unit cell. 68 3 The structures of simple solids E X A M PL E 3 .1 Identifying lattice types S2– Determine the translational symmetry present in the structure of cubic ZnS (Fig. 3.6) and identify the lattice type to which this structure belongs. Zn2+ Answer We need to identify the displacements that, when applied to the entire cell, results in every atom arriving at an equivalent location (same atom type with the same coordination environment). In this case, the displacements (0, + 21 , + 21 ), ( + 21 , + 21 ,0), and ( + 21 ,0, − 21 ) have this effect. For example starting at the Zn2 ion on the bottom left-hand corner of the unit cell, which is surrounded by four S2 ions at the corners of a tetrahedron, and applying the translation ( + 21 ,0, − 21 ) we arrive at the Zn2 ion at the lower right-hand corner, which has an identical coordination to sulfur. These translations correspond to those of the face-centred lattice, so the lattice type is F. Fig 3.6 The cubic ZnS structure. Self-test 3.1 Determine the lattice type of CsCl (Fig. 3.7). Cl– Cs+ (b) Fractional atomic coordinates and projections Key point: Structures may be drawn in projection, with atom positions denoted by fractional coordinates. Fig 3.7 The cubic CsCl structure. W (a) The position of an atom in a unit cell is normally described in terms of fractional coordinates, coordinates expressed as a fraction of the length of a side of the unit cell. Thus, the position of an atom located at xa parallel to a, yb parallel to b, and zc parallel to c is denoted (x,y,z), with 0 x, y, z 1. Three-dimensional representations of complex structures are often difficult to draw and to interpret in two dimensions.2 A clearer method of representing three-dimensional structures on a two-dimensional surface is to draw the structure in projection by viewing the unit cell down one direction, typically one of the axes of the unit cell. The positions of the atoms relative to the projection plane are denoted by the fractional coordinate above the base plane and written next to the symbol defining the atom in the projection. If two atoms lie above each other, then both fractional coordinates are noted in parentheses. For example, the structure of body-centred tungsten, shown in three dimensions in Fig. 3.8a, is represented in projection in Fig. 3.8b. (0,1) E X A M PL E 3 . 2 Drawing a three-dimensional representation in projection 1/2 Convert the face-centred cubic lattice shown in Fig. 3.5 into a projection diagram. (b) Fig. 3.8 (a) The structure of metallic tungsten and (b) its projection representation. (0,1) (0,1) 1/2 Answer We need to identify the locations of the lattice points by viewing the cell from a position perpendicular to one of its faces. The faces of the cubic unit cell are square, so the projection diagram viewed from directly above the unit cell is a square. There is a lattice point at each corner of the unit cell, so the points at the corners of the square projection are labelled (0,1). There is a lattice point on each vertical face, which projects to points at fractional coordinate 21 on each edge of the projection square. There is a lattice point on the lower and on the upper horizontal face of the unit cell, which projects to two points at the centre of the square at 0 and 1, respectively, so we place a final point in the centre of a square and label it (0,1). The resulting projection is shown in Fig. 3.9. Self-test 3.2 Convert the projection diagram of the unit cell of the SiS2 structure shown in Fig. 3.10 into a three-dimensional representation. Fig. 3.9 The projection representation of an fcc unit cell. 3.2 The close packing of spheres S (0,1) (0,1) Si ¼ (0,1) ¼ Key points: The close packing of identical spheres can result in a variety of polytypes, of which hexagonal and cubic close-packed structures are the most common. Many metallic and ionic solids can be regarded as constructed from entities, such as atoms and ions, represented as hard spheres. If there is no directional covalent bonding, these spheres are free to pack together as closely as geometry allows and hence adopt a close-packed structure, a structure in which there is least unfilled space. The coordination number (CN) of a sphere ½ Fig. 3.10 The structure of silicon sulfide (SiS2). 2 Nearly all the structures in this text are available as rotatable, three-dimensional versions in the Online Resource Centre for this text. The description of the structures of solids in a close-packed arrangement (the ‘number of nearest neighbours’) is 12, the greatest number that geometry allows.3 When directional bonding is important, the resulting structures are no longer close-packed and the coordination number is less than 12. Consider first a single layer of identical spheres (Fig. 3.11). The greatest number of immediate neighbours is 6 and there is only one way of constructing this close-packed layer.4 A second close-packed layer of spheres is formed by placing spheres in the dips between the spheres of the first layer. (Note that only half the dips in the original layer are occupied, as there is insufficient space to place spheres into all the dips.) The third closepacked layer can be laid in either of two ways and hence can give rise to either of two polytypes, or structures that are the same in two dimensions (in this case, in the planes) but different in the third. Later we shall see that many different polytypes can be formed, but those described here are two very important special cases. In one polytype, the spheres of the third layer lie directly above the spheres of the first. This ABAB ...pattern of layers, where A denotes layers that have spheres directly above each other and likewise for B, gives a structure with a hexagonal unit cell and hence is said to be hexagonally close-packed (hcp, Figs 3.12a and 3.13). In the second polytype, the spheres of the third layer are placed above the gaps in the first layer. The second layer covers half the holes in the first layer and the third layer lies above the remaining holes. This arrangement results in an ABCABC ...pattern, where C denotes a layer that has spheres not directly above spheres of the A or the B layer positions (but they will be directly above another C type layer). This pattern corresponds to a structure with a cubic unit cell and hence it is termed cubic close-packed (ccp, Figs 3.12b and 3.14). Because each ccp unit cell has a sphere at one corner and one at the centre of each face, a ccp unit cell is sometimes referred to as face-centred cubic (fcc). 69 Fig. 3.11 A close-packed layer of hard spheres. B A (a) C B A A A note on good practice The descriptions ccp and fcc are often used interchangeably, although strictly ccp refers only to a close-packed arrangement whereas fcc refers to the lattice type of the common representation of ccp. Throughout this text the term ccp will be used to describe this close-packing arrangement. It will be drawn as the cubic unit cell, with the fcc lattice type, as this representation is easiest to visualize. The unoccupied space in a close-packed structure amounts to 26 per cent of the total volume (see Example 3.3). However, this unoccupied space is not empty in a real solid because electron density of an atom does not end as abruptly as the hard-sphere model suggests. The type and distribution of holes are important because many structures, including those of some alloys and many ionic compounds, can be regarded as formed from an expanded closepacked arrangement in which additional atoms or ions occupy all or some of the holes. E X A MPL E 3 . 3 Calculating the unoccupied space in a close-packed array Calculate the percentage of unoccupied space in a close-packed arrangement of identical spheres. Answer Because the space occupied by hard spheres is the same in the ccp and hcp arrays, we can choose the geometrically simpler structure, ccp, for the calculation. Consider Fig. 3.15. The spheres of radius r are in contact across the face of the cube and so the length of this diagonal is r 2r r 4r. The side of such a cell is 81/2r from Pythagoras’ theorem (the square of the length of the diagonal (4r)2) equals the sum of the squares of the two sides of length a, so 2 a2 (4r)2 giving a 81/2r), so the cell volume is (81/2r)3 83/2r3. The unit cell contains 81 of a sphere at each corner (for 8 81 1 in all) and half a sphere on each face (for 6 21 3 in all), for a total of 4. Because the volume of each sphere is 34 πr3, the total volume occupied by the spheres themselves is 4 34 πr3 163 πr3. The occupied fraction is therefore ( 163 πr3)/(83/2r3) 163 π/83/2, which evaluates to 0.740. The unoccupied fraction is therefore 0.260, corresponding to 26.0 per cent. (b) Fig. 3.12 The formation of two close-packed polytypes. (a) The third layer reproduces the first to give an ABA structure. (b) The third layer lies above the gaps in the first layer, giving an ABC structure. The different colours identify the different layers of identical spheres. Self-test 3.3 Calculate the fraction of space occupied by identical spheres in a primitive cubic unit cell. 3 That this arrangement, where each sphere has 12 nearest-neighbours, is the highest possible density of packing spheres was conjectured by Johannes Kepler in 1611; the proof was found only in 1998. 4 A good way of showing this yourself is to get a number of identical coins and push them together on a flat surface; the most efficient arrangement for covering the area is with six coins around each coin. This simple modelling approach can be extended to three dimensions by using any collection of identical spherical objects such as balls, oranges, or marbles. Fig. 3.13 The hexagonal close-packed (hcp) unit cell of the ABAB . . . polytype. The colours of the spheres correspond to the layers in Fig. 3.12a. 70 3 The structures of simple solids Fig. 3.16 The structure of solid C60 showing the packing of C60 polyhedra on an fcc unit cell. Fig. 3.14 The cubic close-packed (fcc) unit cell of the ABC . . . polytype. The colours of the spheres correspond to the layers in Fig 3.12b. 1 1 2 4r 8 2r 8 r 1 8 2r Fig. 3.15 The dimensions involved in the calculation of the packing fraction in a closepacked arrangement of identical spheres of radius r. (a) The ccp and hcp arrangements are the most efficient simple ways of filling space with identical spheres. They differ only in the stacking sequence of the close-packed layers and other, more complex, close-packed layer sequences may be formed by locating successive planes in different positions relative to their neighbours (Section 3.4). Any collection of identical atoms, such as those in the simple picture of an elemental metal, or of approximately spherical molecules, is likely to adopt one of these close-packed structures unless there are additional energetic reasons—specifically covalent interactions—for adopting an alternative arrangement. Indeed, many metals adopt such close-packed structures (Section 3.4), as do the solid forms of the noble gases (which are ccp). Almost spherical molecules, such as C60, in the solid state also adopt the ccp arrangement (Fig. 3.16), and so do many small molecules that rotate around their centres and thus appear spherical, such as H2, F2, and one form of solid oxygen, O2. 3.3 Holes in close-packed structures Key points: The structures of many solids can be discussed in terms of close-packed arrangements of one atom type in which the tetrahedral or octahedral holes are occupied by other atoms or ions. The ratio of spheres to octahedral holes to tetrahedral holes in a close-packed structure is 1:1:2. The feature of a close-packed structure that enables us to extend the concept to describe structures more complicated than elemental metals is the existence of two types of hole, or unoccupied space between the spheres. An octahedral hole lies between two triangles of spheres on adjoining layers (Fig. 3.17). For a crystal consisting of N spheres in a close-packed structure, there are N octahedral holes. The distribution of these holes in an hcp unit cell is shown in Fig. 3.18a and those in a ccp unit cell Fig. 3.18b. This illustration also shows that the hole has local octahedral symmetry in the sense that it is surrounded by six nearest-neighbour spheres with their centres at the corners of an octahedron. If each hard sphere has radius r, and if the close-packed spheres are to remain in contact, then each octahedral hole can accommodate a hard sphere representing another type of atom with a radius no larger than 0.414r. (b) Fig. 3.17 (a) An octahedral hole and (b) a tetrahedral hole formed in an arrangement of close-packed spheres. E X A M PL E 3 . 4 Calculating the size of an octahedral hole Calculate the maximum radius of a sphere that may be accommodated in an octahedral hole in a closepacked solid composed of spheres of radius r. Answer The structure of a hole, with the top spheres removed, is shown in Fig. 3.19a. If the radius of a sphere is r and that of the hole is rh, it follows from Pythagoras’ theorem that (r rh)2 (r rh)2 (2r)2 and therefore that (r rh)2 2r2, which implies that r rh 21/2r. That is, rh (21/2 1)r, which evaluates to 0.414r. Note that this is the permitted maximum size subject to keeping the close-packed The structures of metals and alloys spheres in contact; if the spheres are allowed to separate slightly while maintaining their relative positions, then the hole can accommodate a larger sphere. ( 14 , 34 ) Self-test 3.4 Show that the maximum radius of a sphere that can fit into a tetrahedral hole (see below) is rh 0.225r; base your calculation on Fig. 3.19b. A tetrahedral hole, T, Figs 3.17b and 3.20, is formed by a planar triangle of touching spheres capped by a single sphere lying in the dip between them. The tetrahedral holes in any closepacked solid can be divided into two sets: in one the apex of the tetrahedron is directed up (T) and in the other the apex points down (T). In an arrangement of N close-packed spheres there are N tetrahedral holes of each set and 2N tetrahedral holes in all. In a closepacked structure of spheres of radius r, a tetrahedral hole can accommodate another hard sphere of radius no greater than 0.225r (see Self-test 3.4). The location of tetrahedral holes, and the four nearest-neighbour spheres for one hole, in the hcp arrangement is shown in Fig. 3.20a and for a ccp arrangement in Fig. 3.20b. Individual tetrahedral holes in ccp and hcp structures are identical (because they are properties of two neighbouring close-packed layers) but in the hcp arrangement neighbouring T and T holes share a common tetrahedral face and are so close together that they are never occupied simultaneously. Where two types of sphere of different radius pack together (for instance, when cations and anions stack together), the larger spheres (normally the anions) can form a closepacked array and the smaller spheres occupy the octahedral or tetrahedral holes. Thus simple ionic structures can be described in terms of the occupation of holes in close-packed arrays (Section 3.9). 71 (a) 1 2 1 2 (0,1) The structures of metals and alloys X-ray diffraction studies (Section 8.1) reveal that many metallic elements have close-packed structures, indicating that the bonds between the atoms have little directional covalent character (Table 3.2, Fig. 3.21). One consequence of this close-packing is that metals often have high densities because the most mass is packed into the smallest volume. Indeed, the elements deep in the d block, near iridium and osmium, include the densest solids known under normal conditions of temperature and pressure. Osmium has the highest density of all the elements at 22.61 g cm3 and the density of tungsten, 19.25 g cm3, which is almost twice that of lead (11.3 g cm3), results in it being used as weighting material in fishing equipment and as ballast in high performance cars. (b) Fig. 3.18 (a) The location (represented by a hexagon) of the two octahedral holes in the hcp unit cell and (b) the locations (represented by hexagons) of the octahedral holes in the ccp unit cell. E X A MPL E 3 . 5 Calculating the density of a substance from a structure r + rh Calculate the density of gold, with a cubic close-packed array of atoms of molar mass M 196.97 g mol1 and a cubic lattice parameter a 409 pm. Answer Density is an intensive property; therefore the density of the unit cell is the same as the density of any macroscopic sample. We represent the ccp arrangement as a face-centred lattice with a sphere at each lattice point; there are four spheres associated with the unit cell. The mass of each atom is M/NA, where NA is Avogadro’s constant, and the total mass of the unit cell is 4M/NA. The volume of the cubic unit cell is a3. The mass density of the cell is 4M/NAa3. At this point we insert the data: a) 2r 4 (196.97 103 kg mol−1) 1.91104 kg m3 ρ= (6.0221023 mol1) (409 1012 m)3 That is, the density of the unit cell, and therefore of the bulk metal, is 19.1 g cm3. The experimental value is 19.2 g cm3, in good agreement with this calculated value. Self-test 3.5 Calculate the lattice parameter of silver assuming that it has the same structure as elemental gold but a density of 10.5 g cm3. A note on good practice It is always best to proceed symbolically with a calculation for as long as possible: that reduces the risk of numerical error and gives an expression that can be used in other circumstances. 2r r + rh (b) Fig. 3.19 The distances used to calculate the size of (a) an octahedral hole and (b) a tetrahedral hole. 72 3 The structures of simple solids 3.4 Polytypism Key point: Polytypes involving complex stacking arrangements of close-packed layers occur for some metals. ( 18 , 78 ) ( 38 , 58 ) (a) Which of the common close-packed polytypes, hcp or ccp, a metal adopts depends on the details of the electronic structure of its atoms, the extent of interaction between secondnearest-neighbours, and the potential for some directional character in the bonding. Indeed, a close-packed structure need not be either of the common ABAB ...or ABCABC ... polytypes. An infinite range of close-packed polytypes can in fact occur, as the layers may stack in a more complex repetition of A, B, and C layers or even in some permissible random sequence. The stacking cannot be a completely random choice of A, B, and C sequences, however, because adjacent layers cannot have exactly the same sphere positions; for instance, AA, BB, and CC cannot occur because spheres in one layer must occupy dips in the adjacent layer. Cobalt is an example of a metal that displays this more complex polytypism. Above 500ºC, cobalt is ccp but it undergoes a transition when cooled. The structure that results is a nearly randomly stacked set (for instance, ABACBABABC ...) of close-packed layers of Co atoms. In some samples of cobalt the polytypism is not random, as the sequence of planes of atoms repeats after several hundred layers. The long-range repeat may be a consequence of a spiral growth of the crystal that requires several hundred turns before a stacking pattern is repeated. 3.5 Nonclose-packed structures Key points: A common nonclose-packed metal structure is body-centred cubic; a primitive cubic structure is occasionally encountered. Metals that have structures more complex than those described so far can sometimes be regarded as slightly distorted versions of simple structures. Table 3.2 The crystal structures adopted by metals under normal conditions ( 14 , 34 ) 1 Crystal structure Element Hexagonal close-packed (hcp) Be, Ca, Co, Mg, Ti, Zn Cubic close-packed (ccp) Ag, Al, Au, Cd, Cu, Ni, Pb, Pt Body-centred cubic (bcc) Ba, Cr, Fe, W, alkali metals Primitive cubic (cubic-P) Po hcp ccp bcc 2 2 13 14 3 3 (b) Fig. 3.20 (a) The locations (represented by triangles) of the tetrahedral holes in the hcp unit cell and (b) the locations of the tetrahedral holes in the ccp unit cell. 4 5 6 7 8 9 10 11 12 4 5 6 7 Fig. 3.21 The structures of the metallic elements at room temperature. Elements with more complex structures are left blank. The structures of metals and alloys Not all elemental metals have structure based on close-packing and some other packing patterns use space nearly as efficiently. Even metals that are close-packed may undergo a phase transition to a less closely packed structure when they are heated and their atoms undergo large-amplitude vibrations. One commonly adopted arrangement has the translational symmetry of the bodycentred cubic lattice and is known as the body-centred cubic structure (cubic-I or bcc) in which a sphere is at the centre of a cube with spheres at each corner (Fig. 3.22a). Metals with this structure have a coordination number of 8 because the central atom is in contact with the atoms at the corners of the unit cell. Although a bcc structure is less closely packed than the ccp and hcp structures (for which the coordination number is 12), the difference is not very great because the central atom has six second-nearest neighbours, at the centres of the adjacent unit cells, only 15 per cent further away. This arrangement leaves 32 per cent of the space unfilled compared with 26 per cent in the close-packed structures (see Example 3.3). A bcc structure is adopted by 15 of the elements under standard conditions, including all the alkali metals and the metals in Groups 5 and 6. Accordingly, this simple arrangement of atoms is sometimes referred to as the ‘tungsten type’. The least common metallic structure is the primitive cubic (cubic-P) structure (Fig. 3.23), in which spheres are located at the lattice points of a primitive cubic lattice, taken as the corners of the cube. The coordination number of a cubic-P structure is 6. One form of polonium (-Po) is the only example of this structure among the elements under normal conditions. Solid mercury (-Hg), however, has a closely related structure: it is obtained from the cubic-P arrangement by stretching the cube along one of its body diagonals (Fig. 3.24a); a second form of solid mercury (-Hg) has a structure based on the bcc arrangement but compressed along one cell direction (Fig. 3.24b). Although antimony and bismuth normally have structures based on layers of atoms, both convert to a cubic-P structure under pressure and then to close-packed structures at even higher pressures. Metals that have structures more complex than those described so far can sometimes be regarded, like solid mercury, as having slightly distorted versions of simple structures. Zinc and cadmium, for instance, have almost hcp structures, but the planes of close-packed atoms are separated by a slightly greater distance than in perfect hcp. This difference suggests stronger bonding between the close-packed atoms in the plane than between the planes: the bonding draws these atoms together and, in doing so, squeezes out the atoms of the neighbouring layers. 73 (a) 1 2 (b) (0,1) Fig. 3.22 (a) A bcc structure unit cell and (b) its projection representation. (a) (b) (0,1) Fig. 3.23 (a) A primitive cubic unit cell and (b) its projection representation. Hg 3.6 Polymorphism of metals Key points: Polymorphism is a common consequence of the low directionality of metallic bonding. At high temperatures a bcc structure is common for metals that are close-packed at low temperatures on account of the increased amplitude of atomic vibrations. The low directionality of the bonds that metal atoms may form accounts for the wide occurrence of polymorphism, the ability to adopt different crystal forms under different conditions of pressure and temperature. It is often, but not universally, found that the most closely packed phases are thermodynamically favoured at low temperatures and that the less closely packed structures are favoured at high temperatures. Similarly, the application of high pressure leads to structures with higher packing densities, such as ccp and hcp. The polymorphs of metals are generally labelled , , ,...with increasing temperature. Some metals revert to a low-temperature form at higher temperatures. Iron, for example, shows several solid–solid phase transitions; -Fe, which is bcc, occurs up to 906°C, -Fe, which is ccp, occurs up to 1401°C, and then -Fe occurs again up to the melting point at 1530°C. The hcp polymorph, -Fe, is formed at high pressures and was to believed to be the form that exists at the Earth’s core, but recent studies indicate that a bcc polymorph is more likely (Box 3.1). The bcc structure is common at high temperatures for metals that are close-packed at low temperatures because the increased amplitude of atomic vibrations in the hotter solid results in a less close-packed structure. For many metals (among them Ca, Ti, and Mn) the transition temperature is above room temperature; for others (among them Li and Na), the transition temperature is below room temperature. It is also found empirically that a bcc structure is favoured by metals with a small number of valence electrons per orbital. (a) (b) Fig 3.24 The structures of (a) -mercury and (b) -mercury that are closely related to the unit cells with primitive cubic and body-centred cubic lattices, respectively. 74 3 The structures of simple solids B OX 3 .1 Metals under pressure The Earth has an innermost core about 1200 km in diameter that consists of solid iron and is responsible for generating the planet’s powerful magnetic field. The pressure at the centre of the Earth has been calculated to be around 370 GPa (about 3.7 million atm) at a temperature of 50006500°C. The polymorph of iron that exists under these conditions has been much debated with information from theoretical calculations and measurements using seismology. The current thinking is that the iron core consists of the bodycentred cubic polymorph. It has been proposed that this exists either as a giant crystal or a large number of oriented crystals such that the long diagonal of the bcc unit cell aligns along the Earth’s axis of rotation (Fig. B3.1, left). The study of the structures and polymorphism of an element and compounds under high pressure conditions goes beyond the study of the Earth’s core. Hydrogen, when subjected to pressures similar to those at the Earth’s core, is predicted to become a metallic solid, similar to the alkali 6378 km 1278 km 0 Inner core Fig. B3.1 metals, and the cores of planets such as Jupiter have been predicted to contain hydrogen in this form. When pressures of over 55 GPa are applied to iodine the I2 molecules dissociate and adopt the simple face-centred cubic structure; the element becomes metallic and is a superconductor below 1.2 K. 3.7 Atomic radii of metals Key point: The Goldschmidt correction converts atomic radii of metals to the value they would have in a close-packed structure with 12-fold coordination. Table 3.3 The variation of radius with coordination number Coordination number 12 Relative radius 1 8 0.97 6 0.96 4 0.88 An informal definition of the atomic radius of a metallic element was given in Section 1.9 as half the distance between the centres of adjacent atoms in the solid. However, it is found that this distance generally increases with the coordination number of the lattice. The same atom in structures with different coordination numbers may therefore appear to have different radii, and an atom of an element with coordination number 12 appears bigger than one with coordination number 8. In an extensive study of internuclear separations in a wide variety of polymorphic elements and alloys, V. Goldschmidt found that the average relative radii are related as shown in Table 3.3. It is desirable to put all elements on the same footing when comparing trends in their characteristics; that is when comparing the intrinsic properties of their atoms rather than the properties that stem from their environment. Therefore, it is common to adjust the empirical internuclear separation to the value that would be expected if the element were in fact close-packed (with coordination number 12). ■ A brief illustration. The empirical atomic radius of Na is 185 pm, but that is for the bcc structure in which the coordination number is 8. To adjust to 12-coordination we multiply this radius by 1/0.97 1.03 and obtain 191 pm as the radius that a Na atom would have if it were in a close-packed structure. ■ Goldschmidt radii of the elements were in fact the ones listed in Table 1.4 as ‘metallic radii’ and used in the discussion of the periodicity of atomic radius (Section 1.9). The essential features of that discussion to bear in mind now, with ‘atomic radius’ interpreted as Goldschmidtcorrected metallic radius in the case of metallic elements, are that metallic radii generally increase down a group and decrease from left to right across a period. As remarked in Section 1.9, trends in atomic radii reveal the presence of the lanthanide contraction in Period 6, with atomic radii of the elements that follow the lanthanoids found to be smaller than simple extrapolation from earlier periods would suggest. As also remarked there, this contraction can be traced to the poor shielding effect of f electrons. A similar contraction occurs across each row of the d block. E X A M PL E 3 .6 Calculating a metallic radius The cubic unit cell parameter, a, of polonium (-Po) is 335 pm. Use the Goldschmidt correction to calculate a metallic radius for this element. Answer We need to infer the radius of the atoms from the dimensions of the unit cell and the coordination number, and then apply a correction to coordination number 12. Because the Po atoms of radius r are in contact along the unit cell edges, the length of the primitive cubic unit cell is 2r. Thus, the metallic radius of 6-coordinate Po is a/2 with a 335 pm. The conversion factor from 6-fold to 12-fold coordination from Table 3.3 (1/0.960) gives the metallic radius of Po as 12 335 pm 1/0.960 174 pm. Self-test 3.6 Predict the lattice parameter for Po when it adopts a bcc structure. 75 The structures of metals and alloys 3.8 Alloys 3 2 Δ An alloy is a blend of metallic elements prepared by mixing the molten components and then cooling the mixture to produce a metallic solid. Alloys may be homogeneous solid solutions, in which the atoms of one metal are distributed randomly among the atoms of the other, or they may be compounds with a definite composition and internal structure. Alloys typically form from two electropositive metals, so they are likely to be located towards the bottom left-hand corner of a Ketelaar triangle (Fig. 3.25). Solid solutions are classified as either ‘substitutional’ or ‘interstitial’. A substitutional solid solution is a solid solution in which atoms of the solute metal occupy some of the locations of the solvent metal atoms (Fig. 3.26a). An interstitial solid solution is a solid solution in which the solute atoms occupy the interstices (the holes) between the solvent atoms (Fig. 3.26b). However, this distinction is not particularly fundamental because interstitial atoms often lie in a definite array (Fig. 3.26c), and hence can be regarded as a substitutional version of another structure. Some of the classic examples of alloys are brass (up to 38 atom per cent Zn in Cu), bronze (a metal other than Zn or Ni in Cu; casting bronze, for instance, is 10 atom per cent Sn and 5 atom per cent Pb), and stainless steel (over 12 atom per cent Cr in Fe). 1 Alloys 0 1 2 3 4 mean Fig. 3.25 The approximate locations of alloys in a Ketelaar triangle. (a) Substitutional solid solutions Key point: A substitutional solid solution involves the replacement of one type of metal atom in a structure by another. (a) Substitutional solid solutions are generally formed if three criteria are fulfilled: 1. The atomic radii of the elements are within about 15 per cent of each other. 2. The crystal structures of the two pure metals are the same; this similarity indicates that the directional forces between the two types of atom are compatible with each other. 3. The electropositive characters of the two components are similar; otherwise compound formation, where electrons are transferred between species, would be more likely. Thus, although sodium and potassium are chemically similar and have bcc structures, the atomic radius of Na (191 pm) is 19 per cent smaller than that of K (235 pm), and the two metals do not form a solid solution. Copper and nickel, however, two neighbours late in the d block, have similar electropositive character, similar crystal structures (both ccp), and similar atomic radii (Ni 125 pm, Cu 128 pm, only 2.3 per cent different), and form a continuous series of solid solutions, ranging from pure nickel to pure copper. Zinc, copper’s other neighbour in Period 4, has a similar atomic radius (137 pm, 7 per cent larger), but it is hcp, not ccp. In this instance, zinc and copper are partially miscible and form solid solutions known as ‘-brasses’ of composition Cu1xZnx with 0 x 0.38 and the same structural type as pure copper. (b) Interstitial solid solutions of nonmetals Key point: In an interstitial solid solution, additional small atoms occupy holes within the lattice of the original metal structure. Interstitial solid solutions are often formed between metals and small atoms (such as boron, carbon, and nitrogen) that can inhabit the interstices in the structure. The small atoms enter the host solid with preservation of the crystal structure of the original metal and without the transfer of electrons and formation of ionic species. There is either a simple whole-number ratio of metal and interstitial atoms (as in tungsten carbide, WC) or the small atoms are distributed randomly in the available spaces or holes in the structure between the packed atoms. The former substances are true compounds and the latter can be considered as interstitial solid solutions or, on account of the variation in the atomic ratio of the two elements, nonstoichiometric compounds (Section 3.17). Considerations of size can help to decide where the formation of an interstitial solid solution is likely to occur. Thus, the largest solute atom that can enter a close-packed solid without distorting the structure appreciably is one that just fits an octahedral hole, which as we have seen has radius 0.414r. For small atoms such as B, C, or N the atomic radii of the possible host metal atom structures include those of the d-metals such as Fe, Co, and Ni. One important class of materials of this type consists of carbon steels in which C atoms occupy some of the octahedral holes in the Fe bcc lattice. Carbon steels typically contain (b) (c) Fig. 3.26 (a) Substitutional and (b) interstitial alloys. (c) In some cases, an interstitial alloy may be regarded as a substitutional alloy derived from another lattice. 76 3 The structures of simple solids B OX 3 . 2 Steels Steels are alloys of iron, carbon, and other elements. They are classified as mild and medium-or high-carbon steels according to the percentage of carbon they contain. Mild steels contain up to 0.25 atom per cent C, medium-carbon steels contain 0.25 to 0.45 atom per cent, and high-carbon steels contain 0.45 to 1.50 atom per cent. The addition of other metals to these carbon steels can have a major effect on the structure, properties, and therefore applications of the steel. Examples of metals added to carbon steels, so forming ‘stainless steels’, are listed in the table. Stainless steels are also classified by their crystalline structures, which are controlled by factors such as the rate of cooling following their formation in a furnace and the type of added metal. Thus pure iron adopts different polymorphs (Section 3.6) depending on temperature and some of these high-temperature structures can be stabilized at room temperature in steels or by quenching (cooling very rapidly). Stainless steels with the austenite structure comprise over 70 per cent of total stainless steel production. Austenite is a solid solution of carbon and iron that exists in steel above 723°C and is ccp iron with about 2 per cent of the octahedral holes filled with carbon. As it cools, it breaks down into other materials including ferrite and martensite as the solubility of carbon in the iron drops to below 1 atom per cent. The rate of cooling determines the relative proportions of these materials and therefore the mechanical properties (for example, hardness and tensile strength) of the steel. The addition of certain other metals, such as Mn, Ni, and Cr, can allow the austenitic structure to survive cooling to room temperature. These steels contain a maximum of 0.15 atom per cent C and typically 1020 atom per cent Cr plus Ni or Mn as a substitutional solid solution; they can retain an austenitic structure at all temperatures from the cryogenic region to the melting point of the alloy. Metal Atom percentage added Effect on properties Copper 0.21.5 Improves atmospheric corrosion resistance Nickel 0.11 Benefits surface quality Niobium 0.02 Increases tensile strength and yield point Nitrogen 0.0030.012 Improves strength Manganese 0.21.6 Improves strength Vanadium Up to 0.12 Increases strength between 0.2 and 1.6 atom per cent C. With increasing carbon content they become harder and stronger but less malleable (Box 3.2). 3 (c) Intermetallic compounds Δ 2 1 A typical composition, 18 atom per cent Cr and 8 atom per cent Ni, is known as 18/8 stainless. Ferrite is -Fe with only a very small level of carbon, less than 0.1 atom per cent, with a bcc iron crystal structure. Ferritic stainless steels are highly corrosion resistant, but far less durable than austenitic grades. They contain between 10.5 and 27 atom per cent Cr and some Mo, Al, or W. Martensitic stainless steels are not as corrosion resistant as the other two classes, but are strong and tough as well as highly machineable, and can be hardened by heat treatment. They contain 11.5 to 18 atom per cent Cr and 12 atom per cent C which is trapped in the iron structure as a result of quenching compositions with the austenite structure type. The martensitic crystal structure is closely related to that of ferrite but the unit cell is tetragonal rather than cubic. KGe Key point: Intermetallic compounds are alloys in which the structure adopted is different from the structures of either component metal. Zintl Alloys 0 1 2 3 4 mean Fig. 3.27 The approximate locations of Zintl phases in a Ketelaar triangle. The point marks the location of one exemplar, KGe. There are materials formed between two metals that are best regarded as actual compounds despite the similarity of their electropositive nature. For instance, when some liquid mixtures of metals are cooled, they form phases with definite structures that are often unrelated to the parent structure. These phases are called intermetallic compounds. They include -brass (CuZn) and compounds of composition MgZn2, Cu3Au, NaTl, and Na5Zn21. Note that some of these intermetallic compounds contain a very electropositive metal in combination with a less electropositive metal (for example, Na and Zn), and in a Ketelaar triangle lie above the true alloys (Fig. 3.27). Such combinations are called Zintl phases. These compounds are not fully ionic (although they are often brittle) and have some metallic properties, including lustre. A classic example of a Zintl phase is KGe with the structure shown in Fig. 3.28. E X A M PL E 3 .7 Composition, lattice type and unit cell content of iron and its alloys K+ What are the lattice types and unit cell contents of (a) iron metal (Fig. 3.29a) and (b) the iron/chromium alloy, FeCr (Fig. 3.29b)? Ge4– 4 Fig. 3.28 The structure of the Zintl phase KGe showing the Ge44 tetrahedral units and interspersed K ions. Answer We need to identify the translation symmetry of the unit cell and to count the net numbers of atoms present. (a) The iron structure consists of Fe atoms distributed over the sites at the centre and corners of a cubic unit cell with eightfold coordination to the nearest neighbours. All the occupied sites are equivalent, so the structure has the translational symmetry of a bcc lattice. The structure type is the bcc structure. The Fe atom at the centre counts 1 and the eight Fe atoms at the cell corners count 8 81 1, 77 Ionic solids so there are two Fe atoms in the unit cell. (b) For FeCr, the atom at the centre of the unit cell (Cr) is different from the one on the corner (Fe) and thus the translational symmetry present is that of the entire unit cell (not half unit cell displacements characteristic of a bcc structure), so the lattice type is primitive, P. There is one Cr atom and 8 81 1 Fe atom in the unit cell, in accord with the stoichiometry FeCr. Fe Self-test 3.7 What are the stoichiometry and lattice type of the iron/chromium alloy shown in Fig. 3.29c ? (a) Ionic solids Fe Cr Key points: The ionic model treats a solid as an assembly of oppositely charged spheres that interact by nondirectional electrostatic forces; if the thermodynamic properties of the solid calculated on this model agree with experiment, then the compound is normally considered to be ionic. Ionic solids, such as NaCl and KNO3, are often recognized by their brittleness because the electrons made available by cation formation are localized on a neighbouring anion instead of contributing to an adaptable, mobile electron sea. Ionic solids also commonly have high melting points and most are soluble in polar solvents, particularly water. However, there are exceptions: CaF2, for example, is a high-melting ionic solid but it is insoluble in water. Ammonium nitrate, NH4NO3, is ionic in terms of its interactions between the ammonium and nitrate ions, but melts at 170°C. Binary ionic materials are typical of elements with large electronegativity differences, typically ∆ 3, and such compounds are therefore likely to be found at the top corner of a Ketelaar triangle (Fig. 3.27). The classification of a solid as ionic is based on comparison of its properties with those of the ionic model, which treats the solid as an assembly of oppositely charged, hard spheres that interact primarily by nondirectional electrostatic forces (Coulombic forces) and repulsions between complete shells in contact. If the thermodynamic properties of the solid calculated on this model agree with experiment, then the solid may be ionic. However, it should be noted that many examples of coincidental agreement with the ionic model are known, so numerical agreement alone does not imply ionic bonding. The nondirectional nature of electrostatic interactions between ions in an ionic solid contrast with those present in a covalent solid, where the symmetries of the atomic orbitals play a strong role in determining the geometry of the structure. However, the assumption that ions can be treated as perfectly hard spheres (of fixed radius for a particular ion type) that have no directionality in their bonding, is far from true for real ions. For example, with halide anions some directionality might be expected in their bonding that results from the orientations of their p orbitals, and large ions, such as Cs and I, are easily polarizable so do not behave as hard spheres. Even so, the ionic model is a useful starting point for describing many simple structures. We start by describing some common ionic structures in terms of the packing of hard spheres of different sizes and opposite charges. After that, we see how to rationalize the structures in terms of the energetics of crystal formation. The structures described have been obtained by using X-ray diffraction (Section 8.1), and were among the first substances to be examined in this way. 3.9 Characteristic structures of ionic solids The ionic structures described in this section are prototypes of a wide range of solids. For instance, although the rock-salt structure takes its name from a mineral form of NaCl, it is characteristic of numerous other solids (Table 3.4). Many of the structures can be regarded as derived from arrays in which the larger of the ions, usually the anions, stack together in ccp or hcp patterns and the smaller counter-ions (usually the cations) occupy the octahedral or tetrahedral holes in the lattice (Table 3.5). Throughout the following discussion, it will be helpful to refer back to Figs 3.18 and 3.20 to see how the structure being described is related to the hole patterns shown there. The close-packed layers usually need to expand to accommodate the counter-ions but this expansion is often a minor perturbation of the anion arrangement, which will still be referred to as ccp and hcp. This expansion avoids some of the strong repulsion between the identically charged ions and also allows larger species to be inserted into the holes between larger ions. Overall, examining the opportunities for hole-filling in a close-packed array of the larger ion type provides an excellent starting point for the descriptions of many simple ionic structures. (b) Fe Cr (c) Fig. 3.29 The structures of (a) iron, (b) FeCr, and (c) as Fe, Cr alloy, see Self-test 3.7. 78 3 The structures of simple solids Table 3.4 The crystal structures of compounds Crystal structure Example Antifluorite K2O, K2S, Li2O, Na2O, Na2Se, Na2S Caesium chloride CsCl, TlI, CsAu, CsCN, CuZn , NbO Fluorite CaF2,UO2, HgF2, LaH2, PbO2 Nickel arsenide NiAs, NiS, FeS, PtSn, CoS Perovskite CaTiO3 (distorted), SrTiO3, PbZrO3, LaFeO3, LiSrH3, KMnF3 Rock salt NaCl, KBr, RbI, AgCl, AgBr, MgO, CaO, TiO, FeO, NiO, SnAs, UC, ScN Rutile TiO2, MnO2, SnO2,WO2, MgF2, NiF2 Sphalerite (zinc blende) ZnS, CuCl, CdS, HgS, GaP, InAs Spinel MgAl2O4, ZnFe2O4, ZnCr2S4 Wurtzite ZnS, ZnO, BeO, MnS, AgI, AlN, SiC, NH4F The substance in bold type is the one that gives its name to the structure. Table 3.5 The relation of structure to the filling of holes Na+ Close-packing type Hole filling Structure type (exemplar) Cubic (ccp) All octahedral Rock salt (NaCl) Cl– Hexagonal (hcp) (a) Half tetrahedral Sphalerite (ZnS) All octahedral Nickel arsenide (NiAs); with some distortion from perfect hcp CdI2 Half octahedral Rutile (TiO2); with some distortion from perfect hcp All tetrahedral No structure exists: tetrahedral holes share faces Half tetrahedral Wurtzite (ZnS) Key points: Important structures that can be expressed in terms of the occupation of holes include the rock-salt, caesium-chloride, sphalerite, fluorite, wurtzite, nickel-arsenide, and rutile structures. (0,1) (0,1) Fluorite (CaF2) CdCl2 (a) Binary phases, AXn 1 2 1 2 All tetrahedral Half octahedral 1 2 (0,1) (b) Fig. 3.30 (a) The rock-salt structure and (b) its projection representation. Note the relation of this structure to the fcc structure in Fig. 3.18 with an atom in each octahedral hole. The simplest ionic compounds contain just one type of cation (A) and one type of anion (X) present in various ratios covering compositions such as AX and AX2. Several different structures may exist for each of these compositions, depending on the relative sizes of the cations and anions and which holes are filled and to what degree in the close-packed array (Table 3.5). We start by considering compositions AX with equal numbers of cations and anions and then consider AX2, the other commonly found stoichiometry. The rock-salt structure is based on a ccp array of bulky anions with cations in all the octahedral holes (Fig. 3.30). Alternatively, it can be viewed as a structure in which the anions occupy all the octahedral holes in a ccp array of cations. As the number of octahedral holes in a close-packed array is equal to the number of ions forming the array (the X ions), then filling them all with A ions yields the stoichiometry AX. Because each ion is surrounded by an octahedron of six counter-ions, the coordination number of each type of ion is 6 and the structure is said to have (6,6)-coordination. In this notation, the first number in parentheses is the coordination number of the cation, and the second number is the coordination number of the anion. The rock-salt structure can still be described as having a face-centred cubic lattice after this hole filling because the Ionic solids translational symmetry demanded by this lattice type is preserved when all the octahedral sites are occupied. To visualize the local environment of an ion in the rock-salt structure, we should note that the six nearest neighbours of the central ion of the cell shown in Fig. 3.30 lie at the centres of the faces of the cell and form an octahedron around the central ion. All six neighbours have a charge opposite to that of the central ion. The 12 second-nearest neighbours of the central ion, those next further away, are at the centres of the edges of the cell, and all have the same charge as the central ion. The eight third-nearest neighbours are at the corners of the unit cell, and have a charge opposite to that of the central ion. We can use the rules described in Section 3.1 to determine the composition of the unit cell, the number of atoms or ions of each type present. ■ A brief illustration. In the unit cell shown in Fig. 3.30, there are the equivalent of 8 8 6 4 Na ions and 12 1 4 Cl ions. Hence, each unit cell contains four NaCl formula units. The number of formula units present in the unit cell is commonly denoted Z, so in this case Z 4. ■ 1 1 4 2– C2 79 Ca2+ 1 2 The rock-salt arrangement is not just formed for simple monatomic species such as M and X but also for many 1:1 compounds in which the ions are complex units such as [Co(NH3)6][TlCl6]. The structure of this compound can be considered as an array of closepacked octahedral [TlCl6]3 ions with [Co(NH3)6]3 ions in all the octahedral holes. Similarly, compounds such as CaC2, CsO2, KCN, and FeS2 all adopt structures closely related to the rock-salt structure with alternating cations and complex anions (C22, O2, CN, and S22, respectively), although the orientation of these linear diatomic species can lead to elongation of the unit cell and elimination of the cubic symmetry (Fig. 3.31). Further compositional flexibility, but retaining a rock-salt type of structure, can come from having more than one cation or anion type while maintaining the overall 1:1 ratio between opposite ions of opposite charge, as in LiNiO2, which is equivalent to (Li1/2 Ni1/2)O. Much less common than the rock-salt structure for compounds of stoichiometry AX is the caesium-chloride structure (Fig. 3.32), which is possessed by CsCl, CsBr, and CsI, as well as some other compounds formed of ions of similar radii to these, including TlI (see Table 3.4). The caesium-chloride structure has a cubic unit cell with each corner occupied by an anion and a cation occupying the ‘cubic hole’ at the cell centre (or vice versa); as a result, Z 1. An alternative view of this structure is as two interlocking primitive cubic cells, one of Cs and the other of Cl. The coordination number of both types of ion is 8, so the structure is described as having (8,8)-coordination. The radii are so similar that this energetically highly favourable coordination is feasible, with numerous counter-ions adjacent to a given ion. Note that NH4Cl also forms this structure despite the relatively small size of the NH4 ion because the cation can form hydrogen bonds with four of the Cl ions at the corners of the cube (Fig. 3.33). Many 1:1 alloys, such as AlFe and CuZn, have a caesium-chloride arrangement of the two metal atom types. The sphalerite structure (Fig. 3.34), which is also known as the zinc-blende structure, takes its name from a mineral form of ZnS. Like the rock-salt structure it is based on an expanded ccp anion arrangement but now the cations occupy one type of tetrahedral hole, one half the tetrahedral holes present in a close-packed structure. Each ion is surrounded by four neighbours and so the structure has (4,4)-coordination and Z 4. Fig. 3.31 The structure of CaC2 is based on the rock-salt structure but is elongated in the direction parallel to the axes of the C22 ions. Cl– Cs+ (a) (0,1) 1 2 (b) Fig. 3.32 (a) The caesium-chloride structure. The corner lattice points, which are shared by eight neighbouring cells, are surrounded by eight nearest-neighbour lattice points. The anion occupies a cubic hole in a primitive cubic lattice. (b) Its projection. ■ A brief illustration. To count the ions in the unit cell shown in the sphalerite structure shown in Fig. 3.34, we draw up the following table: Cl– Location (share) Number of cations Number of anions Body (1) 41 0 Face ( 21) 0 6 Edge ( 41) 0 0 Vertex ( 81) 0 8 Total: 4 4 Contribution NH4+ 4 1 2 3 1 8 1 0 8 There are four cations and four anions in the unit cell. This ratio is consistent with the chemical formula ZnS, with Z 4. ■ Fig. 3.33 The structure of ammonium chloride, NH4Cl, reflects the ability of the tetrahedral NH4 ion to form hydrogen bonds to the tetrahedral array of Cl ions around it. 80 3 The structures of simple solids Zn2+ S2– (a) (0,1) 1 4 3 4 1 2 (0,1) 1 4 3 4 (b) Fig. 3.34 (a) The sphalerite (zinc-blende) structure and (b) its projection representation. Note its relation to the ccp lattice in Fig. 3.18a with half the tetrahedral holes occupied by Zn2 ions. Zn2+ S2– (a) (0,1) 1 2 3 8 7 8 (b) The wurtzite structure (Fig. 3.35) takes its name from another polymorph of zinc sulfide. It differs from the sphalerite structure in being derived from an expanded hcp anion array rather than a ccp array, but as in sphalerite the cations occupy half the tetrahedral holes; that is just one of the two types (either T or T as discussed in Section 3.3). This structure, which has (4,4)-coordination, is adopted by ZnO, AgI, and one polymorph of SiC, as well as several other compounds (Table 3.4). The local symmetries of the cations and anions are identical with respect to their nearest neighbours in wurtzite and sphalerite but differ at the second-nearest neighbours. The nickel-arsenide structure (NiAs, Fig. 3.36) is also based on an expanded, distorted hcp anion array, but the Ni atoms now occupy the octahedral holes and each As atom lies at the centre of a trigonal prism of Ni atoms. This structure is adopted by NiS, FeS, and a number of other sulfides. The nickel-arsenide structure is typical of MX compounds that contain polarizable ions and are formed from elements with smaller electronegativity differences than elements that, as ions, adopt the rock-salt structure. Compounds that form this structure type lie in the ‘polarized ionic salt area’ of a Ketelaar triangle (Fig. 3.37). There is also potential for some degree of metal–metal bonding between metal atoms in adjacent layers and this structure type (or distorted forms of it) is also common for a large number of alloys based on d- and p-block elements. A common AX2 structural type is the fluorite structure, which takes its name from its exemplar, the naturally occurring mineral fluorite, CaF2. In fluorite, the Ca2 ions lie in an expanded ccp array and the F ions occupy all the tetrahedral holes (Fig. 3.38). In this description it is the cations that are close-packed because the F anions are small. The lattice has (8,4)-coordination, which is consistent with there being twice as many anions as cations. The anions in their tetrahedral holes have four nearest neighbours and the cation site is surrounded by a cubic array of eight anions. An alternative description of the structure is that the anions form a nonclose-packed, primitive cubic lattice and the cations occupy half the cubic holes in this lattice. Note the relation of this structure to the caesium-chloride structure, in which all the cubic holes are occupied. The antifluorite structure is the inverse of the fluorite structure in the sense that the locations of cations and anions are reversed. The latter structure is shown by some alkali metal oxides, including Li2O. In it, the cations (which are twice as numerous as the anions) occupy all the tetrahedral holes of a ccp array of anions. The coordination is (4,8) rather than the (8,4) of fluorite itself. The rutile structure (Fig. 3.39) takes its name from rutile, a mineral form of titanium(IV) oxide, TiO2. The structure can also be considered an example of hole filling in an hcp anion arrangement, but now the cations occupy only half the octahedral holes and there is considerable buckling of the close-packed anion layers. This arrangement results in a structure that reflects the strong tendency of a Ti4 ion to acquire octahedral coordination. Each Ti4 atom is surrounded by six O atoms and each O atom is surrounded by three Ti4 ions; hence the rutile structure has (6,3)-coordination. The principal ore of tin, cassiterite SnO2, has the rutile structure, as do a number of metal difluorides (Table 3.4). In the cadmium-iodide structure (as in CdI2, Fig. 3.40), the octahedral holes between every other pair of hcp layers of I ions (that is half of the total number of octahedral Fig. 3.35 (a) The wurtzite structure and (b) its projection representation. As Ni (a) (0,1) Fig. 3.36 (a) The nickel-arsenide structure, (b) the projection representation of the unit cell, and (c) the trigonal prismatic coordination around As. 1 2 ( 14 , 34 ) (c) (b) 81 Ionic solids 3 2 Polarized ionic Δ holes) are filled by Cd2 ions. The CdI2 structure is often referred to as a ‘layer-structure’ as the repeating layers of atoms perpendicular to the close-packed layers form the sequence I–Cd–I ... I–Cd–I ... I–Cd–I with weak van der Waals interactions between the iodine atoms in adjacent layers. The structure has (6,3)- coordination, being octahedral for the cation and trigonal pyramidal for the anion. The structure type is found commonly for many d-metal halides and chalcogenides (for example, FeBr2, MnI2, ZrS2, and NiTe2). The cadmium-chloride structure (as in CdCl2, Fig. 3.41) is analogous to the CdI2 structure but with a ccp arrangement of anions; half the octahedral sites between alternate anion layers are occupied. This layer structure has identical coordination numbers (6,3) and geometries for the ions to those found for the CdI2 structure-type, although it is preferred for a number of d-metal dichlorides, such as MnCl2 and NiCl2. 1 0 1 2 3 4 mean Fig. 3.37 The location of polarized ionic salts in a Ketelaar triangle. E X A MPL E 3 . 8 Determining the stoichiometry of a hole-filled structure Identify the stoichiometries of the following structures based on hole filling using a cation, A, in closepacked arrays of anions, X. (a) An hcp array in which one-third of the octahedral sites are filled. (b) A ccp array in which all the tetrahedral and all the octahedral sites are filled. Answer We need to be aware that in an array of N close-packed spheres there are 2N tetrahedral holes and N octahedral holes (Section 3.3). Therefore, filling all the octahedral holes in a closed-packed array of anions X with cations A would produce a structure in which cations and anions were in the ratio 1:1, 1 corresponding to the stoichiometry AX. (a) As only one-third of the holes are occupied, the A:X ratio is 3 :1, corresponding to the stoichiometry AX3. An example of this type of structure is BiI3. (b) The total number of A species is 2N N with NX species. The A:X ratio is therefore 3:1, corresponding to the stoichiometry A3X. An example of this type of structure is Li3Bi. Ca2+ F– (a) (0,1) Self-test 3.8 Determine the stoichiometry of an hcp array with two-thirds of the octahedral sites occupied. (4 ,4) 3 1 1 2 (b) Ternary phases AaBbXn (0,1) Key point: The perovskite and spinel structures are adopted by many compounds with the stoichiometries ABO3 and AB2O4, respectively. Structural possibilities increase very rapidly once the compositional complexity is increased to three ionic species. Unlike binary compounds, it is difficult to predict the most likely structure type based on the ion sizes and preferred coordination numbers. This section describes two important structures formed by ternary oxides; the O2 ion is the most common anion, so oxide chemistry is central to a significant part of solid-state chemistry. The mineral perovskite, CaTiO3, is the structural prototype of many ABX3 solids (Table 3.4), particularly oxides. In its ideal form, the perovskite structure is cubic with each A cation surrounded by 12 X anions and each B cation surrounded by six X anions (Fig. 3.42). In fact, the perovskite structure may also be described as a close-packed array of A cations and O2 anions (arranged such that each A cation is surrounded by 12 O2 anions from the original close-packed layers) with B cations in all the octahedral holes that are formed from six of the O2 ions, giving Bn/4[AO3]n/4, which is equivalent to ABO3. (b) Fig. 3.38 (a) The fluorite structure and (b) its projection representation. This structure has a ccp array of cations and all the tetrahedral holes are occupied by anions. O a Ti a c (a) (0,1) I– Cl– 1 2 1 2 Cd2+ (0,1) Cd2+ (b) Fig. 3.40 The CdI2 structure (left). Fig. 3.41 The CdCl2 structure (right). Fig. 3.39 (a) The rutile structure and (b) its projection representation. Rutile itself is one polymorph of TiO2. 82 3 The structures of simple solids (0,1) 1 2 B X (0,1) 1 2 A (a) (b) (c) Fig. 3.42 (a) The perovskite structure, ABX3, (b) a display that emphasizes the octahedral shape of the B sites, and (c) its projection representation. In oxides, X O and the sum of the charges on the A and B ions must be 6. That sum can be achieved in several ways (A2B4 and A3B3 among them), including the possibility of mixed oxides of formula A(B0.5B0.5)O3, as in La(Ni0.5Ir0.5)O3. The A-type cation in perovskites is therefore usually a large ion (of radius greater than 110 pm) of lower charge, such as Ba2 or La3, and the B cation is a small ion (of radius less than 100 pm) of higher charge, such as Ti4, Nb5, or Fe3. The perovskite structure is closely related to the materials that show interesting electrical properties, such as piezoelectricity, ferroelectricity, and high-temperature superconductivity (Section 24.8). E X A M PL E 3 . 9 Determining coordination numbers Demonstrate that the coordination number of the Ti4 ion in the perovskite CaTiO3 is 6. Answer We need to imagine eight of the unit cells shown in Fig. 3.42 stacked together with a Ti atom shared by them all. A local fragment of the structure is shown in Fig. 3.43; it shows that there are six O2 ions around the central Ti4 ion, so the coordination number of Ti in perovskite is 6. An alternative way of viewing the perovskite structure is as BO6 octahedra sharing all vertices in three orthogonal directions with the A cations at the centres of the cubes so formed (Fig. 3.42b). Fig. 3.43 The local coordination environment of a Ti atom in perovskite. Self-test 3.9 What is the coordination number of a Ti4 site and the O2 site in rutile? Spinel itself is MgAl2O4, and oxide spinels, in general, have the formula AB2O4. The spinel structure consists of a ccp array of O2 ions in which the A cations occupy one-eighth of the tetrahedral holes and the B cations occupy half the octahedral holes (Fig. 3.44). Spinels are sometimes denoted A[B2]O4, the square brackets denoting the cation type (normally the smaller, higher charged ion of A and B) that occupies the octahedral holes. So, for example, 3 4 1 2 1 8 1 4 7 8 1 2 3 4 1 2 3 8 1 4 (0,1) 5 8 (0,1) 3 8 (0,1) 3 4 1 8 5 8 (0,1) (a) (b) 1 4 1 2 1 4 7 8 1 2 3 4 (c) Fig. 3.44 (a) The spinel structure showing the tetrahedral oxygen environment around the B cations, (b) showing the octahedral oxygen environment around the A cations, and (c) its projection representation with only the cation locations specified. 83 Ionic solids ZnAl2O4 can be written Zn[Al2]O4 to show that all the Al3 cations occupy octahedral sites. Examples of compounds that have spinel structures include many ternary oxides with the stoichiometry AB2O4 that contain a 3d-series metal, such as NiCr2O4 and ZnFe2O4, and some simple binary d-block oxides, such as Fe3O4, Co3O4, and Mn3O4; note that in these structures A and B are the same element but in different oxidation states, as in Fe2[Fe3]2O4. There are also a number of compositions termed inverse spinels, in which the cation distribution is B[AB]O4 and in which the more abundant cation is distributed over both tetrahedral and octahedral sites. Spinels and inverse spinels are discussed again in Sections 20.1 and 24.8. E X A MPL E 3 .10 Predicting possible ternary phases What ternary oxides with the perovskite or spinel structures might it be possible to synthesize that contain the cations Ti4, Zn2, In3, and Pb2? Use the ionic radii given in Resource section 1. Answer We need to consider whether the sizes of the ions permit the occurrence of the two structures. We can predict that ZnTiO3 does not exist as a perovskite as the Zn2 ion is too small for the A-type site; likewise, PbIn2O4 does not adopt the spinel structure as the Pb2 cation is too large for the tetrahedral sites. We conclude that the permitted structures are PbTiO3 (perovskite), TiZn2O4 (spinel), and ZnIn2O4 (spinel). Self-test 3.10 What additional oxide perovskite composition(s) might be obtained if La3 is added to this list of cations? 3.10 The rationalization of structures (a) Ionic radii Cs+ Ionic radius, r/pm The thermodynamic stabilities and structures of ionic solids can be treated very simply using the ionic model. However, a model of a solid in terms of charged spheres interacting electrostatically is crude and we should expect significant departures from its predictions because many solids are more covalent than ionic. Even conventional ‘good’ ionic solids, such as the alkali metal halides, have some covalent character. Nevertheless, the ionic model provides an attractively simple and effective scheme for correlating many properties. 200 Rb+ Tl+ 150 K+ Ag+ 100 Na+ Key point: The sizes of ions, ionic radii, generally increase down a group, decrease across a period, increase with coordination number, and decrease with increasing charge number. A difficulty that confronts us at the outset is the meaning of the term ‘ionic radius’. As remarked in Section 1.9, it is necessary to apportion the single internuclear separation of nearest-neighbour ions between the two different species (for example, an Na ion and a Cl ion in contact). The most direct way to solve the problem is to make an assumption about the radius of one ion, and then to use that value to compile a set of self-consistent values for all other ions. The O2 ion has the advantage of being found in combination with a wide range of elements. It is also reasonably unpolarizable, so its size does not vary much as the identity of the accompanying cation is changed. In a number of compilations, therefore, the values are based on r(O2) 140 pm. However, this value is by no means sacrosanct: a set of values compiled by Goldschmidt was based on r(O2) 132 pm and other values use the F ion as the basis. For certain purposes (such as for predicting the sizes of unit cells) ionic radii can be helpful, but they are reliable only if they are all based on the same fundamental choice (such as the value 140 pm for O2). If values of ionic radii are used from different sources, it is essential to verify that they are based on the same convention. An additional complication that was first noted by Goldschmidt is that, as we have already seen for metals, ionic radii increase with coordination number (Fig. 3.45). Hence, when comparing ionic radii, we should compare like with like, and use values for a single coordination number (typically 6). The problems of the early workers have been resolved only partly by developments in X-ray diffraction (Section 8.1). It is now possible to measure the electron density between two neighbouring ions and identify the minimum as the boundary between them. However, as can be seen from Fig. 3.46, the electron density passes through a very broad minimum, and its exact location may be very sensitive to experimental uncertainties and to the 50 Cu+ Li+ 2 4 6 8 10 12 Coordination number Fig. 3.45 The variation of ionic radius with coordination number. Electron density Li F Minimum 92 G78 S76 P 60 Fig. 3.46 The variation in electron density along the Li–F axis in LiF. The point P denotes the Pauling radii of the ions, G the original (1927) Goldschmidt radii, and S the Shannon radii. 84 3 The structures of simple solids identities of the two neighbours. That being so, it is still probably more useful to express the sizes of ions in a self-consistent manner than to seek calculated values of individual radii in certain combinations. After all, we are interested in compounds where we always have interactions between pairs of ions so we are consistent in the method of determination and application of ionic radii. Very extensive lists of self-consistent values that have been compiled by analysing X-ray data on thousands of compounds, particularly oxides and fluorides, exist and some are given in Table 1.4 and Resource section 1. The general trends for ionic radii are the same as for atomic radii. Thus: 1. Ionic radii increase down a group. (The lanthanide contraction, Section 1.9, restricts the increase between the 4d- and 5d-series metal ions.) 2. The radii of ions of the same charge decrease across a period. 3. If an element can form cations with different charge numbers, then for a given coordination number its ionic radius decreases with increasing charge number. 4. Because a positive charge indicates a reduced number of electrons, and hence a more dominant nuclear attraction, cations are smaller than anions for elements with similar atomic numbers. 5. When an ion can occur in environments with different coordination numbers, the observed radius, as measured by considering the average distances to the nearest neighbours, increases as the coordination number increases. This increase reflects the fact that the repulsions between the surrounding ions are reduced if they move apart, so leaving more room for the central ion. (b) The radius ratio Key point: The radius ratio indicates the likely coordination numbers of the ions in a binary compound. A parameter that figures widely in the literature of inorganic chemistry, particularly in introductory texts, is the ‘radius ratio’, (gamma), of the ions. The radius ratio is the ratio of the radius of the smaller ion (rsmall) to that of the larger (rlarge): = rsmall (3.1) rlarge In most cases, rsmall is the cation radius and rlarge is the anion radius. The minimum radius ratio that can support a given coordination number is then calculated by considering the geometrical problem of packing together spheres of different sizes (Table 3.6). It is argued that, if the radius ratio falls below the minimum given, then ions of opposite charge will not be in contact and ions of like charge will touch. According to a simple electrostatic argument, a lower coordination number, in which the contact of oppositely charged ions is restored, then becomes favourable. Another way of looking at this argument is that as the radius of the M ion increases, more anions can pack around it, so giving a larger number of favourable Coulombic interactions. In this respect, compare CsCl, and its (8,8)coordination, with NaCl, and its (6,6)-coordination. We can use our previous calculations of hole size (Example 3.4), to put these ideas on a firmer footing. A cation of radius between 0.225r and 0.414r can occupy a tetrahedral hole in a close-packed or slightly expanded close-packed array of anions of radius r. Table 3.6 The correlation of structural type with radius ratio Radius ratio ( ) CN for 1:1 and 1:2 stoichiometries 1 12 None known None known 0.7321 8:8 and 8:4 CsCl CaF2 0.4140.732 6:6 and 6:3 NaCl (ccp), NiAs (hcp) TiO2 0.2250. 414 4:4 ZnS (ccp and hcp) CN denotes coordination number. Binary AB structure type Binary AB2 structure type Ionic solids However, once the radius of a cation reaches 0.414r, the anions are forced so far apart that octahedral coordination becomes possible and most favourable. Note that 0.225r represents the size of the smallest ion that will fit in a tetrahedral hole and that cations between 0.225r and 0.414r will push the anions apart. However, the coordination number cannot increase to 6 with good contacts between cation and anions until the radius goes above 0.414r. Similar arguments apply for the tetrahedral holes that can be filled by ions with sizes up to 0.225r. These concepts of ion packing based on radius ratios can often be used to predict which structure is most likely for any particular choice of cation and anion (Table 3.6). In practice, the radius ratio is most reliable when the cation coordination number is 8, and less reliable with six- and four-coordinate cations because directional covalent bonding becomes more important for these lower coordination numbers. ■ A brief illustration. To predict the crystal structure of TlCl we note that the ionic radii are r(Tl) 159 pm and r(Cl) 181 pm, giving 0.88. We can therefore predict that TlCl is likely to adopt a caesium-chloride structure with (8,8)-coordination. That is the structure found in practice. ■ The ionic radii used in these calculations are those obtained by consideration of structures under normal conditions. At high pressures, different structures may be preferred, especially those with higher coordination numbers and greater density. Thus many simple compounds transform between the simple (4,4)-, (6,6)-, and (8,8)-coordination structures under pressure. Examples of this behaviour include most of the lighter alkali metal halides, which change from a (6,6)-coordinate rock-salt structure to an (8,8)coordinate caesium-chloride structure at 5 kbar (the rubidium halides) or 10–20 kbar (the sodium and potassium halides). The ability to predict the structures of compounds under pressure is important for understanding the behaviour of ionic compounds under such conditions. Calcium oxide, for instance, is predicted to transform from the rocksalt to the caesium-chloride structure at around 600 kbar, the pressure in the Earth’s lower mantle. Similar arguments involving the relative ionic radii of cations and anions and their preferred coordination numbers (that is, preferences for octahedral, tetrahedral, or cubic geometries) can be applied throughout structural solid-state chemistry and aid the prediction of which ions might be incorporated into a particular structure type. For more complex stoichiometries, such as the ternary compounds with the perovskite and spinel structure types, the ability to predict which combinations of cations and anions will yield a specific structure type has proved very useful. One example is that for the hightemperature superconducting cuprates (Section 24.8), the design of a particular structure feature, such as Cu2 in octahedral coordination to oxygen, can be achieved using ionic radii considerations. (c) Structure maps Key point: A structure map is a representation of the variation in crystal structure with the character of the bonding. Even though the use of radius ratios is not totally reliable, it is still possible to rationalize structures by collecting enough information empirically and looking for patterns. This approach has motivated the compilation of ‘structure maps’. A structure map is an empirically compiled map that depicts the dependence of crystal structure on the electronegativity difference between the elements present and the average principal quantum number of the valence shells of the two atoms.5 As such, a structure map can be regarded as an extension of the ideas introduced in Chapter 2 in relation to Ketelaar’s triangle. As we have seen, binary ionic salts are formed for large differences in electronegativity ∆, but as this difference is reduced, polarized ionic salts and more covalently bonded networks become preferred. Now we can focus on this region of the triangle and explore how small changes in electronegativity and polarizability affect the choice of ion arrangement. The ionic character of a bond increases with ∆, so moving from left to right along the horizontal axis of a structure map correlates with an increase in ionic character in 5 Structure maps were introduced by E. Mooser and W.B. Pearson, Acta Cryst., 1959, 12, 1015. 85 86 3 The structures of simple solids Fig. 3.47 A structure map for compounds of formula MX. A point is defined by the electronegativity difference (∆) between M and X and their average principal quantum number n. The location in the map indicates the coordination number expected for that pair of properties. (Based on E. Mooser and W.B. Pearson, Acta Crystallogr., 1959, 12, 1015.) Average principal quantum number 6 CN = 4 5 CN = 6 4 3 2 1 0.1 0.3 0.5 0.7 0.9 ∆ 1.1 1.3 1.5 1.7 the bonding. The principal quantum number is an indication of the radius of an ion, so moving up the vertical axis corresponds to an increase in the average radius of the ions. Because atomic energy levels also become closer as the atom expands, the polarizability of the atom increases too (Section 1.9e). Consequently, the vertical axis of a structure map corresponds to increasing size and polarizability of the bonded atoms. Figure 3.47 is an example of a structure map for MX compounds. We see that the structures we have been discussing for MX compounds fall in distinct regions of the map. Elements with large ∆ have (6,6)-coordination, such as is found in the rock-salt structure; elements with small ∆ (and hence where there is the expectation of covalence) have a lower coordination number. In terms of a structure map representation, GaN is in a more covalent region of Fig. 3.47 than ZnO because ∆ is appreciably smaller. ■ A brief illustration. To predict the type of crystal structure that should be expected for magnesium sulfide, MgS, we note that the electronegativities of magnesium and sulfur are 1.3 and 2.6 respectively, so ∆ 1.3. The average principal quantum number is 3 (both elements are in Period 3). The point ∆ 1.3, n 3 lies just in the sixfold coordination region of the structure map in Fig. 3.47. This location is consistent with the observed rock-salt structure of MgS. ■ The energetics of ionic bonding A compound tends to adopt the crystal structure that corresponds to the lowest Gibbs energy. Therefore, if for the process M (g) X (g) → MX(s) the change in standard reaction Gibbs energy, ∆rG O , is more negative for the formation of a structure A rather than B, then the transition from B to A is spontaneous under the prevailing conditions, and we can expect the solid to be found with structure A. The process of solid formation from the gas of ions is so exothermic that at and near room temperature the contribution of the entropy to the change in Gibbs energy (as in ∆G O ∆H O T∆S O ) may be neglected; this neglect is rigorously true at T 0. Hence, discussions of the thermodynamic properties of solids normally focus, initially at least, on changes in enthalpy. That being so, we look for the structure that is formed most exothermically and identify it as the thermodynamically most stable form. Some typical values of lattice enthalpies are given in Table 3.7 for a number of simple ionic compounds. The energetics of ionic bonding 87 Table 3.7 Lattice enthalpies of some simple inorganic solids HLexp (kJ mol− 1 ) Compound type Structure HLexp (kJ mol− 1 SrCl2 Fluorite 2125 Compound Structure type LiF Rock salt 1030 LiI Rock salt 757 LiH Rock salt 858 NaF Rock salt 923 NaH Rock salt 782 NaCl Rock salt 786 KH Rock salt 699 NaBr Rock salt 747 RbH Rock salt 674 NaI Rock salt 704 CsH Rock salt 648 KCl Rock salt 719 BeO Wurtzite 4293 KI Rock salt 659 MgO Rock salt 3795 CsF Rock salt 744 CaO Rock salt 3414 CsCl Caesium chloride 657 SrO Rock salt 3217 CsBr Caesium chloride 632 BaO Rocksalt 3029 CsI Caesium chloride 600 Li2O Antifluorite 2799 MgF2 Rutile 2922 TiO2 Rutile 12150 CaF2 Fluorite 2597 CeO2 Fluorite 9627 ) 3.11 Lattice enthalpy and the Born–Haber cycle Key points: Lattice enthalpies are determined from enthalpy data by using a Born–Haber cycle; the most stable crystal structure of the compound is commonly the structure with the greatest lattice enthalpy under the prevailing conditions. The lattice enthalpy, ∆HL O , is the standard molar enthalpy change accompanying the formation of a gas of ions from the solid: MX(s) → M(g) X(g) ∆HL O A note on good practice The definition of lattice enthalpy as an endothermic (positive) term corresponding to the break up of the lattice is correct but contrary to many school and college texts where it is defined with respect to lattice formation (and listed as a negative quantity). Because lattice disruption is always endothermic, lattice enthalpies are always positive and their positive signs are normally omitted from their numerical values. As remarked above, if entropy considerations are neglected, then the most stable crystal structure of the compound is the structure with the greatest lattice enthalpy under the prevailing conditions. Lattice enthalpies are determined from enthalpy data by using a Born–Haber cycle, a closed path of steps that includes lattice formation as one stage, such as that shown in Fig. 3.48. The standard enthalpy of decomposition of a compound into its elements in their reference states (their most stable states under the prevailing conditions) is the negative of its standard enthalpy of formation, ∆fH O : M(s) X(s, l, g) → MX(s) ∆fH O Likewise, the standard enthalpy of lattice formation from the gaseous ions is the negative of the lattice enthalpy as specified above. For a solid element, the standard enthalpy of atomization, ∆atomH O , is the standard enthalpy of sublimation, as in the process M(s) → M(g) ∆atomH O For a gaseous element, the standard enthalpy of atomization is the standard enthalpy of dissociation, ∆disH O , as in X2(g) → 2 X(g) ∆disH O K+(g) + e–(g) + Cl(g) 122 K+(g) + e–(g) + 12 Cl2(g) K+(g) + Cl–(g) 425 K(g) + 1 2 Cl2(g) 1 2 Cl2(g) 89 K(s) + –355 x 438 KCl(s) Fig. 3.48 The Born—Haber cycle for KCl. The lattice enthalpy is equal to x. All numerical values are in kilojoules per mole. 88 3 The structures of simple solids The standard enthalpy of formation of ions from their neutral atoms is the enthalpy of ionization (for the formation of cations, ∆ionH O ) and the electron-gain enthalpy (for anions, ∆egH O ): M(g) → M(g) e(g) X(g) e(g) → X(g) ∆ionH O ∆egH O The value of the lattice enthalpy—the only unknown in a well-chosen cycle—is found from the requirement that the sum of the enthalpy changes round a complete cycle is zero (because enthalpy is a state property).6 The value of the lattice enthalpy obtained from a Born–Haber cycle depends on the accuracy of all the measurements being combined, and as a result there can be significant variations, typically ±10 kJ mol1, in tabulated values. H O /(kJ mol1) Sublimation of K(s) 89 Ionization of K(g) 425 Dissociation of Cl2(g) 244 Electron gain by Cl(g) 355 Formation of KCl(s) 438 H O /(kJ mol1) Sublimation of Mg(s) 148 Ionization of Mg(g) to Mg2(g) 2187 Vaporization of Br2(l) 31 Dissociation of Br2(g) 193 Electron gain by Br(g) 331 Formation of MgBr2(s) 524 E X A M PL E 3 .11 Using a Born–Haber cycle to determine a lattice enthalpy Calculate the lattice enthalpy of KCl(s) using a Born–Haber cycle and the information in the margin. Answer The required cycle is shown in Fig. 3.48. The sum of the enthalpy changes around the cycle is zero, so ∆HL O /(kJ mol1) 438 425 89 244/2 355 719 Note that the calculation becomes more obvious by drawing an energy level diagram showing the signs of the various steps of the cycle; all lattice enthalpies are positive. Also as only one Cl atom from Cl2(g) is required to produce KCl, half the dissociation energy of Cl2, 21 244 kJ mol1, is used in the calculation. Self-test 3.11 Calculate the lattice enthalpy of magnesium bromide from the data shown in the margin. 3.12 The calculation of lattice enthalpies Once the lattice enthalpy is known, it can be used to judge the character of the bonding in the solid. If the value calculated on the assumption that the lattice consists of ions interacting electrostatically is in good agreement with the measured value, then it may be appropriate to adopt a largely ionic model of the compound. A discrepancy indicates a degree of covalence. As mentioned earlier, it is important to remember that numerical coincidences can be misleading in this assessment. (a) The Born–Mayer equation Key points: The Born–Mayer equation is used to estimate lattice enthalpy for an ionic lattice. The Madelung constant reflects the effect of the geometry of the lattice on the strength of the net Coulombic interaction. To calculate the lattice enthalpy of a supposedly ionic solid we need to take into account several contributions, including the Coulombic attractions and repulsions between the ions and the repulsive interactions that occur when the electron densities of the ions overlap. This calculation yields the Born–Mayer equation for the lattice enthalpy at T 0: H LO N A z A zB e 2 ⎛ d ⎞ 1 A 4 π 0 d ⎜⎝ d ⎟⎠ (3.2) where d r r is the distance between centres of neighbouring cations and anions, and hence a measure of the ‘scale’ of the unit cell (for the derivation, see Further information 3.1). In this expression NA is Avogadro’s constant, zA and zB the charge numbers of the cation and anion, e the fundamental charge, ε0 the vacuum permittivity, and d a constant (typically 34.5 pm) used to represent the repulsion between ions at short range. The quantity A is called the Madelung constant, and depends on the structure (specifically, on the relative distribution of ions, Table 3.8). The Born–Mayer equation in fact gives the lattice energy as distinct from the lattice enthalpy, but the two are identical at T 0 and the difference may be disregarded in practice at normal temperatures. 6 Note that when the lattice enthalpy is known from calculation, a Born–Haber cycle may be used to determine the value of another elusive quantity, the electron-gain enthalpy (and hence the electron affinity). The energetics of ionic bonding ■ A brief illustration. To estimate the lattice enthalpy of sodium chloride, we use z(Na) 1, z(Cl) 1, from Table 3.8, A 1.748, and from Table 1.4 d rNa rCl 283 pm; hence (using fundamental constants from inside the back cover): HL ° (6.022 10 mol ) (1) (1 ) (1.602 10 C ) 4 π (8.854 1012 J1 C 2 m1 ) (2.82 1010 m) 23 1 7.56 105 J mol1 19 2 ⎛ 34.5 pm ⎞ ⎜ 1 1.748 283 pm ⎟⎠ ⎝ or 756 kJ mol1. This value compares reasonably well with the experimental value from the Born– Haber cycle, 788 kJ mol1. ■ The form of the Born–Mayer equation for lattice enthalpies allows us to account for their dependence on the charges and radii of the ions in the solid. Thus, the heart of the equation is H L ° ∝ z A zB d Therefore, a large value of d results in a low lattice enthalpy, whereas high ionic charges result in a high lattice enthalpy. This dependence is seen in some of the values given in Table 3.7. For the alkali metal halides, the lattice enthalpies decrease from LiF to LiI and from LiF to CsF as the halide and alkali metal ion radii increase, respectively. We also note that the lattice enthalpy of MgO (zAzB 4) is almost four times that of NaCl (zAzB 1) due to the increased charges on the ions for a similar value of d, noting that the Madelung constant is the same. The Madelung constant typically increases with coordination number. For instance, A 1.748 for the (6,6)-coordinate rock-salt structure but A 1.763 for the (8,8)coordinate caesium-chloride structure and 1.638 for the (4,4)-coordinate sphalerite structure. This dependence reflects the fact that a large contribution comes from nearest neighbours, and such neighbours are more numerous when the coordination number is large. However, a high coordination number does not necessarily mean that the interactions are stronger in the caesium-chloride structure because the potential energy also depends on the scale of the lattice. Thus, d may be so large in lattices with ions big enough to adopt eightfold-coordination that the separation of the ions reverses the effect of the small increase in the Madelung constant and results in a smaller lattice enthalpy. (b) Other contributions to lattice enthalpies Key point: Nonelectrostatic contributions to the lattice enthalpy include van der Waals interactions, particularly the dispersion interaction. Another contribution to the lattice enthalpy is the van der Waals interaction between the ions and molecules, the weak intermolecular interactions that are responsible for the formation of condensed phases of electrically neutral species. An important and sometimes dominant contribution of this kind is the dispersion interaction (the ‘London interaction’). The dispersion interaction arises from the transient fluctuations in electron density (and, consequently, instantaneous electric dipole moment) on one molecule driving a fluctuation in electron density (and dipole moment) on a neighbouring molecule, and the attractive interaction between these two instantaneous electric dipoles. The molar potential energy of this interaction, V, is expected to vary as V = N AC d6 (3.3) The constant C depends on the substance. For ions of low polarizability, this contribution is only about 1 per cent of the electrostatic contribution and is ignored in elementary lattice enthalpy calculations of ionic solids. However, for highly polarizable ions, such as Tl and I, such terms can make significant contributions of several per cent. Thus, the dispersion interaction for compounds such as LiF and CsBr is estimated to contribute 16 kJ mol1 and 50 kJ mol1 , respectively. Table 3.8 Madelung constants Structural type A Caesium chloride 1.763 Fluorite 2.519 Rock salt 1.748 Rutile 2.408 Sphalerite 1.638 Wurtzite 1.641 89 90 3 The structures of simple solids 3.13 Comparison of experimental and theoretical values Key points: For compounds formed from elements with ∆ 2, the ionic model is generally valid and lattice enthalpy values derived using the Born–Mayer equation and Born–Haber cycles are similar. For structures formed with small electronegativity differences and polarizable ions there may be additional, nonionic contributions to the bonding. The agreement between the experimental lattice enthalpy and the value calculated using the ionic model of the solid (in practice, from the Born–Mayer equation) provides a measure of the extent to which the solid is ionic. Table 3.9 lists some calculated and measured lattice enthalpies together with electronegativity differences. The ionic model is reasonably valid if ∆ 2, but the bonding becomes increasingly covalent if ∆ 2. However, it should be remembered that the electronegativity criterion ignores the role of polarizability of the ions. Thus, the alkali metal halides give fairly good agreement with the ionic model, the best with the least polarizable halide ions (F) formed from the highly electronegative F atom, and the worst with the highly polarizable halide ions (I) formed from the less electronegative I atom. This trend is also seen in the lattice enthalpy data for the silver halides in Table 3.9. The discrepancy between experimental and theoretical values is largest for the iodide, which indicates major deficiencies in the ionic model for this compound. Overall the agreement is much poorer with Ag than with Li as the electronegativity of silver ( 1.93) is much higher than that of lithium ( 0.98) and significant covalent character in the bonding would be expected. It is not always clear whether it is the electronegativity of the atoms or the polarizability of the resultant ions that should be used as a criterion. The worst agreement with the ionic model is for polarizable-cation/polarizable-anion combinations that are substantially covalent. Here again, though, the difference between the electronegativities of the parent elements is small and it is not clear whether electronegativity or polarizability provides the better criterion. E X A M PL E 3 .12 Using the Born–Mayer equation to decide the theoretical stability of unknown compounds Decide whether solid ArCl is likely to exist. Answer The answer hinges on whether the enthalpy of formation of ArCl is significantly positive or negative: if it is significantly positive (endothermic), the compound is unlikely to be stable (there are, of course, exceptions). Consideration of a Born–Haber cycle for the synthesis of ArCl would show two unknowns, the enthalpy of formation of ArCl and its lattice enthalpy. We can estimate the lattice enthalpy of a purely ionic ArCl by using the Born–Mayer equation assuming the radius of Ar to be midway between that of Na and K. That is, the lattice enthalpy is somewhere between the values for NaCl and KCl at about 745 kJ mol1. So, taking the enthalpy of dissociation of 21 Cl2 to form Cl as 122 kJ mol1, the ionization enthalpy of Ar as 1524 kJ mol1, and the electron affinity of Cl as 356 kJ mol1 gives ∆fH O (ArCl, s) 1524 745 356 122 kJ mol1 545 kJ mol1. That is, the compound is predicted to be very unstable with respect to its elements, mainly because the large ionization enthalpy of Ar is not compensated by the lattice enthalpy. Self-test 3.12 Predict whether CsCl2 with the fluorite structure is likely to exist. Table 3.9 Comparison of experimental and theoretical lattice enthalpies for rock-salt structures ∆ HLcalc LiF 1029 (kJ mol ) −1 ∆ HLexp 1030 (kJ mol ) −1 ( ∆H 1 LiCl 834 853 19 LiBr 788 807 19 Lil 730 757 27 AgF 920 953 33 AgCl 832 903 71 AgBr 815 895 80 AgI 777 882 105 exp L − ∆HLcalc ) (kJ mol ) −1 The energetics of ionic bonding Calculations like that in Example 3.12 were used to predict the stability of the first noble gas compounds. The ionic compound O2PtF6 had been obtained from the reaction of oxygen with PtF6. Consideration of the ionization energies of O2 (1176 kJ mol1) and Xe (1169 kJ mol1) showed them to be almost identical and the sizes of Xe and O2 would be expected to be similar, implying similar lattice enthalpies for their compounds. Hence once O2 had been found to react with platinum hexafluoride, it could be predicted that Xe should too, as indeed it does, to give an ionic compound which is believed to contain XeF and PtF6 ions. Similar calculations may be used to predict the stability, or otherwise, of a wide variety of compounds, for example the stability of alkaline earth monohalides, such as MgCl. Calculations based on Born–Mayer lattice enthalpies and Born–Haber cycles show that such a compound would be expected to disproportionate into Mg and MgCl2. Note that calculations of this type provide only an estimate of the enthalpies of formation of ionic compounds and, hence, some idea of the thermodynamic stability of a compound. It may still be possible to isolate a thermodynamically unstable compound if its decomposition is very slow. Indeed, a compound containing Mg(I) was reported in 2007 (Section 12.13). 3.14 The Kapustinskii equation Key point: The Kapustinskii equation is used to estimate lattice enthalpies of ionic compounds and to give a measure of the thermochemical radii of the constituent ions. A.F. Kapustinskii observed that, if the Madelung constants for a number of structures are divided by the number of ions per formula unit, Nion, then approximately the same value is obtained for them all. He also noted that the value so obtained increases with the coordination number. Therefore, because ionic radius also increases with coordination number, the variation in A/Niond from one structure to another can be expected to be fairly small. This observation led Kapustinskii to propose that there exists a hypothetical rock-salt structure that is energetically equivalent to the true structure of any ionic solid and therefore that the lattice enthalpy can be calculated by using the rock-salt Madelung constant and the appropriate ionic radii for (6,6)-coordination. The resulting expression is called the Kapustinskii equation: ∆H L° N ion z A zB ⎛ d ⎞ 1 ⎜ ⎝ d d ⎟⎠ (3.4) In this equation 1.21 105 kJ pm mol1. The Kapustinskii equation can be used to ascribe numerical values to the ‘radii’ of nonspherical molecular ions, as their values can be adjusted until the calculated value of the lattice enthalpy matches that obtained experimentally from the Born–Haber cycle. The self-consistent parameters obtained in this way are called thermochemical radii (Table 3.10). They may be used to estimate lattice enthalpies, and hence enthalpies of formation, of a wide range of compounds without needing to know the structure, assuming that the bonds are essentially ionic. ■ A brief illustration. To estimate the lattice enthalpy of potassium nitrate, KNO3 we need the number of ions per formula unit (Nion 2), their charge numbers, z(K) 1, z(NO3 ) 1, and the sum of their thermochemical radii, 138 pm 189 pm 327 pm. Then, with d 34.5 pm, 2 \| (1)(1) \| ⎛ 34.5 pm ⎞ ⎜ 1 (1.21105 kJ pm mol−1 ) 327 pm 327 pm ⎠⎟ ⎝ = 622 kJ mol−1 ∆HL° ■ 3.15 Consequences of lattice enthalpies The Born–Mayer equation shows that, for a given lattice type (a given value of A), the lattice enthalpy increases with increasing ion charge numbers (as zAzB). The lattice enthalpy also increases as the ions come closer together and the scale of the lattice decreases. Energies that vary as the electrostatic parameter, (xi), != z A zB d (3.5) 91 92 3 The structures of simple solids Table 3.10 The thermochemical radii of ions, r/pm Main-group elements BeF22− BF4− CO2− 3 NO3− OH 245 228 185 189 140 NO 182 155 CN PO − 3 O2− 2 180 3− 4 238 AsO 3− 4 SO2− 4 ClO 4− 230 236 SeO 2− 4 248 243 SbO3− 4 TeO2− 4 IO 4− 260 254 249 IO3− 182 Complex ions d-Metal oxoanions [TiCl6]2 [IrCl6]2 [SiF6]2 [GeCl6]2 CrO2− MnO4− 4 248 254 194 243 230 2 [TiBr6] [PtCl6] 261 259 2 2 2 [GeF6] [SnCl6] MoO 201 247 254 [ZrCl6]2 [PbCl6]2 247 248 240 2− 4 Source: A.F. Kaputinskii, Q. Rev. Chem. Soc., 1956, 10, 283. (which is often written more succinctly as z2/d) are widely adopted in inorganic chemistry as indicative that an ionic model is appropriate. In this section we consider three consequences of lattice enthalpy and its relation to the electrostatic parameter. (a) Thermal stabilities of ionic solids Key point: Lattice enthalpies may be used to explain the chemical properties of many ionic solids, including their thermal decomposition. The particular aspect we consider here is the temperature needed to bring about thermal decomposition of carbonates (although the arguments can easily be extended to many inorganic solids): MCO3 (s) → MO (s) + CO2 (g) Magnesium carbonate, for instance, decomposes when heated to about 300°C, whereas calcium carbonate decomposes only if the temperature is raised to over 800°C. The decomposition temperatures of thermally unstable compounds (such as carbonates) increase with cation radius (Table 3.11). In general, large cations stabilize large anions (and vice versa). Table 3.11 Decomposition data for carbonates MgCO3 CaCO3 SrCO3 BaCO3 −1 48.3 130.4 183.8 218.1 −1 100.6 178.3 234.6 269.3 175.0 160.6 171.0 172.1 300 840 1100 1300 ( ) ∆H ° (kJ mol ) ∆S ° (kJ mol ) ∆G ° kJ mol −1 decomp/°C Data are for the reaction MCO3(s) → MO(s) CO2(g) at 298 K. is the temperature required to reach p(CO2) 1 bar and has been estimated from the thermodynamic data at 298 K. The energetics of ionic bonding The stabilizing influence of a large cation on an unstable anion can be explained in terms of trends in lattice enthalpies. First, we note that the decomposition temperatures of solid inorganic compounds can be discussed in terms of their Gibbs energies of decomposition into specified products. The standard Gibbs energy for the decomposition of a solid, ∆G O ∆H O T∆S O , becomes negative when the second term on the right exceeds the first, which is when the temperature exceeds ∆H ° T ∆S ° M2+(g) + O2–(g) + CO2(g) ∆decompH(CO3 ) 2– –∆HL(MO) M2+(g) + CO32–(g) MO(s) + CO2(g) (3.6) In many cases it is sufficient to consider only trends in the reaction enthalpy, as the reaction entropy is essentially independent of M because it is dominated by the formation of gaseous CO2. The standard enthalpy of decomposition of the solid is then given by ∆HL(MCO3) where ∆decompH O is the standard enthalpy of decomposition of CO32 in the gas phase (Fig. 3.49): CO32 − (g) → O2 − (g) CO2 (g) Because ∆decompH O is large and positive, the overall reaction enthalpy is positive (decomposition is endothermic), but it is less strongly positive if the lattice enthalpy of the oxide is markedly greater than that of the carbonate because then ∆HL O (MCO3, s) – ∆HL O (MO, s) is negative. It follows that the decomposition temperature will be low for oxides that have relatively high lattice enthalpies compared with their parent carbonates. The compounds for which this is true are composed of small, highly charged cations, such as Mg2, which explains why a small cation increases the lattice enthalpy of an oxide more than that of a carbonate. Figure 3.50 illustrates why a small cation has a more significant influence on the change in the lattice enthalpy as the cation size is varied. The change in separation is relatively small when the parent compound has a large cation initially. As the illustration shows in an exaggerated way, when the cation is very big, the change in size of the anion barely affects the scale of the lattice. Therefore, with a given unstable polyatomic anion, the lattice enthalpy difference is more significant and favourable to decomposition when the cation is small than when it is large. The difference in lattice enthalpy between MO and MCO3 is magnified by a larger charge on the cation as ∆HL O ∝ zAzB/d. As a result, thermal decomposition of a carbonate will occur at lower temperatures if it contains a higher charged cation. One consequence of this dependence on cation charge is that alkaline earth carbonates (M2) decompose at lower temperatures than the corresponding alkali metal carbonates (M). E X A MPL E 3 .13 Assessing the dependence of stability on ionic radius Present an argument to account for the fact that, when they burn in oxygen, lithium forms the oxide Li2O but sodium forms the peroxide Na2O2. Answer We need to consider the role of the relative sizes of cations and ions in determining the stability of a compound. Because the small Li ion results in Li2O having a more favourable lattice enthalpy (in comparison with M2O2) than Na2O, the decomposition reaction M2O2(s) → M2O(s) 21 O2(g) is thermodynamically more favourable for Li2O2 than for Na2O2. Self-test 3.13 Predict the order of decomposition temperatures of alkaline earth metal sulfates in the reaction MSO4(s) → MO(s) SO3(g). The use of a large cation to stabilize a large anion that is otherwise susceptible to decomposition, forming a smaller anionic species, is widely used by inorganic chemists to prepare compounds that are otherwise thermodynamically unstable. For example, the interhalogen anions, such as ICl4, are obtained by the oxidation of I ions by Cl2 but are susceptible to decomposition to iodine monochloride and Cl: –∆rH° MCO3(s) ∆H O ≈ ∆decompH O ∆HL O (MCO3, s) ∆HL O (MO, s) MI(s) 2Cl2 (g) → MICl4 (s) → MCl(s) ICl(g) Cl2 (g) 93 Fig 3.49 A thermodynamic cycle showing the enthalpy changes involved in the decomposition of a solid carbonate MCO3. (a) (b) Fig. 3.50 A greatly exaggerated representation of the change in lattice parameter d for cations of different radii. (a) When the anion changes size (as when CO2 decomposes into O2 and CO2, for 3 instance) and the cation is large, the lattice parameter changes by a small amount. (b) If the cation is small, however, the relative change in lattice parameter is large and decomposition is thermodynamically more favourable. 94 3 The structures of simple solids To disfavour the decomposition, a large cation is used to reduce the lattice enthalpy difference between MICl4 and MCl/MI. The larger alkali metal cations such as K, Rb, and Cs can be used in some cases, but it is even better to use a really bulky alkylammonium ion, such as NtBu4 (with tBu C(CH3)3). (b) The stabilities of oxidation states Key point: The relative stabilities of different oxidation states in solids can often be predicted from considerations of lattice enthalpies. A similar argument can be used to account for the general observation that high metal oxidation states are stabilized by small anions. In particular, F has a greater ability than the other halogens to stabilize the high oxidation states of metals. Thus, the only known halides of Ag(II), Co(III), and Mn(IV) are the fluorides. Another sign of the decrease in stability of the heavier halides of metals in high oxidation states is that the iodides of Cu(II) and Fe(III) decompose on standing at room temperature (to CuI and FeI2). Oxygen is also a very effective species for stabilizing the highest oxidation states of elements because of the high charge and small size of the O2 ion. To explain these observations consider the reaction MX(s) 12 X2 (g) → MX2 (s) where X is a halogen. The aim is to show why this reaction is most strongly spontaneous for X F. If we ignore entropy contributions, we must show that the reaction is most exothermic for fluorine. One contribution to the reaction enthalpy is the conversion of 12 X2 to X. Despite F having a lower electron affinity than Cl, this step is more exothermic for X F than for X Cl because the bond enthalpy of F2 is lower than that of Cl2. The lattice enthalpies, however, play the major role. In the conversion of MX to MX2, the charge number of the cation increases from 1 to 2, so the lattice enthalpy increases. As the radius of the anion increases, however, this difference in the two lattice enthalpies diminishes, and the exothermic contribution to the overall reaction decreases too. Hence, both the lattice enthalpy and the X formation enthalpy lead to a less exothermic reaction as the halogen changes from F to I. Provided entropy factors are similar, which is plausible, we expect an increase in thermodynamic stability of MX relative to MX2 on going from X F to X I down Group 17. Thus many iodides do not exist for metals in their higher oxidation states and compounds such as Cu2(I)2 , Tl3(I)3 and VI5 are unknown, whereas the corresponding fluorides CuF2, TlF3, and VF5 are easily obtained. In effect, the high-oxidation-state metal oxidizes I ions to I2, leading to formation of a lower metal oxidation state such as Cu(I), Tl(I), and V(III) in the iodides of these metals. (c) Solubility Key point: The solubilities of salts in water can be rationalized by considering lattice and hydration enthalpies. Lattice enthalpies play a role in solubilities, as the dissolution involves breaking up the lattice, but the trend is much more difficult to analyse than for decomposition reactions. One rule that is reasonably well obeyed is that compounds that contain ions with widely different radii are soluble in water. Conversely, the least water-soluble salts are those of ions with similar radii. That is, in general, difference in size favours solubility in water. It is found empirically that an ionic compound MX tends to be very soluble when the radius of M is smaller than that of X by about 80 pm. Two familiar series of compounds illustrate these trends. In gravimetric analysis, Ba2 is used to precipitate SO42, and the solubilities of the Group 2 sulfates decrease from MgSO4 to BaSO4. In contrast, the solubility of the Group 2 hydroxides increases down the group: Mg(OH)2 is the sparingly soluble ‘milk of magnesia’ but Ba(OH)2 can be used as a soluble hydroxide for preparation of solutions of OH. The first case shows that a large anion requires a large cation for precipitation. The second case shows that a small anion requires a small cation for precipitation. Before attempting to rationalize the observations, we should note that the solubility of an ionic compound depends on the standard reaction Gibbs energy for MX(s) → M+ (aq) + X− (aq) 95 In this process, the interactions responsible for the lattice enthalpy of MX are replaced by hydration (and by solvation in general) of the ions. However, the exact balance of enthalpy and entropy effects is delicate and difficult to assess, particularly because the entropy change also depends on the degree of order of the solvent molecules that is brought about by the presence of the dissolved solute. The data in Fig. 3.51 suggest that enthalpy considerations are important in some cases at least, as the graph shows that there is a correlation between the enthalpy of solution of a salt and the difference in hydration enthalpies of the two ions. If the cation has a larger hydration enthalpy than its anion partner (reflecting the difference in their sizes) or vice versa, then the dissolution of the salt is exothermic (reflecting the favourable solubility equilibrium). The variation in enthalpy can be explained using the ionic model. The lattice enthalpy is inversely proportional to the distance between the centres of the ions: H L° ∝ 1 r r However, the hydration enthalpy, with each ion being hydrated individually, is the sum of individual ion contributions: 1 1 hyd H ∝ r r If the radius of one ion is small, the term in the hydration enthalpy for that ion will be large. However, in the expression for the lattice enthalpy one small ion cannot make the denominator of the expression small by itself. Thus, one small ion can result in a large hydration enthalpy but not necessarily lead to a high lattice enthalpy, so ion size asymmetry can result in exothermic dissolution. If both ions are small, then both the lattice enthalpy and the hydration enthalpy may be large, and dissolution might not be very exothermic. E X A MPL E 3 .14 Accounting for trends in the solubility of s-block compounds What is the trend in the solubilities of the Group 2 metal carbonates? Answer We need to consider the role of the relative sizes of cations and anions. The CO2 anion has a 3 large radius and has the same magnitude of charge as the cations M2 of the Group 2 elements. The least soluble carbonate of the group is predicted to be that of the largest cation, Ra2. The most soluble is expected to be the carbonate of the smallest cation, Mg2. (Beryllium has too much covalent character in its bonding for it to be included in this analysis.) Although magnesium carbonate is more soluble than radium carbonate, it is still only sparingly soluble: its solubility constant (its solubility product, Ksp) is only 3 108. Self-test 3.14 Which can be expected to be more soluble in water, NaClO4 or KClO4? Defects and nonstoichiometry Key points: Defects, vacant sites, and misplaced atoms are a feature of all solids as their formation is thermodynamically favourable. All solids contain defects, or imperfections of structure or composition. Defects are important because they influence properties such as mechanical strength, electrical conductivity, and chemical reactivity. We need to consider both intrinsic defects, which are defects that occur in the pure substance, and extrinsic defects, which stem from the presence of impurities. It is also common to distinguish point defects, which occur at single sites, from extended defects, which are ordered in one, two, and three dimensions. Point defects are random errors in a periodic lattice, such as the absence of an atom at its usual site or the presence of an atom at a site that is not normally occupied. Extended defects involve various irregularities in the stacking of the planes of atoms. Enthalpy of dissolution, ∆solH/(kJ mol–1) Defects and nonstoichiometry +40 CsI CsBr RbBr +20 RbI KI CsCl KCl KBr RbCl LiF 0 NaCl NaBr NaF NaI KF –20 RbF LiCl –40 CsF –60 LiBr LiI –200 –100 0 +100 +200 {∆hydH(X–) – ∆hydH(M+)} /(kJ mol–1) Fig. 3.51 The correlation between enthalpies of solution of halides and the differences between the hydration enthalpies of the ions. Dissolution is most exothermic when the difference is large. 96 3 The structures of simple solids 3.16 The origins and types of defects Solids contain defects because they introduce disorder into an otherwise perfect structure and hence increase its entropy. The Gibbs energy, G H – TS, of a solid with defects has contributions from the enthalpy and the entropy of the sample. The formation of defects is normally endothermic because, as the lattice is disrupted, the enthalpy of the solid rises. However, the term –TS becomes more negative as defects are formed because they introduce disorder into the lattice and the entropy rises. Provided T 0, therefore, the Gibbs energy will have a minimum at a nonzero concentration of defects and their formation will be spontaneous (Fig. 3.52a). Moreover, as the temperature is raised, the minimum in G shifts to higher defect concentrations (Fig. 3.52b), so solids have a greater number of defects as their melting points are approached. (a) Intrinsic point defects Key points: Schottky defects are site vacancies, formed in cation/anion pairs, and Frenkel defects are displaced, interstitial atoms; the structure of a solid influences the type of defect that occurs, with Frenkel defects forming in solids with lower coordination numbers and more covalency and Schottky defects in more ionic materials. H Energy TS G = H – TS Equilibrium Defect concentration The solid-state physicists W. Schottky and J. Frenkel identified two specific types of point defect. A Schottky defect (Fig. 3.53) is a vacancy in an otherwise perfect arrangement of atoms or ions in a structure. That is, it is a point defect in which an atom or ion is missing from its normal site in the structure. The overall stoichiometry of a solid is not affected by the presence of Schottky defects because, to ensure charge balance, the defects occur in pairs in a compound of stoichiometry MX and there are equal numbers of vacancies at cation and anion sites. In solids of different composition, for example MX2, the defects must occur with balanced charges, so two anion vacancies must be created for each cation lost. Schottky defects occur at low concentrations in purely ionic solids, such as NaCl; they occur most commonly in structures with high coordination numbers, such as close-packed metals, where the enthalpy penalty of reducing the average coordination number of the remaining atoms (from 12 to 11, for instance) is relatively low. A Frenkel defect (Fig. 3.54) is a point defect in which an atom or ion has been displaced onto an interstitial site. For example, in silver chloride, which has the rock-salt structure, a small number of Ag ions reside in tetrahedral sites (1). The stoichiometry of the compound is unchanged when a Frenkel defect forms and it is possible to have Frenkel defects involving either one (M or X displaced) or both (some M and some X interstitials) of the ion types in a binary compound, MX. Thus the Frenkel defects that occur in, for example, PbF2 involve the displacement of a small number of F ions from their normal sites in the fluorite structure, on the tetrahedral holes in the close-packed Pb2 ion array, to sites that Energy H TS G Low T High T Defect concentration Fig. 3.52 (a) The variation of the enthalpy and entropy of a crystal as the number of defects increases. The resulting Gibbs energy G H TS has a minimum at a nonzero concentration, and hence defect formation is spontaneous. (b) As the temperature is increased, the minimum in the Gibbs energy moves to higher defect concentrations, so more defects are present at equilibrium at higher temperatures than at low temperatures. Na+ Cl– Ag+ Fig. 3.53 A Schottky defect is the absence of ions on normally occupied sites; for charge neutrality there must be equal numbers of cation and anion vacancies in a 1:1 compound. Cl– Fig. 3.54 A Frenkel defect forms when an ion moves to an interstitial site. Defects and nonstoichiometry correspond to the octahedral holes. A useful generalization is that Frenkel defects are most often encountered in structures such as wurtzite and sphalerite in which coordination numbers are low (6 or less) and the more open structure provides sites that can accommodate the interstitial atoms. This is not to say that Frenkel defects are exclusive to such structures; as we have seen, the (8,4)-coordination fluorite structure can accommodate such interstitials although some local repositioning of adjacent anions is required to allow for the presence of the displaced anion. The concentration of Schottky defects varies considerably from one type of compound to the next. The concentration of vacancies is very low in the alkali metal halides, being of the order of 106 cm3 at 130°C. That concentration corresponds to about one defect per 1014 formula units. Conversely, some d-metal oxides, sulfides, and hydrides have very high concentrations of vacancies. An extreme example is the high-temperature form of TiO, which has vacancies on both the cation and anion sites at a concentration corresponding to about one defect per 10 formula units. 97 Ag+ Cl– Ag+ 1 Interstitial Ag+ E X A MPL E 3 .15 Predicting defect types What type of intrinsic defect would you expect to find in (a) MgO and (b) CdTe? Answer The type of defect formed depends on factors such as the coordination numbers and the level of covalency in the bonding with high coordination numbers and ionic bonding favouring Schottky defects and low coordination numbers and partial covalency in the bonding favouring Frenkel defects. (a) MgO has the rock-salt structure and the ionic bonding in this compound generally favours Schottky defects. (b) CdTe adopts the wurtzite structure with (4,4)-coordination, favouring Frenkel defects. Self-test 3.15 Predict the most likely type of intrinsic defects for CsF. Schottky and Frenkel defects are only two of the many possible types of defect. Another type is an atom-interchange or anti-site defect, which consists of an interchanged pair of atoms. This type of defect is common in metal alloys with exchange of neutral atoms. It is expected to be very unfavourable for binary ionic compounds on account of the introduction of strongly repulsive interactions between neighbouring similarly charged ions. For example, a copper/gold alloy of exact overall composition CuAu has extensive disorder at high temperatures, with a significant fraction of Cu and Au atoms interchanged (Fig. 3.55). The interchange of similarly charged species on different sites in ternary and compositionally more complex compounds is common; thus in spinels (Section 24.8) the partial swapping of the metal ions between tetrahedral and octahedral sites is often observed. Both Schottky and Frenkel defects are stoichiometry defects in that they do not change the overall composition of the material because the vacancies occur in charge-balanced pairs (Schottky) or each interstitial is derived from one displaced atom or ion (Frenkel, a vacancy-interstitial pair). Similar types of defects, vacancies, and interstitials occur in many inorganic materials and may be balanced by changes in the oxidation number of one component in the system rather than by their creation as charge-balanced pairs. This behaviour, as seen for example in La2CuO4.1 with extra interstitial O2 ions, is discussed more fully in Section 24.8. (b) Extrinsic point defects Key point: Extrinsic defects are defects introduced into a solid as a result of doping with an impurity atom. Extrinsic defects, those resulting from the presence of impurities, are inevitable because perfect purity is unattainable in practice in crystals of any significant size. Such behaviour is commonly seen in naturally occurring minerals. Thus, the incorporation of low levels of Cr into the Al2O3 structure produces the gemstone ruby, whereas replacement of some Al by Fe and Ti results in the blue gemstone sapphire (Box 3.3). The dopant species normally has a similar atomic or ionic radius to the species which it replaces. Thus Cr3 in ruby and Fe3 in sapphire have similar ionic radii to Al3. Impurities can also be introduced intentionally by doping one material with another; an example is the introduction of As into Si to modify the latter’s semiconducting properties. Synthetic equivalents of ruby and sapphire can also be synthesized easily in the laboratory by incorporating small levels of Cr, Fe, and Ti into the Al2O3 structure. Cu Au Fig. 3.55 Atom exchange can also give rise to a point defect as in CuAu. 98 3 The structures of simple solids B OX 3 . 3 Defects and gemstones Defects and dopant ions are responsible for the colours of many gemstones. Whereas aluminium oxide (Al2O3), silica (SiO2), and fluorite (CaF2) in their pure forms are colourless, brightly coloured materials may be produced by substituting in small levels of dopant ions or producing vacant sites that trap electrons. The impurities and defects are often found in naturally occurring minerals on account of the geological and environmental conditions under which they are formed. For example, d-metal ions are often present in the solutions from which the gemstones grew and the presence of ionizing radiation from radioactive species in the natural environment generates electrons that become trapped in their structure. The most common origin of colour in a gemstone is a d-metal ion dopant (see table). Thus ruby is Al2O3 containing around 0.21 atom per cent Cr3 ions in place of the Al3 ions and its red colour results from the absorption of green light in the visible spectrum as a result of the excitation of Cr3d electrons (Section 20.4). The same ion is responsible for the green of emeralds; the different colour reflects a different local coordination environment of the dopant. The host structure is beryl, beryllium aluminium silicate, Be3Al2(SiO3)6, and the Cr3 ion is surrounded by six silicate ions, rather than the six O2 ions in ruby, producing absorption at a different energy. Other d-metal ions are responsible for the colours of other gemstones. Iron(II) produces the red of garnets and the yellow-green of peridots. Manganese(II) is responsible for the pink colour of tourmaline. In ruby and emerald the colour is caused by excitation of electrons on a single dopant d-metal ion, Cr3. When more than one dopant species, which may be of different type or oxidation state, is present it is possible to transfer an electron between them. One example of this behaviour is sapphire. Sapphire, like ruby, is alumina but in this gemstone some adjacent pairs of Al3 ions are replaced by Fe2 and Ti4 pairs. This material absorbs visible radiation in the yellow as an electron is transferred from Fe2 to Ti4, so producing a brilliant blue colour (the complementary colour of yellow). In other gemstones and minerals, colour is a result of doping a host structure with a species that has a different charge from the ion that it replaces or by the presence of a vacancy (Schottky-type defect). In both cases a colour-centre or F-centre (F from the German word farbe for colour) is formed. As the charge at an F-centre is different from that of a normally occupied site in the same structure, it can easily supply an electron to or receive an electron from another ion. This electron can then be excited by absorbing visible light, so producing colour. For instance, in purple fluorite, CaF2, an F-centre is formed from a vacancy on a normally occupied F ion site. This site then traps an electron, generated by exposure of the mineral to ionizing radiation in the natural environment. Excitation of the electron, which acts like a particle in a box, absorbs visible light in the wavelength range 530—600 nm, producing the violet/purple colours of this mineral. In amethyst, the purple derivative of quartz, SiO2, some Si4 ions are substituted by Fe3 ions. This replacement leaves a hole (one missing electron) and excitation of this hole, by ionizing radiation for instance, traps it by forming Fe4 or O in the quartz matrix. Further excitation of the electrons in this material now occurs by the absorption of visible light at 540 nm, producing the observed purple colour. If an amethyst crystal is heated to 450°C the hole is freed from its trap. The colour of the crystal reverts to that typical of iron-doped silica and is a characteristic of the yellow semi-precious gemstone citrine. If citrine is irradiated the trappedhole is regenerated and the original colour restored. Colour centres can also be produced by nuclear transformations. An example of such a transformation is the -decay of 14C in a diamond. This decay produces a 14N atom, with an additional valence electron, embedded in the diamond structure. The electron energy levels associated with these N atoms allow absorption in the visible region of the spectrum and produce the colouration of blue and yellow diamonds. Table B3.1 Gemstones and the origin of their colours Mineral or gemstone Colour Parent formula Dopant or defect responsible for the colour Ruby Red Al2O3 Cr3 replacing Al3 in octahedral sites Emerald Green Be3Al2(SiO3)6 Cr3 replacing Al3 in octahedral sites Tourmaline Green or pink Na3Li3Al6(BO3)3(SiO3)6F4 Cr3 or Mn2 replacing Li and Al3 in octahedral sites respectively Garnet Red Mg3Al2(SiO4)3 Fe2 replacing Mg2 in 8-coordinate sites Peridot Yellow-green Mg2SiO4 Fe2 replacing Mg2 in 6-coordinate sites Sapphire Blue Al2O3 Electron transfer between Fe2 and Ti4 replacing Al3 in adjacent octahedral sites Diamond Colourless, pale blue or yellow C Colour centres from N Amethyst Purple SiO2 Colour centre based on Fe3/Fe4 Fluorite Purple CaF2 Colour centre based on trapped electron When the dopant species is introduced into the host the latter’s structure remains essentially unchanged. If attempts are made to introduce high levels of the dopant species a new structure incorporating all the elements present often forms or the dopant species is not incorporated. This behaviour usually limits the level of extrinsic point defects to low levels. Defects and nonstoichiometry The composition of ruby is typically (Al0.998Cr0.002)2O3, with 0.2 per cent of metal sites as extrinsic Cr3 dopant ions. Some solids may tolerate much higher levels of defects (Section 3.17a). Dopants often modify the electronic structure of the solid. Thus, when an As atom replaces an Si atom, the additional electron from each As atom enters the conduction band. In the more ionic substance ZrO2, the introduction of Ca2 impurities in place of Zr4 ions is accompanied by the formation of an O2 ion vacancy to maintain charge neutrality (Fig. 3.56). Another example of an extrinsic point defect is a colour centre, a generic term for defects responsible for modifications to the IR, visible, and UV absorption characteristics of solids that have been irradiated or exposed to chemical treatment. One type of colour centre is produced by heating an alkali metal halide crystal in the vapour of the alkali metal and gives a material with a colour characteristic of the system: NaCl becomes orange, KCl violet, and KBr blue-green. The process results in the introduction of an alkali metal cation at a normal cation site and the associated electron from the metal atom occupies a halide ion vacancy. A colour centre consisting of an electron in a halide ion vacancy is called an F-centre (Fig. 3.57).5 The colour results from the excitation of the electron in the localized environment of its surrounding ions. An alternative method of producing F-centres involves exposing a material to an X-ray beam that ionizes electrons into anion vacancies. F-centres and extrinsic defects are important in producing colour in gemstones, Box 3.3. Zr 99 Vacancy Ca O Fig. 3.56 Introduction of a Ca2 ion into the ZrO2 lattice produces a vacancy on the O2 sublattice. This substitution helps to stabilize the cubic fluorite structure for ZrO2. E X A MPL E 3 .16 Predicting possible dopant ions What ions might substitute for Al3 in beryl, Be3Al2(SiO3)6, forming extrinsic defects? e– Answer We need to identify ions of similar charge and size. Ionic radii are listed in Resource section 1. Triply charged cations with ionic radii similar to Al3 (r 53 pm) should prove to be suitable dopant ions. Candidates could be Fe3 (r 55 pm), Mn3 (r 65 pm), and Cr3 (r 62 pm). Indeed when the extrinsic defect is Cr3 the material is a bright green beryl, the gemstone emerald. For Mn3 the material is a red or pink beryl and for Fe3 it is the yellow beryl heliodor. Self-test 3.16 What elements other than As might be used to form extrinsic defects in silicon? 3.17 Nonstoichiometric compounds and solid solutions The statement that the stoichiometry of a compound is fixed by its chemical formula is not always true for solids: differences in the composition of unit cells can occur throughout a solid, perhaps because there are defects at one or more atom sites, interstitial atoms are present, or substitutions have occurred at one position. (a) Nonstoichiometry Key point: Deviations from ideal stoichiometry are common in the solid-state compounds of the d-, f-, and later p-block elements. A nonstoichiometric compound is a substance that exhibits variable composition but retains the same structure type. For example, at 1000°C the composition of ‘iron monoxide’, which is sometimes referred to as wüstite, Fe1xO, varies from Fe0.89O to Fe0.96O. Gradual changes in the size of the unit cell occur as the composition is varied, but all the features of the rock-salt structure are retained throughout this composition range. The fact that the lattice parameter of the compound varies smoothly with composition is a defining criterion of a nonstoichiometric compound because a discontinuity in the value of the lattice parameter indicates the formation of a new crystal phase. Moreover, the thermodynamic properties of nonstoichiometric compounds also vary continuously as the composition changes. For example, as the partial pressure of oxygen above a metal oxide is varied, both the lattice parameter and the equilibrium composition of the oxide change continuously (Figs 3.58 and 3.59). The gradual change in the lattice parameter of a solid as a function of its composition is known as Vegard’s rule. Table 3.12 lists some representative nonstoichiometric hydrides, oxides, and sulfides. Note that as the formation of a nonstoichiometric compound requires overall changes in 5 The name comes from the German word for colour centre, Farbenzentrum. Na+ Cl– Fig. 3.57 An F-centre is an electron that occupies an anion vacancy. The energy levels of the electron resemble those of a particle in a three-dimensional square well. 100 3 The structures of simple solids Table 3.12 Representative composition ranges of nonstoichiometric binary hydrides, oxides, and sufides Partial pressure, p(O2) d block f block Hydrides (a) (b) TiHx 12 ZrHx 1.51.6 Fluorite type Hexagonal GdHx 1.92.3 2.853.0 HfHx NbHx 1.71.8 ErHx 1.952.31 2.823.0 0.641.0 LuHx 1.852.23 1.743.0 Oxides 0 0.5 Composition, x 1 a/pm Fig. 3.58 Schematic representation of the variation of the partial pressure of oxygen with composition at constant pressure for (a) a nonstoichiometric oxide MO1x and (b) a stoichiometric pair of metal oxides MO and MO2. The x-axis is the atom ratio in MO1x. (b) x in MO1+x Rutile type TiOx 0.71.25 1.92.0 VOx 0.91.20 1.82.0 NbOx 0.91.04 Sulfides ZrSx 0.91.0 YSx 0.91.0 Expressed as the range of values x may take. composition, it also requires at least one element to exist in more than one oxidation state. Thus in wüstite, Fe1xO, as x increases some iron(II) must be oxidized to iron(III) in the structure. Hence deviations from stoichiometry are usual only for d- and f-block elements, which commonly adopt two or more oxidation states, and for some heavy p-block metals that have two accessible oxidation states. (a) Two-phase region 0 Rock-salt type (b) Solid solutions 1 Fig. 3.59 Schematic representation of the variation of one lattice parameter with composition for (a) a nonstoichiometric oxide MO1x and (b) a stoichiometric pair of metal oxides MO and MO2 with no intermediate stoichiometric phases (which would produce a two-phase mixture for 0 x 1, each phase in the mixture having the lattice parameter of the end member. Key point: A solid solution occurs where there is a continuous variation in compound stoichiometry without a change in structural type. Because many substances adopt the same structural type, it is often energetically feasible to replace one type of atom or ion with another. Such behaviour is seen in many simple metal alloys such as those discussed in Section 3.8. Thus zinc/copper brasses exist for the complete range of compositions Cu1xZnx with 0 x 0.38, where Cu atoms in the structure are gradually replaced by Zn atoms. This replacement occurs randomly throughout the solid, and individual unit cells contain an arbitrary number of Cu and Zn atoms (but such that the sum of their contents gives the overall brass stoichiometry). Another good example is the perovskite structure adopted by many compounds of stoichiometry ABX3 (Section 3.9), in which the composition can be varied continuously by varying the ions that occupy the A, B, and X sites. For instance, both LaFeO3 and SrFeO3 adopt the perovskite structure and we can consider a perovskite crystal that has, randomly distributed, half SrFeO3 unit cells (with Sr on the A-type cation site) and half LaFeO3 unit cells (with La on the A-site). The overall compound stoichiometry is LaSrFe2O6, which is better written (La0.5Sr0.5) FeO3, to reflect the normal ABO3 perovskite stoichiometry. Other proportions of these unit cells are possible and the series of compounds La1xSrxFeO3 for 0 x 1 can be prepared. This system is called a solid solution because all the phases formed as x is varied have the same perovskite structure. A solid solution occurs when there is a single structural type for a range of compositions and there is a smooth variation in lattice parameter over that range. Solid solutions occur most frequently for d-metal compounds because the change in one component might require a change in the oxidation state of another component to preserve the charge balance. Thus, as x increases in La1xSrxFeO3 and La(III) is replaced by Sr(II), the oxidation state of iron must change from Fe(III) to Fe(IV). This change can occur through a gradual replacement of one exact oxidation state, here Fe(III), by another, Fe(IV), on a proportion of the cation sites within the structure. Alternatively, if the material is metallic and has delocalized electrons, then the change can be accommodated by altering the number of electrons in a conduction band, which corresponds to the delocalization of the change in oxidation state rather than its identification with individual atoms. Some other solid solutions The electronic structures of solids 101 include the high-temperature superconductors of composition La2xBaxCuO4 (0 x 0.4), which are superconducting for 0.12 x 0.25, and the spinels Mn1xFe2xO4 (0 x 1). It is also possible to combine solid-solution behaviour on a cation site with nonstoichiometry caused by defects on a different ion site. An example is the system La1xSrxFeO3y, with 0 x 1.0 and 0.0 y 0.5, which has vacancies on the O2 ion sites. The electronic structures of solids The previous sections have introduced concepts associated with the structures and energetics of ionic solids in which it was necessary to consider almost infinite arrays of ions and the interactions between them. Similarly, an understanding of the electronic structures of solids, and the derived properties such as electric conductivity, magnetism, and many optical effects, needs to consider the interactions of electrons with each other and extended arrays of ions. One simple approach is to regard a solid as a single huge molecule and to extend the ideas of molecular orbital theory introduced in Chapter 2 to very large numbers of orbitals. Similar concepts are used in later chapters to understand other key properties of large three-dimensional arrays of electronically interacting centres such as ferromagnetism, superconductivity, and the colours of solids. 3.18 The conductivities of inorganic solids Key points: A metallic conductor is a substance with an electric conductivity that decreases with increasing temperature; a semiconductor is a substance with an electric conductivity that increases with increasing temperature. The molecular orbital theory of small molecules can be extended to account for the properties of solids, which are aggregations of an almost infinite number of atoms. This approach is strikingly successful for the description of metals; it can be used to explain their characteristic lustre, their good electrical and thermal conductivity, and their malleability. All these properties arise from the ability of the atoms to contribute electrons to a common ‘sea’. The lustre and electrical conductivities stem from the mobility of these electrons in response to either the oscillating electric field of an incident ray of light or to a potential difference. The high thermal conductivity is also a consequence of electron mobility because an electron can collide with a vibrating atom, pick up its energy, and transfer it to another atom elsewhere in the solid. The ease with which metals can be mechanically deformed is another aspect of electron mobility because the electron sea can quickly readjust to a deformation of the solid and continue to bind the atoms together. Electronic conduction is also a characteristic of semiconductors. The criterion for distinguishing between a metallic conductor and a semiconductor is the temperature dependence of the electric conductivity (Fig. 3.60): 108 It is also generally the case (but not the criterion for distinguishing them) that the conductivities of metals at room temperature are higher than those of semiconductors. Typical values are given in Fig. 3.60. A solid insulator is a substance with a very low electrical conductivity. However, when that conductivity can be measured, it is found to increase with temperature, like that of a semiconductor. For some purposes, therefore, it is possible to disregard the classification ‘insulator’ and to treat all solids as either metals or semiconductors. Superconductors are a special class of materials that have zero electrical resistance below a critical temperature. 3.19 Bands formed from overlapping atomic orbitals The central idea underlying the description of the electronic structure of solids is that the valence electrons supplied by the atoms spread through the entire structure. This concept is expressed more formally by making a simple extension of MO theory in which the solid Metal Conductivity/(S cm–1) A metallic conductor is a substance with an electric conductivity that decreases with increasing temperature. A semiconductor is a substance with an electric conductivity that increases with increasing temperature. Superconductor 104 1 10–4 Semiconductor 10–8 1 10 100 1000 T/K Fig. 3.60 The variation of the electrical conductivity of a substance with temperature is the basis of the classification of the substance as a metallic conductor, a semiconductor, or a superconductor. 102 3 The structures of simple solids band Energy band gap is treated like an indefinitely large molecule. In solid-state physics, this approach is called the tight-binding approximation. The description in terms of delocalized electrons can also be used to describe nonmetallic solids. We therefore begin by showing how metals are described in terms of molecular orbitals. Then we go on to show that the same principles can be applied, but with a different outcome, to ionic and molecular solids. (a) Band formation by orbital overlap band band gap band Fig. 3.61 The electronic structure of a solid is characterized by a series of bands of orbitals separated by gaps at energies where orbitals do not occur. Key point: The overlap of atomic orbitals in solids gives rise to bands of energy levels separated by energy gaps. The overlap of a large number of atomic orbitals in a solid leads to a large number of molecular orbitals that are closely spaced in energy and so form an almost continuous band of energy levels (Fig. 3.61). Bands are separated by band gaps, which are values of the energy for which there is no molecular orbital. The formation of bands can be understood by considering a line of atoms, and supposing that each atom has an s orbital that overlaps the s orbitals on its immediate neighbours (Fig. 3.62). When the line consists of only two atoms, there is a bonding and an antibonding molecular orbital. When a third atom joins them, there are three molecular orbitals. The central orbital of the set is nonbonding and the outer two are at low energy and high energy, respectively. As more atoms are added, each one contributes an atomic orbital, and hence one more molecular orbital is formed. When there are N atoms in the line, there are N molecular orbitals. The orbital of lowest energy has no nodes between neighbouring atoms. The orbital of highest energy has a node between every pair of neighbours. The remaining orbitals have successively 1, 2,...internuclear nodes and a corresponding range of energies between the two extremes. The total width of the band, which remains finite even as N approaches infinity (as shown in Fig. 3.63), depends on the strength of the interaction between neighbouring atoms. The greater the strength of interaction (in broad terms, the greater the degree of overlap between neighbours), the greater the energy separation of the non-node orbital and the all-node orbital. However, whatever the number of atomic orbitals used to form the molecular orbitals, there is only a finite spread of orbital energies (as depicted in Fig. 3.63). It follows that the separation in energy between neighbouring orbitals must approach zero as N approaches infinity, otherwise the range of orbital energies could not be finite. That is, a band consists of a countable number but near-continuum of energy levels. The band just described is built from s orbitals and is called an s band. If there are p orbitals available, a p band can be constructed from their overlap as shown in Fig. 3.64. Most antibonding Energy Most antibonding 0 Most antibonding Intermediate orbitals Most bonding Most bonding Fig. 3.62 A band can be thought of as formed by bringing up atoms successively to form a line of atoms. N atomic orbitals give rise to N molecular orbitals. Intermediate orbitals 1 2 3 4 5 6 7 8 9 101112 ∞ Number of atoms, N Fig. 3.63 The energies of the orbitals that are formed when N atoms are brought up to form a one-dimensional array. Most bonding Fig. 3.64 An example of a p band in a one-dimensional solid. The electronic structures of solids p band Energy Because p orbitals lie higher in energy than s orbitals of the same valence shell, there is often an energy gap between the s band and the p band (Fig. 3.65). However, if the bands span a wide range of energy and the atomic s and p energies are similar (as is often the case), then the two bands overlap. The d band is similarly constructed from the overlap of d orbitals. The formation of bands is not restricted to one type of atomic orbital and bands may be formed in compounds by combinations of different orbital types, for example the d orbitals of a metal atom may overlap the p orbitals of neighbouring O atoms. Decide whether any d orbitals on titanium in TiO (with the rock-salt structure) can overlap to form a band. Answer We need to decide whether there are d orbitals on neighbouring metal atoms that can overlap with one another. Figure 3.66 shows one face of the rock-salt structure with the dxy orbital drawn in on each of the Ti atoms. The lobes of these orbitals point directly towards each other and will overlap to give a band. In a similar fashion the dzx and dyz orbitals overlap in the directions perpendicular to the xz and yz faces. Self-test 3.17 Which d orbitals can overlap in a metal having a primitive structure? p band band gap s band (a) E X A MPL E 3 .17 Identifying orbital overlap 103 s band (b) Fig. 3.65 (a) The s and p bands of a solid and the gap between them. Whether or not there is in fact a gap depends on the separation of the s and p orbitals of the atoms and the strength of the interaction between them in the solid. (b) If the interaction is strong, the bands are wide and may overlap. (b) The Fermi level O Key point: The Fermi level is the highest occupied energy level in a solid at T 0. (c) Densities of states Key point: The density of states is not uniform across a band: in most cases, the states are densest close to the centre of the band. The number of energy levels in an energy range divided by the width of the range is called the density of states, (Fig. 3.68). The density of states is not uniform across a band because the energy levels are packed together more closely at some energies than at others. This variation is apparent even in one dimension, for—compared with its edges—the centre of the band is relatively sparse in orbitals (as can be seen in Fig. 3.63). In three dimensions, the variation of density of states is more like that shown in Fig. 3.69, with the greatest density of states near the centre of the band and the lowest density at the edges. The reason for this behaviour can be traced to the number of ways of producing a particular linear combination of atomic orbitals. There is only one way of forming a fully bonding molecular orbital (the lower edge of the band) and only one way of forming a fully antibonding orbital (the upper edge). However, there are many ways (in a three-dimensional array of atoms) of forming a molecular orbital with an energy corresponding to the interior of a band. Fig. 3.66 One face of the TiO rock-salt structure showing how orbital overlap can occur for the dxy, dyz, and dzx orbitals. Empty band Energy At T 0, electrons occupy the individual molecular orbitals of the bands in accordance with the building-up principle. If each atom supplies one s electron, then at T 0 the lowest 12 N orbitals are occupied. The highest occupied orbital at T 0 is called the Fermi level; it lies near the centre of the band (Fig. 3.67). When the band is not completely full, the electrons close to the Fermi level can easily be promoted to nearby empty levels. As a result, they are mobile and can move relatively freely through the solid, and the substance is an electrical conductor. The solid is in fact a metallic conductor. We have seen that the criterion of metallic conduction is the decrease of electrical conductivity with increasing temperature. This behaviour is the opposite of what we might expect if the conductivity were governed by thermal promotion of electrons above the Fermi level. The competing effect can be identified once we recognize that the ability of an electron to travel smoothly through the solid in a conduction band depends on the uniformity of the arrangement of the atoms. An atom vibrating vigorously at a site is equivalent to an impurity that disrupts the orderliness of the orbitals. This decrease in uniformity reduces the ability of the electron to travel from one edge of the solid to the other, so the conductivity of the solid is less than at T 0. If we think of the electron as moving through the solid, then we would say that it was ‘scattered’ by the atomic vibration. This carrier scattering increases with increasing temperature as the lattice vibrations increase, and the increase accounts for the observed inverse temperature dependence of the conductivity of metals. Ti Fermi level Occupied levels Fig. 3.67 If each of the N atoms supplies one s electron, then at T 0 the lower ½N orbitals are occupied and the Fermi level lies near the centre of the band. E + dE E Fig. 3.68 The density of states is the number of energy levels in an infinitesimal range of energies between E and E dE. 104 3 The structures of simple solids The density of states is zero in the band gap itself—there is no energy level in the gap. In certain special cases, however, a full band and an empty band might coincide in energy but with a zero density of states at their conjunction (Fig. 3.70). Solids with this band structure are called semimetals. One important example is graphite, which is a semimetal in directions parallel to the sheets of carbon atoms. Energy p band A note on good practice This use of the term ‘semimetal’ should be distinguished from its other use as a synonym for metalloid. In this text we avoid the latter usage. s band (d) Insulators Fig. 3.69 Typical densities of states for two bands in a three-dimensional metal. Energy p band s band Fig. 3.70 The densities of states in a semimetal. Energy p band Key point: A solid insulator is a semiconductor with a large band gap. A solid is an insulator if enough electrons are present to fill a band completely and there is a considerable energy gap before an empty orbital becomes available (Fig. 3.71). In a sodium chloride crystal, for instance, the N Cl ions are nearly in contact and their 3s and three 3p valence orbitals overlap to form a narrow band consisting of 4N levels. The Na ions are also nearly in contact and also form a band. The electronegativity of chlorine is so much greater than that of sodium that the chlorine band lies well below the sodium band, and the band gap is about 7 eV. A total of 8N electrons are to be accommodated (seven from each Cl atom, one from each Na atom). These 8N electrons enter the lower chlorine band, fill it, and leave the sodium band empty. Because the energy of thermal motion available at room temperature is kT ≈ 0.03 eV (k is Boltzmann’s constant), very few electrons have enough energy to occupy the orbitals of the sodium band. In an insulator the band of highest energy that contains electrons (at T 0) is normally termed the valence band. There next higher band (which is empty at T 0) is called the conduction band. In NaCl the band derived from the Cl orbitals is the valence band and the band derived from the Na orbitals is the conduction band. We normally think of an ionic or molecular solid as consisting of discrete ions or molecules. According to the picture just described, however, they can be regarded as having a band structure. The two pictures can be reconciled because it is possible to show that a full band is equivalent to a sum of localized electron densities. In sodium chloride, for example, a full band built from Cl orbitals is equivalent to a collection of discrete Cl ions. s band 3.20 Semiconduction Fig. 3.71 The structure of a typical insulator: there is a significant gap between the filled and empty bands. The characteristic physical property of a semiconductor is that its electrical conductivity increases with increasing temperature. At room temperature, the conductivities of semiconductors are typically intermediate between those of metals and insulators. The dividing line between insulators and semiconductors is a matter of the size of the band gap (Table 3.13); the conductivity itself is an unreliable criterion because, as the temperature is increased, a given substance may have in succession a low, intermediate, and high conductivity. The values of the band gap and conductivity that are taken as indicating semiconduction rather than insulation depend on the application being considered. (a) Intrinsic semiconductors Table 3.13 Some typical band gaps at 298 K Material Eg/eV Carbon (diamond) 5.47 Silicon carbide 3.00 Silicon 1.11 Germaniun 0.66 Gallium arsenide 1.35 Indium arsenide 0.36 Key point: The band gap in a semiconductor controls the temperature dependence of the conductivity through an Arrhenius-like expression. In an intrinsic semiconductor, the band gap is so small that the energy of thermal motion results in some electrons from the valence band populating the empty upper band (Fig. 3.72). This occupation of the conduction band introduces positive holes, equivalent to an absence of electrons, into the lower band, and as a result the solid is conducting because both the holes and the promoted electrons can move. A semiconductor at room temperature generally has a much lower conductivity than a metallic conductor because only very few electrons and holes can act as charge carriers. The strong, increasing temperature dependence of the conductivity follows from the exponential Boltzmann-like temperature dependence of the electron population in the upper band. The electronic structures of solids It follows from the exponential form of the population of the conduction band that the conductivity of a semiconductor should show an Arrhenius-like temperature dependence of the form where Eg is the width of the band gap. That is, the conductivity of a semiconductor can be expected to be Arrhenius-like with an activation energy equal to half the band gap, Ea ≈ 12Eg. This is found to be the case in practice. (b) Extrinsic semiconductors Key points: p-Type semiconductors are solids doped with atoms that remove electrons from the valence band; n-type semiconductors are solids doped with atoms that supply electrons to the conduction band. An extrinsic semiconductor is a substance that is a semiconductor on account of the presence of intentionally added impurities. The number of electron carriers can be increased if atoms with more electrons than the parent element can be introduced by the process called doping. Remarkably low levels of dopant concentration are needed—only about one atom per 109 of the host material—so it is essential to achieve very high purity of the parent element initially. If arsenic atoms ([Ar]4s24p3) are introduced into a silicon crystal ([Ne]3s23p2), one additional electron will be available for each dopant atom that is substituted. Note that the doping is substitutional in the sense that the dopant atom takes the place of an Si atom in the silicon structure. If the donor atoms, the As atoms, are far apart from each other, their electrons will be localized and the donor band will be very narrow (Fig. 3.73a). Moreover, the foreign atom levels will lie at higher energy than the valence electrons of the host structure and the filled dopant band is commonly near the empty conduction band. For T 0, some of its electrons will be thermally promoted into the empty conduction band. In other words, thermal excitation will lead to the transfer of an electron from an As atom into the empty orbitals on a neighbouring Si atom. From there it will be able to migrate through the structure in the band formed by Si–Si overlap. This process gives rise to n-type semiconductivity, the ‘n’ indicating that the charge carriers are negatively charged (that is, electrons). An alternative substitutional procedure is to dope the silicon with atoms of an element with fewer valence electrons on each atom, such as gallium ([Ar]4s24p1). A dopant atom of this kind effectively introduces holes into the solid. More formally, the dopant atoms form a very narrow, empty acceptor band that lies above the full Si band (Fig. 3.73b). At T 0 the acceptor band is empty but at higher temperatures it can accept thermally excited electrons from the Si valence band. By doing so, it introduces holes into the latter and hence allows the remaining electrons in the band to be mobile. Because the charge carriers are now effectively positive holes in the lower band, this type of semiconductivity is called p-type semiconductivity. Semiconductor materials are essential components of all modern electronic circuits and some devices based on them are described in Box 3.4. p band Energy (3.7) s band Fig. 3.72 In an intrinsic semiconductor, the band gap is so small that the Fermi distribution results in the population of some orbitals in the upper band. p band Energy Eg / 2 kT 0 e 105 Donor band Acceptor band s band (a) (b) Fig. 3.73 The band structure in (a) an n-type semiconductor and (b) a p-type semiconductor. B OX 3 . 4 Applications of semiconductors Semiconductors have many applications because their properties can be easily modified by the addition of impurities to produce, for example, nand p-type semiconductors. Furthermore, their electrical conductivities can be controlled by application of an electric field, by exposure to light, by pressure, and by heat; as a result, they can be used in many sensor devices. Diodes and photodiodes When the junction of a p-type and an n-type semiconductor is under ‘reverse bias’ (that is, with the p-side at a lower electric potential), the flow of current is very small, but it is high when the junction is under ‘forward bias’ (with the p-side at a higher electric potential). The exposure of a semiconductor to light can generate electron-hole pairs, which increases its conductivity through the increased number of free carriers (electrons or holes). Diodes that use this phenomenon are known as photodiodes. Compound semiconductor diodes can also be used to generate light, as in light-emitting diodes and laser diodes (Section 24.28). Transistors Bipolar junction transistors (BJT) are formed from two p–n junctions, in either an npn or a pnp configuration, with a narrow central region termed the base. The other regions, and their associated terminals, are known as the emitter and the collector. A small potential difference applied across the base and the emitter junction changes the properties of the base–collector junction so that it can conduct current even though it is reverse biased. Thus a transistor allows a current to be controlled by a small change in potential difference and is consequently used in amplifiers. Because the current flowing through a BJT is dependent on temperature they can be used as temperature sensors. Another type of transistor, the field effect transistor (FET) operates on the principle that semiconductor conductivity can be increased or decreased by the presence of an electric field. The electric field increases the number of charge carriers, thereby changing its conductivity. These FETs are used in both digital and analogue circuits to amplify or switch electronic signals. 106 ZnO1–x Mn1–xO Energy MO 3 The structures of simple solids (a) (b) (c) Fig 3.74 The band structure in (a) a stoichiometric oxide, (b) an anion-deficient oxide, and (c) an anion-excess oxide. Several d-metal oxides, including ZnO and Fe2O3, are n-type semiconductors. In their case, the property is due to small variations in stoichiometry and a small deficit of O atoms. The electrons that should be in localized O atomic orbitals (giving a very narrow oxide band, essentially localized individual O2 ions) occupy a previously empty conduction band formed by the metal orbitals (Fig. 3.74). The electrical conductivity decreases after the solids have been heated in oxygen and cooled slowly back to room temperature because the deficit of O atoms is partly replaced and, as the atoms are added, electrons are withdrawn from the conduction band to form oxide ions. However, when measured at high temperatures the conductivity of ZnO increases as further oxygen is lost from the structure, so increasing the number of electrons in the conduction band. p-Type semiconduction is observed for some low oxidation number d-metal chalcogenides and halides, including Cu2O, FeO, FeS, and CuI. In these compounds, the loss of electrons can occur through a process equivalent to the oxidation of some of the metal atoms, with the result that holes appear in the predominantly metal band. The conductivity increases when these compounds are heated in oxygen (or sulfur and halogen sources for FeS and CuI, respectively) because more holes are formed in the metal band as oxidation progresses. n-Type semiconductivity, however, tends to occur for oxides of metals in higher oxidation states, as the metal can be reduced to a lower oxidation state by occupation of a conduction band formed from the metal orbitals. Thus typical n-type semiconductors include Fe2O3, MnO2, and CuO. By contrast, p-type semiconductivity occurs when the metal is in a low oxidation state, such as MnO and Cr2O3. E X A M PL E 3 .18 Predicting extrinsic semiconducting properties Which of the oxides WO3, MgO, and CdO are likely to show p- or n-type extrinsic semiconductivity? Answer The type of semiconductivity depends on the defect levels that are likely to be introduced which is, in turn, determined by whether the metal present can be easily oxidized or reduced. If the metal can easily be oxidized (which may be the case if it has a low oxidation number), then n-type semiconductivity is expected. On the other hand, if the metal can easily be reduced (which may be the case if it has a high oxidation number), then p-type semiconductivity is expected. Thus, WO3, with tungsten present in the high oxidation state W(VI), is readily reduced and accepts electrons from the O2– ions, which escape as elemental oxygen. The excess electrons enter a band formed from the W d orbitals, resulting in n-type semiconductivity. Similarly, CdO, like ZnO, readily loses oxygen and is predicted to be an n-type semiconductor. In contrast, Mg2 ions are neither easily oxidized nor reduced, therefore MgO does not lose or gain even small quantities of oxygen and is an insulator. Self-test 3.18 Predict p- or n-type extrinsic semiconductivity for V2O5 and CoO. Further information 3.1 The Born–Mayer equation Consider a one-dimensional line of alternating cations A and anions B of charges e and e separated by a distance d. The Coulomb potential energy of a single cation is the sum of its interactions with all the other ions: V e2 4 0 2 2 2e2 ⎛ 2 ⎞ ⎜⎝ d 2d 3d ⎟⎠ 4 d 0 1 1 ⎛ ⎞ ⎜⎝ 1 2 3 ⎟⎠ The sum of the series in parentheses is ln 2, so for this arrangement of ions V 2e2 ln 2 4 0 d The total molar contribution of all the ions is this potential energy multiplied by Avogadro’s constant NA (to convert to a molar value) and divided by 2 (to avoid counting each interaction twice): V N A e2 A 4 0d Further information 107 The factor A ln 2 is an example of a Madelung constant, a constant that represents the geometrical distribution of the ions (here, a straight line of constant separation). Two- and three-dimensional arrays of ions may be treated similarly, and give the values of A listed in Table 3.8. The total molar potential energy includes the repulsive interaction between the ions. We can model that by a short-range exponential function of the form Bed / d , with d a constant that defines the range of the repulsive interaction and B a constant that defines its magnitude. The total molar potential energy of interaction is therefore V N A e2 4 0 d A + Bed / d This potential energy passes through a minimum when dV/dd 0, which occurs at N A e2 B dV A ed / d = 0 d dd 4 0 d 2 It follows that, at the minimum, /d Bed N A e2d 4 0 d 2 A This relation can be substituted into the expression for V, to give V N A e2 ⎛ d ⎞ A 1 ⎜ d ⎟⎠ 4 0 d ⎝ On identifying V with the lattice enthalpy (more precisely, with the lattice energy at T 0), we obtain the Born–Mayer equation (eqn 3.2) for the special case of singly charged ions. The generalization to other charge types is straightforward. If a different expression for the repulsive interaction between the ions is used then this expression will be modified. One alternative is to use an expression such as 1/rn with a large n, typically 6 n 12, which then gives rise to a slightly different expression for V known as the Born–Landé equation: V N A e2 ⎛ 1 ⎞ 1 ⎟ A ⎜ ⎝ n ⎠ 4 0 d The semiempirical Born–Mayer expression, with d 34.5 pm determined from the best agreement with experimental data, is generally preferred to the Born–Landé equation. FURTHER READING R.D. Shannon in Encyclopaedia of inorganic chemistry (ed. R.B. King). Wiley, New York (2005). A survey of ionic radii and their determination. A.F. Wells, Structural inorganic chemistry. Oxford University Press (1985). The standard reference book, which surveys the structures of a huge number of inorganic solids. S.E. Dann, Reactions and characterization of solids. Royal Society of Chemistry, Cambridge (2000). L.E. Smart and E.A. Moore, Solid state chemistry: an introduction. Taylor and Francis, CRC Press (2005). P.A. Cox, The electronic structure and chemistry of solids. Oxford University Press (1987). J.K. Burdett, Chemical bonding in solids. Oxford University Press (1995). Further details of the electronic structures of solids. Two very useful texts on the application of thermodynamic arguments to inorganic chemistry are: Some introductory texts on solid-state inorganic chemistry are: W.E. Dasent, Inorganic energetics. Cambridge University Press (1982). U. Müller, Inorganic structural chemistry. Wiley, New York (1993). D.A. Johnson, Some thermodynamic aspects of inorganic chemistry. Cambridge University Press (1982). A.R. West, Basic solid state chemistry. Wiley, New York (1999). 108 3 The structures of simple solids EXERCISES 3.1 What are the relationships between the unit cell parameters in the orthorhombic crystal system? the cation and anion in each of these structures? (b) In which of these structures will Rb have the larger apparent radius? 3.2 What are the fractional coordinates of the lattice points shown in the face-centred cubic unit cell (Fig. 3.5)? Confirm by counting lattice points and their contributions to the cubic unit cell that a face-centred, F, lattice contains four lattice points in the unit cell and a body-centred one two lattice points. 3.11 Consider the structure of caesium chloride. How many Cs ions occupy second-nearest-neighbour locations of a Cs ion? 3.3 Which of the following schemes for the repeating pattern of close-packed planes are not ways of generating close-packed lattices? (a) ABCABC ... , (b) ABAC ... , (c) ABBA ... , (d) ABCBC ... , (e) ABABC ... , (f) ABCCB ... 1 3.4 Determine the formula of a compound produced by filling 4 of the tetrahedral holes with cations X in a hexagonal close-packed array of anions A. 3.5 Potassium reacts with C60 (Fig. 3.16) to give a compound in which all the octahedral and tetrahedral holes are filled by potassium ions. Derive a stoichiometry for this compound. 3.6 Calculate a value for the atomic radius of the Cs atom with a coordination number of 12 given that the empirical atomic radius of Cs in the metal with a bcc structure is 272 pm. 3.7 Metallic sodium adopts a bcc structure with density 970 kg m3. What is the length of the edge of the unit cell? 3.8 An alloy of copper and gold has the structure shown in Fig. 3.75. Calculate the composition of this unit cell. What is the lattice type of this structure? Given that 24 carat gold is pure gold, what carat gold does this alloy represent? 3.12 Describe the coordination around the anions in the perovskite structure in terms of coordination to the A- and B-type cations. 3.13 Use radius-ratio rules and the ionic radii given in Resource section 1 to predict structures of (a) PuO2, (b) FrI, (c) BeO, (d) InN. 3.14 Based on the variation of ionic radii down a group, what structure would you predict for FrBr? 3.15 What are the most significant terms in the Born–Haber cycle for the formation of Ca3N2? 3.16 By considering the parameters that change in the Born–Mayer expression estimate lattice enthalpies for MgO and AlN given that MgO and AlN both adopt the rock-salt structure with similar lattice parameters and H L° NaCl 786 kJ mol 1 . ( ) 3.17 Use the Kapustinskii equation and the ionic and thermochemical radii given in Resource section 1 and Table 3.10, and r(Bk4) 96 pm to calculate lattice enthalpies of (a) BkO2, (b) K2SiF6, and (c) LiClO4. 3.18 Which member of each pair is likely to be more soluble in water: (a) SrSO4 or MgSO4, (b) NaF or NaBF4? 3.19 On the basis of the factors that contribute to lattice enthalpies place LiF, CaO, RbCl, AlN, NiO, and CsI, all of which adopt the rocksalt structure, in order of increasing lattice energy. 3.20 Recommend a specific cation for the quantitative precipitation of carbonate ion in water. Justify your recommendations. Au Cu 3.21 Predict what type of intrinsic defect is most likely to occur in (a) Ca3N2, (b) HgS. 3.22 Explain why the number of defects in a solid increases as it is heated. 3.23 By considering which dopant ions produce the blue of sapphires provide an explanation for the origin of the colour in the blue form of beryl known as aquamarine. Fig 3.75 The structure of Cu3Au. 3.24 For which of the following compounds might nonstoichiometry be found: magnesium oxide, vanadium carbide, manganese oxide? 3.25 Would VO or NiO be expected to show metallic properties? 3.9 Using Ketelaar’s triangle would you classify Sr2Ga ((Sr) 0.95; (Ga) 1.81) as an alloy or a Zintl phase? 3.26 Describe the difference between a semiconductor and a semimetal. 3.10 Depending on temperature, RbCl can exist in either the rock-salt or caesium-chloride structure. (a) What is the coordination number of 3.27 Classify the following as to whether they are likely to show n- or p-type semiconductivity: Ag2S, VO2, CuBr. PROBLEMS Demonstrate, by considering two adjacent unit cells, that a tetragonal face-centred lattice of dimensions a and c can always be redrawn as a body-centred tetragonal lattice with dimensions a/ 21/2 and c. 3.1 Draw a cubic unit cell (a) in projection (showing the fractional heights of the atoms) and (b) as a three-dimensional representation 1 1 1 1 1 that has atoms at the following positions: Ti at ( 2 , 2 , 2 ), O at ( 2 , 2 ,0), 1 1 1 1 (0, 2 , 2 ), and ( 2 ,0, 2 ), and Ba at (0,0,0). Remember that a cubic unit cell with an atom on the cell face, edge, or corner will have equivalent atoms displaced by the unit cell repeat in any direction. Of what structural type is the cell? 3.3 Draw one layer of close-packed spheres. On this layer mark the positions of the centres of the B layer atoms using the symbol ⊗ and, with the symbol ○, mark the positions of the centres of the C layer atoms of an fcc lattice. 3.2 Draw a tetragonal unit cell and mark on it a set of points that would define (a) a face-centred lattice and (b) a body-centred lattice. 3.4 In the structure of MoS2, the S atoms are arranged in close-packed layers that repeat themselves in the sequence AAA ... The Mo atoms Problems 109 occupy holes with coordination number 6. Show that each Mo atom is surrounded by a trigonal prism of S atoms. Tables 1.5 and 1.6. (b) What factor prevents the formation of this compound despite the favourable lattice enthalpy? 3.5 The ReO3 structure is cubic with an Re atom at each corner of the unit cell and one O atom on each unit cell edge midway between the Re atoms. Sketch this unit cell and determine (a) the coordination numbers of the ions and (b) the identity of the structure type that would be generated if a cation were inserted in the centre of each ReO3 unit cell. 3.15 The common oxidation number for an alkaline earth metal is 2. Using the Born–Mayer equation and a Born–Haber cycle, show that CaCl is an exothermic compound. Use a suitable analogy to estimate an ionic radius for Ca. The sublimation enthalpy of Ca(s) is 176 kJ mol1. Show that an explanation for the nonexistence of CaCl can be found in the enthalpy change for the reaction 2 CaCl(s) → Ca(s) CaCl2(s). 3.6 Consider the structure of rock salt. (a) How many Na ions occupy second-nearest-neighbour locations of an Na ion? (b) Pick out the closest-packed plane of Cl ions. (Hint: This hexagonal plane is perpendicular to a threefold axis.) 3.16 The Coulombic attraction of nearest-neighbour cations and anions accounts for the bulk of the lattice enthalpy of an ionic compound. With this fact in mind, estimate the order of increasing lattice enthalpy of (a) MgO, (b) NaCl, (c) AlN, all of which crystallize in the rock-salt structure. Give your reasoning. 3.7 Imagine the construction of an MX2 structure from the CsCl structure by removal of half the Cs ions to leave tetrahedral coordination around each Cl ion. Identify this MX2 structure. 3.8 Obtain formulae (MXn or MnX) for the following structures derived from hole filling in close-packed arrays with (a) half the octahedral holes filled, (b) one-quarter of the tetrahedral holes filled, and (c) two-thirds of the octahedral holes filled. What are the average coordination numbers of M and X in (a) and (b)? 3.9 Given the following data for the length of a side of the unit cell for compounds that crystallize in the rock-salt structure, determine the cation radii: MgSe (545 pm), CaSe (591 pm), SrSe (623 pm), BaSe (662 pm). (Hint: To determine the radius of Se2, assume that the Se2 ions are in contact in MgSe.) 3.10 Use the structure map in Fig. 3.47 to predict the coordination numbers of the cations and anions in (a) LiF, (b) RbBr, (c) SrS, (d) BeO. The observed coordination numbers are (6,6) for LiF, RbBr, and SrS and (4,4) for BeO. Propose a possible reason for the discrepancies. 3.11 Describe how the structures of the following can be described in terms of simple structure types of Table 3.4 but with complex ions K2PtCl6, [Ni(H2O)6].[SiF6], CsCN. 3.12 The structure of calcite CaCO3 is shown in Fig. 3.76. Describe how this structure is related to that of NaCl. 2– CO3 Ca2+ 3.17 There are two common polymorphs of zinc sulfide: cubic and hexagonal. Based on the analysis of Madelung constants alone, predict which polymorph should be more stable. Assume that the Zn–S distances in the two polymorphs are identical. 3.18 (a) Explain why lattice energy calculations based on the Born– Mayer equation reproduce the experimentally determined values to within 1 per cent for LiCl but only 10 per cent for AgCl given that both compounds have the rock-salt structure. (b) Identify a pair of compounds containing M2 ions that might be expected to show similar behaviour. 3.19 Which of the following pairs of isostructural compounds are likely to undergo thermal decomposition at lower temperature? Give your reasoning. (a) MgCO3 and CaCO3 (decomposition products MO CO2). (b) CsI3 and N(CH3)4I3 (both compounds contain I3; decomposition products MII2; the radius of N(CH3)4 is much greater than that of Cs). 3.20 The Kapustinskii equation shows that lattice enthalpies are inversely proportional to the ion separations. Later work has shown that further simplification of the Kapustinskii equation allows lattice enthalpies to be estimated from the molecular (formula) unit volume (the unit cell volume divided by the number of formula units, Z, it contains) or the mass density (see, for example, H.D.B. Jenkins and D. Tudela, J. Chem. Educ., 2003, 80, 1482). How would you expect the lattice enthalpy to vary as a function of (a) the molecular unit volume and (b) the mass density? Given the following unit cell volumes (all in cubic angstrom, Å3; 1 Å 1010 m) for the alkaline earth carbonates MCO3 and oxides, predict the observed decomposition behaviour of the carbonates. MgCO3 47 MgO CaCO3 61 CaO SrCO3 64 SrO BaCO3 76 BaO 19 28 34 42 3.21 By considering the rock-salt structure and the distances and charges around one central ion show that the first six terms of the Na Madelung series are Fig 3.76 The structure of CaCO3. 6 1 3.13 Using the accepted ionic radius of the ammonium ion, NH4, NH4Br is predicted to have a rock-salt structure with (6,6)-coordination. However, at room temperature NH4Br has a caesium-chloride structure. Explain this observation. 3.14 (a) Calculate the enthalpy of formation of the hypothetical compound KF2 assuming a CaF2 structure. Use the Born–Mayer equation to obtain the lattice enthalpy and estimate the radius of K2 by extrapolation of trends in Table 1.4 and Resource section 1. Ionization enthalpies and electron gain enthalpies are given in 12 2 8 3 6 4 24 5 24 6 Discuss methods for showing this series converges to 1.748 by reference to R.P. Grosso, J.T. Fermann, and W.J. Vining, J. Chem. Educ., 2001, 78, 1198. 3.22 Explain why higher levels of defects are found in solids at high temperatures and close to their melting points. How would pressure affect the equilibrium number of defects in a solid? 3.23 By considering the effect on the lattice energies of incorporating large numbers of defects and the resultant changes in oxidation 110 3 The structures of simple solids numbers of the ions making up the structure, predict which of the following systems should show nonstoichiometry over a large range of x: Zn1xO, Fe1xO, UO2x. 3.24 Graphite is a semimetal with a band structure of the type shown in Fig. 3.70. Reaction of graphite with potassium produces C8K while reaction with bromine yields C8Br. Assuming the graphite sheets remain intact and potassium and bromine enter the graphite structure as K and Br ions respectively, discuss whether you would expect the compounds C8K and C8Br to exhibit metallic, semimetallic, semiconducting, or insulating properties. Acids and bases This chapter focuses on the wide variety of species that are classified as acids and bases. The acids and bases described in the first part of the chapter take part in proton transfer reactions. Proton transfer equilibria can be discussed quantitatively in terms of acidity constants, which are a measure of the tendency for species to donate protons. In the second part of the chapter, we broaden the definition of acids and bases to include reactions that involve electron-pair sharing between a donor and an acceptor. This broadening enables us to extend our discussion of acids and bases to species that do not contain protons and to nonaqueous media. Because of the greater diversity of these species, a single scale of strength is not appropriate. Therefore, we describe two approaches: in one, acids and bases are classified as ‘hard’ or ‘soft’; in the other, thermochemical data are used to obtain a set of parameters characteristic of each species. 4 Brønsted acidity 4.1 Proton transfer equilibria in water 4.2 Solvent levelling 4.3 The solvent system definition of acids and bases Characteristics of Brønsted acids 4.4 Periodic trends in aqua acid strength 4.5 Simple oxoacids The original distinction between acids and bases was based, hazardously, on criteria of taste and feel: acids were sour and bases felt soapy. A deeper chemical understanding of their properties emerged from Arrhenius’s (1884) conception of an acid as a compound that produced hydrogen ions in water. The modern definitions that we consider in this chapter are based on a broader range of chemical reactions. The definition due to Brønsted and Lowry focuses on proton transfer, and that due to Lewis is based on the interaction of electron pair acceptor and electron pair donor molecules and ions. Acid–base reactions are common, although we do not always immediately recognize them as such, especially if they involve more subtle definitions of what it is to be an acid or base. For instance, production of acid rain begins with a very simple reaction between sulfur dioxide and water: SO2(g) H2O(l) ➝ HOSO2(aq) H(aq) This will turn out to be a type of acid–base reaction. Saponification is the process used in soapmaking: NaOH(aq) RCOOR(aq) ➝ NaRCO2(aq) ROH(aq) This too is a type of acid–base reaction. There are many such reactions, and in due course we shall see why they should be regarded as reactions between acids and bases. Brønsted acidity Key points: A Brønsted acid is a proton donor and a Brønsted base is a proton acceptor. A proton has no separate existence in chemistry and it is always associated with other species. A simple representation of a hydrogen ion in water is as the hydronium ion, H3O. Johannes Brønsted in Denmark and Thomas Lowry in England proposed (in 1923) that the essential feature of an acid–base reaction is the transfer of a hydrogen ion, H, from one species to another. In the context of this definition, a hydrogen ion is often referred to as a proton. They suggested that any substance that acts as a proton donor should be classified as an acid, and any substance that acts as a proton acceptor should be classified as a base. Substances that act in this way are now called ‘Brønsted acids’ and ‘Brønsted bases’, respectively: A Brønsted acid is a proton donor. A Brønsted base is a proton acceptor. 4.6 Anhydrous oxides 4.7 Polyoxo compound formation 4.8 Nonaqueous solvents Lewis acidity 4.9 Examples of Lewis acids and bases 4.10 Group characteristics of Lewis acids Reactions and properties of Lewis acids and bases 4.11 The fundamental types of reaction 4.12 Hard and soft acids and bases 4.13 Thermodynamic acidity parameters 4.14 Solvents as acids and bases Applications of acid–base chemistry 4.15 Superacids and superbases 4.16 Heterogeneous acid–base reactions FURTHER READING EXERCISES PROBLEMS 112 4 Acids and bases The definitions make no reference to the environment in which proton transfer occurs, so they apply to proton transfer behaviour in any solvent and even in no solvent at all. An example of a Brønsted acid is hydrogen fluoride, HF, which can donate a proton to another molecule, such as H2O, when it dissolves in water: HF(g) H2O(l) ➝ H3O(aq) F(aq) An example of a Brønsted base is ammonia, NH3, which can accept a proton from a proton donor: H2O(l) NH3(aq) ➝ NH4 (aq) OH(aq) + O 101 pm H 100–120° 1 260 pm 116° H 2O 116° H2O 105° 260 pm 2 H9O4+ OH2 260 pm As these two examples show, water is an example of an amphiprotic substance, a substance that can act as both a Brønsted acid and a Brønsted base. When an acid donates a proton to a water molecule, the latter is converted into a hydronium ion, H3O (1; the dimensions are taken from the crystal structure of H3OClO4). However, the entity H3O is almost certainly an oversimplified description of the proton in water, for it participates in extensive hydrogen bonding, and a better representation is H9O4 (2). Gasphase studies of water clusters using mass spectrometry suggest that a cage of H2O molecules can condense around one H3O ion in a regular pentagonal dodecahedral arrangement, resulting in the formation of the species H(H2O)21. As these structures indicate, the most appropriate description of a proton in water varies according to the environment and the experiment under consideration; for simplicity, we shall use the representation H3O throughout. 4.1 Proton transfer equilibria in water Proton transfer between acids and bases is fast in both directions, so the dynamic equilibria HF(aq) H2O(l) H3O(aq) F(aq) H2O(l) NH3(aq) NH4 (aq) OH(aq) give a more complete description of the behaviour of the acid HF and the base NH3 in water than the forward reaction alone. The central feature of Brønsted acid–base chemistry in aqueous solution is that of rapid attainment of equilibrium in the proton transfer reaction, and we concentrate on this aspect. (a) Conjugate acids and bases Key points: When a species donates a proton, it becomes the conjugate base; when a species gains a proton, it becomes the conjugate acid. Conjugate acids and bases are in equilibrium in solution. The form of the two forward and reverse reactions given above, both of which depend on the transfer of a proton from an acid to a base, is expressed by writing the general Brønsted equilibrium as Acid1 Base2 Acid2 Base1 The species Base1 is called the conjugate base of Acid1, and Acid2 is the conjugate acid of Base2. The conjugate base of an acid is the species that is left after a proton is lost. The conjugate acid of a base is the species formed when a proton is gained. Thus, F– is the conjugate base of HF and H3O is the conjugate acid of H2O. There is no fundamental distinction between an acid and a conjugate acid or a base and a conjugate base: a conjugate acid is just another acid and a conjugate base is just another base. E X A M PL E 4 .1 Identifying acids and bases Identify the Brønsted acid and its conjugate base in the following reactions: (a) HSO4 (aq) OH(aq) ➝ H2O(l) SO42(aq) (b) PO43(aq) H2O(l) ➝ HPO42(aq) OH Answer We need to identify the species that loses a proton and its conjugate partner. (a) The hydrogensulfate ion, HSO4 , transfers a proton to hydroxide; it is therefore the acid and the SO42 ion produced is its conjugate Brønsted acidity base. (b) The H2O molecule transfers a proton to the phosphate ion acting as a base; thus H2O is the acid and the OH ion is its conjugate base. Self-test 4.1 Identify the acid, base, conjugate acid, and conjugate base in the following reactions: (a) HNO3(aq) H2O(l) ➝ H3O(aq) NO3(aq) (b) CO32(aq) H2O(l) ➝ HCO3 (aq) OH (c) NH3(aq) H2S(aq) ➝ NH4(aq) HS(aq) (b) The strengths of Brønsted acids Key points: The strength of a Brønsted acid is measured by its acidity constant, and the strength of a Brønsted base is measured by its basicity constant; the stronger the base, the weaker is its conjugate acid. Throughout this discussion, we shall need the concept of pH, which we assume to be familiar from introductory chemistry: pH –log [H3O], and hence [H3O] 10–pH (4.1) The strength of a Brønsted acid, such as HF, in aqueous solution is expressed by its acidity constant (or ‘acid ionization constant’), Ka: HF(aq) H2O(l) H3O(aq) F–(aq) Ka = [ H3O+ ][ F − ] [HF] More generally: HX(aq) H2O(l) H3O(aq) X–(aq) Ka = [ H3O+ ][ X − ] [HX] (4.2) In this definition, [X–] denotes the numerical value of the molar concentration of the species X– (so, if the molar concentration of HF molecules is 0.001 mol dm–3, then [HF] 0.001). A value Ka 1) is a strong acid. Such acids are commonly regarded as being fully deprotonated in solution (but it must never be forgotten that that is only an approximation). For example, hydrochloric acid is regarded as a solution of H3O and Cl– ions, and a negligible concentration of HCl molecules. A substance with pKa > 0 (corresponding to Ka < 1) is classified as a weak acid; for such species, the proton transfer equilibrium lies in favour of nonionized acid. Hydrogen fluoride is a weak acid in water, and hydrofluoric acid consists of hydronium ions, fluoride ions, and a high proportion of HF molecules. A strong base reacts with water to become almost fully protonated. An example is the oxide ion, O2–, which is immediately converted into OH– ions in water. A weak base is only partially protonated in water. An example is NH3, which dissolves in water to give only a small proportion of NH4 ions. The conjugate base of any strong acid is a weak base because it is thermodynamically unfavourable for such a base to accept a proton. (d) Polyprotic acids Key points: A polyprotic acid loses protons in succession, and successive deprotonations are progressively less favourable; a distribution diagram summarizes how the fraction of each species present depends on the pH of the solution. A polyprotic acid is a substance that can donate more than one proton. An example is hydrogen sulfide, H2S, a diprotic acid. For a diprotic acid, there are two successive proton donations and two acidity constants: H 2 S(aq) + H 2O(l) HS− (aq) + H3O+ (aq) Ka1 = HS− (aq) + H 2O(l) S2− (aq) + H3O+ (aq) Ka 2 = [ H3O+ ][ HS− ] [ H 2 S] [ H3O+ ][S2− ] [ HS− ] Table 4.1 Acidity constants for species in aqueous solution at 25°C Acid HA A Ka pKa Acid HA A– Ka Hydriodic HI I 1011 –11 Ethanoic CH3COOH CH3CO2 1.74 × 105 Perchloric HCIO4 ClO4− 10 –10 Pyridinium ion HC5H5N Hydrobromic HBr Br 109 –9 Carbonic H2CO3 Hydrochloric HCl Cl 7 10 –7 Hydrogen sulfide H2S HS Sulfuric H2SO4 HSO−4 102 –2 Boric acid B(OH)3 B(OH)−4 –2 Ammonium ion NH+4 0.0 Hydrocyanic HCN HCO3− HAsO2− 4 3 Nitric HNO3 NO Hydronium ion H3O H2O Chloric HCIO3 Sulfurous H2SO3 Hydrogensulfate ion HSO−4 Phosphoric H3PO4 ClO3− HSO3− SO2− 4 H2PO−4 10 10 2 1 –1 10 1 Hydrogencarbonate ion 1.5 × 102 1.81 Hydrogenarsenate ion C5H5N 5.6 × 10 5.25 HCO3− 4.3 × 107 6.37 8 9.1 × 10 7.04 7.2 × 1010 9.14 NH3 10 5.6 × 10 9.25 CN 4.9 × 1010 9.31 CO 2 3 AsO3− 4 1.2 × 10 1.92 Hydrogensulfide ion HS 7.5 × 103 2.12 Hydrogenphosphate ion HPO2− 4 PO3− 4 Dihydrogenphosphate ion H2PO−4 HPO2− 4 Hydrofluoric HF F 3.5 × 10 3.45 Formic HCOOH HCO2 1.8 × 104 3.75 The proton transfer equilibrium is B(OH)3(aq) 2H2O(l) H3O(aq) B(OH)−4 (aq). 4.76 6 2 4 pKa S 2 11 4.8 × 10 10.32 3.0 × 1012 11.53 19 1.1 × 10 2.2 × 1013 6.2 × 10 8 19 12.67 7.21 116 4 Acids and bases From Table 4.1, Ka1 9.1 × 10–8 (pKa1 7.04) and Ka2 ≈ 1.1 × 10–19 (pKa2 19). The second acidity constant, Ka2, is almost always smaller than Ka1 (and hence pKa2 is generally larger than pKa1). The decrease in Ka is consistent with an electrostatic model of the acid in which, in the second deprotonation, a proton must separate from a centre with one more negative charge than in the first deprotonation. Because additional electrostatic work must be done to remove the positively charged proton, the deprotonation is less favourable. E X A M PL E 4 . 3 Calculating the concentration of ions in polyprotic acids Calculate the concentration of carbonate ions in 0.10 M H2CO3(aq). Ka1 is given in Table 4.1; Ka2 4.6 × 1011. Answer We need to consider the equilibria for the successive deprotonation steps with their acidity constants: H2CO3 (aq) + H2O(l) HCO3− (aq) + H3O+ (aq) HCO3− (aq) + H2O(l) CO32− (aq) + H3O+ (aq) [H3O+ ][HCO3− ] [H2CO3 ] [H3O+ ][CO32− ] K a2 = [HCO3− ] K a1 = We suppose that the second deprotonation is so slight that it has no effect on the value of [H3O] arising from the first deprotonation, in which case we can write [H3O] [HCO3] in Ka2. These two terms therefore cancel in the expression for Ka2, which results in Ka2 [CO32–] independent of the initial concentration of the acid. It follows that the concentration of carbonate ions in the solution is 4.6 × 10–11 mol dm–3. Self-test 4.3 Calculate the pH of 0.20 Ka2 4.6 × 105. M H2C4H4O6(aq) (tartaric acid), given Ka1 1.0 ×103 and The clearest representation of the concentrations of the species that are formed in the successive proton transfer equilibria of polyprotic acids is a distribution diagram, a diagram showing the fraction of solute present as a specified species X, f(X), plotted against the pH. Consider, for instance, the triprotic acid H3PO4, which releases three protons in succession to give H2PO4–, HPO42–, and PO43–. The fraction of solute present as intact H3PO4 molecules is ( ) f H3 PO4 = [ H3 PO4 ] [ H3 PO4 ] + [ H 2 PO4− ] + [ HPO24− ] + [ PO34− ] (4.7) The concentration of each solute at a given pH can be calculated from the pKa values.1 Figure 4.1 shows the fraction of all four solute species as a function of pH and hence summarizes the relative importance of each acid and its conjugate base at each pH. Conversely, the diagram indicates the pH of the solution that contains a particular fraction of the species. We see, for instance, that if pH < pKa1, corresponding to high hydronium ion concentrations, then the dominant species is the fully protonated H3PO4 molecule. However, if pH > pKa3, corresponding to low hydronium ion concentrations, then the dominant species is the fully deprotonated PO43– ion. The intermediate species are dominant when pH values lie between the relevant pKas. (e) Factors governing the strengths of acids and bases Key points: Proton affinity is the negative of the gas phase proton-gain enthalpy. The proton affinities of p-block conjugate bases decrease to the right along a period and down a group. Solution proton affinities for binary acids are lower than gas phase proton affinities. Small highly charged ions are stabilized in polar solvents. A quantitative understanding of the relative acidities of X–H protons can be obtained by considering the enthalpy changes accompanying proton transfer. We shall consider gas-phase proton transfer reactions first and then consider the effects of the solvent. 1 For the calculations involved, see P. Atkins and L. Jones, Chemical principles. W.H. Freeman & Co. (2010). Brønsted acidity pKa1 Fraction of species f (X) 1.0 H3PO4 pKa2 pKa3 – 4 2– 4 PO3– 4 HPO H2PO 0.8 0.6 0.4 0 2 12.67 7.21 0 2.12 0.2 4 6 8 10 12 14 pH Figure 4.1 The distribution diagram for the various forms of the triprotic acid phosphoric acid in water, as a function of pH. The simplest reaction of a proton is its attachment to a base, A– (which, although denoted here as a negatively charged species, could be a neutral molecule, such as NH3), in the gas phase: A–(g) H(g) → HA(g) The standard enthalpy of this reaction is the proton-gain enthalpy, ∆pgH O . The negative of this quantity is often reported as the proton affinity, Ap (Table 4.2). When ∆pgH O is large and negative, corresponding to an exothermic proton attachment, the proton affinity is high, indicating strongly basic character in the gas phase. If the proton gain enthalpy is only slightly negative, then the proton affinity is low, indicating a weaker basic (or more acidic) character. The proton affinities of the conjugate bases of p-block binary acids HA decrease to the right along a period and down a group, indicating an increase in gas-phase acidity. Thus, HF is a stronger acid than H2O and HI is the strongest acid of the hydrogen halides. In other words, the order of proton affinities of their conjugate bases is I– < OH– < F–. These trends can be explained by using a thermodynamic cycle such as that shown in Fig. 4.2, in which proton gain can be thought of as the outcome of three steps: Electron loss from A−: A−(g) → A(g) e−(g) –∆egH O (A) Ae(A) (the reverse of electron gain by A) Table 4.2 Gas phase and solution proton affinities Conjugate acid Base Ap /kJ mol−1 A′p /kJ mol−1 HF F− 1553 1150 HCl Cl− 1393 1090 HBr Br − 1353 1079 HI I− 1314 1068 H2O OH − 1643 1188 HCN CN− 1476 1183 H2O 723 1130 NH3 865 1182 H3O NH4 Ap is the gas phase proton affinity, A′p is the effective proton affinity for the base in water. 117 118 4 Acids and bases Electron gain by H: H+(g) + A–(g) H(g) e−(g) → H(g) –∆iH O (H) −I(H) (the reverse of the ionization of H) ∆egH°(A) H+(g) + A(g) + e–(g) Combination of H and A: H(g) A(g) → HA(g) −B(H−A) (the reverse of H−A bond dissociation) I(H) The proton-gain enthalpy of the conjugate base A− is the sum of these enthalpy changes: ∆pgH°(A–) Overall: H(g) A−(g) → HA(g) ∆pgH O (A−) Ae(A) − I(H) − B(H−A) Therefore, the proton affinity of A− is H(g) + A(g) Ap(A−) B(H−A) I(H) − Ae(A) B(H–A) HA(g) Figure 4.2 Thermodynamic cycle for a proton gain reaction. (4.8) The dominant factor in the variation in proton affinity across a period is the trend in electron affinity of A, which increases from left to right and hence lowers the proton affinity of A−. Thus, because the proton affinity of A− decreases, the gas-phase acidity of HA increases across a period as the electron affinity of A increases. Because increasing electron affinity correlates with increasing electronegativity (Section 1.9), the gas-phase acidity of HA also increases as the electronegativity of A increases. The dominant factor when descending a group is the decrease in the H−A bond dissociation enthalpy, which lowers the proton affinity of A− and therefore results in an increase in the gas-phase acid strength of HA. The overall result of these effects is a decrease in gas-phase proton affinity of A−, and therefore an increase in the gas-phase acidity of HA, from the top left to bottom right of the p block. On this basis we see that HI is a much stronger acid than CH4. The correlation we have described is modified when a solvent (typically water) is present. The gas-phase process A–(g) H(g) → AH(g) becomes A–(aq) H(aq) → HA(aq) and the negative of the accompanying proton-gain enthalpy is called the effective proton affinity, Ap, of A–(aq). If the species A– denotes H2O itself, the effective proton affinity of H2O is the enthalpy change accompanying the process H2O(l) H(aq) → H3O(aq) The energy released as water molecules are attached to a proton in the gas phase, the process n H2O(g) H(g) → H(H2O)n(g) can be measured by mass spectrometry and used to assess the energy change for the hydration process in solution. It is found that the energy released passes through a maximum value of 1130 kJ mol−1 as n increases, and this value is taken to be the effective proton affinity of H2O in bulk water. The effective proton affinity of the ion OH– in water is simply the negative of the enthalpy of the reaction OH–(g) H(aq) → H2O(l) which can be measured by conventional means (such as the temperature dependence of its equilibrium constant, Kw). The value found is 1188 kJ mol–1. The reaction HA(aq) H2O(l) → H3O(aq) A–(aq) is exothermic if the effective proton affinity of A–(aq) is lower than that of H2O(l) (less than 1130 kJ mol–1) and—provided entropy changes are negligible and enthalpy changes are a guide to spontaneity—will give up protons to the water and be strongly acidic. Likewise, the reaction A–(aq) H2O(l) → HA(aq) OH–(aq) is exothermic if the effective proton affinity of A–(aq) is higher than that of OH–(aq) (1188 kJ mol–1). Provided enthalpy changes are a guide to spontaneity, A–(aq) will accept protons and will act as a strong base. 119 Brønsted acidity ■ A brief illustration. The effective proton affinity of I− in water is 1068 kJ mol−1 compared to 1314 kJ mol−1 in the gas phase, showing that the I− ion is stabilized by hydration. The effective proton affinity is also smaller than the effective proton affinity of water (1130 kJ mol−1), which is consistent with the fact that HI is a strong acid in water. All the halide ions except F− have effective proton affinities smaller than that of water, which is consistent with all the hydrogen halides except HF being strong acids in water. ■ The effects of solvation can be rationalized in terms of an electrostatic model in which the solvent is treated as a continuous dielectric medium. The solvation of a gas-phase ion is always strongly exothermic. The magnitude of the enthalpy of solvation ∆solvH O (the enthalpy of hydration in water, ∆hydH O ) depends on the radius of the ions, the relative permittivity of the solvent, and the possibility of specific bonding (especially hydrogen bonding) between the ions and the solvent. When considering the gas phase we assume that entropy contributions for the proton transfer process are small and so ∆G O ≈ ∆H O . In solution, entropy effects cannot be ignored and we must use ∆G O . The Gibbs energy of solvation of an ion can be identified as the energy involved in transferring the anion from a vacuum into a solvent of relative permittivity εr. The Born equation can be derived using this model:2 ∆solvG O = − N A z 2e2 ⎛ 1⎞ ⎜1 − ⎟ 8 0 r ⎝ r ⎠ (4.9) where z is the charge number of the ion, r is its effective radius, which includes part of the radii of solvent molecules, NA is Avogadro’s constant, ε0 is the vacuum permittivity, and εr is the relative permittivity (the dielectric constant). Because z2/r , the electrostatic parameter of the ion (Section 3.15), this expression can be written N A e 2! ⎛ 1⎞ ⎜1 − ⎟ 8 0 ⎝ r ⎠ (4.10) The Gibbs energy of solvation is proportional to , so small, highly charged ions are stabilized in polar solvents (Fig. 4.3). The Born equation also shows that the larger the relative permittivity the more negative the value of ∆solvG O . This stabilization is particularly important for water, for which εr 80 (and the term in parentheses is close to 1), compared with nonpolar solvents for which εr may be as low as 2 (and the term in parentheses is close to 0.5). Because ∆solvG O is the change in molar Gibbs energy when an ion is transferred from the gas phase into aqueous solution, a large, negative value of ∆solvG O favours the formation of ions in solution compared with the gas phase (Fig. 4.3). The interaction of the charged ion with the polar solvent molecules stabilizes the conjugate base A− relative to the parent acid HA, and as a result the acidity of HA is enhanced by the polar solvent. On the other hand, the effective proton affinity of a neutral base B is higher than in the gas phase because the conjugate acid HB is stabilized by solvation. Because cationic acids, such as NH4, are stabilized by solvation, their effective proton affinity is higher than in the gas phase and their acidity is lowered by a polar solvent. The Born equation ascribes stabilization to Coulombic interactions. However, hydrogen bonding is an important factor in protic solvents such as water and leads to the formation of hydrogen-bonded clusters around some solutes. As a result, water has a greater stabilizing effect on small, highly charged ions than the Born equation predicts. This stabilizing effect is particularly great for F−, OH−, and Cl−, with their high charge densities and for which water acts as a hydrogen-bond donor. Because water has lone pairs of electrons on O, it can also be a hydrogen-bond acceptor. Acidic ions such as NH4 are stabilized by hydrogen bonding and consequently have a lower acidity than predicted by the Born equation. 4.2 Solvent levelling Key point: A solvent with a large autoprotolysis constant can be used to discriminate between a wide range of acid and base strengths. An acid that is weak in water may appear strong in a solvent that is a more effective proton acceptor, and vice versa. Indeed, in sufficiently basic solvents (such as liquid ammonia), it 2 For the derivation of the Born equation see P. Atkins and J. de Paula, Physical chemistry, Oxford University Press and W.H. Freeman & Co. (2010). Gibbs energy of hydration, –∆hydG°/(1000 kJ mol–1) ∆solvG O = − 3 PO3– 4 2.5 2 1.5 SO2– 4 S22– 1 CN– 0.5 Cl– F– OH– I– 0 0 1 2 3 Electrostatic parameter, 4 Figure 4.3 The correlation between ∆solvG O and the electrostatic parameter of selected anions. To obtain as a small dimensionless number we have used 100z2/(r/pm). 120 4 Acids and bases may not be possible to discriminate between their strengths because all of them will be fully deprotonated. Similarly, bases that are weak in water may appear strong in a more strongly proton-donating solvent (such as anhydrous acetic acid). It may not be possible to arrange a series of bases according to strength, for all of them will be effectively fully protonated in acidic solvents. We shall now see that the autoprotolysis constant of a solvent plays a crucial role in determining the range of acid or base strengths that can be distinguished for species dissolved in it. Any acid stronger than H3O in water donates a proton to H2O and forms H3O. Consequently, no acid significantly stronger than H3O can remain protonated in water. No experiment conducted in water can tell us which of HBr and HI is the stronger acid because both transfer their protons essentially completely to give H3O. In effect, solutions of the strong acids HX and HY behave as though they are solutions of H3O ions regardless of whether HX is intrinsically stronger than HY. Water is therefore said to have a levelling effect that brings all stronger acids down to the acidity of H3O. The strengths of such acids can be distinguished by using a less basic solvent. For instance, although HBr and HI have indistinguishable acid strengths in water, in acetic acid HBr and HI behave as weak acids and their strengths can be distinguished: in this way it is found that HI is a stronger proton donor than HBr. The levelling effect can be expressed in terms of the pKa of the acid. An acid such as HCN dissolved in a solvent, HSol, is classified as strong if pKa < 0, where Ka is the acidity constant of the acid in the solvent Sol: HCN(sol) + HSol(l) H 2Sol + (sol) + CN − (sol) Ka = [ H 2Sol + ][CN − ] [ HCN ] That is, all acids with pKa < 0 (corresponding to Ka > 1) display the acidity of H2Sol when they are dissolved in the solvent HSol. An analogous effect can be found for bases in water. Any base that is strong enough to undergo complete protonation by water produces an OH ion for each molecule of base added. The solution behaves as though it contains OH ions. Therefore, we cannot distinguish the proton-accepting power of such bases, and we say that they are levelled to a common strength. Indeed, the OH– ion is the strongest base that can exist in water because any species that is a stronger proton acceptor immediately forms OH ions by proton transfer from water. For this reason, we cannot study NH2 or CH3 in water by dissolving alkali metal amides or methides because both anions generate OH– ions and are fully protonated to NH3 and CH4: KNH2(s) H2O(l) → K(aq) OH–(aq) NH3(aq) Li4(CH3)4(s) 4 H2O(l) → 4 Li(aq) 4 OH–(aq) 4 CH4(g) The base levelling effect can be expressed in terms of the pKb of the base. A base dissolved in HSol is classified as strong if pKb < 0, where Kb is the basicity constant of the base in HSol: NH3 (sol) + HSol(l) NH 4+ (sol) + Sol − (sol) Kb = [ NH 4+ ][Sol − ] [ NH3 ] That is, all bases with pKb < 0 (corresponding to Kb > 1) display the basicity of Sol– in the solvent HSol. Now, because pKa pKb pKsol, this criterion for levelling may be expressed as follows: all bases with pKa > pKsol give a negative value for pKb and behave like Sol– in the solvent HSol. It follows from this discussion of acids and bases in a common solvent HSol that, because any acid is levelled if pKa < 0 in HSol and any base is levelled if pKa > pKsol in the same solvent, then the window of strengths that are not levelled in the solvent is from pKa 0 to pKsol. For water, pKw 14. For liquid ammonia, the autoprotolysis equilibrium is 2 NH3(l) NH4(sol) NH2(sol) pKam 33 It follows from these figures that acids and bases are discriminated much less in water than they are in ammonia. The discrimination windows of a number of solvents are shown in Fig. 4.4. The window for dimethylsulfoxide (DMSO, (CH3)2SO) is wide because pKDMSO 37. Brønsted acidity Fluorosulfuric acid, H2SO3F Hydrofluoric acid, HF Sulfuric acid, H2SO4 Methanoic acid, HCOOH Ethanoic acid, CH3COOH Ethanol, CH3CH2OH Water, H2O Dimethylsulfoxide (DMSO) Ammonia, NH3 –20 –10 0 10 20 Effective pH in water 30 40 Figure 4.4 The acid–base discrimination window for a variety of solvents. The width of each window is proportional to the autoprotolysis constant of the solvent. Consequently, DMSO can be used to study a wide range of acids (from H2SO4 to PH3). Water has a narrow window compared to some of the other solvents shown in the illustration. One reason is the high relative permittivity of water, which favours the formation of H3O and OH– ions. Permittivity is a measure of the ability of a material to resist the formation of an electric field within it. E X A MPL E 4 . 4 Differentiating acidities in different solvents Which of the solvents given in Fig. 4.4 could be used to differentiate the acidities of HCl (pKa ≈ –6) and HBr (pKa ≈ –9)? Answer We need to look for a solvent with an acid–base discrimination window between –6 and –9. The only solvents in the table for which the window covers the range –6 to –9 are methanoic (formic) acid, HCOOH, and hydrofluoric acid, HF. Self-test 4.4 Which of the solvents given in Fig. 4.4 could be used to discriminate the acidities of PH3 (pKa ≈ 27) and GeH4 (pKa ≈ 25)? 4.3 The solvent system definition of acids and bases Key point: The solvent system definition of acids and bases extends the Brønsted–Lowry definition to include species that do not participate in proton transfer. The Brønsted–Lowry definition of acids and bases describes acids and bases in terms of the proton. This system can be extended to species that cannot participate in proton transfer by recognizing an analogy with the autoprotolysis reaction of water: 2 H2O(l) H3O(aq) OH–(aq) An acid increases the concentration of H3O ions and a base increases the concentration of OH– ions. We can recognize a similar structure in the autoionization reaction of some aprotic solvents, such as bromine trifluoride, BrF3: 2 BrF3(l) BrF2(sol) BrF4–(sol) where sol denotes solution in the non-ionized species (BrF3 in this case). In the solventsystem definition, any solute that increases the concentration of the cation generated by autoionization of the solvent is defined as an acid and any that increases the concentration of the corresponding anion is defined as a base. This solvent system definition can be applied to any solvent that autoionizes and is particularly useful when discussing reactions in nonaqueous solvents (Section 4.8). 121 122 4 Acids and bases E X A M PL E 4 . 5 Identifying acids and bases using the solvent system method The salt BrF2AsF6 is soluble in BrF3. Is it an acid or base in this solvent? Answer We need to identify the autoionization products of the solvent and then decide whether the solute increases the concentration of the cation (an acid) or the anion (a base). The autoionization products of BrF3 are BrF2 and BrF4. The solute produces BrF2 and AsF6 ions when it dissolves. As the salt increases the concentration of the cations it is defined as an acid in the solvent system. Self-test 4.5 Is KBrF4 an acid or a base in BrF3? Characteristics of Brønsted acids 3+ H2O Key point: Aqua acids, hydroxoacids, and oxoacids are typical of specific regions of the periodic table. We shall now concentrate on Brønsted acids and bases in water. We have focused the discussion so far on acids of the type HX. However, the largest class of acids in water consists of species that donate protons from an –OH group attached to a central atom. A donatable proton of this kind is called an acidic proton to distinguish it from other protons that may be present in the molecule, such as the nonacidic methyl protons in CH3COOH. There are three classes of acids to consider: Fe 3 [Fe(OH2)6]3+ 1) An aqua acid, in which the acidic proton is on a water molecule coordinated to a central metal ion. E(OH2)(aq) H2O(l) E(OH)–(aq) H3O(aq) An example is OH eT [Fe(OH2)6]3(aq) H2O(l) [Fe(OH2)5OH]2(aq) H3O(aq) The aqua acid, the hexaaquairon(III) ion, is shown as (3). 2) A hydroxoacid, in which the acidic proton is on a hydroxyl group without a neighbouring oxo group (O). An example is Te(OH)6 (4). 3) An oxoacid, in which the acidic proton is on a hydroxyl group with an oxo group attached to the same atom. 4 Te(OH)6 OH S O Sulfuric acid, H2SO4 (O2S(OH)2; 5), is an example of an oxoacid. The three classes of acids can be regarded as successive stages in the deprotonation of an aqua acid: H H aqua acid hydroxoacid oxoacid An example of these successive stages is provided by a d-block metal in an intermediate oxidation state, such as Ru(IV): 5 O2S(OH)2, H2SO4 4+ OH2 L L Ru L L OH2 2+ OH −2 H+ +2 H L L Ru L L OH O − H+ +H L L Ru L L OH Aqua acids are characteristic of central atoms in low oxidation states, of s- and d-block metals, and of metals on the left of the p block. Oxoacids are commonly found where the central element is in a high oxidation state. An element from the right of the p block may also produce an oxoacid in one of its intermediate oxidation states (HClO2 is an example). 4.4 Periodic trends in aqua acid strength Key points: The strengths of aqua acids typically increase with increasing positive charge of the central metal ion and with decreasing ionic radius; exceptions are commonly due to the effects of covalent bonding. 123 Characteristics of Brønsted acids E X A MPL E 4 .6 Accounting for trends in aqua acid strength Account for the trend in acidity [Fe(OH2)6]2 < [Fe(OH2)6]3 < [Al(OH2)6]3 ≈ [Hg(OH2)]2. Answer We need to consider the charge density on the metal centre and its effect on the ease with which the H2O ligands can be deprotonated. The weakest acid is the Fe2 complex on account of its relatively large ionic radius and low charge. The increase of charge to 3 increases the acid strength. The greater acidity of Al3 can be explained by its smaller radius. The anomalous ion in the series is the Hg2 complex. This complex reflects the failure of an ionic model because in the complex there is a large transfer of positive charge to oxygen as a result of covalent bonding. Self-test 4.6 Arrange [Na(OH2)6], [Sc(OH2)6]3, [Mn(OH2)6]2, and [Ni(OH2)6]2 in order of increasing acidity. 4.5 Simple oxoacids The simplest oxoacids are the mononuclear acids, which contain one atom of the parent element. They include H2CO3, HNO3, H3PO4, and H2SO4.3 These oxoacids are formed by the electronegative elements at the upper right of the periodic table and by other elements in high oxidation states (Table 4.3). One interesting feature in the table is the occurrence of planar H2CO3 and HNO3 molecules but not their analogues in later periods. As we saw in Chapter 2, π bonding is more important among the Period 2 elements, so their atoms are more likely to be constrained to lie in a plane. (a) Substituted oxoacids Key points: Substituted oxoacids have strengths that may be rationalized in terms of the electronwithdrawing power of the substitutent; in a few cases, a nonacidic H atom is attached directly to the central atom of an oxoacid. One or more –OH groups of an oxoacid may be replaced by other groups to give a series of substituted oxoacids, which include fluorosulfuric acid, O2SF(OH), and aminosulfuric acid, O2S(NH2)OH (6). Because fluorine is highly electronegative, it withdraws electrons 3 These acids are more helpfully written as (HO)2CO, HONO2, (HO)3PO, and (HO)2SO2, and boric acid written as B(OH)3 rather than H3BO3. In this text we use both forms of notation depending upon the properties being explained. 3+ Tl 2 3+ Sn2+ 4 Hg 2+ 3+ Fe 4+ 3+ Cr Th Sc 6 pKa The strengths of aqua acids typically increase with increasing positive charge of the central metal ion and with decreasing ionic radius. This variation can be rationalized to some extent in terms of an ionic model, in which the metal cation is represented by a sphere of radius r+ carrying z positive charges. Because protons are more easily removed from the vicinity of cations of high charge and small radius, the model predicts that the acidity should increase with increasing z and with decreasing r+. The validity of the ionic model of acid strengths can be judged from Fig. 4.5. Aqua ions of elements that form ionic solids (principally those from the s block) have pKa values that are quite well described by the ionic model. Several d-block ions (such as Fe2 and Cr3) lie reasonably near the same straight line, but many ions (particularly those with low pKa, corresponding to high acid strength) deviate markedly from it. This deviation indicates that the metal ions repel the departing proton more strongly than is predicted by the ionic model. This enhanced repulsion can be rationalized by supposing that the positive charge of the cation is not confined to the central ion but is delocalized over the ligands and hence is closer to the departing proton. The delocalization is equivalent to attributing covalence to the element–oxygen bond. Indeed, the correlation is worst for ions that are disposed to form covalent bonds. For the later d- and the p-block metal ions (such as Cu2 and Sn2, respectively), the strengths of the aqua acids are much greater than the ionic model predicts. For these species, covalent bonding is more important than ionic bonding and the ionic model is unrealistic. The overlap between metal orbitals and the orbitals of an oxygen ligand increases from left to right across a period. It also increases down a group, so aqua ions of heavier d-block metals tend to be stronger acids. Cd 2+ 3+ Lu Cu2+ 8 Nd3+ 2+ Fe 10 Zn Au Ca 12 2+ 2+ 2+ Mg2+ Ba 2+ + 14 Li Sr Na+ 0 8 16 4 12 Electrostatic parameter, Figure 4.5 The correlation between acidity constant and the electrostatic parameter of aqua ions. 124 4 Acids and bases Table 4.3 The structure and pKa values of oxoacids p0 p1 p2 O HO——Cl O C N HO 7.2 O OH 3.6 Si OH OH 10 P HO OH OH HO HO S OH OH O 2.1, 7.4, 12.7 Te 2.0 HO HO I OH OH OH HO OH 7.8, 11.2 1.6, 7.0 OH Se O B HO A s OH 9.1 HO OH O OH O OH H 1.8, 6.6 O l C O –10 Cl O P OH OH –1.9 O O OH OH O O l C O HO OH –1.4 O OH p3 –1.0 OH OH OH OH 2.3, 6.9, 11.5 2.6, 8.0 p is the number of nonprotonated O atoms. † Boric acid is a special case; see Section 13.5. S O NH2 OH 6 O2S(NH2)OH A note on good practice The structures of oxoacids are drawn with double bonds to oxo groups, O. This representation indicates the connectivity of the O atom to the central atom but in reality resonance lowers the calculated energy of the molecule and distributes the bonding character of the electrons over the molecule. OH O P from the central S atom and confers on S a higher effective positive charge. As a result, the substituted acid is stronger than O2S(OH)2. Another electron acceptor substituent is –CF3, as in the strong acid trifluoromethylsulfonic acid, CF3SO3H (that is, O2S(CF3)(OH)). By contrast, the –NH2 group, which has lone pair electrons, can donate electron density to S by π bonding. This transfer of charge reduces the positive charge of the central atom and weakens the acid. A trap for the unwary is that not all oxoacids follow the familiar structural pattern of a central atom surrounded by OH and O groups. Occasionally an H atom is attached directly to the central atom, as in phosphonic (phosphorous) acid, H3PO3. Phosphonic acid is in fact only a diprotic acid, as the substitution of two OH groups leaves a P–H bond (7) and consequently a nonacidic proton. This structure is consistent with NMR and vibrational spectra, and the structural formula is OPH(OH)2. The nonacidity of the H–P bond reflects the much lower electron-withdrawing ability of the central P atom compared to O (Section 4.1e). Substitution for an oxo group (as distinct from a hydroxyl group) is another example of a structural change that can occur. An important example is the thiosulfate ion, S2O32– (8), in which an S atom replaces an O atom of a sulfate ion. H (b) Pauling’s rules Key point: The strengths of a series of oxoacids containing a specific central atom with a variable number of oxo and hydroxyl groups are summarized by Pauling’s rules. 7 OPH(OH)2, H3PO3 For a series of mononuclear oxoacids of an element E, the strength of the acids increases with increasing number of O atoms. This trend can be explained qualitatively 125 Characteristics of Brønsted acids by considering the electron-withdrawing properties of oxygen. The O atoms withdraw electrons, so making each O–H bond weaker. Consequently, protons are more readily released. In general, for any series of oxoacids, the one with the most O atoms is the strongest. For example, the acid strengths of the oxoacids of chlorine decrease in the order HOCl4 > HClO3 > HClO2 > HClO. Similarly, H2SO4 is stronger than H2SO3 and HNO3 is stronger than HNO2. Another important factor is the degree to which differing numbers of terminal oxo groups stabilize the deprotonated (conjugate) base by resonance. For example, the conjugate base of H2SO4, the HSO4– anion, can be described as a resonance hybrid of three contributions (9), whereas the conjugate base of H2SO3, the HSO3– anion, has only two resonance contributions (10). Consequently, H2SO4 is a stronger acid than H2SO3. The trends can be systematized semiquantitatively by using two empirical rules devised by Linus Pauling, where p is the number of oxo groups and q is the number of hydroxyl groups: 1. For the oxoacid OpE(OH)q, pKa ≈ 8 – 5p. 2– S O 8 S2O2– 3 O– O The successive pKa values of polyprotic acids (those with q > 1), increase by 5 units for each successive proton transfer. Rule 1 predicts that neutral hydroxoacids with p 0 have pKa ≈ 8, acids with one oxo group have pKa ≈ 3, and acids with two oxo groups have pKa ≈ –2. For example, sulfuric acid, O2S(OH)2, has p 2 and q 2, and pKa1 ≈ –2 (signifying a strong acid). Similarly, pKa2 is predicted to be 3, although comparison with the experimental value of 1.9 reminds us that these rules are only approximations. The success of Pauling’s rules may be gauged by inspection of Table 4.3, in which acids are grouped according to p. The variation in strengths down a group is not large, and the complicated, and perhaps cancelling, effects of changing structures allow the rules to work moderately well. The more important variation across the periodic table from left to right and the effect of change of oxidation number are taken into account by the number of oxo groups. In Group 15, the oxidation number 5 requires one oxo group (as in OP(OH)3) whereas in Group 16 the oxidation number 6 requires two (as in O2S(OH)2). (c) Structural anomalies Key point: In certain cases, notably H2CO3 and H2SO3, a simple molecular formula misrepresents the composition of aqueous solutions of nonmetal oxides. An interesting use of Pauling’s rules is to detect structural anomalies. For example, carbonic acid, OC(OH)2, is commonly reported as having pKa1 6.4, but the rules predict pKa1 3. The anomalously low acidity indicated by the experimental value is the result of treating the concentration of dissolved CO2 as if it were all H2CO3. However, in the equilibrium CO2(aq) H2O(l) OC(OH)2(aq) only about 1 per cent of the dissolved CO2 is present as OC(OH)2, so the actual concentration of acid is much less than the concentration of dissolved CO2. When this difference is taken into account, the true pKa1 of H2CO3 is about 3.6, as Pauling’s rules predict. The experimental value pKa1 1.8 reported for sulfurous acid, H2SO3, suggests another anomaly, in this case acting in the opposite way. In fact, spectroscopic studies have failed to detect the molecule OS(OH)2 in solution, and the equilibrium constant for SO2(aq) H2O(l) H2SO3(aq) is less than 10–9. The equilibria of dissolved SO2 are complex, and a simple analysis is inappropriate. The ions that have been detected include HSO3 and S2O52, and there is evidence for an SH bond in the solid salts of the hydrogensulfite ion. This discussion of the composition of aqueous solutions of CO2 and SO2 calls attention to the important point that not all nonmetal oxides react fully with water to form acids. Carbon monoxide is another example: although it is formally the anhydride of methanoic acid, HCOOH, carbon monoxide does not in fact react with water at room temperature to give the acid. The same is true of some metal oxides: OsO4, for example, can exist as dissolved neutral molecules. S OH O O – O S O O S OH O OH O– 9 OH O OH – S O– O S O 10 126 4 Acids and bases E X A M PL E 4 .7 Using Pauling’s rules Identify the structural formulas that are consistent with the following pKa values: H3PO4, 2.12; H3PO3, 1.80; H3PO2, 2.0. Answer We can use Pauling’s rules to use the pKa values to predict the number of oxo groups. All three values are in the range that Pauling’s first rule associates with one oxo group. This observation suggests the formulas (HO)3PO, (HO)2HPO, and (HO)H2PO. The second and the third formulas are derived from the first by replacement of –OH by H bound to P (as in structure 7). Self-test 4.7 Predict the pKa values of (a) H3PO4, (b) H2PO4, (c) HPO42. 4.6 Anhydrous oxides We have treated oxoacids as being derived by deprotonation of their parent aqua acids. It is also useful to take the opposite viewpoint and to consider aqua acids and oxoacids as being derived by hydration of the oxides of the central atom. This approach emphasizes the acid and base properties of oxides and their correlation with the location of the element in the periodic table. (a) Acidic and basic oxides Key points: Metallic elements typically form basic oxides; nonmetallic elements typically form acidic oxides. An acidic oxide is an oxide that, on dissolution in water, binds an H2O molecule and releases a proton to the surrounding solvent: CO2(g) H2O(l) OC(OH)2(aq) OC(OH)2(aq) H2O(l) H3O(aq) O2COH–(aq) An equivalent interpretation is that an acidic oxide is an oxide that reacts with an aqueous base (an alkali): CO2(g) OH–(aq) → O2C(OH)–(aq) A basic oxide is an oxide to which a proton is transferred when it dissolves in water: BaO(s) H2O(l) → Ba2(aq) 2 OH–(aq) The equivalent interpretation in this case is that a basic oxide is an oxide that reacts with an acid: BaO(s) 2 H3O(aq) → Ba2(aq) 3 H2O(l) Because acidic and basic oxide character often correlates with other chemical properties, a wide range of properties can be predicted from a knowledge of the character of oxides. In a number of cases the correlations follow from the basic oxides being largely ionic and of acidic oxides being largely covalent. For instance, an element that forms an acidic oxide is likely to form volatile, covalent halides. By contrast, an element that forms a basic oxide is likely to form solid, ionic halides. In short, the acidic or basic character of an oxide is a chemical indication of whether an element should be regarded as a metal or a nonmetal. Generally, metals form basic oxides and nonmetals form acidic oxides. (b) Amphoterism Key points: The frontier between metals and nonmetals in the periodic table is characterized by the formation of amphoteric oxides; amphoterism also varies with the oxidation state of the element. An amphoteric oxide is an oxide that reacts with both acids and bases.4 Thus, aluminium oxide reacts with acids and alkalis: Al2O3(s) 6 H3O(aq) 3 H2O(l) → 2 [Al(OH2)6]3(aq) Al2O3(s) 2 OH–(aq) 3 H2O(l) → 2 [Al(OH)4]–(aq) 4 The word ‘amphoteric’ is derived from the Greek word for ‘both’. Characteristics of Brønsted acids Amphoterism is observed for the lighter elements of Groups 2 and 13, as in BeO, Al2O3, and Ga2O3. It is also observed for some of the d-block elements in high oxidation states, such as MoO3 and V2O5, in which the central atom is very electron withdrawing, and some of the heavier elements of Groups 14 and 15, such as SnO2 and Sb2O5. Figure 4.6 shows the location of elements that in their characteristic group oxidation states have amphoteric oxides. They lie on the frontier between acidic and basic oxides, and hence serve as an important guide to the metallic or nonmetallic character of an element. The onset of amphoterism correlates with a significant degree of covalent character in the bonds formed by the elements, either because the metal ion is strongly polarizing (as for Be) or because the metal ion is polarized by the O atom attached to it (as for Sb). An important issue in the d block is the oxidation number necessary for amphoterism. Figure 4.7 shows the oxidation number for which an element in the first row of the block has an amphoteric oxide. We see that on the left of the block, from titanium to manganese and perhaps iron, oxidation state 4 is amphoteric (with higher values on the border of acidic and lower values of the border of basic). On the right of the block, amphoterism occurs at lower oxidation numbers: the oxidation states 3 for cobalt and nickel and 2 for copper and zinc are fully amphoteric. There is no simple way of predicting the onset of amphoterism. However, it presumably reflects the ability of the metal cation to polarize the oxide ions that surround it—that is, to introduce covalence into the metaloxygen bond. The degree of covalence typically increases with the oxidation number of the metal as the increasingly positively charged cation becomes more strongly polarizing (Section 1.9e). 1 2 13 14 15 16 127 17 Be Al Ga Ge As In Sn Sb Pb Bi Basic Acidic Figure 4.6 The location of elements having amphoteric oxides. The circled elements have amphoteric oxides in all oxidation states. The elements in boxes have acidic oxides in the highest oxidation state and amphoteric oxides in lower oxidation states. E X A MPL E 4 . 8 Using oxide acidity in qualitative analysis Answer When the oxidation number of the metal is 3, all the metal oxides are sufficiently basic to be insoluble in a solution with pH ≈ 10. Aluminium(III) oxide is amphoteric and redissolves in alkaline solution to give aluminate ions, [Al(OH)4]. Vanadium(III) and chromium(III) oxides are oxidized by H2O2 to give vanadate ions, [VO4]3, and chromate ions, [CrO4]2, which are the anions derived from the acidic oxides V2O5 and CrO3, respectively. 7 Oxidation number In the traditional scheme of qualitative analysis, a solution of metal ions is oxidized and then aqueous ammonia is added to raise the pH. The ions Fe3, Ce3, Al3, Cr3, and V3 precipitate as hydrous oxides. The addition of H2O2 and NaOH redissolves the aluminium, chromium, and vanadium oxides. Discuss these steps in terms of the acidities of oxides. 6 Acidic 5 4 Amphoteric 3 Self-test 4.8 If Ti(IV) ions were present in the sample, how would they behave? 2 4.7 Polyoxo compound formation Key points: Acids containing the OH group condense to form polyoxoanions; polycation formation from simple aqua cations occurs with the loss of H2O. Oxoanions form polymers as the pH is lowered whereas aqua ions form polymers as the pH is raised. As the pH of a solution is increased, the aqua ions of metals that have basic or amphoteric oxides generally undergo polymerization and precipitation. Because the precipitation occurs quantitatively at a pH characteristic of each metal, one application of this behaviour is the separation of metal ions,. With the exception of Be2 (which is amphoteric), the elements of Groups 1 and 2 have no important solution species beyond the aqua ions M(aq) and M2(aq). By contrast, the solution chemistry of the elements becomes very rich as the amphoteric region of the periodic table is approached. The two most common examples are polymers formed by Fe(III) and Al(III), both of which are abundant in the Earth’s crust. In acidic solutions, both form octahedral hexaaqua ions, [Al(OH2)6]3 and [Fe(OH2)6]3. In solutions of pH > 4, both precipitate as gelatinous hydrous oxides: [Fe(OH2)6]3(aq) (3 n) H2O(l) → Fe(OH)3.nH2O(s) 3 H3O(aq) [Al(OH2)6]3(aq) (3 n) H2O(l) → Al(OH)3.nH2O(s) 3 H3O(aq) The precipitated polymers, which are often of colloidal dimensions (between 1 nm and 1 µm), slowly crystallize to stable mineral forms. The extensive network structure of Basic Sc Ti V Cr MnFe Co Ni Cu Zn Figure 4.7 The oxidation numbers for which elements in the first row of the d block have amphoteric oxides. Predominantly acidic oxides are shown shaded pink, predominantly basic oxides are shaded blue. 128 4 Acids and bases aluminium polymers, which are neatly packed in three dimensions, contrasts with the linear polymers of their iron analogues. Polyoxoanion formation from oxoanions occurs by protonation of an O atom and its departure as H2O: 2 [CrO4]2–(aq) 2 H3O(aq) → [O3CrOCrO3]2–(aq) 3 H2O(l) The importance of polyoxo anions can be judged by the fact that they account for most of the mass of oxygen in the Earth’s crust, as they include almost all silicate minerals. They also include the phosphate polymers (such as ATP (11)) used for energy transfer in living cells. The formation of polyoxoanions is important for early d-block ions, particularly V(V), Mo(VI), W(VI), and (to a lesser extent) Nb(V), Ta(V), and Cr(VI); see Section 19.8. They are formed when base is added to aqueous solutions of the ions or oxides in high oxidation states. Polyoxoanions are also formed by some nonmetals, but their structures are different from those of their d-metal analogues. The common species in solution are rings and chains. The silicates are very important examples of polymeric oxoanions, and we discuss them in detail in Chapter 14. One example of a polysilicate mineral is MgSiO3, which contains an infinite chain of SiO32⫺ units. In this section we illustrate some features of polyoxoanions using phosphates as examples. The simplest condensation reaction, starting with the orthophosphate ion, PO43–, is O − HO 2 PO4 + 6 H P O OH O 4− P OH H 2O OH The elimination of water consumes protons and decreases the average charge number of each P atom to –2. If each phosphate group is represented as a tetrahedron with the O atoms located at the corners, the diphosphate ion, P2O74⫺ (12), can be drawn as (13). Phosphoric acid can be prepared by hydrolysis of the solid phosphorus(V) oxide, P4O10. An initial step using a limited amount of water produces a metaphosphate ion with the formula 4⫺ (14). This reaction is only the simplest among many, and the separation of products P4O12 from the hydrolysis of phosphorus(V) oxide by chromatography reveals the presence of chain species with from one to nine P atoms. Higher polymers are also present and can be removed from the column only by hydrolysis. Figure 4.8 is a schematic representation of a two-dimensional paper chromatogram: the upper spot sequence corresponds to linear polymers and the lower sequence corresponds to rings. Chain polymers of formula Pn with n 10 to 50 can be isolated as mixed amorphous glasses analogous to those formed by silicates (Section 14.15). The polyphosphates are biologically important. At physiological pH (close to 7.4), the P–O–P entity is unstable with respect to hydrolysis. Consequently, its hydrolysis can serve as a mechanism for providing the energy to drive a reaction (the Gibbs energy). Similarly, the formation of the P–O–P bond is a means of storing Gibbs energy. The key to energy NH2 11b ADP 3– N O– –O P O O– O P O O– O P N N 4– N O O O O H P H H OH H OH 11a ATP 4– 12 P2O74– Characteristics of Brønsted acids 129 exchange in metabolism is the hydrolysis of adenosine triphosphate, ATP (11a), to adenosine diphosphate, ADP (11b): 4– ATP4– 2 H2O → ADP3– HPO42– H3O ∆rG O –41 kJ mol–1 at pH 7.4 Energy flow in metabolism depends on the subtle construction of pathways to make ATP from ADP. The energy is used metabolically by pathways that have evolved to exploit the delivery of a thermodynamic driving force resulting from the hydrolysis of ATP. 13 P2O4– 7 4.8 Nonaqueous solvents Not all proton transfer reactions take place in aqueous media. Nonaqueous solvents can be selected for reactions of molecules that are readily hydrolyzed, to avoid levelling by water, or to enhance the solubility of a solute. Nonaqueous solvents are often selected on the basis of their liquid range and relative permittivity. Some physical properties of some common nonaqueous solvents are given in Table 4.4. The solvent system definition of acids and bases applies to both protic and aprotic nonaqueous solvents. 4– (a) Liquid ammonia Key points: Liquid ammonia is a useful nonaqueous solvent. Many reactions in liquid ammonia are analogous to those in water. 4– 14 P4O12 Liquid ammonia is widely used as a nonaqueous solvent. It boils at –33°C at 1 atm and, despite a somewhat lower relative permittivity (εr 22) than that of water, it is a good solvent for inorganic compounds such as ammonium salts, nitrates, cyanides, and thiocyanides, and organic compounds such as amines, alcohols, and esters. It closely resembles the aqueous system as can be seen from the autoionization s in a h C Orth o te sa o h p 2 NH3(l) NH4(sol) NH2(sol) Py ro 4 lyT o rip ids c A t n e lv o Solutes that increase the concentration of NH , the solvated proton, are acids. Solutes that decrease the concentration of NH4 or increase the concentration of NH2 are defined as bases. Thus, ammonium salts are acids in liquid ammonia and amines are bases. Liquid ammonia is a more basic solvent than water and enhances the acidity of many compounds that are weak acids in water. For example, acetic acid is almost completely ionized in liquid ammonia: lyT o p tra e CH3COOH(sol) NH3(l) → NH4(sol) CH3CO2–(aq) 5 Many reactions in liquid ammonia are analogous to those in water. The following acid– base neutralization can be carried out: NH4Cl(sol) NaNH2(sol) → NaCl(sol) 2NH3(l) Table 4.4 Physical properties of some nonaqueous solvents Solvent Melting point/°C Boiling point/°C Relative permittivity Liquid ammonia 77.7 33.5 23.9 (at –33°C) 16.7 Sulfuric acid 10.4 Hydrogen fluoride Ethanol Dinitrogen tetroxide Bromine trifluoride Dimethyl sulfoxide (DMSO) 117.9 290 (decomposes) 6.15 100 83.4 19.5 80 114.5 78.3 24.55 11.2 21.1 8.8 125.8 18.5 189 6 m tra e T - -5 6 sicso a B t n e lv Liquid ammonia is a very good solvent for alkali and alkali earth metals, with the exception of beryllium. The alkali metals are particularly soluble and 336 g of caesium can be dissolved in 100 g of liquid ammonia at –50°C. The metals can be recovered by evaporating the ammonia. These solutions are very conducting and are blue when dilute and bronze when concentrated. Electron paramagnetic resonance spectra (Section 8.6) show that the Glacial acetic acid s g in R taT e rim 2.42 107 46.45 Figure 4.8 A representation of a twodimensional paper chromatogram of a complex mixture of phosphates formed by condensation reactions. The sample spot was placed at the lower left corner. Basic solvent separation was used first, followed by acidic solvent perpendicular to the basic one. This separates open chains from rings. The upper spot sequence corresponds to linear polymers and the lower sequence corresponds to rings. 130 4 Acids and bases solutions contain unpaired electrons. The blue colour typical of the solutions is the outcome of a very broad optical absorption band in the near IR with a maximum near 1500 nm. The metal is ionized in ammonia solution to give ‘solvated electrons’: NH (l) 3 ⎯ → Na(sol) e–(sol) Na(s) ⎯⎯⎯ The blue solutions survive for long times at low temperature but decompose slowly to give hydrogen and sodium amide, NaNH2. The exploitation of the blue solutions to produce compounds called ‘electrides’ is discussed in Section 11.13. (b) Hydrogen fluoride Key point: Hydrogen fluoride is a reactive toxic solvent that is highly acidic. Liquid hydrogen fluoride (bp 19.5ºC) is an acidic solvent with a relative permittivity (εr 84 at 0°C) comparable to that of water (εr 78 at 25°C). It is a good solvent for ionic substances. However, as it is both highly reactive and toxic, it presents handling problems, including its ability to etch glass. In practice, liquid hydrogen fluoride is usually contained in polytetrafluoroethylene and polychlorotrifluoroethylene vessels. Hydrogen fluoride is particularly hazardous because it penetrates tissue rapidly and interferes with nerve function. Consequently, burns may go undetected and treatment may be delayed. It can also etch bone and reacts with calcium in the blood. Liquid hydrogen fluoride is a highly acidic solvent as it has a high autoprotolysis constant and produces solvated protons very readily (Section 4.1(b)): 3 HF(l) → H2F(sol) HF2–(sol) Although the conjugate base of HF is formally F–, the ability of HF to form a strong hydrogen bond to F– means that the conjugate base is better regarded as the bifluoride ion, HF2–. Only very strong acids are able to donate protons and function as acids in HF, for example fluorosulfonic acid: HSO3F(sol) HF(l) H2F(sol) SO3F–(sol) Organic compounds such as acids, alcohols, ethers, and ketones can accept a proton and act as bases in HF(l). Other bases increase the concentration of HF2 to produce basic solutions: CH3COOH(l) 2 HF(l) CH3C(OH)2(sol) HF2(sol) In this reaction acetic acid, an acid in water, is acting as a base. Many fluorides are soluble in liquid HF as a result of the formation of the HF2– ion; for example LiF(s) HF(l) → Li(sol) HF2–(sol) (c) Anhydrous sulfuric acid Key point: The autoionization of anhydrous sulfuric acid is complex, with several competing side reactions. Anhydrous sulfuric acid is an acidic solvent. It has a high relative permittivity and is viscous because of extensive hydrogen bonding (Section 10.6). Despite this association the solvent is appreciably autoionized at room temperature. The major autoionization is 2 H2SO4(l) H3SO4(sol) HSO4–(sol) However, there are secondary autoionizations and other equilibria, such as H2SO4(l) H2O(sol) SO3(sol) H2O(sol) H2SO4(l) H3O(sol) HSO4–(sol) SO3(sol) H2SO4(l) H2S2O7(sol) H2S2O7(sol) H2SO4(l) H3SO4(sol) HS2O7–(sol) The high viscosity and high level of association through hydrogen bonding would usually lead to low ion mobilities. However, the mobilities of H3SO4 and HSO4– are comparable to Lewis acidity those of H3O and OH– in water, indicating that similar proton transfer mechanisms are taking place. The main species taking part are H3SO4 and HSO4–: O O O O − O S H O O O S+ H O O H H O O S O O H H O O O S H H O O S O − O O H O H H O O H O H O +S S O O S O H H O O H Most strong oxo acids accept a proton in anhydrous sulfuric acid and are thus bases: H3PO4(sol) H2SO4(l) H4PO4(sol) HSO4–(sol) An important reaction is that of nitric acid with sulfuric acid to generate the nitronium ion, NO2, which is the active species in aromatic nitration reactions: HNO3(sol) 2 H2SO4(l) NO2(sol) H3O(sol) 2 HSO4–(sol) Some acids that are very strong in water act as weak acids in anhydrous sulfuric acids, for example perchloric acid, HClO4, and fluorosulfuric acid, HFSO3. (d) Dinitrogen tetroxide Key point: Dinitrogen tetroxide autoionizes by two reactions. The preferred route can be enhanced by addition of electron-pair donors or acceptors. Dinitrogen tetroxide, N2O4, has a narrow liquid range with a freezing point at –11.2°C and boiling point of 21.2°C. Two autoionization reactions occur: N2O4(l) NO(sol) NO3–(sol) N2O4(l) NO2(sol) NO2–(sol) The first autoionization is enhanced by addition of electron pair donors (which in the next section we see to be ‘Lewis bases’), such as diethyl ether: N2O4(l) :X XNO(sol) NO3–(sol) Electron pair acceptors (‘Lewis acids’; see the next section) such as BF3 enhance the second autoionization reaction: N2O4(l) BF3(sol) NO2(sol) F3BNO2–(sol) Dinitrogen tetroxide has a low relative permittivity and is not a very useful solvent for inorganic compounds. It is, however, a good solvent for many esters, carboxylic acids, halides, and organic nitro compounds. Lewis acidity Key points: A Lewis acid is an electron pair acceptor; a Lewis base is an electron pair donor. The Brønsted–Lowry theory of acids and bases focuses on the transfer of a proton between species. The solvent system generalizes the Brønsted–Lowry theory to include the transfer of cationic and anionic species other than protons. Whereas both definitions are more general than any that preceded them, they still fail to take into account reactions between substances that show similar features but in which no proton or other charged species is transferred. This deficiency was remedied by a more general theory of acidity introduced by G.N. Lewis in the same year as Brønsted and Lowry introduced theirs (1923). Lewis’s approach became influential only in the 1930s. A Lewis acid is a substance that acts as an electron pair acceptor. A Lewis base is a substance that acts as an electron pair donor. We denote a Lewis acid by A and a Lewis base 131 132 4 Acids and bases by :B, often omitting any other lone pairs that may be present. The fundamental reaction of Lewis acids and bases is the formation of a complex (or adduct), A–B, in which A and :B bond together by sharing the electron pair supplied by the base. A note on good practice The terms ‘Lewis acid’ and ‘Lewis base’ are used in discussions of the equilibrium properties of reactions. In the context of reaction rates, an electron pair donor is called a nucleophile and an electron acceptor is called an electrophile. 4.9 Examples of Lewis acids and bases Key points: Brønsted acids and bases exhibit Lewis acidity and basicity; the Lewis definition can be applied to aprotic systems. A proton is a Lewis acid because it can attach to an electron pair, as in the formation of NH4 from NH3. It follows that any Brønsted acid, as it provides protons, exhibits Lewis acidity too. Note that the Brønsted acid HA is the complex formed by the Lewis acid H with the Lewis base A–. We say that a Brønsted acid exhibits Lewis acidity rather than that a Brønsted acid is a Lewis acid. All Brønsted bases are Lewis bases because a proton acceptor is also an electron pair donor: an NH3 molecule, for instance, is a Lewis base as well as a Brønsted base. Therefore, the whole of the material presented in the preceding sections of this chapter can be regarded as a special case of Lewis’s approach. However, because the proton is not essential to the definition of a Lewis acid or base, a wider range of substances can be classified as acids and bases in the Lewis scheme than can be classified in the Brønsted scheme. We meet many examples of Lewis acids later, but we should be alert to the following possibilities: 1. A molecule with an incomplete octet of valence electrons can complete its octet by accepting an electron pair. A prime example is B(CH3)3, which can accept the lone pair of NH3 and other donors: Me Me N B Me Me H H H Me H B N Me H H Hence, B(CH3)3 is a Lewis acid. 2. A metal cation can accept an electron pair supplied by the base in a coordination compound. This aspect of Lewis acids and bases is treated at length in Chapters 7 and 20. An example is the hydration of Co2, in which the lone pairs of H2O (acting as a Lewis base) donate to the central cation to give [Co(OH2)6]2. The Co2 cation is therefore the Lewis acid. 3. A molecule or ion with a complete octet may be able to rearrange its valence electrons and accept an additional electron pair. For example, CO2 acts as a Lewis acid when it forms HCO3– by accepting an electron pair from an O atom in an OH– ion: O O + C – OH – C O H O O A molecule or ion may be able to expand its valence shell (or simply be large enough) to accept another electron pair. An example is the formation of the complex [SiF6]2– when two F– ions (the Lewis bases) bond to SiF4 (the acid). F F i S F F F +2F – F F –2 F i S F F This type of Lewis acidity is common for the halides of the heavier p-block elements, such as SiX4, AsX3, and PX5 (with X a halogen). Lewis acidity 133 E X A MPL E 4 . 9 Identifying Lewis acids and bases Identify the Lewis acids and bases in the reactions (a) BrF3 F → BrF4, (b) KH H2O → KOH H2. Answer We need to identify the electron pair acceptor (the acid) and the electron pair donor (the base). (a) The acid BrF3 accepts a pair of electrons from the base F. Therefore BrF3 is a Lewis acid and F is a Lewis base. (b) The ionic hydride complex KH provides H, which displaces H from water to give H2 and OH. The net reaction is H– H2O → H2 OH– If we think of this reaction as H– H:OH– → HH :OH– we see that H provides a lone pair and is therefore a Lewis base. It reacts with H2O to drive out OH, another Lewis base. Self-test 4.9 Identify the acids and bases in the reactions (a) FeCl3 Cl → FeCl4, (b) I I2 → I3. 4.10 Group characteristics of Lewis acids An understanding of the trends in Lewis acidity and basicity enables us to predict the outcome of many reactions of the s- and p-block elements. (a) Lewis acids and bases of the s-block elements Key point: Alkali metal ions act as Lewis acids with water, forming hydrated ions. The existence of hydrated alkali metal ions in water can be regarded as an aspect of their Lewis acid character, with H2O the Lewis base. Alkali metal ions do not act as Lewis bases but their fluorides act as a source of the Lewis base F– and form fluoride complexes with Lewis acids, such as SF4: CsF SF4 → Cs[SF5]– The Be atom in beryllium dihalides acts as a Lewis acid by forming a polymeric chain structure in the solid state (15). In this structure, a bond is formed when a lone pair of electrons of a halide ion, acting as a Lewis base, is donated into an empty sp3 hybrid orbital on the Be atom. The Lewis acidity of beryllium chloride is also demonstrated by the formation of adducts such as BeCl42– (16). (b) Group 13 Lewis acids Be Hal 15 BeHal2 Key points: The ability of boron trihalides to act as Lewis acids generally increases in the order BF3 < BCl3 < BBr3; aluminium halides are dimeric in the gas phase and are used as catalysts in solution. 2– The planar molecules BX3 and AlX3 have incomplete octets, and the vacant p orbital perpendicular to the plane (17) can accept a lone pair from a Lewis base: X Me N B X X Me Me X X X Me B N Me Me The acid molecule becomes pyramidal as the complex is formed and the B–X bonds bend away from their new neighbours. The order of thermodynamic stability of complexes of :N(CH3)3 with BX3 is BF3 < BCl3 < BBr3. This order is opposite to that expected on the basis of the relative electronegativities of the halogens: an electronegativity argument would suggest that F, the most electronegative halogen, ought to leave the B atom in BF3 most electron deficient and hence able to form the strongest bond to the incoming base. The currently accepted explanation is that the halogen atoms in the BX3 molecule can form π bonds with the empty B2p orbital (18), and that these π bonds must be disrupted to make the acceptor orbital available for complex formation. The π bond also favours the planar structure of the molecule, a structure that must be converted into tetrahedral in the adduct. The small F atom forms the 16 BeCl2– 4 17 AlX3 and BX3 134 4 Acids and bases strongest π bonds with the B2p orbital: recall that p–p π bonding is strongest for Period 2 elements, largely on account of the small atomic radii of these elements and the significant overlap of their compact 2p orbitals (Section 2.5). Thus, the BF3 molecule has the strongest π bond to be broken when the amine forms an N–B bond. Boron trifluoride is widely used as an industrial catalyst. Its role is to extract bases bound to carbon and hence to generate carbocations: F F B X C F 18 R R – B X R R F C R Boron trifluoride is a gas at room temperature and pressure, but it dissolves in diethyl ether to give a solution that is convenient to use. This dissolution is also an aspect of Lewis acid character because, as BF3 dissolves, it forms a complex with the :O atom of a solvent molecule. Aluminium halides are dimers in the gas phase; aluminium chloride, for example, has molecular formula Al2Cl6 in the vapour state (19). Each Al atom acts as an acid towards a Cl atom initially belonging to the other Al atom. Aluminium chloride is widely used as a Lewis acid catalyst for organic reactions. The classic examples are Friedel–Crafts alkylation (the attachment of R to an aromatic ring) and acylation (the attachment of RCO) during which AlCl4– is formed. The catalytic cycle is shown in Fig. 4.9. Al Cl 20 Al2Cl6 RCH2Cl: AlCl3 CH2R F F R (c) Group 14 Lewis acids RCH2Cl–AlCl3 Key points: Group 14 elements other than carbon exhibit hypervalence and act as Lewis acids by becoming five- or six-coordinate; tin(II) chloride is both a Lewis acid and a Lewis base. HCl + Unlike carbon, a Si atom can expand its valence shell (or is simply large enough) to become hypervalent. For example, a five-coordinate trigonal bipyramidal structure is possible (20). A representative Lewis acid–base reaction is that of SiF4 with two F– ions: – AlCl lCl4 lCl + CH2R C H F F i S F Figure 4.9 The catalytic cycle for the Friedel–Crafts alkylation reaction. – O i S F F 2F – F F 2– F i F S F Germanium and tin fluorides can react similarly. Because the Lewis base F–, aided by a proton, can displace O2– from silicates, hydrofluoric acid is corrosive towards glass (SiO2). The trend in acidity for SiX4, which follows the order SiI4 < SiBr4 < SiCl4 < SiF4, correlates with the increase in the electron-withdrawing power of the halogen from I to F and is the reverse of that for BX3. Tin(II) chloride is both a Lewis acid and a Lewis base. As an acid, SnCl2 combines with Cl– to form SnCl3– (21). This complex retains a lone pair, and it is sometimes more revealing to write its formula as :SnCl3–. It acts as a base to give metal–metal bonds, as in the complex (CO)5Mn–SnCl3 (22). Compounds containing metal–metal bonds are currently the focus of much attention in inorganic chemistry, as we see later in the text (Section 19.11). Tin(IV) halides are Lewis acids. They react with halide ions to form SnX62–: SnCl4 2 Cl– → SnCl62– The strength of the Lewis acidity follows the order SnF4 > SnCl4 > SnBr4 > SnI4. C6H5 20 [Si(C6H5)(OC6H4O)2]– – Sn Cl 21 SnCl3– E X A M PL E 4 .10 Predicting the relative Lewis basicity of compounds Rationalize the following relative Lewis basicities: (a) (H3Si)2O < (H3C)2O; (b) (H3Si)3N < (H3C)3N. Answer Nonmetallic elements in Period 3 and later can expand their valence shells by delocalization of the O or N lone pairs to create multiple bonds (O and N are thus acting as π-electron donors). The silyl ether and silyl amine are therefore the weaker Lewis bases in each pair. Self-test 4.10 Given that π bonding between Si and the lone pairs of N is important, what difference in structure between (H3Si)3N and (H3C)3N do you expect? Lewis acidity 135 (d) Group 15 Lewis acids Key points: Oxides and halides of the heavier Group 15 elements act as Lewis acids. Phosphorus pentafluoride is a strong Lewis acid and forms complexes with ethers and amines. The heavier elements of the nitrogen group (Group 15) form some of the most important Lewis acids, SbF5 being one of the most widely studied compounds. The reaction with HF produces a superacid (Section 4.15) F F F Sb F F 2 HF F F F F Sb F – Sn + H 2F + Cl F Mn (e) Group 16 Lewis acids CO Key points: Sulfur dioxide can act as a Lewis acid by the formation of a complex; to act as a Lewis base, the SO2 molecule can donate either its S or its O lone pair to a Lewis acid. Sulfur dioxide is both a Lewis acid and a Lewis base. Its Lewis acidity is illustrated by the formation of a complex with a trialkylamine acting as a Lewis base: 22 [Mn(CO)5(SnCl3)] R O O S R R O R S N N O R R To act as a Lewis base, the SO2 molecule can donate either its S or its O lone pair to a Lewis acid. When SbF5 is the acid, the O atom of SO2 acts as the electron pair donor, but when Ru(II) is the acid, the S atom acts as the donor (23). Sulfur trioxide is a strong Lewis acid and a very weak (O donor) Lewis base. Its acidity is illustrated by the reaction O O O R N S O O R R R S O N R R Ru A classic aspect of the acidity of SO3 is its highly exothermic reaction with water in the formation of sulfuric acid. The resulting problem of having to remove large quantities of heat from the reactor used for the commercial production of sulfuric acid is alleviated by exploiting the Lewis acidity of sulfur trioxide further to carry out the hydration by a two-stage process. Before dilution, sulfur trioxide is dissolved in sulfuric acid to form the mixture known as oleum. This reaction is an example of Lewis acid–base complex formation: O O S O O O S HO O H Cl NH3 SO2 23 [RuCl(NH3)4(SO2)]+ O O O S S HO O O O H The resulting H2S2O7 can then be hydrolyzed in a less exothermic reaction: H2S2O7 H2O → 2 H2SO4 (f) Halogens as Lewis acids Key point: Bromine and iodine molecules act as mild Lewis acids. Lewis acidity is expressed in an interesting and subtle way by Br2 and I2, which are both strongly coloured. The strong visible absorption spectra of Br2 and I2 arise from transitions to low-lying unfilled antibonding orbitals. The colours of the species therefore suggest that the empty orbitals may be low enough in energy to serve as acceptor orbitals in 136 4 Acids and bases (CH3)2CO Br2 228 282 Br2 σ 282 125° O–Br–Br–O O...O CT (a) sp C σ 2 2 sp + sp O 2 O sp (b) sp2 – sp2 2 C (c) Br2σ Figure 4.10 The interaction of Br2 with the carbonyl group of propanone. (a) The structure of (CH3)2COBr2 shown by X-ray diffraction. (b) The orbital overlap responsible for the complex formation. (c) A partial molecular orbital energy level diagram for the σ and σ orbitals of Br2 with the appropriate combinations of the sp2 orbitals on the two O atoms. The charge transfer transition is labelled CT. Lewis acid–base complex formation.5 Iodine is violet in the solid and gas phases, and in nondonor solvents such as trichloromethane. In water, propanone (acetone), or ethanol, all of which are Lewis bases, iodine is brown. The colour changes because a solvent–solute complex is formed from the lone pair of donor molecule O atoms and a low-lying orbital of the dihalogen. The interaction of Br2 with the carbonyl group of propanone is shown in Fig. 4.10. The illustration also shows the transition responsible for the new absorption band observed when a complex is formed. The orbital from which the electron originates in the transition is predominantly the lone pair orbital of the base (the ketone). The orbital to which the transition occurs is predominantly the LUMO of the acid (the dihalogen). Thus, to a first approximation, the transition transfers an electron from the base to the acid and is therefore called a charge-transfer transition. The triiodide ion, I3–, is an example of a complex between a halogen acid (I2) and a halide base (I–). One of the applications of its formation is to render molecular iodine soluble in water so that it can be used as a titration reagent: I2(s) I–(aq) I3–(aq) K 725 The triiodide ion is one example of a large class of polyhalide ions (Section 17.8). Reactions and properties of Lewis acids and bases Reactions of Lewis acids and bases are widespread in chemistry, the chemical industry, and biology. For example, cement is made by grinding together limestone (CaCO3) and a source of aluminosilicates, such as clay, shale, or sand, which are then heated to 1500˚C in a rotary cement kiln. The limestone is heated and decomposes to lime (CaO), which reacts with the silicates to form molten calcium silicates of varying compositions such as Ca2SiO4, Ca3SiO5, and Ca3Al2O6. 2 CaO(s) SiO2(s) → Ca2SiO4(s) In industry carbon dioxide is removed from flue gas in order to reduce atmospheric emissions and to supply the demands of the soft drinks industry. This is achieved by using liquid amine scrubbers. 2 RNH2(aq) CO2(g) H2O(l) → (RNH3)2CO3(aq) 5 The terms donor–acceptor complex and charge-transfer complex were at one time used to denote these complexes. However, the distinction between these complexes and the more familiar Lewis acid–base complexes is arbitrary and in the current literature the terms are used more or less interchangeably. Reactions and properties of Lewis acids and bases 137 The toxicity of carbon monoxide to animals is an example of a Lewis acid–base reaction. Normally, oxygen forms a bond to the Fe(II) atom of haemoglobin and does so reversibly. Carbon monoxide is a much better Lewis acid than O2 and forms a strong, almost irreversible, bond to the iron(II) site of haemoglobin: Hb–FeII CO → Hb–FeIICO All reactions between d-block metal atoms or ions to form coordination compounds (Chapter 7) are examples of reactions between a Lewis acid and a Lewis base: Ni2(aq) 6 NH3 → [Ni(NH3)6]2 Friedel–Crafts alkylations and acylations are widely used in synthetic organic chemistry. They require a strong Lewis acid catalyst such as AlCl3 or FeCl3. + RCl catalyst HCl The first step is the reaction between the Lewis acid and the alkyl halide: RCl AlCl3 → R [AlCl4]– 4.11 The fundamental types of reaction Lewis acids and bases undergo a variety of characteristic reactions. The simplest Lewis acid–base reaction in the gas phase or noncoordinating solvents is complex formation: A :B → AB Two examples are F F N F O O F F H + B S O H H F Me O O O H B H H Me S O Me N O Me Both reactions involve Lewis acids and bases that are independently stable in the gas phase or in solvents that do not form complexes with them. Consequently, the individual species (as well as the complexes) may be studied experimentally. Figure 4.11 shows the interaction of orbitals responsible for bonding in Lewis complexes. The exothermic character of the formation of the complex stems from the fact that the newly formed bonding orbital is populated by the two electrons supplied by the base whereas the newly formed antibonding orbital is left unoccupied. As a result, there is a net lowering of energy when the bond forms. (a) Displacement reactions Energy A A–B LUMO B HOMO Key point: In a displacement reaction, an acid or base drives out another acid or base from a Lewis complex. A displacement of one Lewis base by another is a reaction of the form B–A :B → B: A–B An example is Acid Me O Me F F B F F N F F Me B N O Me Complex Base Figure 4.11 The molecular orbital representation of the orbital interactions responsible for formation of a complex between Lewis acid A and Lewis base :B. 138 4 Acids and bases All Brønsted proton transfer reactions are of this type, as in HS–(aq) H2O(l) → S2–(aq) H3O(aq) In this reaction, the Lewis base H2O displaces the Lewis base S2 from its complex with the acid H. Displacement of one acid by another, A BA → AB A is also possible, as in the reaction F + N H 4Cl B F F F F H B F N HCl H H In the context of d-metal complexes, a displacement reaction in which one ligand is driven out of the complex and is replaced by another is generally called a substitution reaction (Section 21.1). (b) Metathesis reactions Key point: A metathesis reaction is a displacement reaction assisted by the formation of another complex. A metathesis reaction (or ‘double displacement reaction’) is an interchange of partners:6 A–B A–B → A–B A–B The displacement of the base :B by :B is assisted by the extraction of :B by the acid A. An example is the reaction e M e M M e e M i S I +g B A()sr M e e M i S Br Is()g A Here the base Br– displaces I–, and the extraction is assisted by the formation of the less soluble AgI. 4.12 Hard and soft acids and bases The proton (H) was the key electron pair acceptor in the discussion of Brønsted acid and base strengths. When considering Lewis acids and bases we must allow for a greater variety of acceptors and hence more factors that influence the interactions between electron pair donors and acceptors in general. (a) The classification of acids and bases Key points: Hard and soft acids and bases are identified empirically by the trends in stabilities of the complexes that they form: hard acids tend to bind to hard bases and soft acids tend to bind to soft bases. It proves helpful when considering the interactions of Lewis acids and bases containing elements drawn from throughout the periodic table to consider at least two main classes of substance. The classification of substances as ‘hard’ and ‘soft’ acids and bases was introduced by R.G. Pearson; it is a generalization—and a more evocative renaming—of the distinction between two types of behaviour that were originally named simply ‘class a’ and ‘class b’ respectively, by S. Ahrland, J. Chatt, and N.R. Davies. The two classes are identified empirically by the opposite order of strengths (as measured by the equilibrium constant, Kf, for the formation of the complex) with which they form complexes with halide ion bases: r Hard acids bond in the order: I– < Br– < Cl– < F–. r Soft acids bond in the order: F– < Cl– < Br– < I–. 6 The name metathesis comes from the Greek word for exchange. 139 Reactions and properties of Lewis acids and bases r Hard acids bond in the order: R3P PF5. Superacids are known that can protonate almost any organic compound. In the 1960s, George Olah and his colleagues found that carbonium ions were stabilized when hydrocarbons were dissolved in superacids.7 In inorganic chemistry, superacids have been used to observe a wide variety of reactive cations such as S82, H3O2, Xe2, and HCO, some of which have been isolated for structural characterization. A superbase is a compound that is a more efficient proton acceptor than the OH– ion, the strongest base that can exist in aqueous solution. Superbases react with water to produce the OH– ion. Inorganic superbases are usually salts of Group 1 or Group 2 cations with small, highly charged anions. The highly charged anions are attracted to acid solvents such as water and ammonia. For example, lithium nitride, Li3N, reacwts violently with water: Li3N(s) 3 H2O(l) → 3 LiOH(aq) NH3(g) The nitride anion is a stronger base than the hydride ion and deprotonates hydrogen: Li3N(s) 2 H2(g) → LiNH2(s) 2 LiH(s) Lithium nitride is a possible hydrogen storage material as this reaction is reversible at 270°C (Box 10.4). Sodium hydride is a superbase that is used in organic chemistry to deprotonate carboxylic acids, alcohols, phenols, and thiols. Calcium hydride reacts with water to liberate hydrogen: CaH2(s) 2 H2O(l) → Ca(OH)2(s) 2 H2(g) Calcium hydride is used as a dessicant, to inflate weather balloons, and as a laboratory source of pure hydrogen. 4.16 Heterogeneous acid–base reactions Key point: The surfaces of many catalytic materials and minerals have Brønsted and Lewis acid sites. Some of the most important reactions involving the Lewis and Brønsted acidity of inorganic compounds occur at solid surfaces. For example, surface acids, which are solids with a high surface area and Lewis acid sites, are used as catalysts in the petrochemical industry for the interconversion of hydrocarbons. The surfaces of many materials that are important in the chemistry of soil and natural waters also have Brønsted and Lewis acid sites. Silica surfaces do not readily produce Lewis acid sites because –OH groups remain tenaciously attached at the surface of SiO2 derivatives; as a result, Brønsted acidity is dominant. The Brønsted acidity of silica surfaces themselves is only moderate (and comparable to that of acetic acid). However, as already remarked, aluminosilicates display strong Brønsted acidity. When surface OH groups are removed by heat treatment, the aluminosilicate surface possesses strong Lewis acid sites. The best-known class of aluminosilicates is the zeolites (Section 14.15), which are widely used as environmentally benign heterogeneous catalysts (Chapter 26). The catalytic activity of zeolites arises from their acidic nature and they are known as solid acids. Other solid acids include supported heteropoly acids and acidic clays. Some reactions occurring at these catalysts are very sensitive to the presence of Brønsted or Lewis acid sites. For example, toluene can be subjected to Friedel–Crafts alkylation over a bentonite clay catalyst: C H 2X + H 3C + HX When the reagent is benzyl chloride Lewis acid sites are involved in the reaction, and when the reagent is benzyl alcohol Brønsted sites are involved. 7 Carbocations could not be studied before Olah’s experiments, and he won the 1994 Nobel Prize for Chemistry for this work. 143 144 4 Acids and bases Surface reactions carried out using the Brønsted acid sites of silica gels are used to prepare thin coatings of a wide variety of organic groups using surface modification reactions such as OH O OSiR3 Si O O +HOSiR 3 O Si O +H 2O OSiR3 OH O Si O O +C lSiR O 3 O Si O O +HC l Thus, silica gel surfaces can be modified to have affinities for specific classes of molecules. This procedure greatly expands the range of stationary phases that can be used for chromatography. The surface –OH groups on glass can be modified similarly, and glassware treated in this manner is sometimes used in the laboratory when proton-sensitive compounds are being studied. Solid acids are finding new applications in green chemistry. Traditional industrial processes generate large volumes of hazardous waste during the final stages of the process when the product is separated from the reagents and byproducts. Solid catalysts are easily separated from liquid products and reactions can often operate under milder conditions and give greater selectivity. FURTHER READING W. Stumm and J.J. Morgan, Aquatic chemistry: chemical equilibria and rates in natural waters. Wiley, New York (1995). The classic text on the chemistry of natural waters. N. Corcoran, Chemistry in non-aqueous solvents. Kluwer Academic Publishers, Dordrecht, The Netherlands (2003). A comprehensive account. J. Chipperfield, Non-aqueous solvents. Oxford University Press (1999). A readable introduction to the topic. J. Burgess, Ions in solution. Ellis Horwood, Chichester (1988). A readable account of solvation with an introduction to acidity and polymerization. J. Burgess, Ions in solution: basic principles of chemical interactions. Ellis Horwood, Chichester (1999). G.A. Olah, G.K. Prakash, and J. Sommer, Superacids. Wiley, New York (1985). G.A. Olah, ‘My search for carbocations and their role in chemistry’, Nobel lectures in chemistry 1991–1995, ed. B.G. Malmstrom. World Scientific Publishing, Singapore (1996). R.J. Gillespie and J. Laing, Superacid solutions in hydrogen fluoride. J. Am. Chem. Soc., 1988, 110, 6053. E.S. Stoyanov, K.-C Kim, and C.A. Reed, A strong acid that does not protonate water. J. Phys. Chem. A., 2004, 108, 9310. EXERCISES 4.1 Sketch an outline of the s and p blocks of the periodic table and indicate on it the elements that form (a) strongly acidic oxides, (b) strongly basic oxides, and (c) show the regions for which amphoterism is common. 4.7 The effective proton affinity of F– in water is 1150 kJ mol–1. Predict whether it will behave as an acid or a base in water. 4.2 Identify the conjugate bases corresponding to the following acids: [Co(NH3)5(OH2)]3, HSO4–, CH3OH, H2PO4–, Si(OH)4, HS–. 4.9 Aided by Fig. 4.2 (taking solvent levelling into account), identify which bases from the following lists are (a) too strong to be studied experimentally, (b) too weak to be studied experimentally, or (c) of directly measurable base strength. (i) CO32–, O2–, ClO4–, and NO3– in water; (ii) HSO4–, NO3–, ClO4– in H2SO4. 4.3 Identify the conjugate acids of the bases C5H5N (pyridine), HPO42–, O2–, CH3COOH, [Co(CO)4]–, CN–. 4.4 Calculate the equilibrium concentration of H3O in a 0.10 m solution of butanoic acid (Ka 1.86 × 10–5). What is the pH of this solution? –5 4.5 The Ka of ethanoic acid, CH3COOH, in water is 1.8 × 10 . Calculate Kb of the conjugate base, CH3CO2–. 4.6 The value of Kb for pyridine, C5H5N, is 1.8 ×10–9. Calculate Ka for the conjugate acid, C5H5NH. 4.8 Draw the structures of chloric acid and chlorous acid, and predict their pKa values using Pauling’s rules. 4.10 The aqueous solution pKa values for HOCN, H2NCN, and CH3CN are approximately 4, 10.5, and 20 (estimated), respectively. Explain the trend in these cyano derivatives of binary acids and compare them with H2O, NH3, and CH4. Is the CN group electron donating or withdrawing? Problems 4.11 The pKa value of HAsO42– is 11.6. Is this value consistent with Pauling’s rules? 4.12 Use Pauling’s rules to place the following acids in order of increasing acid strength: HNO2, H2SO4, HBrO3, and HClO4 in a nonlevelling solvent. 4.13 Draw the structures and indicate the charges of the tetraoxoanions of X Si, P, S, and Cl. Summarize and account for the trends in the pKa values of their conjugate acids. 4.14 Which member of the following pairs is the stronger acid? Give reasons for your choice. (a) [Fe(OH2)6]3 or [Fe(OH2)6]2, (b) [Al(OH2)6]3 or [Ga(OH2)6]3, (c) Si(OH)4 or Ge(OH)4, (d) HClO3 or HClO4, (e) H2CrO4 or HMnO4, (f) H3PO4 or H2SO4. 4.15 Arrange the oxides Al2O3, B2O3, BaO, CO2, Cl2O7, SO3 in order from the most acidic through amphoteric to the most basic. 4.16 Arrange the acids HSO4–, H3O, H4SiO4, CH3GeH3, NH3, HSO3F in order of increasing acid strength. 4.17 The ions Na and Ag have similar radii. Which aqua ion is the stronger acid? Why? 4.18 Which of the elements Al, As, Cu, Mo, Si, B, Ti form oxide polyanions and which form oxide polycations? 4.19 When a pair of aqua cations forms an M–O–M bridge with the elimination of water, what is the general rule for the change in charge per M atom on the ion? 4.20 Write a balanced equation for the formation of P2O74– from PO43–. Write a balanced equation for the condensation of [Fe(OH2)6]3 to give [(H2O)4Fe(OH)2Fe(OH2)4]4. 4.21 Write balanced equations for the main reaction occurring when (a) H3PO4 and Na2HPO4 and (b) CO2 and CaCO3 are mixed in aqueous media. 4.22 Hydrogen fluoride acts as an acid in anhydrous sulfuric acid and as a base in liquid ammonia. Give the equations for both reactions. 4.23 Explain why hydrogen selenide is a stronger acid than hydrogen sulfide. 4.24 Sketch the p block of the periodic table. Identify as many elements as you can that act as Lewis acids in one of their lower oxidation states and give the formula of a representative Lewis acid for each element. 4.25 For each of the following processes identify the acids and bases involved and characterize the process as complex formation or acid– base displacement. Identify the species that exhibit Brønsted acidity as well as Lewis acidity. (a) SO3H2O → HSO4– H (b) CH3[B12] Hg2 → [B12] CH3Hg; [B12] designates the coporphyrin, vitamin B12. (c) KCl SnCl2 → K [SnCl3]– (d) AsF3(g)SbF5(l) → [AsF2][SbF6]–(s) (e) Ethanol dissolves in pyridine to produce a nonconducting solution. 4.26 Select the compound on each line with the named characteristic and state the reason for your choice. (a) Strongest Lewis acid: BCl3 BF3 BeCl2 BCl3 B(n-Bu)3 145 BBr3 B(t-Bu)3 (b) More basic towards B(CH3)3 Me3N Et3N 4-CH3C5H4N 2-CH3C5H4N 4.27 Using hard–soft concepts, which of the following reactions are predicated to have an equilibrium constant greater than 1? Unless otherwise stated, assume gas-phase or hydrocarbon solution and 25ºC. (a) R3PBBr3 R3NBF3 R3PRBF3 R3NBBr3 (b) SO2 (C6H5)3 P:HOC(CH3)3 (C6H5)3PSO2 HOC(CH3)3 (c) CH3HgI HCl CH3HgCl HI (d) [AgCl2]2–(aq) 2 CN–(aq) [Ag(CN)2]–(aq) 2 Cl–(aq) 4.28 The molecule (CH3)2N–PF2 has two basic atoms, P and N. One is bound to B in a complex with BH3, the other to B in a complex with BF3. Decide which is which and state the reason for your decision. 4.29 The enthalpies of reaction of trimethylboron with NH3CH3, NH2, (CH3)2NH, and (CH3)3N are –58, –74, –81, and –74 kJ mol–1, respectively. Why is trimethylamine out of line? 4.30 With the aid of the table of E and C values (Table 4.6), discuss the relative basicity in (a) acetone and dimethylsulfoxide, (b) dimethylsulfide and dimethylsulfoxide. Comment on a possible ambiguity for dimethylsulfoxide. 4.31 Give the equation for the dissolution of SiO2 glass by HF and interpret the reaction in terms of Lewis and Brønsted acid–base concepts. 4.32 Aluminium sulfide, Al2S3, gives off a foul odour characteristic of hydrogen sulfide when it becomes damp. Write a balanced chemical equation for the reaction and discuss it in terms of acid–base concepts. 4.33 Describe the solvent properties that would (a) favour displacement of Cl– by I– from an acid centre, (b) favour basicity of R3As over R3N, (c) favour acidity of Ag over Al3, (d) promote the reaction 2 FeCl3 ZnCl2 → Zn2 2 [FeCl4]–. In each case, suggest a specific solvent that might be suitable. 4.34 The Lewis acid AlCl3 catalysis of the acylation of benzene was described in Section 4.10b. Propose a mechanism for a similar reaction catalysed by an alumina surface. 4.35 Use acid–base concepts to comment on the fact that the only important ore of mercury is cinnabar, HgS, whereas zinc occurs in nature as sulfides, silicates, carbonates, and oxides. 4.36 Write balanced Brønsted acid–base equations for the dissolution of the following compounds in liquid hydrogen fluoride: (a) CH3CH2OH, (b) NH3, (c) C6H5COOH. 4.37 Is the dissolution of silicates in HF a Lewis acid–base reaction, a Brønsted acid–base reaction, or both? 4.38 The f-block elements are found as M(III) lithophiles in silicate minerals. What does this indicate about their hardness? 4.39 Use the data in Table 4.6 to calculate the enthalpy change for the reaction of iodine with phenol. PROBLEMS 4.1 In analytical chemistry a standard procedure for improving the detection of the stoichiometric point in titrations of weak bases with strong acids is to use acetic acid as a solvent. Explain the basis of this approach. 4.2 In the gas phase, the base strength of amines increases regularly along the series NH3 < CH3NH2 < (CH3)2NH < (CH3)3N. Consider the role of steric effects and the electron-donating ability of CH3 in determining this order. In aqueous solution, the order is 146 4 Acids and bases reversed. What solvation effect is likely to be responsible for this change? oxides in Fig. 4.6. Does this analysis agree with describing S2– as a softer base than O2–? 4.3 The hydroxoacid Si(OH)4 is weaker than H2CO3. Write balanced equations to show how dissolving a solid, M2SiO4, can lead to a reduction in the pressure of CO2 over an aqueous solution. Explain why silicates in ocean sediments might limit the increase of CO2 in the atmosphere. 4.9 The compounds SO2 and SOCl2 can undergo an exchange of radioactively labelled sulfur. The exchange is catalysed by Cl– and SbCl5. Suggest mechanisms for these two exchange reactions with the first step being the formation of an appropriate complex. 4.4 The precipitation of Fe(OH)3 discussed in the chapter is used to clarify waste waters because the gelatinous hydrous oxide is very efficient at coprecipitating some contaminants and entrapping others. The solubility constant of Fe(OH)3 is Ks [Fe3][OH–]3 ≈ 1.0 × 10–38. As the autoprotolysis constant of water links [H3O] to [OH–] by Kw [H3O][OH–] 1.0 × 10–14, we can rewrite the solubility constant by substitution as [Fe3]/[H]3 1.0 × 104. (a) Balance the chemical equation for the precipitation of Fe(OH)3 when iron(III) nitrate is added to water. (b) If 6.6 kg of Fe(NO3)3.9H2O is added to 100 dm3 of water, what is the final pH of the solution and the molar concentration of Fe3, neglecting other forms of dissolved Fe(III)? Give formulas for two Fe(III) species that have been neglected in this calculation. 4.5 The frequency of the symmetrical M–O stretching vibration of the octahedral aqua ions [M(OH2)6]2 increases along the series, Ca2 < Mn2 < Ni2. How does this trend relate to acidity? 4.6 An electrically conducting solution is produced when AlCl3 is dissolved in the basic polar solvent CH3CN. Give formulas for the most probable conducting species and describe their formation using Lewis acid–base concepts. 4.7 The complex anion [FeCl4]– is yellow whereas [Fe2Cl6] is reddish. Dissolution of 0.1 mol FeCl3(s) in 1 dm3 of either POCl3 or PO(OR)3 produces a reddish solution that turns yellow on dilution. Titration of red solutions in POCl3 with Et4NCl solutions leads to a sharp colour change (from red to yellow) at a 1:1 mole ratio of FeCl3/Et4NCl. Vibrational spectra suggest that oxochloride solvents form adducts with typical Lewis acids by coordination of oxygen. Compare the following two sets of reactions as possible explanations of the observations. (a) Fe2Cl6 2 POCl3 2 [FeCl4]– 2 [POCl2] POCl2 Et4NCl Et4N POCl3 (b) Fe2Cl6 4 POCl3 [FeCl2(OPCl3)4] [FeCl4]– Both sets of equilibria are shifted to products by dilution. 4.8 In the traditional scheme for the separation of metal ions from solution that is the basis of qualitative analysis, ions of Au, As, Sb, and Sn precipitate as sulfides but redissolve on addition of excess ammonium polysulfide. By contrast, ions of Cu, Pb, Hg, Bi, and Cd precipitate as sulfides but do not redissolve. In the language of this chapter, the first group is amphoteric for reactions involving SH– in place of OH–. The second group is less acidic. Locate the amphoteric boundary in the periodic table for sulfides implied by this information. Compare this boundary with the amphoteric boundary for hydrous 4.10 In the reaction of t-butyl bromide with Ba(NCS)2, the product is 91 per cent S-bound t-Bu-SCN. However, if Ba(NCS)2 is impregnated into solid CaF2, the yield is higher and the product is 99 per cent t-Bu-NCS. Discuss the effect of alkaline earth metal salt support on the hardness of the ambident nucleophile SCN–. (See T. Kimura, M. Fujita, and T. Ando, J. Chem Soc., Chem. Commun., 1990, 1213.) 4.11 Pyridine forms a stronger Lewis acid–base complex with SO3 than with SO2. However, pyridine forms a weaker complex with SF6 than with SF4. Explain the difference. 4.12 Predict whether the equilibrium constants for the following reactions should be greater than 1 or less than 1: (a) CdI2(s) CaF2(s) CdF2(s) CaI2(s) (b) [CuI4]2–(aq) [CuCl4]3–(aq) [CuCl4]2–(aq) [CuI4]3–(aq) (c) NH2–(aq) H2O(l) NH3(aq) OH–(aq) 4.13 For parts (a), (b), and (c), state which of the two solutions has the lower pH: (a) 0.1 m Fe(ClO4)2(aq) or 0.1 m Fe(ClO4)3(aq) (b) 0.1 m Ca(NO3)2(aq) or 0.1 m Mg(NO3)2(aq) (c) 0.1 m Hg(NO3)2(aq) or 0.1 m Zn(NO3)2(aq) 4.14 A paper by Gillespie and Liang entitled ‘Superacid solutions in hydrogen fluoride’ (J. Am. Chem. Soc., 1988, 110, 6053) discusses the acidity of various solutions of inorganic compounds in HF. (a) Give the order of acid strength of the pentafluorides determined during the investigation. (b) Give the equations for the reactions of SbF5 and AsF5 with HF. (c) SbF5 forms a dimer, Sb2F11 , in HF. Give the equation for the equilibrium between the monomeric and the dimeric species. 4.15 Why are strongly acidic solvents (e.g. SbF5/HSO3F) used in the preparation of cations such as I2 and Se82, whereas strongly basic solvents are needed to stabilize anionic species such as S42– and Pb94–? 4.16 In their paper ‘The strengths of the hydrohalic acids’ (J. Chem. Educ., 2001, 78, 116), R. Schmid and A. Miah discuss the validity of literature values of the pKas for HF, HCl, HBr, and HI. (a) On what basis have the literature values been estimated? (b) To what is the low acid strength of HF relative to HCl usually attributed? (c) What reason do the authors suggest for the high acid strength of HCl? 4.17 Superacids are well known. Superbases also exist and are usually based on hydrides of Group 1 and Group 2 elements. Write an account of the chemistry of superbases. Oxidation and reduction Oxidation is the removal of electrons from a species; reduction is the addition of electrons. Almost all elements and their compounds can undergo oxidation and reduction reactions and the element is said to exhibit one or more different oxidation states. In this chapter we present examples of this ‘redox’ chemistry and develop concepts for understanding why oxidation and reduction reactions occur, considering mainly their thermodynamic aspects. We discuss the procedures for analysing redox reactions in solution and see that the electrode potentials of electrochemically active species provide data that are useful for determining and understanding the stability of species and solubility of salts. We describe procedures for displaying trends in the stabilities of various oxidation states, including the influence of pH. Next, we describe the applications of this information to environmental chemistry, chemical analysis, and inorganic synthesis. The discussion concludes with a thermodynamic examination of the conditions needed for some major industrial oxidation and reduction processes, particularly the extraction of metals from their ores. 5 Reduction potentials 5.1 Redox half-reactions 5.2 Standard potentials and spontaneity 5.3 Trends in standard potentials 5.4 The electrochemical series 5.5 The Nernst equation Redox stability 5.6 The influence of pH 5.7 Reactions with water 5.8 Oxidation by atmospheric oxygen A large class of reactions of inorganic compounds can be regarded as occurring by the transfer of electrons from one species to another. Electron gain is called reduction and electron loss is called oxidation; the joint process is called a redox reaction. The species that supplies electrons is the reducing agent (or ‘reductant’) and the species that removes electrons is the oxidizing agent (or ‘oxidant’). Many redox reactions release a great deal of energy and they are exploited in combustion or battery technologies. Many redox reactions occur between reactants in the same physical state. Some examples are: in gases: 2 NO(g) O2(g) → 2 NO2(g) 2 C4H10(g) 13 O2(g) → 8 CO2(g) 10 H2O(g) in solution: Fe3 (aq) Cr 2 (aq) → Fe2 (aq) Cr 3 (aq) 3 CH3CH2OH(aq) 2 CrO42(aq) 10 H(aq) → 3 CH3CHO(aq) 2 Cr3(aq) 8 H2O(l) in biological systems: ‘Mn4’(V,IV,IV,IV) 2 H2O(l) → ‘Mn4’(IV,III,III,III) 4 H(aq) O2(g) in solids: LiCoO2(s) C(s) → Li@C(s) CoO2(s) The biological example refers to the production of O2 from water by an ‘Mn4’ cofactor contained in one of the photosynthetic complexes of plants (Section 27.10). In the solid-state example the symbol Li@C(s) indicates that a Li ion has penetrated between the graphene sheets of graphite to form an intercalation compound. The reaction takes place in a lithium-ion battery during charging and its reverse takes place during discharge. Redox reactions can also occur at interfaces (phase boundaries), such as a gas/solid or a 5.9 Disproportionation and comproportionation 5.10 The influence of complexation 5.11 The relation between solubility and standard potentials The diagrammatic presentation of potential data 5.12 Latimer diagrams 5.13 Frost diagrams 5.14 Pourbaix diagrams 5.15 Natural waters Chemical extraction of the elements 5.16 Chemical reduction 5.17 Chemical oxidation 5.18 Electrochemical extraction FURTHER READING EXERCISES PROBLEMS 148 5 Oxidation and reduction solid/liquid interface. Examples include the dissolution of a metal and reactions occurring at an electrode. Because of the diversity of redox reactions it is often convenient to analyse them by applying a set of formal rules expressed in terms of oxidation numbers (Section 2.1) and not to think in terms of actual electron transfers. Oxidation then corresponds to an increase in the oxidation number of an element and reduction corresponds to a decrease in its oxidation number. If no element in a reaction undergoes a change in oxidation number, then the reaction is not redox. We shall adopt this approach when we judge it appropriate. ■ A brief illustration. The simplest redox reactions involve the formation of cations and anions from the elements. Examples include the oxidation of lithium to Li ions when it burns in air to form Li2O and the reduction of chlorine to Cl when it reacts with calcium to form CaCl2. For the Group 1 and 2 elements the only oxidation numbers commonly encountered are those of the element (0) and of the ions, 1 and 2, respectively. However, many of the other elements form compounds in more than one oxidation state. Thus lead is commonly found in its compounds as Pb(II), as in PbO, and as Pb(IV), as in PbO2. ■ The ability to exhibit multiple oxidation numbers is seen at its fullest in d-metal compounds, particularly in Groups 6, 7, and 8; osmium, for instance, forms compounds that span oxidation numbers between 2, as in [Os(CO)4]2, and 8, as in OsO4. Because the oxidation state of an element is often reflected in the properties of its compounds, the ability to express the tendency of an element to form a compound in a particular oxidation state quantitatively is very useful in inorganic chemistry. Reduction potentials Because electrons are transferred between species in redox reactions, electrochemical methods (using pairs of electrodes to measure electron transfer reactions under controlled thermodynamic conditions) are of major importance and lead to the construction of tables of ‘standard potentials’. The tendency of an electron to migrate from one species to another is expressed in terms of the differences between their standard potentials. 5.1 Redox half-reactions Key point: A redox reaction can be expressed as the difference of two reduction half-reactions. It is convenient to think of a redox reaction as the combination of two conceptual halfreactions in which the electron loss (oxidation) and gain (reduction) are displayed explicitly. In a reduction half-reaction, a substance gains electrons, as in 2 H(aq) 2 e → H2(g) In an oxidation half-reaction, a substance loses electrons, as in Zn(s) → Zn2(aq) 2 e Electrons are not ascribed a state in the equation of a half-reaction: they are ‘in transit’. The oxidized and reduced species in a half-reaction constitute a redox couple. A couple is written with the oxidized species before the reduced, as in H/H2 and Zn2/Zn, and typically the phases are not shown. For reasons that will become clear, it is useful to represent oxidation half-reactions by the corresponding reduction half-reaction. To do so, we simply reverse the equation for the oxidation half-reaction. Thus, the reduction half-reaction associated with the oxidation of zinc is written Zn2(aq) 2 e → Zn(s) A redox reaction in which zinc is oxidized by hydrogen ions, Zn(s) 2 H(aq) → Zn2(aq) H2(g) is then written as the difference of the two reduction half-reactions. In some cases it may be necessary to multiply each half-reaction by a factor to ensure that the numbers of electrons released and used match. Reduction potentials E X A MPL E 5 .1 Combining half-reactions Write a balanced equation for the oxidation of Fe2 by permanganate ions (MnO4 ) in acid solution. Answer Balancing redox reactions often requires additional attention to detail because species other than products and reactants, such as electrons and hydrogen ions, often need to be considered. A systematic approach is as follows: 1. 2. 3. 4. Write the unbalanced half-reactions for the two species as reductions. Balance the elements other than hydrogen. Balance O atoms by adding H2O to the other side of the arrow. If the solution is acidic, balance the H atoms by adding H; if the solution is basic, balance the H atoms by adding OH to one side and H2O to the other. 5. Balance the charge by adding e. 6. Multiply each half-reaction by a factor to ensure that the numbers of e match. 7. Subtract one half-reaction from the other and cancel redundant terms. The half-reaction for the reduction of Fe3 is straightforward as it involves only the balance of charge: Fe3(aq) e → Fe2(aq) The unbalanced half-reaction for the reduction of MnO4 is MnO4 (aq) → Mn2(aq) Balance the O with H2O: MnO4 (aq) → Mn2(aq) 4 H2O(l) Balance the H with H(aq): MnO4 (aq) 8 H(aq) → Mn2(aq) 4 H2O(l) Balance the charge with e: MnO4 (aq) 8 H(aq) 5 e → Mn2(aq) 4 H2O(l) To balance the number of electrons in the two half-reactions the first is multiplied by 5 and the second by 2 to give 10 e in each case. Then subtracting the iron half-reaction from the permanganate half-reaction and rearranging so that all stoichiometric coefficients are positive gives MnO4 (aq) 8 H(aq) 5 Fe2(aq) → Mn2(aq) 5 Fe3(aq) 4 H2O(l) Self-test 5.1 Use reduction half-reactions to write a balanced equation for the oxidation of zinc metal by permanganate ions in acid solution. 5.2 Standard potentials and spontaneity Key point: A reaction is thermodynamically favourable (spontaneous) in the sense K 1, if E ° 0, where E ° is the difference of the standard potentials corresponding to the half-reactions into which the overall reaction may be divided. Thermodynamic arguments can be used to identify which reactions are spontaneous (that is, have a natural tendency to occur). The thermodynamic criterion of spontaneity is that, at constant temperature and pressure, the reaction Gibbs energy change, ∆rG, is negative. It is usually sufficient to consider the standard reaction Gibbs energy, ∆r G ° , which is related to the equilibrium constant through ∆r G ° RT ln K (5.1) A negative value of ∆rG ° corresponds to K 1 and therefore to a ‘favourable’ reaction in the sense that the products dominate the reactants at equilibrium. It is important to realize, however, that ∆rG depends on the composition and that all reactions are spontaneous (that is, have ∆rG 0) under appropriate conditions. Because the overall chemical equation is the difference of two reduction half-reactions, the standard Gibbs energy of the overall reaction is the difference of the standard Gibbs energies of the two half-reactions. However, because reduction half-reactions always occur in pairs in any actual chemical reaction, only the difference in their standard 149 150 5 Oxidation and reduction Gibbs energies has any significance. Therefore, we can choose one half-reaction to have ∆r G ° 0, and report all other values relative to it. By convention, the specially chosen half-reaction is the reduction of hydrogen ions: H (aq) e → 12 H 2 (g) ∆r G ° 0 at all temperatures. E ■ A brief illustration. The standard Gibbs energy for the reduction of Zn2 ions is found by determining experimentally that Salt bridge H2(g) Zn2 (aq) H2 ( g) → Zn(s) 2H+ ( aq) ∆ rG ° 147 kJ mol1 Then, because the H reduction half-reaction makes zero contribution to the reaction Gibbs energy (according to our convention), it follows that Zn2 (aq) 2 e → Zn(s) Pt Zn(s) H+(aq) Zn2+(aq) Fig. 5.1 A schematic diagram of a galvanic cell. The standard potential, E cell ° , is the potential difference when the cell is not generating current and all the substances are in their standard states. ∆ rG ° 147 kJ mol1 ■ Standard reaction Gibbs energies may be measured by setting up a galvanic cell, an electrochemical cell in which a chemical reaction is used to generate an electric current, in which the reaction driving the electric current through the external circuit is the reaction of interest (Fig. 5.1). The potential difference between its electrodes is then measured. The cathode is the electrode at which reduction occurs and the anode is the site of oxidation. In practice, we must ensure that the cell is acting reversibly in a thermodynamic sense, which means that the potential difference must be measured with no current flowing. If desired, the measured potential difference can be converted to a reaction Gibbs energy by using ∆rG FE, where is the stoichiometric coefficient of the electrons transferred when the half-reactions are combined and F is Faraday’s constant (F 96.48 kC mol1). Tabulated values, normally for standard conditions, are usually kept in the units in which they were measured, namely volts (V). ■ A brief comment. Standard conditions are all substances at 1 bar and unit activity. For reactions involving H ions, standard conditions correspond to pH 0, approximately 1 M acid. Pure solids and liquids have unit activity. Although we use (nu) for the stoichiometric coefficient of the electron, electrochemical equations in inorganic chemistry are also commonly written with n in its place; we use to emphasize that it is a dimensionless number, not an amount in moles. ■ The potential that corresponds to the ∆r G ° of a half-reaction is written E ° , with ∆r G ° FE ° (5.2) The potential E ° is called the standard potential (or ‘standard reduction potential’, to emphasize that, by convention, the half-reaction is a reduction and written with the oxidized species and electrons on the left). Because ∆rG ° for the reduction of H is arbitrarily set at zero, the standard potential of the H/H2 couple is also zero at all temperatures: H + (aq) e → 12 H 2 (g) E ° (H, H2) 0 ■ A brief illustration. For the Zn2/Zn couple, for which 2, it follows from the measured value of ∆rG ° that at 25C: Zn2 (aq) 2e → Zn(s) E ° (Zn2+ , Zn) 0.76 V ■ Because the standard reaction Gibbs energy is the difference of the ∆rG ° values for the two contributing half-reactions, Ecell ° for an overall reaction is also the difference of the two standard potentials of the reduction half-reactions into which the overall reaction can be divided. Thus, from the half-reactions given above it follows that the difference is 2 H (aq) Zn(s) → Zn2 (aq) H 2 (g) E cell ° 0.76V Note that the E ° values for couples (and their half-reactions) are called standard potentials and that their difference is denoted Ecell ° and called the standard cell potential. The consequence of the negative sign in eqn 5.2 is that a reaction is favourable (in the sense K 1) if the corresponding standard cell potential is positive. Because E ° 0 for the reaction in the illustration (E ° 0.76 V), we know that zinc has a thermodynamic tendency to reduce H ions under standard conditions (aqueous, pH 0, and Zn2 at unit activity); that is zinc metal dissolves in acids. The same is true for any metal that has a couple with a negative standard potential. 151 Reduction potentials A note on good practice The cell potential used to be called (and in practice is still widely called) the electromotive force (emf). However, a potential is not a force, and IUPAC favours the name ‘cell potential’. E X A MPL E 5 . 2 Calculating a standard cell potential Use the following standard potentials to calculate the standard potential of a copperzinc cell. Cu2 (aq) 2 e → Cu(s) Zn2 (aq) 2 e → Zn(s) E ° (Cu2 , Cu) 0.34 V E ° ( Zn2 , Zn) 0.76 V Answer For this calculation we note from the standard potentials that Cu2 is the more oxidizing species (the couple with the higher potential), and will be reduced by the species with the lower potential (Zn in this case). The spontaneous reaction is therefore Cu2(aq) Zn(s) → Zn2(aq) Cu(s), and the cell potential is the difference of the two half-reactions (copperzinc), E cell ° E ° ( Cu2, Cu) E ° (Zn2, Zn) 0.34 V (0.76 V) 1.10 V The cell will produce a potential difference of 1.1 V (under standard conditions). Self-test 5.2 Is copper metal expected to dissolve in dilute hydrochloric acid? Is copper metal expected to be oxidized by dilute hydrochloric acid? Combustion is a familiar type of redox reaction, and the energy that is released can be exploited in heat engines. A fuel cell converts a chemical fuel directly into electrical power (Box 5.1). 5.3 Trends in standard potentials Key point: The atomization and ionization of a metal and the hydration enthalpy of its ions all contribute to the value of the standard potential. The factors that contribute to the standard potential of the couple M/M can be identified by consideration of a thermodynamic cycle and the corresponding changes in Gibbs energy that contribute to the overall reaction M (aq) 12 H 2 (g) → H (aq) M(s) The thermodynamic cycle shown in Fig. 5.2 has been simplified by ignoring the reaction entropy, which is largely independent of the identity of M. The entropy contribution T S ° lies in the region of 20 to 40 kJ mol1, which is small in comparison with the reaction enthalpy, the difference between the standard enthalpies of formation of H(aq) and M(aq). In this analysis we use the absolute values of the enthalpies of formation of M and H, not the values based on the convention f H ° (H, aq) 0. Thus, we use f H ° (H, aq) 445 kJ mol1, which is obtained by considering the formation of an H atom from 12 H2(g) (218 kJ mol1), ionization to H(g) (1312 kJ mol1), and hydration of H(g) (approximately 1085 kJ mol1). The analysis of the cell potential into its thermodynamic contributions allows us to account for trends in the standard potentials. For instance, the variation of standard potential down Group 1 seems contrary to expectation based on electronegativities insofar as Cs/Cs ( 0.79, E ° 2.94 V) has a less negative standard potential than Li/Li ( 2.20, E ° 3.04 V) despite Li having a higher electronegativity than Cs. Lithium has a higher enthalpy of sublimation and ionization energy than Cs, and in isolation this difference would imply a less negative standard potential as formation of the ion is less favourable. However, Li has a large negative enthalpy of hydration, which results from its small size (its ionic radius is 90 pm) compared with Cs (181 pm) and its consequent strong electrostatic interaction with water molecules. Overall, the favourable enthalpy of hydration of Li outweighs terms relating to the formation of Li(g) and gives rise to a more negative standard potential. The relatively less negative standard potential for Na/Na (2.71 V) in comparison with the rest of Group 1 (close to 2.9 V) is a result of a combination of a fairly high sublimation enthalpy and moderate hydration enthalpy (Table 5.1). The value of E ° (Na, Na) 2.71 V may also be compared with that for E ° (Ag, Ag) 0.80 V. The (6-coordinate) ionic radii of these ions (r 102 pm and rAg 115 pm) Na M+(g) + e–(g) + H+(g) –∆hydH°(H+) –I(H) M+(g) + H(g) M+(g) + e–(g) + H+(aq) – 12 D(H–H) I(M) M+(g) + 12 H2(g) M(g) + H+(aq) ∆hydH°(M+) ∆subH°(M) M(s) + H+(aq) ∆rH° M+(aq) + 1 2 H2(g) Fig. 5.2 A thermodynamic cycle showing the properties that contribute to the standard potential of a metal couple. Endothermic processes are drawn with upward-pointing arrows and exothermic contributions with downward pointing arrows. 152 5 Oxidation and reduction B OX 5 .1 Fuel cells A fuel cell converts a chemical fuel, such as hydrogen (used for larger power requirements) or methanol (a convenient fuel for small applications), directly into electrical power, using O2 or air as the oxidant. As power sources, fuel cells offer several advantages over rechargeable batteries or combustion engines, and their use is steadily increasing. Compared to batteries, which have to be replaced or recharged over a significant period of time, a fuel cell operates as long as fuel is supplied. Furthermore, a fuel cell does not contain large amounts of environmental contaminants such as Ni and Cd, although relatively small amounts of Pt and other metals are required as electrocatalysts. The operation of a fuel cell is more efficient than combustion devices, with near-quantitative conversion of fuel to H2O and (for methanol) CO2. Fuel cells are also much less polluting because nitrogen oxides are not produced at the relatively low temperatures that are used. Because an individual cell potential is less than about 1 V, fuel cells are connected in series known as ‘stacks’ in order to produce a useful voltage. Important classes of hydrogen fuel cell are the proton-exchange membrane fuel cell (PEMFC), the alkaline fuel cell (AFC), and the solid oxide fuel cell (SOFC), which differ in their mode of electrode reactions, chemical charge transfer, and operational temperature. Details are included in the table. Fuel cell Reaction at anode Electrolyte Transfer ion Reaction at cathode Temp. range/oC Pressure/atm Efficiency/% PEMFC H2 → 2 H 2 e− H-conducting polymer (PEM) H 2 H 21 O2 2 e− → H2O 80100 18 35−40 AFC H2 → 2 H 2 e− Aqueous alkali OH− H2O 21 O2 2 e− → 2 OH− 80250 110 50−60 SOFC H2 O2− → H2O 2 e− Solid oxide O2− 8001000 1 50−55 DMFC CH3OH H2O → CO2 6 H 6 e− H-conducting polymer H 040 1 20−40 1 2 O2 2 e− → O2− 2H 21 O2 2 e− → H2O The basic principles of fuel cells are illustrated by a PEMFC (Fig. B5.1), which operates at modest temperatures (80100oC) and is suitable as an on-board power supply for road vehicles. At the anode, a continuous supply of H2 is oxidized and the resulting H ions, the chemical charge carriers, pass through a membrane to the cathode, at which O2 is reduced to H2O. This process produces a flow of electrons from anode to cathode (the current) that is directed through the load (typically an electric motor). The anode (the site of H2 oxidation) and the cathode (the site of O2 reduction) are both loaded with a Pt catalyst to obtain efficient electrochemical conversions of fuel and oxidant. The major factor limiting the efficiency of PEMFC and other fuel cells is the sluggish reduction of O2 at the cathode, which involves expenditure of a few tenths of a volt (the ‘overpotential’) just to drive this reaction at a practical rate. The operating voltage is usually about 0.7 V. The membrane is composed of an H-conducting polymer, sodium perfluorosulfonate (invented by Du Pont and known commercially as Nafion®). An AFC is more efficient than a PEMFC because the reduction of O2 at the Pt cathode is much easier under alkaline conditions. Hence the operating voltage is typically greater than about 0.8 V. The membrane of the PEMFC is now replaced by a pumped flow of hot aqueous alkali between the two electrodes. Alkaline fuel cells were used to provide power for the pioneering Apollo spacecraft moon missions. An SOFC operates at much higher temperatures (8001100oC) and is used to provide electricity and heating in buildings (in the arrangement called combined heat and power, CHP). The cathode is typically a complex metal oxide based on LaCoO3, such as La(1x)SrxMn(1y)CoyO3, whereas the anode is typically NiO mixed with RuO2 and a lanthanoid oxide such as Ce(1x)GdxO1.95. The chemical charge is carried by a ceramic oxide such as ZrO2 doped with yttrium, which allows conduction by O2− ion transfer at high temperatures (Section 23.4). The high operating temperature relaxes the requirement for such an efficient catalyst as Pt. Methanol is used as a fuel in either of two ways. One exploits methanol as an ‘H2 carrier’, because the reforming reaction (see Chapter 10) is used to generate H2 which is then supplied in situ to a normal hydrogen fuel cell as mentioned above. This indirect method avoids the need to store H2 under pressure. The other is the direct methanol fuel cell (DMFC) which incorporates anode and cathode each loaded with Pt or a Pt alloy, and a PEM. The methanol is supplied Load e– Anode Cathode Protonexchange membrane H2 O2 H+ H2O Fig. B5.1 A schematic diagram of a proton-exchange membrane (PEM) fuel cell. The anode and cathode are loaded with a catalyst (Pt) to convert fuel (H2) and oxidant (O2) into H and H2O, respectively. The membrane (usually a material called Nafion®) allows the H ions produced at the anode to be transferred to the cathode. to the anode as an aqueous solution (at 1 mol dm−3). The DMFC is particularly suitable for small low-power devices such as mobile phones and portable electronic processors and it provides a promising alternative to the Li-ion battery. The principal disadvantage of the DMFC is its relatively low efficiency. This inefficiency arises from two factors that lower the operating voltage: the sluggish kinetics at the anode (oxidation of CH3OH to CO2 and H2O) in addition to the poor cathode kinetics already mentioned, and transfer of methanol across the membrane to the cathode (‘crossover’), which occurs because methanol permeates the hydrophilic PEM easily. A 50/50 Pt/Ru mixture supported on carbon is used as the anode catalyst to improve the rate of methanol oxidation. Reduction potentials Table 5.1 Thermodynamic contributions to E ° for a selection of metals at 298 K ∆ subH ° /(kJ mol−1) Li Na Cs Ag 161 109 79 284 1 I /(kJ mol ) ∆ hydH ° /(kJ mol1) 526 502 382 735 520 406 264 468 ∆ fH ° (M , aq) /(kJ mol1) 167 206 197 551 ∆ rH ° /(kJ mol ) 278 240 248 106 T ∆ r S ° /(kJ mol1) 16 22 34 29 ∆ rG ° /(kJ mol ) 294 262 282 77 E °/V 3.04 2.71 2.92 0.80 1 1 ∆ fH ° (H ,aq) 455kJ mol1. Table 5.2 Selected standard potentials at 298 K; further values are included in Resource section 3 E °/ V Couple F2(g) 2 e → 2 F(aq) 2.87 Ce (aq) e → Ce (aq) 1.76 MnO4 (aq) 8H (aq) 5e → Mn2 (aq) 4H2O(l) 1.51 4 3 Cl2(g)2 e → 2 Cl(aq) 1.36 O2(g) 4 H(aq) 4 e → 2 H2O(l) 1.23 [IrCl6]2(aq) e → [IrCl6]3(aq) 0.87 Fe3(aq) e → Fe2(aq) 0.77 [PtCl4] (aq) 2 e → Pt(s) 4 Cl (aq) 0.60 I3 (aq) 2e → 3 I (aq) 0.54 [Fe(CN)6]3(aq) e → [Fe(CN)6]4(aq) 0.36 AgCl(s) e → Ag(s) Cl(aq) 0.22 2 2 H (aq) 2 e → H2(g) Agl(s) e → Ag(s) l(aq) 0 0.15 Zn2(aq) 2 e → Zn(s) 0.76 Al3(aq) 3 e → Al(s) 1.68 Ca2(aq) 2 e → Ca(s) 2.84 Li(aq) e → Li(s) 3.04 are similar, and consequently their ionic hydration enthalpies are also similar. However, the much higher enthalpy of sublimation of silver, and particularly its high ionization energy, which is due to the poor screening by the 4d electrons, results in a positive standard potential. This difference is reflected in the very different behaviour of the metals when treated with a dilute acid: sodium reacts and dissolves explosively, producing hydrogen, whereas silver is unreactive. Similar arguments can be used to explain many of the trends observed in the standard potentials given in Table 5.2. For example, the positive potentials characteristic of the noble metals result in large part from their very high sublimation enthalpies. 5.4 The electrochemical series Key points: The oxidized member of a couple is a strong oxidizing agent if E ° is positive and large; the reduced member is a strong reducing agent if E ° is negative and large. A negative standard potential (E ° 0) signifies a couple in which the reduced species (the Zn in Zn2/Zn) is a reducing agent for H ions under standard conditions in aqueous solution. That is, if E ° (Ox, Red) 0, then the substance ‘Red’ is a strong enough reducing 153 154 5 Oxidation and reduction agent to reduce H ions (in the sense that K 1 for the reaction). A short compilation of E ° values at 25C is given in Table 5.2. The list is arranged in the order of the electrochemical series: Ox/Red couple with strongly positive E °[Ox is strongly oxidizing ] Ox/Red couple with strongly negative E ° [Red is strongly reducing] An important feature of the electrochemical series is that the reduced member of a couple has a thermodynamic tendency to reduce the oxidized member of any couple that lies above it in the series. Note that the classification refers only to the thermodynamic aspect of the reaction—its spontaneity under standard conditions and the value of K, not its rate. Thus even reactions that are found to be thermodynamically favourable from the electrochemical series may not progress, or progress only extremely slowly, if the kinetics of the process are unfavourable. E X A M PL E 5 . 3 Using the electrochemical series Among the couples in Table 5.2 is the permanganate ion, MnO4 , the common analytical reagent used in redox titrations of iron. Which of the ions Fe2, Cl, and Ce3 can permanganate oxidize in acidic solution? Answer We need to note that a reagent that is capable of reducing MnO4 ions must be the reduced form of a redox couple having a more negative standard potential than the couple MnO4 /Mn2. The standard potential of the couple MnO4 /Mn2 in acidic solution is 1.51 V. The standard potentials of Fe3/Fe2, Cl2 /Cl, and Ce4/Ce3 are 0.77, 1.36, and 1.76 V, respectively. It follows that MnO4 ions are sufficiently strong oxidizing agents in acidic solution (pH 0) to oxidize Fe2 and Cl, which have less positive standard potentials. Permanganate ions cannot oxidize Ce3, which has a more positive standard potential. It should be noted that the presence of other ions in the solution can modify the potentials and the conclusions (Section 5.10); this variation with conditions is particularly important in the case of H ions, and the influence of pH is discussed in Section 5.6. The ability of MnO4 ions to oxidize Cl means that HCl cannot be used to acidify redox reactions involving permanganate but H2SO4 is used instead. Self-test 5.3 Another common analytical oxidizing agent is an acidic solution of dichromate ions, Cr2O2 , 7 3 2 3 for which E ° (Cr2O2 , Cr ) 1.38 V. Is the solution useful for a redox titration of Fe to Fe ? Could 7 there be a side reaction when Cl is present? 5.5 The Nernst equation Key point: The cell potential at an arbitrary composition of the reaction mixture is given by the Nernst equation. To judge the tendency of a reaction to run in a particular direction at an arbitrary composition, we need to know the sign and value of ∆rG at that composition. For this information, we use the thermodynamic result that rG rG ° RT ln Q (5.3a) where Q is the reaction quotient1 a Ox A b RedB → a′ Red A b ′ Ox B Q [Red A ]a′ [Ox B ]b′ [Ox A ]a [R RedB ]b (5.3b) The reaction quotient has the same form as the equilibrium constant K but the concentrations refer to an arbitrary stage of the reaction; at equilibrium, Q K. When evaluating Q and K, the quantities in square brackets are to be interpreted as the numerical values of the molar concentrations. Both Q and K are therefore dimensionless quantities. The reaction is spontaneous at an arbitrary stage if ∆rG 0. This criterion can be expressed 1 For reactions involving gas-phase species, the molar concentrations of the latter are replaced by partial pressures relative to p ° 1 bar. Reduction potentials 155 in terms of the potential of the corresponding cell by substituting Ecell ∆rG/F and Ecell ° rG ° / ␯F into eqn 5.3a, which gives the Nernst equation: Ecell Ecell ° RT ln Q ␯F (5.4) A reaction is spontaneous if, under the prevailing conditions, Ecell 0, for then ∆rG 0. At equilibrium Ecell 0 and Q K, so eqn 5.4 implies the following very important relationship between the standard potential of a cell and the equilibrium constant of the cell reaction at a temperature T: ln K ␯FEcell ° (5.5) RT Table 5.3 lists the values of K that correspond to cell potentials in the range 2 to 2 V, with 1 and at 25C. The table shows that, although electrochemical data are often compressed into the range 2 to 2 V, this narrow range corresponds to 68 orders of magnitude in the value of the equilibrium constant for 1. If we regard the cell potential Ecell as the difference of two reduction potentials, just as E cell ° is the difference of two standard reduction potentials, then the potential of each couple, E, that contributes to the cell reaction can be written like eqn 5.4, RT EE° ln Q ␯F (5.6a) but with [Red] [Ox ] a′ a Ox e → a Red Q (5.6b) a By convention, the electrons do not appear in the expression for Q. The temperature dependence of a standard cell potential (eqns 5.6a and 5.6b) provides a straightforward way to determine the standard entropy of many redox reactions. From eqn 5.2, we can write ␯FEcell ° rG ° r H ° T r S ° (5.7a) Then, if we suppose that r H ° and r S ° are independent of temperature over the small range usually of interest, it follows that { } ( ) ␯FEcell ° (T2 ) ␯FEcell ° (T1 ) T2 T1 r S ° and therefore that r S ° { } ␯F Ecell ° (T2 ) Ecell ° (T1 ) T2 T1 (5.7b) In other words, r S ° is proportional to the slope of a graph of a plot of the standard cell potential against temperature. The standard reaction entropy r S ° often reflects the change in solvation accompanying a redox reaction: for each half-cell reaction a positive entropy contribution is expected when the corresponding reduction results in a decrease in electric charge (solvent molecules are less tightly bound and more disordered). Conversely, a negative contribution is expected when there is an increase in charge. As discussed in Section 5.3, entropy contributions to standard potentials are usually very similar when comparing redox couples involving the same change in charge. E X A MPL E 5 . 4 The potential generated by a fuel cell Calculate the cell potential (measured using a load of such high resistance that negligible current flows) produced by a fuel cell in which the overall reaction is 2 H2(g) O2(g) → 2 H2O(l) with H2 and O2 each at 1 bar and 25C. (Note that in a working PEM fuel cell the temperature is usually 80100C to improve performance.) Table 5.3 The relationship between K and E ° E °/ V K 2 1034 1 1017 0 1 1 1017 2 1034 156 5 Oxidation and reduction Answer We note that under zero-current conditions, the cell potential is given by the difference of standard potentials of the two redox couples. For the reaction as stated, we write Right : O2 (g) 4H (aq) 4 e → 2H2O(l) E ° 1.23 V E ° 0 Left : 2 H (aq) 2 e → H2 (g) Overall(Right Left ) : 2H2 (g) O2 (g) → 2H2O(l) The standard potential of the cell is therefore E cell ° (1.23 V) 0 1.23 V The reaction is spontaneous as written, and the right-hand electrode is the cathode (the site of reduction). Self-test 5.4 What potential difference would be produced in a fuel cell operating with oxygen and hydrogen with both gases at 5.0 bar? Redox stability When assessing the thermodynamic stability of a species in solution, we must bear in mind all possible reactants: the solvent, other solutes, the species itself, and dissolved oxygen. In the following discussion, we focus on the types of reaction that result from the thermodynamic instability of a solute. We also comment briefly on kinetic factors, but the trends they show are generally less systematic than those shown by stabilities. 5.6 The influence of pH Key point: Many redox reactions in aqueous solution involve transfer of H as well as electrons and the electrode potential therefore depends on the pH. For many reactions in aqueous solution the electrode potential varies with pH because reduced species of a redox couple are usually much stronger Brønsted bases than the oxidized species. For a redox couple in which there is transfer of e electrons and H protons, it follows from eqn 5.6b that Ox e e H → RedH H Q [RedH ␯ ] H ␯ + H [Ox ][ H ] and EE° [RedH ␯ ] RT [RedH ␯H ] ␯ H+ RT RT H E ln[ H ] ln ° ln [Ox ] ␯ e F [Ox ][ H + ]␯ H ␯e F ␯e F (We have used ln x ln 10 log x.) If the concentrations of Red and Ox are combined with E ° we define E as E′ E ° RT [RedH ␯H ] ln ␯e F [Ox] and if we use ln [H] ln 10 log [H] with pH log [H], the potential of the electrode can be written E E′ ␯ H+ RT ln10 ␯e F pH (5.8a) At 25C, E E′ (0.059 V)␯ ␯e H+ pH (5.8b) That is, the potential decreases (becoming more negative) as the pH increases and the solution becomes more basic. 157 Redox stability − − ■ A brief illustration. The half-reaction for the perchlorate/chlorate (ClO4 ClO3 ) couple is ClO4 (aq) 2 H(aq) 2 e → ClO3 (aq) H2O(l) − − Therefore whereas at pH 0, E ° 1.201 V, at pH 7 the reduction potential for the ClO4 ClO3 couple is 1.201 (2/2)(7 0.059) V 0.788 V. The perchlorate anion is a stronger oxidant under acid conditions. ■ A note on good practice Always include the sign of a reduction potential, even when it is positive. ■ A brief illustration. To determine the reduction potential of the H/H2 couple at pH 7.0, the 1 other species being present in their standard states, we note that E E ° (H ,H2 ) 0. The reduction halfreaction is 2H(aq) 2e → H2(g), so e 2 and H 2. The biological standard potential is therefore Potential, E/V E ⊕ 0 (2/2)(7 0.059) V 0.41 V ■ 5.7 Reactions with water Water may act as an oxidizing agent, when it is reduced to H2: 0.5 O2/H2O Natural waters 0 pH = 9 1.5 pH = 4 Standard potentials in neutral solution (pH 7) are denoted EW ° . These potentials are particularly useful in biochemical discussions because cell fluids are buffered near pH 7. The condition pH 7 (with unit activity for the other electroactive species present) corresponds to the so-called biological standard state; in biochemical contexts they are sometimes denoted either E⊕ or Em7, the ‘m7’ denoting the ‘midpoint’ potential at pH 7. H 2O(l) e → 12 H 2 (g) OH (aq) For the equivalent reduction of hydronium ions in water at any pH (and partial pressure of H2 of 1 bar) we have seen that the Nernst equation gives H (aq) e → 12 H 2 (g) E 0.059 V pH (5.9) This is the reaction that chemists typically have in mind when they refer to ‘the reduction of water’. Water may also act as a reducing agent, when it is oxidized to O2: 2H 2O(l) → O2 (g) 4 H (aq) 4 e When the partial pressure of O2 is 1 bar, the Nernst equation for the O2,2H/2H2O halfreaction becomes E 1.23 V (0.059 V pH) H+/H2 –0.5 (5.10) because H/e 4/4 1. Both H and O2 therefore have the same pH dependence for their reduction half-reactions. The variation of these two potentials with pH is shown in Fig. 5.3. (a) Oxidation by water Key point: For metals with large, negative standard potentials, reaction with aqueous acids leads to the production of H2 unless a passivating oxide layer is formed. The reaction of a metal with water or aqueous acid is in fact the oxidation of the metal by water or hydrogen ions because the overall reaction is one of the following processes (and their analogues for more highly charged metal ions): M(s) H 2O(l) → M (aq) 12 H 2 (g) OH (aq) M(s) H (aq) → M (aq) 12 H 2 (g) These reactions are thermodynamically favourable when M is an s-block metal, a 3d-series metal from Group 3 to at least Group 8 or 9 and beyond (Ti, V, Cr, Mn, Ni), or a lanthanoid. An example from Group 3 is 2 Sc(s) 6 H (aq) → 2 Sc3 (aq) 3 H 2 (g) When the standard potential for the reduction of a metal ion to the metal is negative, the metal should undergo oxidation in 1 m acid with the evolution of hydrogen. Although the reactions of magnesium and aluminium with moist air are spontaneous, both metals can be used for years in the presence of water and oxygen. They survive because they –1 0 4 8 pH 12 Fig. 5.3 The variation of the reduction potentials of water with pH. The sloping lines defining the upper and lower limits of thermodynamic water stability are the potentials for the O2 /H2O and H/H2 couples, respectively. The central zone represents the stability range of natural waters. 158 5 Oxidation and reduction are passivated, or protected against reaction, by an impervious film of oxide. Magnesium oxide and aluminium oxide both form a protective skin on the parent metal beneath. A similar passivation occurs with iron, copper, and zinc. The process of ‘anodizing’ a metal, in which the metal is made an anode in an electrolytic cell, is one in which partial oxidation produces a smooth, hard passivating film on its surface. Anodizing is especially effective for the protection of aluminium by the formation of an inert, cohesive, and impenetrable Al2O3 layer. Production of H2 by electrolysis or photolysis of water is widely viewed as one of the renewable energy solutions for the future and is discussed in more detail in Chapter 10. (b) Reduction by water Key point: Water can act as a reducing agent, that is be oxidized by other species. The strongly positive potential of the O2, H/H2O couple (eqn 5.10) shows that acidified water is a poor reducing agent except towards strong oxidizing agents. An example of the latter is Co3(aq), for which E ° (Co3, Co2) 1.92 V. It is reduced by water with the evolution of O2 and Co3 does not survive in aqueous solution: 4 Co3(aq) 2 H2O(l) → 4 Co2(aq) O2(g) 4 H(aq) Ecell ° 0.69 V Because H ions are produced in the reaction, lower acidity (higher pH) favours the oxidation; lowering the concentration of H ions encourages the formation of the products. Only a few oxidizing agents (Ag2 is another example) can oxidize water rapidly enough to give appreciable rates of O2 evolution. Standard potentials greater than 1.23 V occur for several redox couples that are regularly used in aqueous solution, including Ce4/Ce3 (E ° 1.76 V), the acidified dichromate ion couple Cr2O72/Cr3 (E ° 1.38 V), and the acidified permanganate couple MnO4/Mn2 (E ° 1.51 V). The origin of the barrier to reaction is a kinetic one, stemming from the need to transfer four electrons and to form an OO bond. Given that the rates of redox reactions are often controlled by the slow rate at which an OO bond can be formed, it remains a challenge for inorganic chemists to find good catalysts for O2 evolution. The importance of this process is not due to any economic demand for O2 but because of the desire to generate H2 (a ‘green’ fuel) from water by electrolysis or photolysis. Some progress has been made using Co, Ru, and Ir complexes. Existing catalysts include the relatively poorly understood coatings that are used in the anodes of cells for the commercial electrolysis of water. They also include the enzyme system found in the O2 evolution apparatus of the plant photosynthetic centre. This system is based on a special cofactor containing four Mn atoms and one Ca atom (Section 27.10). Although Nature is elegant and efficient, it is also complex, and the photosynthetic process is only slowly being elucidated by biochemists and bioinorganic chemists. (c) The stability field of water Key point: The stability field of water shows the region of pH and reduction potential where couples are neither oxidized by nor reduce hydrogen ions. A reducing agent that can reduce water to H2 rapidly, or an oxidizing agent that can oxidize water to O2 rapidly, cannot survive in aqueous solution. The stability field of water, which is shown in Fig. 5.3, is the range of values of potential and pH for which water is thermodynamically stable towards both oxidation and reduction. The upper and lower boundaries of the stability field are identified by finding the dependence of E on pH for the relevant half-reactions. As we have seen above, both oxidation (to O2) and reduction of water have the same pH dependence (a slope of 0.059 V when E is plotted against pH at 25C) and the stability field is confined within the boundaries of a pair of parallel lines of that slope. Any species with a potential more negative than that given in eqn 5.9 can reduce water (specifically, can reduce H) with the production of H2; hence the lower line defines the low-potential boundary of the stability field. Similarly, any species with a potential more positive than that given in eqn 5.10 can liberate O2 from water and the upper line gives the high-potential boundary. Couples that are thermodynamically unstable in water lie outside (above or below) the limits defined by the sloping lines in Fig. 5.3: species that are oxidized by water have potentials lying below the H2 production line and species that are reduced by water have potentials lying above the O2 production line. Redox stability The stability field in ‘natural’ water is represented by the addition of two vertical lines at pH 4 and pH 9, which mark the limits on pH that are commonly found in lakes and streams. A diagram like that shown in the illustration is known as a Pourbaix diagram and is widely used in environmental chemistry, as we shall see in Section 5.14. 5.8 Oxidation by atmospheric oxygen Key point: The oxygen present in air and dissolved in water can oxidize metals and metal ions in solution. The possibility of reaction between the solutes and dissolved O2 must be considered when a solution is contained in an open beaker or is otherwise exposed to air. As an example, consider an aqueous solution containing Fe2 in contact with an inert atmosphere such as N2. Because E ° (Fe3, Fe2) 0.77 V, which lies within the stability field of water, we expect Fe2 to survive in water. Moreover, we can also infer that the oxidation of metallic iron by H(aq) should not proceed beyond Fe(II) because further oxidation to Fe(III) is unfavourable (by 0.77 V) under standard conditions. However, the picture changes considerably in the presence of O2. In nature, many elements are found as oxidized species, either as soluble oxoanions such as SO42, NO3, and MoO42 or as ores such as Fe2O3. In fact, Fe(III) is the most common form of iron in the Earth’s crust, and most iron in sediments that have been deposited from aqueous environments is present as Fe(III). The reaction 4 Fe2 (aq) O2 (g) 4 H (aq) → 4 Fe3 (aq) 2 H 2O(l) is the difference of the following two half-reactions: O2(g) 4 H(aq) 4 e → 2 H2O E ° 1.23 V Fe3(aq) e → Fe2(aq) E ° 0.77 V which implies that Ecell ° 0.46 V at pH 0. The oxidation of Fe2(aq) by O2 is therefore spontaneous (in the sense K 1) at pH 0 and also at higher pH, although Fe(III) aqua species are hydrolysed and are precipitated as ‘rust’ (Section 5.13). E X A MPL E 5 . 5 Judging the importance of atmospheric oxidation The oxidation of copper roofs to a green substance (typically ‘basic copper carbonate’) is an example of atmospheric oxidation in a damp environment. Estimate the potential for oxidation of copper by oxygen in acid-to-neutral aqueous solution. Cu2(aq) is not deprotonated between pH 0 and 7, so we may assume no hydrogen ions are involved in the half-reaction. Answer We need to consider the reaction between Cu metal and atmospheric O2 in terms of the two relevant reduction half-reactions: O2(g) 4 H(aq) 4 e → 2 H2O Cu2(aq) 2 e → Cu(s) E 1.23 V (0.059 V) pH E ° 0.34 V The difference is Ecell 0.89 V (0.059 V) pH Therefore, Ecell 0.89 V at pH 0 and 0.48 V at pH 7, so atmospheric oxidation by the reaction 2 Cu(s) O2(g) 4 H(aq) → 2 Cu2(aq) 2 H2O(l) has K 1 in both neutral and acid environments. Nevertheless, copper roofs do last for more than a few minutes: their familiar green surface is a passive layer of an almost impenetrable hydrated copper(II) carbonate, sulfate, or, near the sea, chloride. These compounds are formed from oxidation in the presence of atmospheric CO2, SO2, or salt water and the anion is also involved in the redox chemistry. Self-test 5.5 The standard potential for the conversion of sulfate ions, SO2 , to SO2(aq) by the reaction 4 SO2 (aq) 4 H(aq) 2 e → SO2(aq) 2 H2O(l) is 0.16 V. What is the thermodynamically expected 4 fate of SO2 emitted into fog or clouds? 159 160 5 Oxidation and reduction 5.9 Disproportionation and comproportionation Key point: Standard potentials can be used to define the inherent stability and instability of different oxidation states in terms of disproportionation and comproportionation. Because E ° (Cu, Cu) 0.52 V and E ° (Cu2, Cu) 0.16 V, and both potentials lie within the stability field of water, Cu ions neither oxidize nor reduce water. Nevertheless, Cu(I) is not stable in aqueous solution because it can undergo disproportionation, a redox reaction in which the oxidation number of an element is simultaneously raised and lowered. In other words, the element undergoing disproportionation serves as its own oxidizing and reducing agent: 2 Cu(aq) → Cu2(aq) Cu(s) This reaction is the difference of the following two half-reactions: Cu(aq) e → Cu(s) E ° 0.52 V Cu (aq) e → Cu (aq) E ° 0.16 V 2 Because Ecell ° 0.52 V 0.16 V 0.36 V for the disproportionation reaction, K 1.3 × 106 at 298 K, so the reaction is highly favourable. Hypochlorous acid also undergoes disproportionation: 5 HClO(aq) → 2 Cl2(g) ClO3(aq) 2 H2O(l) H(aq) This redox reaction is the difference of the following two half-reactions: 4 HClO(aq) 4 H + (aq) 4 e → 2 Cl2 (g) 4 H 2O(l) E ° 1.63 V ClO3 (aq) 5 H (aq) 4 e → HClO(aq) 2 H 2O(l) E ° 1.43 V So overall Ecell ° 1.63 V 1.43 V 0.20 V, and K 3 × 1013 at 298 K. E X A M PL E 5 .6 Assessing the likelihood of disproportionation Show that Mn(VI) is unstable with respect to disproportionation into Mn(VII) and Mn(II) in acidic aqueous solution. Answer To answer this question we need to consider the two half-reactions, one an oxidation, the other a reduction, that involve the species Mn(VI). The overall reaction (noting, from Pauling’s rules, Section 4.5, that the Mn(VI) oxoanion MnO2 should be protonated at pH 0) 4 5HMnO4 (aq) 3 H (aq) → 4 MnO4 (aq) Mn2 (aq) 4 H2O(l) is the difference of the following two half-reactions HMnO4 (aq) 7H (aq) 4 e → Mn2 (aq) 4 H2O(l) E ° 1.63 V 4 MnO4 (aq) 4 H (aq) 4 e → 4 HMnO4 (aq) E ° 0.90 V The difference of the standard potentials is 0.73 V, so the disproportionation is essentially complete (K 1050 at 298 K). A practical consequence of the disproportionation is that high concentrations of Mn(VI) ions cannot be obtained in acidic solution; they can, however, be obtained in basic solution, as we see in Section 5.12. Self-test 5.6 The standard potentials for the couples Fe2/Fe and Fe3/Fe2 are 0.41 V and 0.77 V, respectively. Should we expect Fe2 to disproportionate in aqueous solution? In comproportionation, the reverse of disproportionation, two species with the same element in different oxidation states form a product in which the element is in an intermediate oxidation state. An example is Ag 2 (aq) Ag(s) → 2 Ag (aq) Ecell ° 1.18 V The large positive potential indicates that Ag(II) and Ag(0) are completely converted to Ag(I) in aqueous solution (K 1 × 1020 at 298 K). Redox stability 161 5.10 The influence of complexation Key points: The formation of a more thermodynamically stable complex when the metal is in the higher oxidation state of a couple favours oxidation and makes the standard potential more negative; the formation of a more stable complex when the metal is in the lower oxidation state of the couple favours reduction and the standard potential becomes more positive. The formation of metal complexes (see Chapter 7) affects standard potentials because the ability of a complex (ML) formed by coordination of a ligand (L) to accept or release an electron differs from that of the corresponding aqua ion (M). M(aq) e → M(1)(aq) E ° (M) ML(aq) e → ML(1)(aq) E ° (ML) The change in standard potential for the ML redox couple relative to that of M reflects the degree to which the ligand L coordinates more strongly to the oxidized or reduced form of M. The change in standard potential is analysed by considering the thermodynamic cycle shown in Fig. 5.4. Because the sum of reaction Gibbs energies round the cycle is zero, we can write FE ° (M) RT ln K ox FE ° (ML) RT ln K red 0 ox (5.11) red (1) , respectively where K and K are equilibrium constants for L binding to M and M (of the form K [ML]/[M][L]), and we have used ∆ rG ° RT ln K in each case. This expression rearranges to E ° (M) E ° (ML) RT K ox ln red F K (5.12a) L – = –RT ln Kred ∆rG°(2) ML(ν–1)+(aq) – = –FE°(M) – ∆rG°(1) – = +FE°(ML) – ∆rG°(3) Mν+(aq) + e– – = ∆rG°(4) +RT ln Kox At 25C and with ln x ln 10 log x K ox E ° (M) E ° (ML) (0.059 V ) log red K M(ν–1)+(aq) L MLν+(aq) + e– (5.12b) Thus, every ten-fold increase in the equilibrium constant for ligand binding to M compared to M(1) decreases the reduction potential by 0.059 V. Fig. 5.4 Thermodynamic cycle showing how the standard potential of the couple M/M is altered by the presence of a ligand L. ■ A brief illustration. The standard potential for the half-reaction [Fe(CN)6]3(aq) e → [Fe(CN)6]4(aq) is 0.36 V; that is, 0.41 V more negative than that of the aqua redox couple [Fe(OH2)6]3(aq) e → [Fe(OH2)6]2(aq). This equates to CN having a 107-fold greater affinity (in the sense K ox ≈ 107K red) for Fe(III) compared to Fe(II). ■ E X A MPL E 5 .7 Interpreting potential data to identify bonding trends in complexes Ruthenium is located immediately below iron in the periodic table. The following reduction potentials have been measured for species of Ru in aqueous solution. What do these values suggest when compared to their Fe counterparts ? [Ru(OH2 )6 ]3 e → [Ru(OH2 )6 ]2 E ° 0.25 V [Ru(CN)6 ]3 e → [Ru(CN)6 ]4 E ° 0.85 V Answer We can answer this question by noting that if complexation by a certain ligand causes the reduction potential of a metal ion to shift in a positive direction, then the new ligand must be stabilizing the reduced metal ion. In this case we see that CN stabilizes Ru(II) with respect to Ru(III). This behaviour is in stark contrast to the behaviour of Fe (see the preceding brief illustration) where we noted that CN stabilizes Fe(III), a result more in keeping with FeCN bonds being more ionic. The contrasting effects for species having identical charges suggest that the bonding between CN and Ru(II) is particularly strong. Self-test 5.7 The ligand bpy (1) forms complexes with Fe(III) and Fe(II). The standard potential of the [Fe(bpy)3]3/[Fe(bpy)3]2 couple is 1.02 V. Does bpy bind preferentially to Fe(III) or Fe(II)? N N 1 2,2´ -bipyridine (bpy) 162 5 Oxidation and reduction 5.11 The relation between solubility and standard potentials Key points: The standard cell potential can be used to determine the solubility product. The solubility of sparingly soluble compounds is expressed by an equilibrium constant known as the solubility product, Ksp . The approach is analogous to that introduced above for relating complexation equilibria to standard potentials. For metal ions M forming a precipitate MX with anions X, M(aq) X(aq) MX(s) Ksp [M][X] (5.13) To generate the overall (non-redox) solubility reaction we use the difference of the two reduction half-reactions M(aq) e → M(s) E ° (M ␯ /M) MX(s) e → M(s) X(aq) E ° (MX ␯ / M,X) From which it follows that ln Ksp ␯F {E ° (MX/M,X ) E ° (M ␯ /M)} RT (5.14) E X A M PL E 5 . 8 Determining a solubility product from standard potentials The possibility of plutonium waste leaking from nuclear facilities is a serious environmental problem. Calculate the solubility product of Pu(OH)4 based on the following potentials measured in acid or basic solution. Hence, comment on the consequences of Pu(IV) waste leaking into environments of low pH as compared to high pH. Pu4(aq) 4 e− → Pu(s) E ° 1.28 V Pu(OH)4(s) 4 e− → Pu(s) 4 OH(aq) E 2.06 V at pH 14 Answer We need to consider a thermodynamic cycle that combines the changes in Gibbs energy for the electrode reactions at pH 0 and 14 using the potentials given, and the standard Gibbs energy for the reaction Pu4(aq) with OH(aq). The solubility product for Pu(OH)4 is Ksp [Pu4][OH]4 , so the corresponding Gibbs energy term is RT ln Ksp. For the thermodynamic cycle ∆G 0, so we obtain RT ln Ksp 4FE ° (Pu4/Pu) 4FE ° (Pu(OH)4 /Pu) and therefore lnK sp 4F {(2.06 V ) (1.28 V )} RT It follows that Ksp 1.7 1053. Self-test 5.8 Given that the standard potential for the Ag/Ag couple is 0.80 V, calculate the potential of the AgCl/Ag,Cl couple under conditions of [Cl] 1.0 mol dm3, given that Ksp 1.77 1010. The diagrammatic presentation of potential data There are several useful diagrammatic summaries of the relative stabilities of different oxidation states in aqueous solution. Latimer diagrams are useful for summarizing quantitative data for individual elements. Frost diagrams are useful for the qualitative portrayal of the relative and inherent stabilities of oxidation states of a range of elements. We use them frequently in this context in the following chapters to convey the sense of trends in the redox properties of the members of a group. 5.12 Latimer diagrams In a Latimer diagram (also known as a reduction potential diagram) for an element, the numerical value of the standard potential (in volts) is written over a horizontal line (or arrow) connecting species with the element in different oxidation states. The most highly oxidized form of the element is on the left, and in species to the right the element is in successively lower oxidation states. The diagrammatic presentation of potential data A Latimer diagram summarizes a great deal of information in a compact form and (as we explain) shows the relationships between the various species in a particularly clear manner. (a) Construction Key points: In a Latimer diagram, oxidation numbers decrease from left to right and the numerical values of E ° in volts are written above the line joining the species involved in the couple. The Latimer diagram for chlorine in acidic solution, for instance, is 1.67 1.36 1.20 1.18 1.65 ClO4 ⎯⎯⎯ → ClO3 ⎯⎯⎯ → HClO2 ⎯⎯⎯ ⎯→ HClO ⎯⎯⎯ → Cl2 ⎯⎯⎯ → Cl 7 3 5 1 0 1 As in this example, oxidation numbers are sometimes written under (or over) the species. Conversion of a Latimer diagram to a half-reaction often involves balancing elements by including the predominant species present in acidic aqueous solution (H and H2O). The procedure for balancing redox equations was shown in Section 5.1. The standard state for this couple includes the condition that pH 0. For example: the notation .67 HClO ⎯1 ⎯⎯ → Cl2 denotes 2 HClO(aq) 2 H (aq) 2 e → Cl2 (g) 2 H 2O(l) E ° 1.67 V Similarly, 1.20 ClO4 ⎯⎯⎯ → ClO3 denotes ClO4 (aq) 2 H (aq) 2 e → ClO3 (aq) H 2O(l) E ° 1.20 0V Note that both of these half-reactions involve hydrogen ions, and therefore the potentials depend on pH. In basic aqueous solution (corresponding to pOH 0 and therefore pH 14), the Latimer diagram for chlorine is 0.42 1.36 0.37 0.30 0.68 → ClO3 ⎯⎯⎯ → ClO2 ⎯⎯⎯ ⎯→ ClO ⎯⎯⎯ → Cl2 ⎯⎯⎯ → Cl ClO4 ⎯⎯⎯ 7 5 3 1 0 1 Note that the value for the Cl2/Cl couple is the same as in acidic solution because its halfreaction does not involve the transfer of protons. (b) Nonadjacent species Key points: The standard potential of a couple that is the combination of two other couples is obtained by combining the standard Gibbs energies, not the standard potentials, of the half-reactions. The Latimer diagram given above includes the standard potential for two nonadjacent species (the couple ClO/Cl). This information is redundant in the sense that it can be inferred from the data on adjacent species, but it is often included for commonly used couples as a convenience. To derive the standard potential of a nonadjacent couple when it is not listed explicitly we cannot in general just add their standard potentials but must make use of eqn 5.2 (∆r G ° ␯FE ° ) and the fact that the overall ∆r G ° for two successive steps a and b is the sum of the individual values: ∆r G ° (a b) ∆r G ° (a) ∆r G ° (b) To find the standard potential of the composite process, we convert the individual E ° values to ∆r G ° by multiplication by the relevant factor F, add them together, and then convert the sum back to E ° for the nonadjacent couple by division by F for the overall electron transfer: ␯FE °(a b) ␯(a)FE °(a) ␯(b)FE °(b) Because the factors F cancel and (a) (b), the net result is E °(a b) ␯(a)E °(a) ␯(b)E °(b) ␯(a) ␯(b) (5.15) 163 164 5 Oxidation and reduction ■ A brief example. To use the Latimer diagram to calculate the value of E ° for the ClO2/Cl2 couple in basic aqueous solution we note the following two standard potentials: ClO2 (aq) 2H (aq) 2e → ClO (aq) H2O(l) ClO (aq) e → Cl2 (aq) E ° (a ) 0.6 68 V E ° (b ) 0.42 V 1 2 Their sum, ClO2 (aq) 2 H(aq) 3 e → 1 2 Cl2(g) H2O(l) is the half-reaction for the couple we require. We see that (a) 2 and (b) 1. It follows from eqn 5.15 that the standard potential of the ClO/Cl couple is E ° (2)(0.68 V ) (1)(0.42 V ) 0.59 V ■ 3 (c) Disproportionation Key point: A species has a tendency to disproportionate into its two neighbours if the potential on the right of the species in a Latimer diagram is higher than that on the left. Consider the disproportionation 2 M (aq) → M(s) M 2 (aq) This reaction has K 1 if E ° 0. To analyse this criterion in terms of a Latimer diagram, we express the overall reaction as the difference of two half-reactions: M (aq) e → M(s) E °(R) M 2 (aq) e → M (aq) E °(L) The designations L and R refer to the relative positions, left and right respectively, of the couples in a Latimer diagram (recall that the more highly oxidized species lies to the left). The standard potential for the overall reaction is E ° E ° (R) E ° (L), which is positive if E ° (R) E ° (L). We can conclude that a species is inherently unstable (that is, it has a tendency to disproportionate into its two neighbours) if the potential on the right of the species is higher than the potential on the left. E X A M PL E 5 . 9 Identifying a tendency to disproportionate A part of the Latimer diagram for oxygen is 0.70 1.76 O2 ⎯⎯⎯ → H2O2 ⎯⎯⎯ → H2O Does hydrogen peroxide have a tendency to disproportionate in acid solution? Answer We can approach this question by reasoning that if H2O2 is a stronger oxidant than O2, then it should react with itself to produce O2 by oxidation and 2 H2O by reduction. The potential to the right of H2O2 is higher than that to its left, so we anticipate that H2O2 should disproportionate into its two neighbours under acid conditions. From the two half-reactions 2 H (aq) 2 e H2O2 (aq) → 2 H2O E ° 1.76 V O2 2 H (aq) 2 e → H2O2 (aq) E ° 0.70 V we conclude that for the overall reaction 2 H2O2 (aq) → 2 H2O(l) O2 (g) E ° 1.06 V NE° Increasing stability Most stable oxidation state and is spontaneous (in the sense K 1). Self-test 5.9 Use the following Latimer diagram (acid solution) to discuss whether (a) Pu(IV) disproportionates to Pu(III) and Pu(V) in aqueous solution; (b) Pu(V) disproportionates to Pu(VI) and Pu(IV). 1.02 1.04 1.01 PuO22 ⎯⎯⎯ → PuO2 ⎯⎯⎯ → Pu4 ⎯⎯⎯ ⎯ → Pu3+ 6 5 4 +3 5.13 Frost diagrams Oxidation number, N Fig. 5.5 Oxidation state stability as viewed in a Frost diagram. A Frost diagram (also known as an oxidation state diagram) of an element X is a plot of NE ° for the couple X(N)/X(0) against the oxidation number, N, of the element. The general form of a Frost diagram is given in Fig. 5.5. Frost diagrams depict whether a particular species The diagrammatic presentation of potential data 7 NO3– 6 N2O4 5 HNO2 NE / V 4 NH2OH 3 NH3 2 NO Fig. 5.6 The Frost diagram for nitrogen: the steeper the slope of a line, the higher the standard potential for the couple. The red line refers to standard (acid) conditions (pH 0), the blue line refers to pH 14. Note that because HNO3 is a strong acid, even at pH 0 it is present as its conjugate base NO3. N2H4 N2O4 N 2O NH3OH NO 1 N2H N2 NH4+ –2 NO3– NO2– N2O 5 0 –1 –3 165 –1 0 +1 +2 +3 +4 +5 Oxidation number, N X(N) is a good oxidizing agent or reducing agent. They also provide an important guide for identifying the oxidation states of an element that are inherently stable or unstable. (a) Gibbs energies of formation for different oxidation states Key point: A Frost diagram shows how the Gibbs energies of formation of different oxidation states of an element vary with oxidation number. The most stable oxidation state of an element corresponds to the species that lies lowest in its Frost diagram. For a half-reaction in which a species X with oxidation number N is converted to its elemental form, the reduction half-reaction is written X(N) N e → X(0) +1 HO2– 0 NE/ V Because NE ° is proportional to the standard reaction Gibbs energy for the conversion of the species X(N) to the element (explicitly, NE ° ∆r G ° /F, where ∆r G ° is the standard reaction Gibbs energy for the half-reaction given above), a Frost diagram can also be regarded as a plot of standard reaction Gibbs energy against oxidation number. Consequently, the most stable states of an element in aqueous solution correspond to species that lie lowest in its Frost diagram. The example given in Fig. 5.6 shows data for nitrogen species formed in aqueous solution at pH 0 and pH 14. Only NH4(aq) is exergonic (∆f G ° 0); all other species are endergonic (∆f G ° 0). The diagram shows that the higher oxides and oxoacids are highly endergonic in acid solution but relatively stabilized in basic solution. The opposite is generally true for species with N 0, except that hydroxylamine is particularly unstable regardless of pH. –1 Answer We begin by placing the element in its zero oxidation state (O2) at the origin for the NE ° and N axes. For the reduction of O2 to H2O2 (for which N 1), E ° 0.70 V, so NE ° 0.70 V. Because the oxidation number of O in H2O is 2 and E ° for the O2 /H2O couple is 1.23 V, NE ° at N 2 E ° 2.46 V. These results are plotted in Fig. 5.7. Self-test 5.10 Construct a Frost diagram from the Latimer diagram for Tl: 1.25 0.34 Tl3 ⎯⎯⎯ → Tl ⎯⎯⎯ → Tl (b) Construction and interpretation Key point: Frost diagrams are conveniently constructed by using electrode potential data. They may be used to gauge the inherent stabilities of different oxidation states of an element and to decide whether particular species are good oxidizing or reducing agents. H2O2 –2 E X A MPL E 5 .10 Constructing a Frost diagram Construct a Frost diagram for oxygen from the Latimer diagram in Example 5.9. OH– O2 –3 –2 H2O –1 Oxidation number, N Fig. 5.7 The Frost diagram for oxygen in acidic solution (red line, pH 0) and alkaline solution (blue line, pH 14). 0 166 5 Oxidation and reduction N‘E °’ To interpret the qualitative information contained in a Frost diagram it will be useful to keep the following features in mind. NE ° The slope of the line joining any two points in a Frost diagram is equal to the standard potential of the couple formed by the two species that the points represent (Fig. 5.8). It follows that the steeper the line joining two points (left to right) in a Frost diagram, the higher the standard potential of the corresponding couple (Fig. 5.9a). ■ A brief illustration. Refer to the oxygen diagram in Fig. 5.7. At the point corresponding to N 1 N‘’E °’’ (for H2O2), (1) E ° 0.70 V, and at N 2 (for H2O), (2) E ° 2.46 V. The difference of the two values is 1.76 V. The change in oxidation number of oxygen on going from H2O2 to H2O is 1. Therefore, the slope of the line is (1.76 V) /(1) 1.76 V, in accord with the value for the H2O2 /H2O couple in the Latimer diagram. ■ N‘’ N‘ Oxidation number, N Fig. 5.8 The general structure of a region of a Frost diagram used to establish the relationship between the slope of a line and the standard potential of the corresponding couple. The oxidizing agent in the couple with the more positive slope (the more positive E ° ) is liable to undergo reduction (Fig. 5.9b). 3. The reducing agent of the couple with the less positive slope (the most negative E ° ) is liable to undergo oxidation (Fig. 5.9b). For example, the steep slope connecting NO3 to lower oxidation numbers in Fig. 5.6 shows that nitrate is a good oxidizing agent under standard conditions. We saw in the discussion of Latimer diagrams that a species is liable to undergo disproportionation if the potential for its reduction from X(N) to X(N 1) is greater than its potential for oxidation from X(N) to X(N 1). The same criterion can be expressed in terms of a Frost diagram (Fig. 5.9c): 4. A species in a Frost diagram is unstable with respect to disproportionation if its point lies above the line connecting the two adjacent species (on a convex curve). When this criterion is satisfied, the standard potential for the couple to the left of the species is greater than that for the species on the right. A specific example is NH2OH; as can be seen in Fig. 5.6, this compound is unstable with respect to disproportionation into NH3 and N2. The origin of this rule is illustrated in Fig. 5.9d, where we show geometrically that the reaction Gibbs energy of a species with intermediate oxidation number lies above the average value for the two species on either side. As a result, there is a tendency for the intermediate species to disproportionate into the two other species. The criterion for comproportionation to be spontaneous can be stated analogously (Fig. 5.9e): 5. Two species will tend to comproportionate into an intermediate species that lies below the straight line joining the terminal species (on a concave curve). Oxidized Tends to disproportionate NE° Lower standard potential Reduced NE° NE° Higher standard potential (a) (b) (c) Oxidation number, N Oxidation number, N Oxidation number, N Fig. 5.9 The interpretation of a Frost diagram to gauge (a) reduction potential, (b) tendency towards oxidation and reduction, (c, d) disproportionation, and (e, f) comproportionation. (d) Oxidation number, N NE° NE° ∆G < 0 NE° Tend to comproportionate (e) Oxidation number, N Mean ∆G < 0 (f) Oxidation number, N The diagrammatic presentation of potential data 167 A substance that lies below the line connecting its neighbours in a Frost diagram is inherently more stable than they are because their average molar Gibbs energy is higher (Fig. 5.9f) and hence comproportionation is thermodynamically favourable. The nitrogen in NH4NO3, for instance, has two ions with oxidation numbers 3 (NH4) and 5 (NO3). Because N2O lies below the line joining NH4 to NO3, their comproportionation is spontaneous: NH4 (aq) NO3 (aq) → N 2O(g) 2 H 2O(l) However, although the reaction is expected to be spontaneous on thermodynamic grounds under standard conditions, the reaction is kinetically inhibited in solution and does not ordinarily occur. The corresponding reaction NH 4 NO3 (s) → N 2O(g) 2 H 2O(g) in the solid state is both thermodynamically spontaneous (∆r G ° 168 kJ mol1) and, once initiated by a detonation, explosively fast. Indeed, ammonium nitrate is often used in place of dynamite for blasting rocks. Frost diagrams can equally well be constructed for other conditions. The potentials at pH 14 are denoted EB ° and the blue line in Fig. 5.6 is a ‘basic Frost diagram’ for nitrogen. The important difference from the behaviour in acidic solution is the stabilization of NO2 against disproportionation: its point in the basic Frost diagram no longer lies above the line connecting its neighbours. The practical outcome is that metal nitrites are stable in neutral and basic solutions and can be isolated, whereas HNO2 cannot (although solutions of HNO2 have some short-term stability as their decomposition is kinetically slow). In some cases there are marked differences between strongly acidic and basic solutions, as for the phosphorus oxoanions. This example illustrates an important general point about oxoanions: when their reduction requires removal of oxygen, the reaction consumes H ions, and all oxoanions are stronger oxidizing agents in acidic than in basic solution. E X A MPL E 5 .11 Using a Frost diagram to judge the thermodynamic stability of ions in solution Figure 5.10 shows the Frost diagram for manganese. Comment on the stability of Mn3 in acidic aqueous solution. Answer We approach this question by inspecting how the NE ° value for Mn3 (N 3) compares with the values for species on either side (N 3, N 3)). Because Mn3 lies above the line joining Mn2 to MnO2, it should disproportionate into these two species. The chemical reaction is 2 Mn3 (aq) 2 H2O(l) → Mn2 (aq) MnO2 (s) 4 H (aq) Self-test 5.13 What is the oxidation number of Mn in the product when MnO4 is used as an oxidizing agent in aqueous acid? +6 MnO–4 +5 HMnO4– +4 NE °/ V +3 MnO3– +2 +1 Mn 0 Mn3+ MnO2 –1 Mn2+ –2 –3 0 +1 +2 +3 +5 +4 Oxidation number, N +6 +7 Fig. 5.10 The Frost diagram for manganese in acidic solution (pH 0). Note that because HMnO3 and HMnO4 are strong acids, even at pH 0 they are present as their conjugate bases. 168 5 Oxidation and reduction Modified Frost diagrams summarize potential data under specified conditions of pH; their interpretation is the same as for pH 0, but oxoanions often display markedly different thermodynamic stabilities; all oxoanions are stronger oxidizing agents in acidic than in basic solution. E X A M PL E 5 .12 Application of Frost diagrams at different pH Potassium nitrite is stable in basic solution but, when the solution is acidified, a gas is evolved that turns brown on exposure to air. What is the reaction? Answer To answer this we use the Frost diagram (Fig. 5.6) to compare the inherent stabilities of N(III) in acid and basic solutions. The point representing the NO2 ion in basic solution lies below the line joining NO to NO3 ; the ion therefore is not liable to disproportionation. On acidification, the HNO2 point rises and the straightness of the line through NO, HNO2, and N2O4 (dimeric NO2) implies that all three species are present at equilibrium. The brown gas is NO2 formed from the reaction of NO evolved from the solution with air. In solution, the species of oxidation number 2 (NO) tends to disproportionate. However, the escape of NO from the solution prevents its disproportionation to N2O and HNO2. Self-test 5.12 By reference to Fig. 5.6, compare the strength of NO3 as an oxidizing agent in acidic and basic solution. +0.8 5.14 Pourbaix diagrams Fe3+ O2/H2O E/ V +0.4 Fe(OH)3(s) Fe2+ 0 –0.4 H2O/H2 Fe (O H) 2 –0.8 0 2 4 6 8 10 pH (s) 12 14 Fig. 5.11 A simplified Pourbaix diagram for some important naturally occurring aquaspecies of iron. Key points: A Pourbaix diagram is a map of the conditions of potential and pH under which species are stable in water. A horizontal line separates species related by electron transfer only, a vertical line separates species related by proton transfer only, and sloped lines separate species related by both electron and proton transfer. A Pourbaix diagram (also known as an EpH diagram) indicates the conditions of pH and potential under which a species is thermodynamically stable. The diagrams were introduced by Marcel Pourbaix in 1938 as a convenient way of discussing the chemical properties of species in natural waters and they are particularly useful in environmental and corrosion science. Iron is essential for almost all life forms and the problem of its uptake from the environment is discussed further in Chapter 27. Figure 5.11 is a simplified Pourbaix diagram for iron, omitting such low concentration species as oxygen-bridged Fe(III) dimers. This diagram is useful for the discussion of iron species in natural waters (see Section 5.15) because the total iron concentration is low; at high concentrations complex multinuclear iron species can form. We can see how the diagram has been constructed by considering some of the reactions involved. The reduction half-reaction Fe3 (aq) e → Fe2 (aq) E ° 0.77 V does not involve H ions, so its potential is independent of pH and hence corresponds to a horizontal line on the diagram. If the environment contains a couple with a potential above this line (a more positive, oxidizing couple), then the oxidized species, Fe3, will be the major species. Hence, the horizontal line towards the top left of the diagram is a boundary that separates the regions where Fe3 and Fe2 dominate. Another reaction to consider is Fe3(aq) 3H 2O(l) → Fe(OH)3 (s) 3H(aq) This reaction is not a redox reaction (there is no change in oxidation number of any element), so it is insensitive to the electric potential in its environment and therefore is represented by a vertical line on the diagram. However, this boundary does depend on pH, with Fe3(aq) favoured by low pH and Fe(OH)3(s) favoured by high pH. We adopt the convention that Fe3 is the dominant species in the solution if its concentration exceeds 10 μmol dm3 (a typical freshwater value). The equilibrium concentration of Fe3 varies with pH, and the vertical boundary at pH 3 represents the pH at which Fe3 becomes dominant according to this definition. In general, a vertical line in a Pourbaix diagram does not involve a redox reaction but signifies a pH-dependent change of state of either the oxidized or reduced form. As the pH is increased, the Pourbaix diagram includes reactions such as Fe(OH)3 (s) 3H(aq) e → Fe2(aq) 3H 2O(l) Chemical extraction of the elements (for which the slope of potential against pH, according to eqn 5.8b, is H/e 3(0.059 V)) and eventually Fe2(aq) is also precipitated as Fe(OH)2. Inclusion of the metal dissolution couple (Fe2/Fe(s)) would complete construction of the Pourbaix diagram for well-known aqua species of iron. Key point: Pourbaix diagrams show that iron can exist in solution as Fe2 under acidic reducing conditions such as in contact with soils rich in organic matter. Surface water (lake, stream) +0.8 +0.4 Bog water 0 –0.4 –0.8 0 Ocean water Organic-rich lake water Organic-rich waterlogged soils Organic-rich saline water 2 6 4 8 10 pH Fig. 5.12 The stability field of water showing regions typical of various natural waters. +2 MnO2 O2/H2O +1 E/V The chemistry of natural waters can be rationalized by using Pourbaix diagrams of the kind we have just constructed. Thus, where fresh water is in contact with the atmosphere, it is saturated with O2, and many species may be oxidized by this powerful oxidizing agent. More fully reduced forms are found in the absence of oxygen, especially where there is organic matter to act as a reducing agent. The major acid system that controls the pH of the medium is CO2/H2CO3/HCO3/CO32, where atmospheric CO2 provides the acid and dissolved carbonate minerals provide the base. Biological activity is also important because respiration releases CO2. This acidic oxide lowers the pH and hence makes the potential more positive. The reverse process, photosynthesis, consumes CO2, thus raising the pH and making the potential more negative. The condition of typical natural waters—their pH and the potentials of the redox couples they contain—is summarized in Fig. 5.12. From Fig. 5.11 we see that Fe3 can exist in water if the environment is oxidizing; hence, where O2 is plentiful and the pH is low (below 3), iron will be present as Fe3. Because few natural waters are so acidic, Fe3(aq) is very unlikely to be found in the environment. The iron in insoluble Fe2O3 or insoluble hydrated forms such as FeO(OH) can enter solution as Fe2 if it is reduced, which occurs when the condition of the water lies below the sloping boundary in the diagram. We should observe that, as the pH rises, Fe2 can form only if there are strong reducing couples present, and its formation is very unlikely in oxygen-rich water. Figure 5.11 shows that iron will be reduced and dissolved in the form of Fe2 in both bog waters and organic-rich waterlogged soils (at pH near 4.5 in both cases and with corresponding E values near 0.03 V and 0.1 V, respectively). It is instructive to analyse a Pourbaix diagram in conjunction with an understanding of the physical processes that occur in water. As an example, consider a lake where the temperature gradient, cool at the bottom and warmer above, tends to prevent vertical mixing. At the surface, the water is fully oxygenated and the iron must be present in particles of the insoluble FeO(OH); these particles tend to settle. At greater depth, the O2 content is low. If the organic content or other sources of reducing agents are sufficient, the oxide will be reduced and iron will dissolve as Fe2. The Fe(II) ions will then diffuse towards the surface where they encounter O2 and are oxidized to insoluble FeO(OH) again. +1.2 E/ V 5.15 Natural waters 169 E X A MPL E 5 .13 Using a Pourbaix diagram Figure 5.13 is part of a Pourbaix diagram for manganese. Identify the environment in which the solid MnO2 or its corresponding hydrous oxides are important. Is Mn(III) formed under any conditions? Answer We approach this problem by locating the zone of stability for MnO2 on the Pourbaix diagram and inspecting its position relative to the boundary between O2 and H2O. Manganese dioxide is the thermodynamically favoured state in well-oxygenated water under all pH conditions with the exception of strong acid (pH 1). Under mildly reducing conditions, in waters having neutral-to-acidic pH, the stable species is Mn2(aq). Manganese(III) species are stabilized only in oxygenated waters at higher pH. Self-test 5.13 Use Fig. 5.11 to evaluate the possibility of finding Fe(OH)3(s) in a waterlogged soil. Chemical extraction of the elements The original definition of ‘oxidation’ was a reaction in which an element reacts with oxygen and is converted to an oxide. ‘Reduction’ originally meant the reverse reaction, in which an oxide of a metal is converted to the metal. Although both terms have been generalized and Mn2O3 Mn2+ Mn3O4 0 Mn(OH)2 H2O/H2 0 2 6 4 8 10 pH Fig. 5.13 A section of the Pourbaix diagram for manganese. The broken black vertical lines represent the normal pH range in natural waters. 170 5 Oxidation and reduction expressed in terms of electron transfer and changes in oxidation state, these special cases are still the basis of a major part of chemical industry and laboratory chemistry. In the following sections we discuss the extraction of the elements in terms of changing their oxidation number from its value in a naturally occurring compound to zero (corresponding to the element). 5.16 Chemical reduction Only a few metals, such as gold, occur in nature as their elements. Most metals are found as their oxides, such as Fe2O3, or as ternary compounds, such as FeTiO3. Sulfides are also common, particularly in mineral veins where deposition occurred under water-free and oxygen-poor conditions. Slowly, prehistoric humans learned how to transform ores to produce metals for making tools and weapons. Copper could be extracted from its ores by aerial oxidation at temperatures attainable in the primitive hearths that became available about 6000 years ago. 2Cu2 S(s) 3O2 (g) → 2Cu2O(s) 2 SO2 (g) 2Cu2O(s) Cu2 S(s) → 6Cu(s) SO2 (g) It was not until nearly 3000 years ago that higher temperatures could be reached and less readily reduced elements, such as iron, could be extracted, leading to the Iron Age. These elements were produced by heating the ore to its molten state with a reducing agent such as carbon. This process is known as smelting. Carbon remained the dominant reducing agent until the end of the nineteenth century, and metals that needed higher temperatures for their production remained unavailable even though their ores were reasonably abundant. The availability of electric power expanded the scope of carbon reduction because electric furnaces can reach much higher temperatures than carbon-combustion furnaces, such as the blast furnace. Thus, magnesium is a metal of the twentieth century because one of its modes of recovery, the Pidgeon process, involves the very high temperature, electrothermal reduction of the oxide by carbon: → Mg(l) CO(g) MgO(s) C(s) ⎯⎯ Note that the carbon is oxidized only to carbon monoxide, the product favoured thermodynamically at the very high reaction temperatures used. The technological breakthrough in the nineteenth century that resulted in the conversion of aluminium from a rarity into a major construction metal was the introduction of electrolysis, the driving of a nonspontaneous reaction (including the reduction of ores) by the passage of an electric current. (a) Thermodynamic aspects Key points: An Ellingham diagram summarizes the temperature dependence of the standard Gibbs energies of formation of metal oxides and is used to identify the temperature at which reduction by carbon or carbon monoxide becomes spontaneous. As we have seen, the standard reaction Gibbs energy, ∆r G ° , is related to the equilibrium constant, K, through ∆r G ° RT ln K, and a negative value of ∆r G ° corresponds to K 1. It should be noted that equilibrium is rarely attained in commercial processes as many such systems involve dynamic stages where, for example, reactants and products are in contact only for short times. Furthermore, even a process at equilibrium for which K 1 can be viable if the product (particularly a gas) is swept out of the reaction chamber and the reaction continues to chase the ever-vanishing equilibrium composition. In principle, we also need to consider rates when judging whether a reaction is feasible in practice, but reactions are often fast at high temperature and thermodynamically favourable reactions are likely to occur. A fluid phase (typically a gas or solvent) is usually required to facilitate what would otherwise be a sluggish reaction between coarse particles. To achieve a negative ∆rG ° for the reduction of a metal oxide with carbon or carbon monoxide, one of the following reactions (a) C(s) 12 O2 (g) → CO (g) ∆ rG ° (C,CO) (b) C(s) O2 (g) → CO2 (g) ∆ rG ° (C,CO2 ) (c) CO(g) 12 O2 (g) → CO2 (g) ∆ rG ° (CO,CO2 ) 1 2 1 2 1 2 Chemical extraction of the elements 171 must have a more negative ∆rG ° than a reaction of the form (d) x M(s or l) 12 O2 (g) → M xO ( s ) (a d) M xO(s) C(s) → x M(s or l) CO(g) ∆ rG ° (C,CO) ∆ rG ° (M, M xO) (b d) M xO(s) C(s) → x M(s or l) CO2 (g) ∆ rG ° (C,CO2 ) ∆ rG ° (M, M xO) (c d) M xO(s) CO(g) → x M(s or l) CO2 (g) ∆ rG ° (CO,CO2 ) ∆ rG ° (M, M xO) 1 2 1 2 will have a negative standard reaction Gibbs energy, and therefore have K 1. The procedure followed here is similar to that adopted with half-reactions in aqueous solution (Section 5.1), but now all the reactions are written as oxidations with 12 O2 in place of e, and the overall reaction is the difference of reactions with matching numbers of oxygen atoms. The relevant information is commonly summarized in an Ellingham diagram (Fig. 5.14), which is a graph of ∆rG ° against temperature. We can understand the appearance of an Ellingham diagram by noting that ∆r G ° ∆r H ° T∆r S ° and using the fact that the enthalpy and entropy of reaction are, to a reasonable approximation, independent of temperature. That being so, the slope of a line in an Ellingham diagram should therefore be equal to ∆r S ° for the relevant reaction. Because the standard molar entropies of gases are much larger than those of solids, the reaction entropy of (a), in which there is a net formation of gas (because 1 mol CO replaces 12 mol O2), is positive, and its line therefore has a negative slope. The standard reaction entropy of (b) is close to zero as there is no net change in the amount of gas, so its line is horizontal. Reaction (c) has a negative reaction entropy because 32 mol of gas molecules is replaced by 1 mol CO2; hence the line in the diagram has a positive slope. The standard reaction entropy of (d), in which there is a net consumption of gas, is negative, and hence the plot has a positive slope (Fig. 5.15). The kinks in the lines, where the slope of the metal oxidation line changes, are where the metal undergoes a phase change, particularly melting, and the reaction entropy changes accordingly. At temperatures for which the C/CO line (a) lies above the metal oxide line (d), ∆r G ° (M, MxO) is more negative than ∆r G ° (C, CO). At these temperatures, ∆r G ° (C, CO) ∆r G ° (M, MxO) is positive, so the reaction (a d) has K 1. However, for temperatures for which the C/CO line lies below the metal oxide line, the reduction of the metal oxide by carbon has K 1. Similar remarks apply to the temperatures at which the other two carbon oxidation lines (b) and (c) lie above or below the metal oxide lines. In summary: r For temperatures at which the C/CO line lies below the metal oxide line, carbon can be used to reduce the metal oxide and itself is oxidized to carbon monoxide. r For temperatures at which the C/CO2 line lies below the metal oxide line, carbon can be used to achieve the reduction, but is oxidized to carbon dioxide. r For temperatures at which the CO/CO2 line lies below the metal oxide line, carbon monoxide can reduce the metal oxide to the metal and is oxidized to carbon dioxide. Figure 5.16 shows an Ellingham diagram for a selection of common metals. In principle, production of all the metals shown in the diagram, even magnesium and calcium, could be accomplished by pyrometallurgy, heating with a reducing agent. However, there are severe practical limitations. Efforts to produce aluminium by pyrometallurgy (most notably in Japan, where electricity is expensive) were frustrated by the volatility of Al2O3 at the very high temperatures required. A difficulty of a different kind is encountered in the pyrometallurgical extraction of titanium, where titanium carbide, TiC, is formed instead of the metal. In practice, pyrometallurgical extraction of metals is confined principally to magnesium, iron, cobalt, nickel, zinc, and a variety of ferroalloys (alloys with iron). E X A MPL E 5 .14 Using an Ellingham diagram What is the lowest temperature at which ZnO can be reduced to zinc metal by carbon? What is the overall reaction at this temperature? Answer To answer this question we examine the Ellingham diagram in Fig. 5.16 and estimate the temperature at which the ZnO line crosses the C, CO line. The C, CO line lies below the ZnO line at approximately ∆rG (M, MxO) ∆rG (C, CO) CO C can reduce MxO to M Temperature Fig. 5.14 The variation of the standard reaction Gibbs energies for the formation of a metal oxide and carbon monoxide with temperature. The formation of carbon monoxide from carbon can reduce the metal oxide to the metal at temperatures higher than the point of intersection of the two lines. More specifically, at the intersection the equilibrium constant changes from K 1 to K 1. This type of display is an example of an Ellingham diagram. ∆rG (CO, CO2) Reaction Gibbs energy under the same reaction conditions. If that is so, then one of the reactions Reaction Gibbs energy ∆ rG ° (M, M xO) (a) (b) ∆rG (C, CO2) (c) ∆rG (C, CO) (d) ∆rG (M, MxO) Temperature Fig. 5.15 Part of an Ellingham diagram showing the standard Gibbs energy for the formation of a metal oxide and the three-carbon oxidation Gibbs energies. The slopes of the lines are determined largely by whether or not there is net gas formation or consumption in the reaction. A phase change generally results in a kink in the graph (because the entropy of the substance changes). 5 Oxidation and reduction 1200C; above this temperature reduction of the metal oxide is spontaneous. The contributing reactions are reaction (a) and the reverse of Zn(g) 21 O2 (g) → ZnO(s) so the overall reaction is the difference, or C(s) ZnO(s) → CO(g) Zn(g) The physical state of zinc is given as a gas because the element boils at 907C (the corresponding inflection in the ZnO line in the Ellingham diagram can be seen in Fig. 5.16). Self-test 5.14 What is the minimum temperature for reduction of MgO by carbon? Similar principles apply to reductions using other reducing agents. For instance, an Ellingham diagram can be used to explore whether a metal M can be used to reduce the oxide of another metal M. In this case, we note from the diagram whether at a temperature of interest the M/MO line lies below the M/MO line, as M is now taking the place of C. When ∆ rG ° ∆ rG ° (M ′, M ′O) ∆ rG ° (M, MO) is negative, where the Gibbs energies refer to the reactions (a) M ′(s or l) 12 O2 (g) → M ′O(s) ∆ rG ° (M ′, M ′O) (b) M(s or l) 12 O2 (g) → MO(s) ∆ rG ° (M, MO) the reaction (a b) MO(s) M′(s or l) → M(s or l) M′O(s) and its analogues for MO2, and so on) is feasible (in the sense K 1). For example, because in Fig. 5.16 the line for MgO lies below the line for SiO2 at temperatures below 2400C, magnesium may be used to reduce SiO2 below that temperature. This reaction has in fact been used to produce low-grade silicon as discussed in the following section. +100 Ag2O CuO 0 (c) –100 FeO ZnO –1 ∆rG °/(kJ mol ) 172 (b) –200 1 2 SiO2 –300 (a) 1 2 TiO2 –400 1 3 Al2O3 MgO –500 CaO 0 500 1000 1500 Temperature, 2000 /°C 2500 Fig. 5.16 An Ellingham diagram for the reduction of metal oxides. Chemical extraction of the elements 173 (b) Survey of processes Key points: A blast furnace produces the conditions required to reduce iron oxides with carbon; electrolysis may be used to bring about a nonspontaneous reduction as required for the extraction of aluminium from its oxide. Industrial processes for achieving the reductive extraction of metals show a greater variety than the thermodynamic analysis might suggest. An important factor is that the ore and carbon are both solids, and a reaction between two solids is rarely fast. Most processes exploit gas/solid or liquid/solid heterogeneous reactions. Current industrial processes are varied in the strategies they adopt to ensure economical rates, exploit materials, and avoid environmental problems. We can explore these strategies by considering three important examples that reflect low, moderate, and extreme difficulty of reduction. The least difficult reductions include those of copper ores. Roasting and smelting are still widely used in the pyrometallurgical extraction of copper. However, some recent techniques seek to avoid the major environmental problems caused by the production of the large quantity of SO2, released to the atmosphere, that accompanies roasting. One promising development is the hydrometallurgical extraction of copper, the extraction of a metal by reduction of aqueous solutions of its ions, using H2 or scrap iron as the reducing agent. In this process, Cu2 ions, leached from low-grade ores by acid or bacterial action, are reduced by hydrogen in the reaction or by a similar reduction using iron. This process is less harmful to the environment provided the acid by-product is used or neutralized locally rather than contributing to acidic atmospheric pollutants. It also allows economic exploitation of lower grade ores. Cu2 (aq) H 2 (g) → Cu(s) 2 H (aq) That extraction of iron is of intermediate difficulty is shown by the fact that the Iron Age followed the Bronze Age. In economic terms, iron ore reduction is the most important application of carbon pyrometallurgy. In a blast furnace (Fig. 5.17), which is still the major source of the element, the mixture of iron ores (Fe2O3, Fe3O4), coke (C), and limestone (CaCO3) is heated with a blast of hot air. Combustion of coke in this blast raises the temperature to 2000C, and the carbon burns to carbon monoxide in the lower part of the furnace. The supply of Fe2O3 from the top of the furnace meets the hot CO rising from below. The iron(III) oxide is reduced, first to Fe3O4 and then to FeO at 500700C, and the CO is oxidized to CO2. The final reduction to iron, from FeO, by carbon monoxide occurs between 1000 and 1200C in the central region of the furnace. Thus overall Ore, coke, limestone Fe2O3 (s) 3CO(g) → 2 Fe(l) 3CO2 (g) The function of the lime, CaO, formed by the thermal decomposition of calcium carbonate is to combine with the silicates present in the ore to form a molten layer of calcium silicates (slag) in the hottest (lowest) part of the furnace. Slag is less dense than iron and can be drained away. The iron formed melts at about 400C below the melting point of the pure metal on account of the dissolved carbon it contains. The impure iron, the densest phase, settles to the bottom and is drawn off to solidify into ‘pig iron’, in which the carbon content is high (about 4 per cent by mass). The manufacture of steel is then a series of reactions in which the carbon content is reduced and other metals are used to form alloys with the iron (see Box 3.1). More difficult than the extraction of either copper or iron is the extraction of silicon from its oxide: indeed, silicon is very much an element of the twentieth century. Silicon of 96 to 99 per cent purity is prepared by reduction of quartzite or sand (SiO2) with high purity coke. The Ellingham diagram (Fig. 5.16) shows that the reduction is feasible only at temperatures in excess of about 1700C. This high temperature is achieved in an electric arc furnace in the presence of excess silica (to prevent the accumulation of SiC): SiO2 (l) 2C(s) ⎯1700ºC ⎯⎯⎯ → Si(l) 2CO(g) 2 SiC(s) SiO2 (l) → 3Si(l) 2CO(g) Very pure silicon (for semiconductors) is made by converting crude silicon to volatile compounds, such as SiCl4. These compounds are purified by exhaustive fractional distillation and then reduced to silicon with pure hydrogen. The resulting semiconductor-grade silicon is melted and large single crystals are pulled slowly from the cooled surface of a melt: this procedure is called the Czochralski process. 200°C 3 Fe2O3 + CO 2 Fe 3O4 + CO2 CaCO3 CaO + CO2 700°C Fe3O4 +CO 3 FeO + CO2 1000°C C + CO2 2 CO FeO + CO Fe + CO2 1200°C CaO + SiO2 CaSiO3 Fe(s) Fe(l) Slag forms 2000°C P(V), S(VI) reduced 2 C + O2 2 CO Air Air Slag Iron Fig. 5.17 A schematic diagram of a blast furnace showing the typical composition and temperature profile. 174 5 Oxidation and reduction As we have remarked, the Ellingham diagram shows that the direct reduction of Al2O3 with carbon becomes feasible only above 2400C, which makes it uneconomically expensive and wasteful in terms of any fossil fuels used to heat the system. However, the reduction can be brought about electrolytically (Section 5.18). 5.17 Chemical oxidation Key point: Elements obtained by chemical oxidation include the heavier halogens, sulfur, and (in the course of their purification) certain noble metals. As oxygen is available from fractional distillation of air, chemical methods for its production are not necessary. Sulfur is an interesting mixed case. Elemental sulfur is either mined or produced by oxidation of the H2S that is removed from ‘sour’ natural gas and crude oil. The oxidation is accomplished by the Claus process, which consists of two stages. In the first, some hydrogen sulfide is oxidized to sulfur dioxide: 2 H 2 S(g) 3O2 (g) → 2 SO2 (g) 2 H 2O(l) In the second stage, this sulfur dioxide is allowed to react in the presence of a catalyst with more hydrogen sulfide: Oxide catalyst,300ºC 2 H 2 S(g) SO2 (g) ⎯⎯⎯⎯⎯⎯⎯ → 3S(s) 2 H 2O(l) The catalyst is typically Fe2O3 or Al2O3. The Claus process is environmentally benign; otherwise it would be necessary to burn the toxic hydrogen sulfide to polluting sulfur dioxide. The only important metals extracted in a process using an oxidation stage are the ones that occur in native form (that is, as the element). Gold is an example because it is difficult to separate the granules of metal in low-grade ores by simple ‘panning’. The dissolution of gold depends on oxidation, which is favoured by complexation with CN ions: Au(s) 2CN (aq) → [ Au (CN )2 ] (aq) + e This complex is then reduced to the metal by reaction with another reactive metal, such as zinc: 2[Au (CN )2 ] (aq) Zn(s) → 2 Au(s) + [ Zn(CN)4 ]2 (aq) However, because of the toxicity of cyanide, alternative methods of extracting gold have been used, involving processing by bacteria. The lighter, strongly oxidizing halogens are extracted electrochemically, as described in Section 5.18. The more readily oxidizable halogens, Br2 and I2, are obtained by chemical oxidation of the aqueous halides with chlorine. For example, 2 NaBr(aq) Cl2 (g) → 2 NaCl(aq) Br2 (l) 5.18 Electrochemical extraction Key points: Elements obtained by electrochemical reduction include aluminium; those obtained by electrochemical oxidation include chlorine. The extraction of metals from ores electrochemically is confined mainly to the more electropositive elements, as discussed in the case of aluminium later in this section. For other metals produced in bulk quantities, such as iron and copper, the more energy-efficient and cleaner routes used by industry in practice, using chemical methods of reduction, were described in Section 5.16b. In some specialist cases, electrochemical reduction is used to isolate small quantities of platinum group metals. So, for example, treatment of spent catalytic converters with acids under oxidizing conditions produces a solution containing complexes of Pt(II) and other platinum group metals, which can then be reduced electrochemically. The metals are deposited at the cathode with an overall 80 per cent efficient extraction from the ceramic catalytic converter. As we saw in Section 5.16, an Ellingham diagram shows that the reduction of Al2O3 with carbon becomes feasible only above 2400C, which is uneconomical. However, the reduction can be brought about electrolytically, and all modern production uses the electrochemical HallHéroult process, which was invented in 1886 Further Reading 175 independently by Charles Hall and Paul Héroult. The process requires pure aluminium hydroxide that is extracted from aluminium ores using the Bayer process. In this process the bauxite ore used as a source of aluminium is a mixture of the acidic oxide SiO2 and amphoteric oxides and hydroxides, such as Al2O3, AlOOH, and Fe2O3. The Al2O3 is dissolved in hot aqueous sodium hydroxide, which separates the aluminium from much of the less soluble Fe2O3, although silicates are also rendered soluble in these strongly basic conditions. Cooling the sodium aluminate solution results in the precipitation of Al(OH)3, leaving the silicates in solution. For the final stage, in the HallHéroult process, the aluminium hydroxide is dissolved in molten cryolite (Na3AlF6) and the melt is reduced electrolytically at a steel cathode with graphite anodes. The latter participate in the electrochemical reaction by reacting with the evolved oxygen atoms so that the overall process is 2 Al2O3 (s) 3C(s) → 4 Al(s) 3CO2 (g) As the power consumption of a typical plant is huge, aluminium is often produced where electricity is cheap (for example, from hydroelectric sources in Canada) and not where bauxite is mined (in Jamaica, for example). The lighter halogens are the most important elements extracted by electrochemical oxidation. The standard reaction Gibbs energy for the oxidation of Cl ions in water 2Cl (aq) 2 H 2O(l) → 2OH (aq) H 2 (g) Cl2 (g) ∆ rG ° 422 kJ mol1 is strongly positive, which suggests that electrolysis is required. The minimum potential difference that can achieve the oxidation of Cl is about 2.2 V (from ∆ rG ° ␯FE ° and 2) It may appear that there is a problem with the competing reaction 2 H 2O(l) → 2 H 2 (g) O2 (g) ∆ rG ° 414 kJ mol1 which can be driven forwards by a potential difference of only 1.2 V (in this reaction, 4). However, the rate of oxidation of water is very slow at potentials at which it first becomes favourable thermodynamically. This slowness is expressed by saying that the reduction requires a high overpotential, (eta), the potential that must be applied in addition to the equilibrium value before a significant rate of reaction is achieved. Consequently, the electrolysis of brine produces Cl2, H2, and aqueous NaOH, but not much O2. Oxygen, not fluorine, is produced if aqueous solutions of fluorides are electrolysed. Therefore, F2 is prepared by the electrolysis of an anhydrous mixture of potassium fluoride and hydrogen fluoride, an ionic conductor that is molten above 72C. FURTHER READING P. Zanello, Inorganic electrochemistry: theory, practice and applications. Royal Society of Chemistry (2003). An introduction to electrochemical investigations. A.J. Bard, M. Stratmann, F. Scholtz, and C.J. Pickett, Encyclopedia of Electrochemistry: Inorganic Chemistry, Vol. 7b. Wiley (2006). J.-M. Savéant, Elements of molecular and biomolecular electrochemistry: an electrochemical approach to electron-transfer chemistry. Wiley (2006). R.M. Dell and D.A.J. Rand, Understanding batteries. Royal Society of Chemistry (2001). J. Emsley, The elements. Oxford University Press (1998). Excellent source of data on the elements, including standard potentials. A.G. Howard, Aquatic environmental chemistry. Oxford University Press (1998). Discussion of the compositions of freshwater and marine systems explaining the effects of oxidation and reduction processes. M. Pourbaix, Atlas of electrochemical equilibria in aqueous solution. Pergamon Press, Oxford (1966). The original and still good source of Pourbaix diagrams. A.J. Bard, R. Parsons, and R. Jordan, Standard potentials in aqueous solution. M. Dekker, New York (1985). A collection of cell potential data with discussion. W. Stumm and J.J. Morgan, Aquatic chemistry. Wiley, New York (1996). A standard reference on natural water chemistry. I. Barin, Thermochemical data of pure substances, Vols 1 and 2. VCH, Weinheim (1989). A comprehensive source of thermodynamic data for inorganic substances. J. Larminie and A. Dicks, Fuel cell systems explained. Wiley (2003). C. Spiegel, Design and building of fuel cells. McGraw Hill (2007). 176 5 Oxidation and reduction EXERCISES 5.1 Assign oxidation numbers for each of the elements participating in the following reactions. 10 2 NO(g) O2(g) → 2 NO2(g) ClO4– 8 ClO3– 2 Mn3(aq) 2 H2O → MnO2 Mn2 4 H(aq) LiCoO2(s) C(s) → Li@C(s) CoO2(s) Ca(s) H2(g)→ CaH2(s) 5.3 Use standard potential data from Resource section 3 as a guide to write balanced equations for the reactions that each of the following species might undergo in aerated aqueous acid. If the species is stable, write ‘no reaction’. (a) Cr2, (b) Fe2, (c) Cl, (d) HClO, (e) Zn(s). 5.4 Use the information in Resource section 3 to write balanced equations for the reactions, including disproportionations, that can be expected for each of the following species in aerated acidic aqueous solution: (a) Fe2, (b) Ru2, (c) HClO2, (d) Br2. 5.5 Explain why the standard potentials for the half-cell reactions [Ru(NH3 )6 ] (aq) e → [Ru(NH3 )6 ] (aq) – 2 [ Fe(CN)6 ]3 (aq) e → [ Fe(CN)6 ]4 (aq) vary with temperature in opposite directions. 5.6 Balance the following redox reaction in acid solution: MnO4 H2SO3 → Mn2 HSO4. Predict the qualitative pH dependence on the net potential for this reaction (that is, increases, decreases, remains the same). 5.7 Write the Nernst equation for (a) the reduction of O2(g): O2(g) 4 H(aq) 4 e → 2 H2O(l) (b) the reduction of Fe2O3(s): Fe2O3(s) 6H(aq) 6e → 2Fe(s) 3H2O(l) In each case express the formula in terms of pH. What is the potential for the reduction of O2 at pH 7 and p(O2) 0.20 bar (the partial pressure of oxygen in air)? 5.8 Answer the following questions using the Frost diagram in Fig. 5.18. (a) What are the consequences of dissolving Cl2 in aqueous basic solution? (b) What are the consequences of dissolving Cl2 in aqueous acid? (c) Is the failure of HClO3 to disproportionate in aqueous solution a thermodynamic or a kinetic phenomenon? 5.9 Use standard potentials as a guide to write equations for the main net reaction that you would predict in the following experiments: (a) N2O is bubbled into aqueous NaOH solution, (b) zinc metal is added to aqueous sodium triiodide, (c) I2 is added to excess aqueous HClO3. 5.10 Adding NaOH to an aqueous solution containing Ni2 results in precipitation of Ni(OH)2. The standard potential for the Ni2/Ni NE°/V HClO2 5.2 Use data from Resource section 3 to suggest chemical reagents that would be suitable for carrying out the following transformations and write balanced equations for the reactions: (a) oxidation of HCl to chlorine gas, (b) reduction of Cr(III)(aq) to Cr(II)(aq), (c) reduction of Ag to Ag(s), (d) reduction of I2 to I. 3 6 4 ClO4– – ClO3 2 HClO ClO2– ClO– 0 Cl2 Cl– –2 –1 1 3 5 Oxidation number, N 7 Fig. 5.18 A Frost diagram for chlorine. The red line refers to acid conditions (pH 0) and the blue line to pH 14. Note that because HClO3 and HClO4 are strong acids, even at pH 0 they are present as their conjugate bases. couple is 0.26 V and the solubility product Ksp [Ni2][OH]2 1.5 1016. Calculate the electrode potential at pH 14. 5.11 Characterize the condition of acidity or basicity that would most favour the following transformations in aqueous solution: (a) Mn2 → MnO4, (b) ClO4 → ClO3, (c) H2O2 → O2, (d) I2→ 2 I. 5.12 Use the Latimer diagram for chlorine to determine the potential for reduction of ClO4 to Cl2. Write a balanced equation for this halfreaction. 5.13 Calculate the equilibrium constant of the reaction Au(aq) 2 CN(aq) → Au(CN)2 from the standard potentials Au(aq) e → Au(s) E ° 1.69V Au (CN )2 + e → Au(s) + 2CN(aq) E ° 0.6V 5.14 Use Fig. 5.12 to find the approximate potential of an aerated lake at pH 6. With this information and Latimer diagrams from Resource section 3, predict the species at equilibrium for the elements (a) iron, (b) manganese, (c) sulfur. 5.15 Using the following Latimer diagram, which shows the standard potentials for sulfur species in acid solution (pH 0), construct a Frost diagram and calculate the standard potential for the HSO4−/S8(s) couple. 1.96 0.16 0.40 S2O82 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → HSO ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → H 2 SO3 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 4 0.60 0.14 S2O32 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ →S ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → H2S 5.16 Using the following aqueous acid solution reduction potentials E ° (Pd2, Pd) 0.915 V and E ° ([PdCl4]2, Pd) 0.50 V, calculate the equilibrium constant for the reaction Pd2(aq) 4 Cl(aq) [PdCl4]2(aq) in 1m HCl(aq). Problems 177 5.17 Calculate the reduction potential at 25C for the conversion of MnO4 to MnO2(s) in aqueous solution at pH 9.00 and 1 m MnO4(aq) given that E ° (MnO4, MnO2) 1.69 V. 5.20 Explain why water with high concentrations of dissolved carbon dioxide and open to atmospheric oxygen is very corrosive towards iron. 5.18 Draw a Frost diagram for mercury in acid solution, given the following Latimer diagram: + 0.911 + 0.796 Hg 2 + ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → Hg 2 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → Hg 5.21 The species Fe2 and H2S are important at the bottom of a lake where O2 is scarce. If the pH 6, what is the maximum value of E characterizing the environment? 2 Comment on the tendency of any of the species to act as an oxidizing agent, a reducing agent, or to undergo disproportionation. 5.22 The ligand EDTA forms stable complexes with hard acid centres. How will complexation with EDTA affect the reduction of M2 to the metal in the 3d-series? 5.19 From the following Latimer diagram, calculate the value of E ° for the reaction 2 HO2(aq) → O2(g) H2O2(aq). 5.23 In Fig. 5.11, which of the boundaries depend on the choice of Fe2 concentration as 105 mol dm3? 5.24 Consult the Ellingham diagram in Fig. 5.16 and determine if there are any conditions under which aluminium might be expected to reduce MgO. Comment on these conditions. O2 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → HO2 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → H 2 O2 0.125 1.510 Comment on the thermodynamic tendency of HO2 to undergo disproportionation. PROBLEMS 5.1 Use standard potential data to suggest why permanganate (MnO4−) is not a suitable oxidizing agent for the quantitative estimation of Fe2 in the presence of HCl but becomes so if sufficient Mn2 and phosphate ion are added to the solution. (Hint: Phosphate forms complexes with Fe3, thereby stabilizing it.) 5.2 Many of the tabulated data for standard potentials have been determined from thermochemical data rather than direct electrochemical measurements of cell potentials. Carry out a calculation to illustrate this approach for the half-reaction Sc2O3(s) 3 H2O(l) 6 e → 2 Sc(s) 6 OH(aq). 5.7 Discuss how the equilibrium Cu2(aq) Cu(s) 2 Cu(aq) can be shifted by complexation with chloride ions (see J. Malyyszko and M. Kaczor, J. Chem. Educ., 2003, 80, 1048). Sc3(aq) OH(aq) H2O(l) Sc2O3(s) Sc(s) ∆ f H ° /(kJ mol1) 614.2 230.0 285.8 1908.7 0 S m ° /(J K1 mol1) 255.2 10.75 69.91 77.0 34.76 S m ° is the standard molar entropy 5.3 The reduction potential of an ion such as OH can be strongly influenced by the solvent. (a) From the review article by D.T. Sawyer and J.L. Roberts, Acc. Chem. Res., 1988, 21, 469, describe the magnitude of the change in the potential of the OH/OH couple on changing the solvent from water to acetonitrile, CH3CN. (b) Suggest a qualitative interpretation of the difference in solvation of the OH ion in these two solvents. 5.4 Given the following standard potentials in basic solution CrO24 (aq) 4 H 2O(l) 3 e → Cr(OH)3 (s) 5 OH (aq) E ° 0.11 V [u CH N( aq32) ]e ( ) u Cs H N→ q a E ° 0.10 V 5.6 Using Resource sections 1−3, and data for atomization of the elements ∆f H ° 397 kJ mol−1 (Cr) and 664 kJ mol−1 (Mo), construct thermodynamic cycles for the reactions of Cr or Mo with dilute acids and thus consider the importance of metallic bonding in determining the standard reduction potentials for formation of cations from metals. 5.8 In their article ‘Variability of the cell potential of a given chemical reaction’ (J. Chem. Educ., 2004, 81, 84), L.H. Berka and I. Fishtik conclude that E ° for a chemical reaction is not a state function because the half-reactions are arbitrarily chosen and may contain different numbers of transferred electrons. Discuss this objection. 5.9 The standard potentials for phosphorus species in aqueous solution at pH 0 and pH 14 are represented by the following Latimer diagrams. pH 0 0.276 0.499 ⎯ → H3 PO3 ⎯⎯⎯ ⎯ → H3 PO2 H3 PO4 ⎯⎯⎯ 0.508 0.063 ⎯⎯⎯ ⎯ → P ⎯⎯⎯ ⎯ → PH3 pH 14 1.12 1.57 PO34 ⎯⎯⎯ → HPO32 ⎯⎯⎯ → H 2 PO2 2.05 0.89 ⎯⎯⎯ → P ⎯⎯⎯ → PH3 () 2 (3 ) and assuming that a reversible reaction can be established on a suitable catalyst, calculate ∆ rG ° , and K for the reductions of (a) CrO42 and (b) [Cu(NH3)2] in basic solution. Comment on why ∆ rG ° and K are so different between the two cases despite the values of E ° being so similar. 5.5 Explain the significance of reduction potentials in inorganic chemistry, highlighting their applications in investigations of stability, solubility, and reactivity in water. (a) Account for the difference in reduction potentials between pH 0 and pH 14. (b) Construct a single Frost diagram showing both sets of data. (c) Phosphine (PH3) can be prepared by heating phosphorus with aqueous alkali. Discuss the reactions that are feasible and estimate their equilibrium constants. 5.10 Construct an Ellingham diagram for the thermal splitting of water (H2O(g) → H2(g) 12 O2(g)) by using ∆rH ° 260 kJ mol−1; ∆r S ° 60 J K1 mol−1 (you may assume that ∆H ° is independent of temperature). Hence, calculate the temperature at which H2 may be 178 5 Oxidation and reduction obtained by spontaneous decomposition of water (Chem. Rev. 2007, 107, 4048). Comment on the feasibility of producing H2 by this means. 5.11 Enterobactin (Ent) is a special ligand secreted by some bacteria to sequester Fe from the environment (Fe is an essential nutrient for almost all living species, see Chapter 27). The equilibrium constant for formation of [Fe(III)(Ent)] (K 1052) is at least 40 orders of magnitude higher than the equilibrium constant for the corresponding Fe(II) complex. Determine the feasibility of releasing Fe from [Fe(III)(Ent)] by its reduction to Fe(II) under conditions of neutral pH, noting that the strongest common reducing agent normally available to bacteria is H2. 5.12 Standard potentials at 25oC for indium and thallium in aqueous solution (pH 0) are given below. In3(aq) 3 e− → In(s) E ° −0.338 V In(aq) e− → In(s) E ° −0.126 V Tl3(aq) 3 e− → Tl(s) E ° 0.72 V Tl(aq) e− → Tl(s) E ° −0.336 V Use the data to construct a Frost diagram for the two elements and discuss the relative stabilities of the species. 6 Molecular symmetry Symmetry governs the bonding and hence the physical and spectroscopic properties of molecules. In this chapter we explore some of the consequences of molecular symmetry and introduce the systematic arguments of group theory. We shall see that symmetry considerations are essential for constructing molecular orbitals and analysing molecular vibrations. They also enable us to extract information about molecular and electronic structure from spectroscopic data. An introduction to symmetry analysis The systematic treatment of symmetry makes use of a branch of mathematics called group theory. Group theory is a rich and powerful subject, but we shall confine our use of it at this stage to the classification of molecules in terms of their symmetry properties, the construction of molecular orbitals, and the analysis of molecular vibrations and the selection rules that govern their excitation. We shall also see that it is possible to draw some general conclusions about the properties of molecules without doing any calculations at all. Applications of symmetry 6.1 Symmetry operations, elements and point groups 6.2 Character tables 6.3 Polar molecules 6.4 Chiral molecules 6.5 Molecular vibrations The symmetries of molecular orbitals 6.6 Symmetry-adapted linear combinations 6.7 The construction of molecular orbitals An introduction to symmetry analysis That some molecules are ‘more symmetrical’ than others is intuitively obvious. Our aim though, is to define the symmetries of individual molecules precisely, not just intuitively, and to provide a scheme for specifying and reporting these symmetries. It will become clear in later chapters that symmetry analysis is one of the most pervasive techniques in inorganic chemistry. 6.1 Symmetry operations, elements and point groups 6.8 The vibrational analogy Representations 6.9 The reduction of a representation 6.10 Projection operators FURTHER READING EXERCISES PROBLEMS Key points: Symmetry operations are actions that leave the molecule apparently unchanged; each symmetry operation is associated with a symmetry element. The point group of a molecule is identified by noting its symmetry elements and comparing these elements with the elements that define each group. A fundamental concept of the chemical application of group theory is the symmetry operation, an action, such as rotation through a certain angle, that leaves the molecule apparently unchanged. An example is the rotation of an H2O molecule by 180º around the bisector of the HOH angle (Fig. 6.1). Associated with each symmetry operation there is a symmetry element, a point, line, or plane with respect to which the symmetry operation is performed. Table 6.1 lists the most important symmetry operations and their corresponding elements. All these operations leave at least one point unchanged (the centre of the molecule), and hence they are referred to as the operations of point-group symmetry. The identity operation, E, consists of doing nothing to the molecule. Every molecule has at least this operation and some have only this operation, so we need it if we are to classify all molecules according to their symmetry. The rotation of an H2O molecule by 180º around a line bisecting the HOH angle (as in Fig. 6.1) is a symmetry operation, denoted C2. In general, an n-fold rotation is a symmetry operation if the molecule appears unchanged after rotation by 360º/n. The corresponding symmetry element is a line, an n-fold rotation axis, Cn, about which the rotation is performed. There is only one rotation operation associated with a C2 axis (as in H2O) because clockwise and anticlockwise rotations by 180º are identical. The trigonal-pyramidal NH3 C2 180° Figure 6.1 An H2O molecule may be rotated through any angle about the bisector of the HOH bond angle, but only a rotation of 180 (the C2 operation) leaves it apparently unchanged. 180 6 Molecular symmetry Table 6.1 Symmetry operations and symmetry elements 120° Symmetry operation C3 Symmetry element Symbol Identity ‘whole of space’ E Rotation by 360/n n-fold symmetry axis Cn Reflection mirror plane Inversion centre of inversion i Rotation by 360/n followed by reflection in a plane perpendicular to the rotation axis n-fold axis of improper rotation Sn Note the equivalences S1 = and S2 = i. 120° C3 C32 Figure 6.2 A threefold rotation and the corresponding C3 axis in NH3. There are two rotations associated with this axis, one through 120 (C3) and one through 240 (C32). v h d C4 C2´ C2” molecule has a threefold rotation axis, denoted C3, but there are now two operations associated with this axis, one a clockwise rotation by 120º and the other an anticlockwise rotation by 120º (Fig. 6.2). The two operations are denoted C3 and C32 (because two successive clockwise rotations by 120º are equivalent to an anticlockwise rotation by 120º), respectively. The square-planar molecule XeF4 has a fourfold C4 axis, but in addition it also has two pairs of twofold rotation axes that are perpendicular to the C4 axis: one pair (C2) passes through each trans-FXeF unit and the other pair (C2) passes through the bisectors of the FXeF angles (Fig. 6.3). By convention, the highest order rotational axis, which is called the principal axis, defines the z-axis (and is typically drawn vertically). The reflection of an H2O molecule in either of the two planes shown in Fig. 6.4 is a symmetry operation; the corresponding symmetry element, the plane of the mirror, is a mirror plane, . The H2O molecule has two mirror planes that intersect at the bisector of the HOH angle. Because the planes are ‘vertical’, in the sense of containing the rotational (z) axis of the molecule, they are labelled with a subscript v, as in v and v. The XeF4 molecule in Fig. 6.3 has a mirror plane h in the plane of the molecule. The subscript h signifies that the plane is ‘horizontal’ in the sense that the vertical principal rotational axis of the molecule is perpendicular to it. This molecule also has two more sets of two mirror planes that intersect the fourfold axis. The symmetry elements (and the associated operations) are denoted v for the planes that pass through the F atoms and d for the planes that bisect the angle between the F atoms. The d denotes ‘dihedral’ and signifies that the plane bisects the angle between two C2 axes (the FXeF axes). To understand the inversion operation, i, we need to imagine that each atom is projected in a straight line through a single point located at the centre of the molecule and then out to an equal distance on the other side (Fig. 6.5). In an octahedral molecule such as SF6, with the point at the centre of the molecule, diametrically opposite pairs of atoms at the corners of the octahedron are interchanged. The symmetry element, the point through 1 Figure 6.3 Some of the symmetry elements of a square-planar molecule such as XeF4. 2 5 v i 4 3 6 6 4 ' v 3 5 2 Figure 6.4 The two vertical mirror planes v and v in H2O and the corresponding operations. Both planes cut through the C2 axis. 1 Figure 6.5 The inversion operation and the centre of inversion i in SF6. An introduction to symmetry analysis which the projections are made, is called the centre of inversion, i. For SF6, the centre of inversion lies at the nucleus of the S atom. Likewise, the molecule CO2 has an inversion centre at the C nucleus. However, there need not be an atom at the centre of inversion: an N2 molecule has a centre of inversion midway between the two nitrogen nuclei. An H2O molecule does not possess a centre of inversion. No tetrahedral molecule has a centre of inversion. Although an inversion and a twofold rotation may sometimes achieve the same effect, that is not the case in general and the two operations must be distinguished (Fig. 6.6). An improper rotation consists of a rotation of the molecule through a certain angle around an axis followed by a reflection in the plane perpendicular to that axis (Fig. 6.7). The illustration shows a fourfold improper rotation of a CH4 molecule. In this case, the operation consists of a 90º (that is, 360°/4) rotation about an axis bisecting two HCH bond angles, followed by a reflection through a plane perpendicular to the rotation axis. Neither the 90º (C4) operation nor the reflection alone is a symmetry operation for CH4 but their overall effect is a symmetry operation. A fourfold improper rotation is denoted S4. The symmetry element, the improper-rotation axis, Sn (S4 in the example), is the corresponding combination of an n-fold rotational axis and a perpendicular mirror plane. An S1 axis, a rotation through 360º followed by a reflection in the perpendicular plane, is equivalent to a reflection alone, so S1 and h are the same; the symbol h is generally used rather than S1. Similarly, an S2 axis, a rotation through 180º followed by a reflection in the perpendicular plane, is equivalent to an inversion, i (Fig. 6.8); the symbol i is employed rather than S2. E X A M PL E 6 .1 Identifying symmetry elements Identify the symmetry elements in the eclipsed and staggered conformations of an ethane molecule. Answer We need to identify the rotations, reflections, and inversions that leave the molecule apparently unchanged. Don’t forget that the identity is a symmetry operation. By inspection of the molecular models, we see that the eclipsed conformation of a CH3CH3 molecule (1) has the elements E, C3, C2, h, v, and S3. The staggered conformation (2) has the elements E, C3, d, i, and S6. (a) i C2 i (b) C2 Figure 6.6 Care must be taken not to confuse (a) an inversion operation with (b) a twofold rotation. Although the two operations may sometimes appear to have the same effect, that is not the case in general. H Self-test 6.1 Sketch the S4 axis of an NH4 ion. How many of these axes does the ion possess? C C3 (1) Rotate 1 A C3 axis S1 C4 (2) Reflect H σ (a) C (1) Rotate h S2 (b) Figure 6.7 A fourfold axis of improper rotation S4 in the CH4 molecule. S6 (2) Reflect i Figure 6.8 (a) An S1 axis is equivalent to a mirror plane and (b) an S2 axis is equivalent to a centre of inversion. 181 2 An S6 axis 182 6 Molecular symmetry The assignment of a molecule to its point group consists of two steps: 1. Identify the symmetry elements of the molecule. 2. Refer to Table 6.2. Table 6.2 The composition of some common groups Point group Symmetry elements C1 E SiHClBrF C2 E, C2 H2O2 Cs E, NHF2 C2v E, C2, v, v SO2Cl2, H2O C3v E, 2C3, 3 v NH3, PCl3, POCl3 C∞v E, C2, 2C, ∞ v OCS, CO, HCl D2h E, 3C2, i, 3 N2O4, B2H6 D3h E, 2C3, 3C2, h, 2S3, 3 v BF3, PCl5 E, 2C4, C2, 2C 2′ , 2C 2′′ , i, 2S4, h, 2 v, 2 d XeF4, trans-[MA4B2] D∞h E, ∞C2, 2C, i, ∞ v, 2S CO2, H2, C2H2 Td E, 8C3, 3C2, 6S4, 6 d CH4, SiCl4 Oh E, 8C3, 6C2, 6C4, 3C2, i, 6S4, 8S6, 3 h, 6 d SF6 D4h Shape Examples In practice, the shapes in the table give a very good clue to the identity of the group to which the molecule belongs, at least in simple cases. The decision tree in Fig. 6.9 can also be used to assign most common point groups systematically by answering the questions at each decision point. The name of the point group is normally its Schoenflies symbol, such as C2v for a water molecule. E X A M PL E 6 . 2 Identifying the point group of a molecule To what point groups do H2O and XeF4 belong? Answer We need to work through Fig. 6.9. (a) The symmetry elements of H2O are shown in Fig. 6.10. H2O possesses the identity (E), a twofold rotation axis (C2), and two vertical mirror planes ( v and v). The set An introduction to symmetry analysis of elements (E,C2, v, v) corresponds to the group C2v. (b) The symmetry elements of XeF4 are shown in Fig. 6.3. XeF4 possesses the identity (E), a fourfold axis (C4), two pairs of twofold rotation axes that are perpendicular to the principal C4 axis, a horizontal reflection plane h in the plane of the paper, and two sets of two vertical reflection planes, v and d. This set of elements identifies the point group as D4h. 183 C2 Self-test 6.2 Identify the point groups of (a) BF3, a trigonal-planar molecule, and (b) the tetrahedral SO42– ion. v ' C2 v Y C?N n cu le o M Y Y Y D∞h i? N r? a e in L N Y σ ?N h Y Two or N more C n, n > 2? C∞v Y Linear groups Y C5? i? Y σ ?N h Select Cn with N highest n; then is nC2 ⊥ Cn? Y nσ ? N d Y nσ ? N v N v Figure 6.10 The symmetry elements of H2O. The diagram on the right is the view from above and summarizes the diagram on the left. Y S ? N 2n Dnh N Oh i? N Yσ?N h Dnd Dn Cnh Cnv Cs S2n Ih ' v Y Ci C1 O C Cn Td 3 CO2 (D∞h) Cubic groups Figure 6.9 The decision tree for identifying a molecular point group. The symbols of each point refer to the symmetry elements. It is very useful to be able to recognize immediately the point groups of some common molecules. Linear molecules with a centre of symmetry, such as H2, CO2 (3), and HC⬅CH belong to D"h. A molecule that is linear but has no centre of symmetry, such as HCl or OCS (4) belongs to C"v. Tetrahedral (Td) and octahedral (Oh) molecules have more than one principal axis of symmetry (Fig. 6.11): a tetrahedral CH4 molecule, for instance, has four C3 axes, one along each CH bond. The Oh and Td point groups are known as cubic groups because they are closely related to the symmetry of a cube. A closely related group, the icosahedral group, Ih, characteristic of the icosahedron, has 12 fivefold axes (Fig. 6.12). The icosahedral group is important for boron compounds (Section 13.11) and the C60 fullerene molecule (Section 14.6). The distribution of molecules among the various point groups is very uneven. Some of the most common groups for molecules are the low-symmetry groups C1 and Cs. There are many examples of polar molecules in groups C2v (such as SO2) and C3v (such as NH3). There are many linear molecules, which belong to the groups C"v (HCl, OCS) and D"h (Cl2 and CO2), and a number of planar-trigonal molecules, D3h (such as BF3, 5), trigonal-bipyramidal molecules (such as PCl5, 6), which are D3h, and square-planar molecules, D4h (7). So-called ‘octahedral’ molecules with two identical substituents opposite each other, as in (8), are also D4h. The last example shows that the point-group classification of a molecule is more precise than the casual use of the terms ‘octahedral’ or ‘tetrahedral’ that indicate molecular geometry. For instance, a molecule may be called octahedral (that is, it has octahedral geometry) even if it has six different groups attached to the central atom. However, the ‘octahedral molecule’ belongs to the octahedral point group Oh only if all six groups and the lengths of their bonds to the central atom are identical and all angles are 90º. O C S 4 OCS (C∞v) (a) 6.2 Character tables Key point: The systematic analysis of the symmetry properties of molecules is carried out using character tables. We have seen how the symmetry properties of a molecule define its point group and how that point group is labelled by its Schoenflies symbol. Associated with each point group is (b) Figure 6.11 Shapes having cubic symmetry. (a) The tetrahedron, point group Td. (b) The octahedron, point group Oh. 184 6 Molecular symmetry C5 F Cl P B 5 BF3 (D3h) Figure 6.12 The regular icosahedron, point group Ih, and its relation to a cube. Cl 2– Pt 6 PCl5 (D3h) a character table. A character table displays all the symmetry elements of the point group together with a description, as we explain below, of how various objects or mathematical functions transform under the corresponding symmetry operations. A character table is complete: every possible object or mathematical function relating to the molecule belonging to a particular point group must transform like one of the rows in the character table of that point group. The structure of a typical character table is shown in Table 6.3. The entries in the main part of the table are called characters, (chi). Each character shows how an object or mathematical function, such as an atomic orbital, is affected by the corresponding symmetry operation of the group. Thus: 7 [PtCl4]2– (D4h) Y X M 8 trans-[MX4Y2] (D4h) Character Significance 1 the orbital is unchanged –1 the orbital changes sign 0 the orbital undergoes a more complicated change For instance, the rotation of a pz orbital about the z axis leaves it apparently unchanged (hence its character is 1); a reflection of a pz orbital in the xy plane changes its sign (character –1). In some character tables, numbers such as 2 and 3 appear as characters: this feature is explained later. The class of an operation is a specific grouping of symmetry operations of the same geometrical type: the two (clockwise and anticlockwise) threefold rotations about an axis form one class, reflections in a mirror plane form another, and so on. The number of members of each class is shown in the heading of each column of the table, as in 2C3, denoting that there are two members of the class of threefold rotations. All operations of the same class have the same character. Each row of characters corresponds to a particular irreducible representation of the group. An irreducible representation has a technical meaning in group theory but, broadly speaking, it is a fundamental type of symmetry in the group (like the symmetries represented Table 6.3 The components of a character table Name of point group Symmetry operations R arranged by class (E, Cn, etc.) Functions Further functions Symmetry species (#) Characters () Translations and components of dipole moments (x, y, z), of relevance to IR activity; rotations Quadratic functions such as z2, xy, etc., of relevance to Raman activity Schoenflies symbol. Order of group, h An introduction to symmetry analysis 185 by and π orbitals for linear molecules). The label in the first column is the symmetry species (essentially, a label, like and π) of that irreducible representation. The two columns on the right contain examples of functions that exhibit the characteristics of each symmetry species. One column contains functions defined by a single axis, such as translations or p orbitals (x,y,z) or rotations (Rx,Ry,Rz), and the other column contains quadratic functions such as d orbitals (xy, etc). Character tables for a selection of common point groups are given in Resource section 4. E X A M PL E 6 . 3 Identifying the symmetry species of orbitals Identify the symmetry species of the oxygen valence-shell atomic orbitals in an H2O molecule, which has C2v symmetry. Answer The symmetry elements of the H2O molecule are shown in Fig. 6.10 and the character table for C2v is given in Table 6.4. We need to see how the orbitals behave under these symmetry operations. An s orbital on the O atom is unchanged by all four operations, so its characters are (1,1,1,1) and thus it has symmetry species A1. Likewise, the 2pz orbital on the O atom is unchanged by all operations of the point group and is thus totally symmetric under C2v: it therefore has symmetry species A1. The character of the O2px orbital under C2 is –1, which means simply that it changes sign under a twofold rotation. A px orbital also changes sign (and therefore has character –1) when reflected in the yz-plane (v), but is unchanged (character 1) when reflected in the xz-plane ( v). It follows that the characters of an O2px orbital are (1,–1,1,–1) and therefore that its symmetry species is B1. The character of the O2py orbital under C2 is –1, as it is when reflected in the xz-plane ( v). The O2py is unchanged (character 1) when reflected in the yz-plane (v). It follows that the characters of an O2py orbital are (1,–1,–1,1) and therefore that its symmetry species is B2. Self-test 6.3 Identify the symmetry species of all five d orbitals of the central Xe atom in XeF4 (D4h, Fig. 6.3). The letter A used to label a symmetry species in the group C2v means that the function to which it refers is symmetric with respect to rotation about the twofold axis (that is, its character is 1). The label B indicates that the function changes sign under that rotation (the character is –1). The subscript 1 on A1 means that the function to which it refers is also symmetric with respect to reflection in the principal vertical plane (for H2O this is the plane that contains all three atoms). A subscript 2 is used to denote that the function changes sign under this reflection. Now consider the slightly more complex example of NH3, which belongs to the point group C3v (Table 6.5). An NH3 molecule has higher symmetry than H2O. This higher symmetry is apparent by noting the order, h, of the group, the total number of symmetry operations that can be carried out. For H2O, h = 4 and for NH3, h = 6. For highly symmetric molecules, h is large; for example h = 48 for the point group Oh. Inspection of the NH3 molecule (Fig. 6.13) shows that whereas the N2pz orbital is unique (it has A1 symmetry), the N2px and N2py orbitals both belong to the symmetry representation E. In other words, the N2px and N2py orbitals have the same symmetry characteristics, are degenerate, and must be treated together. The characters in the column headed by the identity operation E give the degeneracy of the orbitals: Symmetry label Degeneracy A, B 1 E 2 T 3 Table 6.5 The C3v character table C3v E 2 C3 3 v h=6 A1 1 1 1 z A2 1 1 –1 Rz E 2 –1 0 (x, y) (Rx, Ry) z2 (zx, yz) (x2 – y2, xy) Table 6.4 The C2v character table C2v E C2 v v h = 4 A1 1 1 1 1 z A2 1 1 –1 –1 Rz B1 1 –1 1 –1 x, Ry xy B2 1 –1 –1 1 y, Rx zx, yz x2, y2, z2 186 6 Molecular symmetry Figure 6.13 The nitrogen 2pz orbital in ammonia is symmetric under all operations of the C3v point group and therefore has A1 symmetry. The 2px and 2py orbitals behave identically under all operations (they cannot be distinguished) and are given the symmetry label E. pz – px – py – A1 E Be careful to distinguish the italic E for the operation and the roman E for the label: all operations are italic and all labels are roman. Degenerate irreducible representations also contain zero values for some operations because the character is the sum of the characters for the two or more orbitals of the set, and if one orbital changes sign but the other does not, then the total character is 0. For example, the reflection through the vertical mirror plane containing the y-axis in NH3 results in no change of the py orbital, but an inversion of the px orbital. E X A M PL E 6 . 4 Determining degeneracy Can there be triply degenerate orbitals in BF3? Answer To decide if there can be triply degenerate orbitals in BF3 we note that the point group of the molecule is D3h. Reference to the character table for this group (Resource section 4) shows that, because no character exceeds 2 in the column headed E, the maximum degeneracy is 2. Therefore, none of its orbitals can be triply degenerate. Self-test 6.4 The SF6 molecule is octahedral. What is the maximum possible degree of degeneracy of its orbitals? Applications of symmetry Important applications of symmetry in inorganic chemistry include the construction and labelling of molecular orbitals and the interpretation of spectroscopic data to determine structure. However, there are several simpler applications, one being to use group theory to decide whether a molecule is polar or chiral. In many cases the answer may be obvious and we do not need to use group theory. However, that is not always the case and the following examples illustrate the approach that can be adopted when the result is not obvious. There are two aspects of symmetry. Some properties require a knowledge only of the point group to which a molecule belongs. These properties include its polarity and chirality. Other properties require us to know the detailed structure of the character table. These properties include the classification of molecular vibrations and the identification of their IR and Raman activity. We illustrate both types of application in this section. 6.3 Polar molecules Key point: A molecule cannot be polar if it belongs to any group that includes a centre of inversion, any of the groups D and their derivatives, the cubic groups (T, O), the icosahedral group (I), and their modifications. A polar molecule is a molecule that has a permanent electric dipole moment. A molecule cannot be polar if it has a centre of inversion. Inversion implies that a molecule has matching charge distributions at all diametrically opposite points about a centre, which rules out a dipole moment. For the same reason, a dipole moment cannot lie perpendicular to any mirror plane or axis of rotation that the molecule may possess. For example, a mirror plane demands identical atoms on either side of the plane, so there can be no dipole moment across the plane. Similarly, a symmetry axis implies the presence of identical atoms at points related by the corresponding rotation, which rules out a dipole moment perpendicular to the axis. Applications of symmetry 187 In summary: 1. A molecule cannot be polar if it has a centre of inversion. 2. A molecule cannot have an electric dipole moment perpendicular to any mirror plane. 3. A molecule cannot have an electric dipole moment perpendicular to any axis of rotation. E X A M PL E 6 . 5 Judging whether or not a molecule can be polar Ru The ruthenocene molecule (9) is a pentagonal prism with the Ru atom sandwiched between two C5H5 rings. Can it be polar? Answer We should decide whether the point group is D or cubic because in neither case can it have a permanent electric dipole. Reference to Fig. 6.9 shows that a pentagonal prism belongs to the point group D5h. Therefore, the molecule must be nonpolar. 9 Self-test 6.5 A conformation of the ferrocene molecule that lies 4 kJ mol1 above the lowest energy configuration is a pentagonal antiprism (10). Is it polar? Fe 10 6.4 Chiral molecules H Key point: A molecule cannot be chiral if it possesses an improper rotation axis (Sn). A chiral molecule (from the Greek word for ‘hand’) is a molecule that cannot be superimposed on its own mirror image. An actual hand is chiral in the sense that the mirror image of a left hand is a right hand, and the two hands cannot be superimposed. A chiral molecule and its mirror image partner are called enantiomers (from the Greek word for ‘both parts’). Chiral molecules that do not interconvert rapidly between enantiomeric forms are optically active in the sense that they can rotate the plane of polarized light. Enantiomeric pairs of molecules rotate the plane of polarization of light by equal amounts in opposite directions. A molecule with an improper rotation axis, Sn, cannot be chiral. A mirror plane is an S1 axis of improper rotation and a centre of inversion is equivalent to an S2 axis; therefore, molecules with either a mirror plane or a centre of inversion have axes of improper rotation and cannot be chiral. Groups in which Sn is present include Dnh, Dnd, and some of the cubic groups (specifically, Td and Oh). Therefore, molecules such as CH4 and Ni(CO)4 that belong to the group Td are not chiral. That a ‘tetrahedral’ carbon atom leads to optical activity (as in CHClFBr) should serve as another reminder that group theory is stricter in its terminology than casual conversation. Thus CHClFBr (11) belongs to the group C1, not to the group Td; it has tetrahedral geometry but not tetrahedral symmetry. When judging chirality, it is important to be alert for axes of improper rotation that might not be immediately apparent. Molecules with neither a centre of inversion nor a mirror plane (and hence with no S1 or S2 axes) are usually chiral, but it is important to verify that a higher-order improper-rotation axis is not also present. For instance, the quaternary ammonium ion (12) has neither a mirror plane (S1) nor an inversion centre (S2), but it does have an S4 axis and so it is not chiral. F Br Cl 11 CHClFBr (C1) H CH3 CH2 N 12 [N(CH2CH(CH3)CH(CH3)CH2)2]+ E X A M PL E 6 .6 Judging whether or not a molecule is chiral The complex [Mn(acac)3], where acac denotes the acetylacetonato ligand (CH3COCHCOCH3), has the structure shown as (13). Is it chiral? Answer We begin by identifying the point group in order to judge whether it contains an improper-rotation axis either explicitly or in a disguised form. The chart in Fig. 6.9 shows that the ion belongs to the point group D3, which consists of the elements (E, C3, 3C2) and hence does not contain an Sn axis either explicitly or in a disguised form. The complex ion is chiral and hence, because it is long-lived, optically active. Self-test 6.6 Is the conformation of H2O2 shown in (14) chiral? The molecule can usually rotate freely about the O–O bond: comment on the possibility of observing optically active H2O2. acac Mn 13 [Mn(acac)3] (D3d) 188 6 Molecular symmetry 6.5 Molecular vibrations Key points: If a molecule has a centre of inversion, none of its modes can be both IR and Raman active; a vibrational mode is IR active if it has the same symmetry as a component of the electric dipole vector; a vibrational mode is Raman active if it has the same symmetry as a component of the molecular polarizability. 14 H2O2 t r A knowledge of the symmetry of a molecule can assist and greatly simplify the analysis of infrared (IR) and Raman spectra (Chapter 8). It is convenient to consider two aspects of symmetry. One is the information that can be obtained directly by knowing to which point group a molecule as a whole belongs. The other is the additional information that comes from knowing the symmetry species of each normal mode. All we need to know at this stage is that the absorption of infrared radiation can occur when a vibration results in a change in the electric dipole moment of a molecule; a Raman transition can occur when the polarizability of a molecule changes during a vibration. For a molecule of N atoms there are 3N displacements to consider as the atoms move. For a nonlinear molecule, three of these displacements correspond to translational motion of the molecule as a whole, and three correspond to an overall rotation, leaving 3N – 6 vibrational modes. There is no rotation around the axis if the molecule is linear, so the molecule has only two rotational degrees of freedom instead of three, leaving 3N – 5 vibrational displacements. (a) The exclusion rule t r t r v The three-atom nonlinear molecule H2O has 3 3 – 6 = 3 vibrational modes (Fig. 6.14). It should be intuitively obvious (and can be confirmed by group theory) that all three vibrational displacements lead to a change in the dipole moment. It follows that all three modes of this C2v molecule are IR active. It is much more difficult to judge intuitively whether or not a mode is Raman active because it is hard to know whether a particular distortion of a molecule results in a change of polarizability (although modes that result in a swelling of the molecule such as the symmetric stretch of CO2 are good prospects). This difficulty is partly overcome by the exclusion rule, which is sometimes helpful: If a molecule has a centre of inversion, none of its modes can be both IR and Raman active. (A mode may be inactive in both.) v E X A M PL E 6 .7 Using the exclusion rule v Figure 6.14 An illustration of the counting procedure for displacements of the atoms in a nonlinear molecule. Symmetric stretch Antisymmetric stretch Bend Bend Figure 6.15 The stretches and bends of a CO2 molecule. There are four vibration modes of the linear triatomic CO2 molecule (Fig. 6.15). Which are IR or Raman active? Answer To establish whether or not a stretch is IR active, we need to consider its effect on the dipole moment of the molecule. If we consider the symmetric stretch 1 we can see it leaves the electric dipole moment unchanged at zero and so it is IR inactive: it may therefore be Raman active (and is). In contrast, for the antisymmetric stretch, 3, the C atom moves opposite to that of the two O atoms: as a result, the electric dipole moment changes from zero in the course of the vibration and the mode is IR active. Because the CO2 molecule has a centre of inversion, it follows from the exclusion rule that this mode cannot be Raman active. Both bending modes cause a departure of the dipole moment from zero and are therefore IR active. It follows from the exclusion rule that the two bending modes are Raman inactive. Self-test 6.7 The bending mode of N2O is active in the IR. Can it also be Raman active? (b) Information from the symmetries of normal modes So far, we have remarked that it is often intuitively obvious whether a vibrational mode gives rise to a changing electric dipole and is therefore IR active. When intuition is unreliable, perhaps because the molecule is complex or the mode of vibration is difficult to visualize, a symmetry analysis can be used instead. We shall illustrate the procedure by considering the two square-planar palladium species (15) and (16). The Pt analogues of these species and the distinction between them are of considerable social and practical significance because the cis isomer is used as a chemotherapeutic agent against certain cancers (whereas the trans isomer is therapeutically inactive (Section 27.18). First, we note that the cis isomer (15) has C2v symmetry, whereas the trans isomer (16) is D2h. Both species have bands in the Pd–Cl stretching region between 200 and 400 cm–1. We know immediately from the exclusion rule that the two modes of the trans isomer Applications of symmetry (which has a centre of symmetry) cannot be active in both IR and Raman. However, to decide which modes are IR active and which are Raman active we consider the characters of the modes themselves. It follows from the symmetry properties of dipole moments and polarizabilities (which we do not verify here) that: Cl H3N E v C2 v 1 1 1 1 The symmetry of this vibration is therefore A1. For the antisymmetric stretch, the identity E leaves the displacement vectors unchanged and the same is true of v, which lies in the plane containing the two Cl atoms. However, both C2 and v interchange the two oppositely directed displacement vectors, and so convert the overall displacement into –1 times itself. The characters are therefore E v C2 v 1 1 1 1 The C2v character table identifies the symmetry species of this mode as B2. A similar analysis of the trans isomer, but using the D2h group, results in the labels Ag and B2u for the symmetric and antisymmetric PdCl stretches, respectively. E X A M PL E 6 . 8 Identifying the symmetry species of vibrational displacements The trans isomer in Fig. 6.16 has D2h symmetry. Verify that the symmetry species of the antisymmetric Pd–Cl stretches is B2u. Answer We need to start by considering the effect of the various elements of the group on the displacement vectors of the Cl ligands. The elements of D2h are E, C2(x), C2(y), C2(z), i, (xy), (yz), and (zx). Of these, E, C2(y), (xy), and (yz) leave the displacement vectors unchanged and so have characters 1. The remaining operations reverse the directions of the vectors, so giving characters of –1: E C2(x) C2(y) C2(z) i (xy) (yz) (zx) 1 1 1 1 1 1 1 1 We now compare this set of characters with the D2h character table and establish that the symmetry species is B2u. Self-test 6.8 Confirm that the symmetry species of the symmetric mode of the Pd–Cl stretches in the trans isomer is Ag. As we have remarked, a vibrational mode is IR active if it has the same symmetry species as the displacements x, y, or z. In C2v, z is A1 and y is B2. Both A1 and B2 vibrations of the cis isomer are therefore IR active. In D2h, x, y, and z are B3u, B2u, and B1u, respectively, and only vibrations with these symmetries can be IR active. The antisymmetric Pd–Cl stretch of the trans isomer has symmetry B2u and is IR active. The symmetric Ag mode of the trans isomer is not IR active. Cl NH3 The symmetry species of the vibration must be the same as that of x, y, or z in the character table for the vibration to be IR active and the same as that of a quadratic function, such as xy or x2, for it to be Raman active. Our first task, therefore, is to classify the normal modes according to their symmetry species, and then to identify which of these modes have the same symmetry species as x, etc. and xy, etc. by referring to the final columns of the character table of the molecular point group. Figure 6.16 shows the symmetric (left) and antisymmetric (right) stretches of the Pd–Cl bonds for each isomer, where the NH3 group is treated as a single mass point. To classify them according to their symmetry species in their respective point groups we use an approach similar to the symmetry analysis of molecular orbitals we will use for determining SALCs (Section 6.10). Consider the cis isomer and its point group C2v (Table 6.4). For the symmetric stretch, we see that the pair of displacement vectors representing the vibration is apparently unchanged by each operation of the group. For example, the twofold rotation simply interchanges two equivalent displacement vectors. It follows that the character of each operation is 1: Pd 15 Cl H3N Pd Cl 16 NH3 189 190 6 Molecular symmetry Figure 6.16 The PdCl stretching modes of cis and trans forms of [Pd(Cl)2(NH3)2]. The motion of the Pd atom (which preserves the centre of mass of the molecule) is not shown. z Cl Cl y Pd NH3 A1 NH3 B2 (a) cis z y Ag B2u (b) trans Pd−N Absorption (a) trans To determine the Raman activity, we note that in C2v the quadratic forms xy, etc. transform as A1, A2, B1, and B2 and therefore in the cis isomer the modes of symmetry A1, A2, B1, and B2 are Raman active. In D2h, however, only Ag, B1g, B2g, and B3g are Raman active. The experimental distinction between the cis and trans isomers now emerges. In the PdCl stretching region, the cis (C2v) isomer has two bands in both the Raman and IR spectra. By contrast, the trans (D2h) isomer has one band at a different frequency in each spectrum. The IR spectra of the two isomers are shown in Fig. 6.17. (c) The assignment of molecular symmetry from vibrational spectra (b) cis Pd−Cl 1000 800 600 400 Wavenumber, ~/cm–1 Figure 6.17 The IR spectra of cis (red) and trans (blue) forms of [Pd(Cl)2(NH3)2]. (R. Layton, D.W. Sink, and J.R. Durig, J. Inorg. Nucl. Chem., 1966, 28, 1965.) An important application of vibrational spectra is the identification of molecular symmetry and hence shape and structure. An especially important example arises in metal carbonyls in which CO molecules are bound to a metal atom. Vibrational spectra are especially useful because the CO stretch is responsible for very strong characteristic absorptions between 1850 and 2200 cm–1 (Section 22.5). The set of characters obtained by considering the symmetries of the displacements of atoms are often found not to correspond to any of the rows in the character table. However, because the character table is a complete summary of the symmetry properties of an object, the characters that have been determined must correspond to a sum of two or more of the rows in the table. In such cases we say that the displacements span a reducible representation. Our task is to find the irreducible representations that they span. To do so, we identify the rows in the character table that must be added together to reproduce the set of characters that we have obtained. This process is called reducing a representation. In some cases the reduction is obvious; in others it may be carried out systematically by using a procedure explained in Section 6.9. E X A M PL E 6 . 9 Reducing a representation One of the first metal carbonyls to be characterized was the tetrahedral (Td) molecule Ni(CO)4. The vibrational modes of the molecule that arise from stretching motions of the CO groups are four combinations of the four CO displacement vectors. Which modes are IR or Raman active? The CO displacements of Ni(CO)4 are shown in Fig. 6.18. Answer We need to consider the motion of the four CO displacement vectors and then consult the character table for Td (Table 6.6). Under the E operation all four vectors remain unchanged, under a C3 only one remains the same, under both C2 and S4 none of the vectors remains unchanged, and under d two remain the same. The characters are therefore: 3 C2 6 S4 E 8 C3 6 d 4 1 0 0 2 191 The symmetries of molecular orbitals This set of characters does not correspond to any one symmetry species. However, it does correspond to the sum of the characters of symmetry species A1 and T2: A1 T2 A1 + T2 E 8 C3 3 C2 6 S4 6 d 1 3 4 1 0 1 1 –1 0 1 –1 0 1 1 2 A1 T2 It follows that the CO displacement vectors transform as A1 + T2. By consulting the character table for Td, we see that the combination labelled A1 transforms like x2 + y2 + z2, indicating that it is Raman active but not IR active. By contrast, x, y, and z and the products xy, yz, and zx transform as T2, so the T2 modes are both Raman and IR active. Consequently, a tetrahedral carbonyl molecule is recognized by one IR band and two Raman bands in the CO stretching region. Self-test 6.9 Show that the four CO displacements in the square-planar (D4h) [Pt(CO)4]2+ cation transform as A1g + B1g + Eu. How many bands would you expect in the IR and Raman spectra for the [Pt(CO)4]2+ cation? T2 T2 Figure 6.18 The modes of Ni(CO)4 that correspond to the stretching of CO bonds. Table 6.6 The Td character table Td E 8 C3 3 C2 6 S4 6 d A1 1 1 1 1 1 h = 24 x2 + y2 + z2 A2 1 1 1 –1 –1 E 2 –1 2 0 0 T1 3 0 –1 1 –1 (Rx, Ry, Rz) T2 3 0 –1 –1 1 (x, y, z) (2x2 – y2 – z2, x2 – y2) (xy, yz, zx) The symmetries of molecular orbitals We shall now see in more detail the significance of the labels used for molecular orbitals introduced in Sections 2.7 and 2.8 and gain more insight into their construction. At this stage the discussion will continue to be informal and pictorial, our aim being to give an introduction to group theory but not the details of the calculations involved. The specific objective here is to show how to identify the symmetry label of a molecular orbital from a drawing like those in Resource section 5 and, conversely, to appreciate the significance of a symmetry label. The arguments later in the book are all based on simply ‘reading’ molecular orbital diagrams qualitatively. (a) s A fundamental principle of the MO theory of diatomic molecules (Section 2.8) is that molecular orbitals are constructed from atomic orbitals of the same symmetry. Thus, in a diatomic molecule, an s orbital may have nonzero overlap with another s orbital or with a pz orbital on the second atom (where z is the internuclear direction, Fig. 6.19), but not with a px or py orbital. Formally, whereas the pz orbital of the second atom has the same rotational symmetry as the s orbital of the first atom and the same symmetry with respect to reflection in a mirror plane containing the internuclear axis, the px and py orbitals do not. The restriction that , π, or δ bonds can be formed from atomic orbitals of the same symmetry species stems from the requirement that all components of the molecular orbital must behave identically under transformation if they are to have nonzero overlap. Exactly the same principle applies in polyatomic molecules, where the symmetry considerations may be more complex and require us to use the systematic procedures provided by group theory. The general procedure is to group atomic orbitals, such as the three H1s orbitals of NH3, together to form combinations of a particular symmetry and then to build molecular orbitals by allowing combinations of the same symmetry on different atoms to overlap, such as an N2s orbital and the appropriate combination of the three H1s orbitals. s 6.6 Symmetry-adapted linear combinations Key point: Symmetry-adapted linear combinations of orbitals are combinations of atomic orbitals that conform to the symmetry of a molecule and are used to construct molecular orbitals of a given symmetry species. – pz s (b) + px s – (c) Figure 6.19 An s orbital can overlap (a) an s or (b) a pz orbital on a second atom with constructive interference. (c) An s orbital has zero net overlap with a px or py orbital because the constructive interference between the parts of the atomic orbitals with the same sign exactly matches the destructive interference between the parts with opposite signs. 192 6 Molecular symmetry Specific combinations of atomic orbitals that are used to build molecular orbitals of a given symmetry are called symmetry-adapted linear combinations (SALCs). A collection of commonly encountered SALCs of orbitals is shown in Resource section 5; it is usually simple to identify the symmetry of a combination of orbitals by comparing it with the diagrams provided there. E X A M PL E 6 .10 Identifying the symmetry species of a SALC Identify the symmetry species of the SALCs that may be constructed from the H1s orbitals of NH3. Answer We start by establishing how the set of H1s orbitals transform under the operations of the appropriate symmetry group of the molecule. An NH3 molecule has symmetry C3v and the three H1s orbitals of all remain unchanged under the identity operation E. None of the H1s orbitals remains unchanged under a C3 rotation, and only one remains unchanged under a vertical reflection v. As a set they therefore span a representation with the characters E 3 2 C3 0 3 v 1 We now need to reduce this set of characters, and by inspection we can see they correspond to A1 + E. It follows that the three H1s orbitals contribute two SALCs, one with A1 symmetry and the other with E symmetry. The SALC with E symmetry has two members of the same energy. In more complicated examples the reduction might not be obvious and we use the systematic procedure discussed in Section 6.10. Self-test 6.10 What is the symmetry label of the SALC = Als + Bls + Cls + Dls in CH4, where Jls is an Hls orbital on atom J? The generation of SALCs of a given symmetry is a task for group theory, as we explain in Section 6.10. However, they often have an intuitively obvious form. For instance, the fully symmetrical A1 SALC of the H1s orbitals of NH3 (Fig. 6.20) is + 1 = A1s + B1s+ C1s To verify that this SALC is indeed of symmetry A1 we note that it remains unchanged under the identity E, each C3 rotation, and any of the three vertical reflections, so its characters are (1,1,1) and hence it spans the fully symmetrical irreducible representation of C3v. The E SALCs are less obvious, but as we shall see are A1 + 2 = 2A1s – B1s– C1s – – 3 = B1s – C1s E E X A M PL E 6 .11 Identifying the symmetry species of SALCs – Figure 6.20 The (a) A1 and (b) E symmetryadapted linear combinations of H1s orbitals in NH3. C2 1 v –1 v –1 Inspection of the character table for C2v shows that these characters correspond to symmetry species A2. N + O – O + Answer To establish the symmetry species of a SALC we need to see how it transforms under the symmetry operations of the group. A picture of the SALC is shown in Fig. 6.21, and we can see that under C2, changes into itself, implying a character of 1. Under v, both atomic orbitals change sign, so is transformed into –, implying a character of –1. The SALC also changes sign under v, so the character for this operation is also –1. The characters are therefore E 1 C2 – Identify the symmetry species of the SALC = O – O in the C2v molecule NO2, where O is a 2px orbital on one O atom and O is a 2px orbital on the other O atom. Self-test 6.11 Identify the symmetry species of the combination = A1s – B1s + C1s – D1s for a squareplanar (D4h) array of H atoms A, B, C, D. 2px 2px Figure 6.21 The combination of O2px orbitals referred to in Example 6.11. 6.7 The construction of molecular orbitals Key point: Molecular orbitals are constructed from SALCs and atomic orbitals of the same symmetry species. The symmetries of molecular orbitals 193 We have seen that the SALC 1 of H1s orbitals in NH3 has A1 symmetry. The N2s and N2pz also have A1 symmetry in this molecule, so all three can contribute to the same molecular orbitals. The symmetry species of these molecular orbitals will be A1, like their components, and they are called a1 orbitals. Note that the labels for molecular orbitals are lowercase versions of the symmetry species of the orbital. Three such molecular orbitals are possible, each of the form = c1N2s + c2N2p + c31 z with ci coefficients that are found by computational methods. They are labelled 1a1, 2a1, and 3a1 in order of increasing energy (the order of increasing number of internuclear nodes), and correspond to bonding, nonbonding, and antibonding combinations (Fig. 6.22). We have also seen (and can confirm by referring to Resource section 5) that in a C3v molecule the SALCs 2 and 3 of the H1s orbitals have E symmetry. The C3v character table shows that the same is true of the N2px and N2py orbitals (Fig. 6.23). It follows that 2 and 3 can combine with these two N2p orbitals to give doubly degenerate bonding and antibonding orbitals of the form 3a1 = c4N2p + c52, and c6N2p + c73 x y These molecular orbitals have E symmetry and are therefore called e orbitals. The pair of lower energy, denoted 1e, are bonding and the upper pair, 2e, are antibonding. 2a1 E X A M PL E 6 .12 Constructing molecular orbitals from SALCs The two SALCs of H1s orbitals in the C2v molecule H2O are 1 = A1s + B1s (17) and 2 = A1s – B1s (18). Which oxygen orbitals can be used to form molecular orbitals with them? Answer We start by establishing how the SALCs transform under the symmetry operations of the group (C2v). Under E neither SALC changes sign, so their characters are 1. Under C2, 1 does not change sign but 2 does; their characters are therefore 1 and –1, respectively. Under v the combination 1 does not change sign but 2 does change sign, so their characters are again +1 and –1, respectively. Under the reflection v neither SALC changes sign, so their characters are 1. The characters are therefore E 1 1 1 2 C2 1 –1 v 1 –1 v 1 1 We now consult the character table and identify their symmetry labels as A1 and B2, respectively. The same conclusion could have been obtained more directly by referring to Resource section 5. According to the entries on the right of the character table, the O2s and O2pz orbitals also have A1 symmetry; O2py has B2 symmetry. The linear combinations that can be formed are therefore 1a1 Figure 6.22 The three a1 molecular orbitals of NH3 as computed by molecular modelling software. A1s B1s 17 = c1O2s + c2O2p + c31 z = c4O2p + c52 y The three a1 orbitals are bonding, intermediate, and antibonding in character according to the relative signs of the coefficients c1, c2, and c3. Similarly, depending on the relative signs of the coefficients c4 and c5, one of the two b2 orbitals is bonding and the other is antibonding. A1s B1s – 2 Self-test 6.12 The four SALCs built from Cl3s orbitals in the square planar (D4h) [PtCl4] anion have symmetry species A1g, B1g, and Eu. Which Pt atomic orbitals can combine with which of these SALCs? A symmetry analysis has nothing to say about the energies of orbitals other than to identify degeneracies. To calculate the energies, and even to arrange the orbitals in order, it is necessary to use quantum mechanics; to assess them experimentally it is necessary to use techniques such as photoelectron spectroscopy. In simple cases, however, we can use the general rules set out in Section 2.8 to judge the relative energies of the orbitals. For example, in NH3, the 1a1 orbital, containing the low-lying N2s orbital, can be expected to lie lowest in energy and its antibonding partner, 3a1, will probably lie highest, with the nonbonding 2a1 approximately half-way between. The 1e bonding orbital is next higher in energy after 1a1, and the 2e correspondingly lower in energy than the 3a1 orbital. This qualitative analysis leads to the energy level scheme shown in Fig. 6.24. These days, there is no difficulty in using one of the widely available software packages to calculate the energies of 18 194 6 Molecular symmetry – – – the orbitals directly by either an ab initio or a semi-empirical procedure; the energies given in Fig. 6.24 have in fact been calculated in this way. Nevertheless, the ease of achieving computed values should not be seen as a reason for disregarding the understanding of the energy level order that comes from investigating the structures of the orbitals. The general procedure for constructing a molecular orbital scheme for a reasonably simple molecule can now be summarized as follows: 1. Assign a point group to the molecule. 2. Look up the shapes of the SALCs in Resource section 5. – – Arrange the SALCs of each molecular fragment in increasing order of energy, first noting whether they stem from s, p, or d orbitals (and put them in the order s < p < d), and then their number of internuclear nodes. 4. Combine SALCs of the same symmetry type from the two fragments, and from N SALCs form N molecular orbitals. 5. Estimate the relative energies of the molecular orbitals from considerations of overlap and relative energies of the parent orbitals, and draw the levels on a molecular orbital energy level diagram (showing the origin of the orbitals). 6. Confirm, correct, and revise this qualitative order by carrying out a molecular orbital calculation by using appropriate software. 6.8 The vibrational analogy Key point: The shapes of SALCs are analogous to stretching displacements. One of the great strengths of group theory is that it enables disparate phenomena to be treated analogously. We have already seen how symmetry arguments can be applied to molecular vibrations, so it should come as no surprise that SALCs have analogies in the normal modes of molecules. In fact, the illustrations of SALCs in the Resource section can be interpreted as contributions to the normal vibrational modes of molecules. The following example illustrates how this is done. Figure 6.23 The two bonding e orbitals of NH3 as schematic diagrams and as computed by molecular modelling software. 3a1 2e Energy Lone pair 2a1 E X A M PL E 6 .13 Predicting the IR and Raman bands of an octahedral molecule Consider an AB6 molecule, such as SF6, that belongs to the Oh point group. Sketch the normal modes of A–B stretches and comment on their activities in IR or Raman spectroscopy. Answer We argue by analogy with the shapes of SALCs and identify the SALCs that can be constructed from s orbitals in an octahedral arrangement (Resource section 4). These orbitals are the analogues of the stretching displacements of the A–B bonds and the signs represent their relative phases. They have the symmetry species A1g, Eg, and T1u. The resulting linear combinations of stretches are illustrated in Fig. 6.25. The A1g (totally symmetric) and Eg modes are Raman active and the T1u mode is IR active. Self-test 6.13 Consider only bands due to stretching vibrations and predict how the IR and Raman spectra of SF5Cl differ from those of SF6. 1e 1a1 Figure 6.24 A schematic molecular orbital energy level diagram for NH3 and an indication of its ground-state electron configuration. Representations We now move on to a more quantitative treatment and introduce two topics that are important for applying symmetry arguments in the treatment of molecular orbitals and spectroscopy systematically. 6.9 The reduction of a representation Key point: A reducible representation can be resolved into its constituent irreducible representations by using the reduction formula. We have seen that the three H1s orbitals of NH3 give rise to—the technical term is ‘span’—two irreducible representations in C3v, one of symmetry species A1 and the other Representations A1g T1u Eg T1u T1u Eg 195 Figure 6.25 The A1g and T1u M–L stretching modes of an octahedral ML6 complex. The motion of the central metal atom M, which preserves the centre of mass of the molecule, is not shown (it is stationary in both the A1g and Eg modes). of symmetry species E. Here we present a systematic way for arriving at the identification of the symmetry species spanned by a set of orbitals or atom displacements. The fact the three H1s orbitals of NH3 span two particular irreducible representations is expressed formally by writing # = A1 + E where # (uppercase gamma) denotes the symmetry species of the reducible representation. In general, we write # c1#1 c 2 #2 .... (6.1a) where the #i denote the various symmetry species of the group and the ci tell us how many times each symmetry species appears in the reduction. A very deep theorem from group theory (see Further reading) provides an explicit formula for calculating the coefficients ci in terms of the characters i of the irreducible representation #i and the corresponding characters of the original reducible representation #: ci 1 h ∑ g(C) (R)(R) i (6.1b) C Here h is the order of the point group (the number of symmetry elements; it is given in the top row of the character table) and the sum is over each class C of the group with g(C) the number of elements in that class. How this expression is used is illustrated by the following example. E X A M PL E 6 .14 Using the reduction formula Consider the molecule cis-[PdCl2(NH3)2], which, if we ignore the hydrogen atoms, belongs to the point group C2v. What are the symmetry species spanned by the displacements of the atoms? Answer To analyse this problem we consider the 15 displacements of the five nonhydrogen atoms (Fig. 6.26) and obtain the characters of what will turn out to be a reducible representation # by examining what happens when we apply the symmetry operations of the group. Then we use eqn 6.1b to identify the symmetry species of the irreducible representations into which that reducible representation can be reduced. To identify the characters of # we note that each displacement that moves to a new location under a particular symmetry operation contributes 0 to the character of that operation; those that remain the same contribute 1; those that are reversed contribute –1. Thus, because all 15 displacements remain unmoved under the identity, (E) = 15. A C2 rotation leaves only the z displacement on Pd unchanged (contributing 1) but reverses the x and y displacements on Pd (contributing –2), so z y x Figure 6.26 The atomic displacements in cis-[PdCl2(NH3)3] with the H atoms ignored. 196 6 Molecular symmetry (C2) = –1. Under the reflection v the z and x displacements on Pd are unchanged (contributing 2) and the y displacement on Pd is reversed (contributing –1), so ( v) = 1. Finally, for any reflection in a vertical plane passing through the plane of the atoms, the five z displacements on these atoms remain the same (contributing 5), so do the five y displacements (another 5), but the five x displacements are reversed (contributing –5); therefore ( v) = 5. The characters of # are therefore , E 15 C2 –1 v 1 v 5 Now we use eqn 6.1b, noting that h = 4 for this group, and noting that g(C) = 1 for all C. To find how many times the symmetry species A1 appears in the reducible representation, we write c1 41 {115 1 (1) 111 5} 5 By repeating this procedure for the other species, we find Γ 5A1 2A 2 3B1 5B2 For C2v, the translations of the entire molecule span A1 + B1 + B2 (as given by the functions x, y, and z in the final column in the character table) and the rotations span A2 + B1 + B2 (as given by the functions Rx, Ry, and Rz in the final column in the character table). By subtracting these symmetry species from the ones we have just found, we can conclude that the vibrations of the molecule span 4A1 + A2 + B1 + 3B2. Self-test 6.14 Determine the symmetries of all the vibration modes of [PdCl4]2, a D4h molecule. Many of the modes of cis-[PdCl2(NH3)2] found in Example 6.14 are complex motions that are not easy to visualize: they include Pd–N stretches and various buckling motions of the plane. However, even without being able to visualize them easily, we can infer at once that the A1, B1, and B2 modes are IR active (because the functions x, y, and z, and hence the components of the electric dipole, span these symmetry species) and all the modes are Raman active (because the quadratic forms span all four species). Cl1s 2– 6.10 Projection operators Key point: A projection operator is used to generate SALCs from a basis of orbitals. To generate an unnormalized SALC of a particular symmetry species from an arbitrary set of basis atomic orbitals, we select any one of the set and form the following sum: (a) ∑ (R)R (6.2) i R A1 where i(R) is the character of the operation R for the symmetry species of the SALC we want to generate. Once again, the best way to illustrate the use of this expression is with an example. E X A M PL E 6 .15 Generating a SALC Generate the SALC of Cl orbitals for [PtCl4]2–. The basis orbitals are denoted 1, 2, 3, and 4 and are shown in Fig. 6.27a. E B1 (b) Figure 6.27 (a) The Cl orbital basis used to construct SALCs in [PtCl4]2–, and (b) the SALCs constructed for [PtCl4]2–. Answer To implement eqn 6.2 we start with one of the basis orbitals and subject it to all the symmetry operations of the D4h point group, writing down the basis function R into which it is transformed. For example, the operation C4 moves 1 into the position occupied by 2, C2 moves it to 3 and C43 moves it to 4. Continuing for all operations we obtain Operation R: E C4 R1 1 2 C43 C2 C2 C2 C 2 C2 i S4 S43 h v v d d 4 1 4 2 4 1 1 3 2 4 3 3 2 3 We now add together all the new basis functions, and for each class of operation we multiply by the character i(R) for the irreducible representation we are interested in. Thus, for A1g (as all characters are 1) we obtain 41 + 42 + 43 + 44. The (unnormalized) SALC is therefore (A1g) = 1 + 2 + 3 + 4 Problems 197 As we continue down the character table using the various symmetry species, the SALCs emerge as follows: (B1g) = 1 – 2 + 3 – 4 (Eu) = 1 – 3 Under all other irreducible representations the projection operators vanish (thus no SALCs exist of those symmetries). We then continue by using 2 as our basis function, whereupon we obtain the same SALCs except for (B1g) = 2 – 1 + 4 – 3 (Eu) = 2 – 4 Completing the process with 3 and 4 gives similar SALCs (only the signs of some of the component orbitals change). The forms of the SALCs are therefore A1g + B1g + Eu (Fig. 6.27b). Self-test 6.15 Use projection operators in SF6 to determine the SALCs for bonding in an octahedral complex. (Use the O point group rather than Oh.) FURTHER READING P. Atkins and J. de Paula, Physical chemistry. Oxford University Press and W.H. Freeman & Co (2010). An account of the generation and use of character tables without too much mathematical background. P. Atkins and R. Friedman, Molecular quantum mechanics. Oxford University Press (2005). For more rigorous introductions, see: J.S. Ogden, Introduction to molecular symmetry. Oxford University Press (2001). EXERCISES 6.1 Draw sketches to identify the following symmetry elements: (a) a C3 axis and a v plane in the NH3 molecule, (b) a C4 axis and a h plane in the square-planar [PtCl4]2– ion. 220, 213, and 83 cm–1. Detailed analysis of the 369 and 295 cm–1 bands show them to arise from totally symmetric modes. Show that the Raman spectrum is consistent with a trigonal-bipyramidal geometry. 6.2 Which of the following molecules and ions has (a) a centre of inversion, (b) an S4 axis: (i) CO2, (ii) C2H2, (iii) BF3, (iv) SO42–? 6.9 How many vibrational modes does an SO3 molecule have (a) in the plane of the nuclei, (b) perpendicular to the molecular plane? 6.3 Determine the symmetry elements and assign the point group of (a) NH2Cl, (b) CO32–, (c) SiF4, (d) HCN, (e) SiFClBrI, (f) BF4–. 6.10 What are the symmetry species of the vibrations of (a) SF6, (b) BF3 that are both IR and Raman active? 6.4 How many planes of symmetry does a benzene molecule possess? What chloro-substituted benzene of formula C6HnCl6–n has exactly four planes of symmetry? 6.11 What are the symmetry species of the vibrational modes of a C6v molecule that are neither IR nor Raman active? 6.5 Determine the symmetry elements of objects with the same shape as the boundary surface of (a) an s orbital, (b) a p orbital, (c) a dxy orbital, (d) a dz2 orbital. 6.6 (a) Determine the symmetry group of an SO32– ion. (b) What is the maximum degeneracy of a molecular orbital in this ion? (c) If the sulfur orbitals are 3s and 3p, which of them can contribute to molecular orbitals of this maximum degeneracy? 6.7 (a) Determine the point group of the PF5 molecule. (Use VSEPR, if necessary, to assign geometry.) (b) What is the maximum degeneracy of its molecular orbitals? (c) Which P3p orbitals contribute to a molecular orbital of this degeneracy? 6.8 Reaction of AsCl3 with Cl2 at low temperature yields a product, believed to be AsCl5, which shows Raman bands at 437, 369, 295, 6.12 The [AuCl4]– ion has D4h symmetry. Determine the representations # of all 3N displacements and reduce it to obtain the symmetry species of the irreducible representations. 6.13 How could IR and Raman spectroscopy be used to distinguish between: (a) planar and pyramidal forms of PF3, (b) planar and 90º-twisted forms of B2F4 (D2h and D2d, respectively). 6.14 (a) Take the four hydrogen 1s orbitals of CH4 and determine how they transform under Td. (b) Confirm that it is possible to reduce this representation to A1 + T2. (c) With which atomic orbitals on C would it be possible to form MOs with H1s SALCs of symmetry A1 + T2? 6.15 Consider CH4. Use the projection operator method to construct the SALCs of A1 + T2 symmetry that derive from the four H1s orbitals. 6.16 Use the projection operator method to determine the SALCs required for formation of bonds in (a) BF3, (b) PF5. PROBLEMS 6.1 Consider a molecule IF3O2 (with I as the central atom). How many isomers are possible? Assign point group designations to each isomer. 6.2 (a) Determine the point group of the most symmetric planar conformation of B(OH)3 and the most symmetric nonplanar conformation of B(OH)3. Assume that the BOH bond angles are 109.5º in all conformations. (b) Sketch a conformation of B(OH)3 that is chiral, once again keeping all three BOH bond angles equal to 109.5º. 198 6 Molecular symmetry 6.3 How many isomers are there for ‘octahedral’ molecules with the formula MA3B3, where A and B are monoatomic ligands? What is the point group of each isomer? Are any of the isomers chiral? Repeat this exercise for molecules with the formula MA2B2C2. 6.4 Group theory is often used by chemists as an aid in the interpretation of infrared spectra. For example, there are four NH bonds in NH4 and four stretching modes are possible: each one is a linear combination of the four stretching modes, and each one has a characteristic symmetry. There is the possibility that several vibrational modes occur at the same frequency, and hence are degenerate. A quick glance at the character table will tell if degeneracy is possible. (a) In the case of the tetrahedral NH4 ion, is it necessary to consider the possibility of degeneracies? (b) Are degeneracies possible in any of the vibrational modes of NH2D2? 6.5 Determine whether the number of IR and Raman active stretching modes could be used to determine uniquely whether a sample of gas is BF3, NF3, or ClF3. 6.6 Figure 6.28 shows the electronic energy levels of CH3. What is the point group used for this illustration? Identify the following contributions: Energy 2a1′ 2e′ 1a2″ 1e′ 1a1′ Figure 6.28 (a) H1s to a1 (b) C2s and C2p to a1 (c) H1s to e (d) C2s and C2p to e1 (e) C2s and C2p to a2 (f) H1s to a2 Now add two H1s orbitals on the z-axis (above and below the plane), modify the linear combinations of each symmetry type accordingly, and construct a new A2 linear combination. Are there bonding and nonbonding (or only weakly antibonding orbitals) that can accommodate ten electrons and allow carbon to become hypervalent? 6.7 Consider the p orbitals on the four Cl atoms of tetrahedral [CoCl4]–, with one p orbital on each Cl pointing directly at the central metal atom. (a) Confirm that the four p orbitals which point at the metal transform in an identical manner to the four s orbitals on the Cl atoms. How might these p orbitals contribute to the bonding of the complex? (b) Take the remaining eight p orbitals and determine how they transform. Reduce the representation you derive to determine the symmetry of the SALCs these orbitals contribute to. Which metal orbitals can these SALCs bond with? (c) Generate the SALCs referred to in (b). 6.8 Consider all 12 of the p orbitals on the four Cl atoms of a square planar complex like [PtCl4]2–. (a) Determine how these p orbitals transform under D4h and reduce the representation. (b) Which metal orbitals can these SALCs bond with? (c) Which SALCs and which metal orbitals contribute to bonds? (d) Which SALCs and which metal orbitals contribute to the in-plane π bonds? (e) Which SALCs and which metal orbitals contribute to the out of plane π bonds? 6.9 Take an octahedral complex and construct all the - and π-bonding SALCs. An introduction to coordination compounds 7 Metal complexes, in which a single central metal atom or ion is surrounded by several ligands, play an important role in inorganic chemistry, especially for elements of the d block. In this chapter, we introduce the common structural arrangements for ligands around a central metal atom and the isomeric forms that are possible. The language of coordination chemistry In the context of metal coordination chemistry, the term complex means a central metal atom or ion surrounded by a set of ligands. A ligand is an ion or molecule that can have an independent existence. Two examples of complexes are [Co(NH3)6]3, in which the Co3 ion is surrounded by six NH3 ligands, and [Na(OH2)6], in which the Na ion is surrounded by six H2O ligands. We shall use the term coordination compound to mean a neutral complex or an ionic compound in which at least one of the ions is a complex. Thus, Ni(CO)4 (1) and [Co(NH3)6]Cl3 (2) are both coordination compounds. A complex is a combination of a Lewis acid (the central metal atom) with a number of Lewis bases (the ligands). The atom in the Lewis base ligand that forms the bond to the central atom is called the donor atom because it donates the electrons used in bond formation. Thus, N is the donor atom when NH3 acts as a ligand, and O is the donor atom when H2O acts as a ligand. The metal atom or ion, the Lewis acid in the complex, is the acceptor atom. All metals, from all blocks of the periodic table, form complexes. The principal features of the geometrical structures of metal complexes were identified in the late nineteenth and early twentieth centuries by Alfred Werner, whose training was in organic stereochemistry. Werner combined the interpretation of optical and geometrical isomerism, patterns of reactions, and conductance data in work that remains a model of how to use physical and chemical evidence effectively and imaginatively. The striking colours of many d- and f-metal coordination compounds, which reflect their electronic structures, were a mystery to Werner. This characteristic was clarified only when the description of electronic structure in terms of orbitals was applied to the problem in the period from 1930 to 1960. We look at the electronic structure of d-metal complexes in Chapter 20 and f-metal complexes in Chapter 23. The geometrical structures of metal complexes can now be determined in many more ways than Werner had at his disposal. When single crystals of a compound can be grown, X-ray diffraction (Section 8.1) gives precise shapes, bond distances, and angles. Nuclear magnetic resonance (Section 8.5) can be used to study complexes with lifetimes longer than microseconds. Very short-lived complexes, those with lifetimes comparable to diffusional encounters in solution (a few nanoseconds), can be studied by vibrational and electronic spectroscopy. It is possible to infer the geometries of complexes with long lifetimes in solution (such as the classic complexes of Co(III), Cr(III), and Pt(II) and many organometallic compounds) by analysing patterns of reactions and isomerism. This method was originally exploited by Werner, and it still teaches us much about the synthetic chemistry of the compounds as well as helping to establish their structures. Constitution and geometry 7.1 Representative ligands 7.2 Nomenclature 7.3 Low coordination numbers 7.4 Intermediate coordination numbers 7.5 Higher coordination numbers 7.6 Polymetallic complexes Isomerism and chirality 7.7 Square-planar complexes 7.8 Tetrahedral complexes 7.9 Trigonal-bipyramidal and squarepyramidal complexes 7.10 Octahedral complexes 7.11 Ligand chirality The thermodynamics of complex formation 7.12 Formation constants 7.13 Trends in successive formation constants 7.14 The chelate and macrocyclic effects 7.15 Steric effects and electron delocalization FURTHER READING EXERCISES PROBLEMS CO Ni The language of coordination chemistry Key points: In an inner-sphere complex, the ligands are attached directly to a central metal ion; outersphere complexes occur where cation and anion associate in solution. 1 Ni(CO)4 200 7 An introduction to coordination compounds 3+ NH3 Cl– Co 2 [Co(NH3)6]Cl3 O S + 2 H2O n M 2– 4 In what we normally understand as a complex, more precisely an inner-sphere complex, the ligands are attached directly to the central metal atom or ion. These ligands form the primary coordination sphere of the complex and their number is called the coordination number of the central metal atom. As in solids, a wide range of coordination numbers can occur, and the origin of the structural richness and chemical diversity of complexes is the ability of the coordination number to range up to 12. Although we shall concentrate on inner-sphere complexes throughout this chapter, we should keep in mind that complex cations can associate electrostatically with anionic ligands (and, by other weak interactions, with solvent molecules) without displacement of the ligands already present. The product of this association is called an outer-sphere complex. With [Mn(OH2)6]2 and SO42 ions, for instance, the equilibrium concentration of the outer-sphere complex {[Mn(OH2)6]2SO42} (3) can, depending on the concentration, exceed that of the inner-sphere complex [Mn(OH2)5SO4] in which the ligand SO42 is directly attached to the metal ion. It is worth remembering that most methods of measuring complex formation equilibria do not distinguish outer-sphere from inner-sphere complex formation but simply detect the sum of all bound ligands. Outer-sphere complexation should be suspected whenever the metal and ligands have opposite charges. A large number of molecules and ions can behave as ligands, and a large number of metal ions form complexes. We now introduce some representative ligands and consider the basics of naming complexes. 7.1 Representative ligands Key points: Polydentate ligands can form chelates; a bidentate ligand with a small bite angle can result in distortions from standard structures. 3 [Mn(OH2)6]SO4 Table 7.1 gives the names and formulas of a number of common simple ligands and Table 7.2 gives the common prefixes used. Some of these ligands have only a single donor pair of electrons and will have only one point of attachment to the metal: such ligands are classified as monodentate (from the Latin meaning ‘one-toothed’). Ligands that have more than one point of attachment are classified as polydentate. Ligands that specifically have two points of attachment are known as bidentate, those with three, tridentate, and so on. Table 7.1 Typical ligands and their names Name Formula Acetylacetonato ⫺ Ammine O NH3 Aqua H2O Abbreviation Donor atoms Number of donors acac O 2 N 1 O 1 N 2 O bpy 2,2-Bipyridine N N Bromido Br Br 1 Carbanato CO2 3 O 1 or 2 Carbonyl CO C 1 Chlorido Cl 1 O 6 Cl 1,4,7,10,13,16-Hexaoxacyclooctadecane O O O 18-crown-6 O O O The language of coordination chemistry Table 7.2 Prefixes used for naming complexes Table 7.1 (Continued) Name Formula 4,7,13,16,21-Pentaoxa-1, 10-diaza-bicyclo [8.8.5]tricosane Abbreviation O O O Diethylenetriamine Bis(diphenylphosphino)ethane Bis(diphenylphosphino)methane CN Number of donors Prefix Meaning N, O 2N, 5O mono- 1 di-, bis- 2 tri-,tris- 3 tetra-, tetrakis- 4 penta- 5 O N Cyanido 2.2.1 crypt N Donor atoms O C NH(CH2CH2NH2)2 Ph2P PPh2 Ph2P hexa- 6 7 dien N 3 hepta- dppe P 2 octa- 8 nona- 9 dppm PPh2 1 P 2 Cyclopentadienyl C 5H5 Cp C 5 Ethylenediamine (1,2-diaminoethane) NH2CH2CH2NH2 en N 2 Ethylenediaminetetraacetato ⫺O edta4 N, O 2N, 4O F 1 N, O 1N, 10 H 1 O 1 ⫺O Fluorido Glycinato Hydrido Hydroxido Iodido Nitrato NitritoκO NitritoκN Oxido CO⫺ 2 2C N 2C N CO⫺ 2 F 2 NH2CH2CO H gly OH I 1 3 O 1 or 2 2 O 1 2 NO N 1 2 O 1 ox O 2 py N 1 S 1 N 4 N 1 S 1 S 1 I NO NO O O O O O⫺ Oxalato ⫺ Pyridine N Sulfido S 2 cyclam Tetraazacyclotetradecane ThiocyanatoκN ThiocyanatoκS N N N N NCS SCN Thiolato RS Triaminotriethylamine N(CH2CH2NH2)3 tren N 4 Tricyclohexylphosphine P(C6H11)3 PCy3 P 1 Trimethylphosphine P(CH3)3 PMe3 P 1 Triphenylphosphine P(C6H5)3 PPh3 P 1 deca- 10 undeca- 11 dodeca- 12 201 202 7 An introduction to coordination compounds Ambidentate ligands have more than one different potential donor atom. An example is the thiocyanate ion (NCS), which can attach to a metal atom either by the N atom, to give thiocyanato-N complexes, or by the S atom, to give thiocyanato-S complexes. Another example of an ambidentate ligand is NO2: as MNO2 (4) the ligand is nitrito-N and as MONO (5) it is nitrito-O. O M N A note on good practice The ‘ terminology’ in which the letter (kappa) is used to indicate the atom of ligation has only recently been introduced, and the old names isothiocyanato, indicating attachment by the N atom, or thiocyanato, indicating attachment by the S atom, are still widely encountered. Similarly, the old names nitro, indicating attachment by the N atom, or nitrito, indicating attachment by the O atom, are also still widely encountered. 4 Nitrito-κN ligand N M O 5 Nitrito-κO ligand CH2 NH2 M 6 Ethylenediamine ligand (en) – O Polydentate ligands can produce a chelate (from the Greek for ‘claw’), a complex in which a ligand forms a ring that includes the metal atom. An example is the bidentate ligand ethylenediamine (1,2-diaminoethane, en, NH2CH2CH2NH2), which forms a five-membered ring when both N atoms attach to the same metal atom (6). It is important to note that normal chelating ligands will attach to the metal only at two adjacent coordination sites, in a cis fashion. The hexadentate ligand ethylene-diaminetetraacetic acid, as its anion (edta4), can attach at six points (at two N atoms and four O atoms) and can form an elaborate complex with five five-membered rings (7). This ligand is used to trap metal ions, such as Ca2 ions, in ‘hard’ water. Complexes of chelating ligands often have additional stability over those of non-chelating ligands—the origin of this so called chelate effect is discussed later in this chapter (Section 7.14). Table 7.1 includes some of the most common chelating ligands. In a chelate formed from a saturated organic ligand, such as ethylenediamine, a fivemembered ring can fold into a conformation that preserves the tetrahedral angles within the ligand and yet still achieve an LML angle of 90°, the angle typical of octahedral complexes. Six-membered rings may be favoured sterically or by electron delocalization through their π orbitals. The bidentate -diketones, for example, coordinate as the anions of their enols in six-membered ring structures (8). An important example is the acetylacetonato anion (acac, 9). Because biochemically important amino acids can form five- or six-membered rings, they also chelate readily. The degree of strain in a chelating ligand is often expressed in terms of the bite angle, the LML angle in the chelate ring (10). C H N Co Key points: The cation and anion of a complex are named according to a set of rules; cations are named first and ligands are named in alphabetical order. 7 [Co(edta)]– H3C CH3 O 7.2 Nomenclature M 8 O H3C CH3 O O 9 Detailed guidance on nomenclature is beyond the scope of this book and only a general introduction is given here. In fact, the names of complexes often become so cumbersome that inorganic chemists often prefer to give the formula rather than spell out the entire name. For compounds that consist of one or more ions, the cation is named first followed by the anion (as for simple ionic compounds), regardless of which ion is complex. Complex ions are named with their ligands in alphabetical order (ignoring any numerical prefixes). The ligand names are followed by the name of the metal with either its oxidation number in parentheses, as in hexaamminecobalt(III) for [Co(NH3)6]3, or with the overall charge on the complex specified in parentheses, as in hexaamminecobalt(3). The suffix -ate is added to the name of the metal (sometimes in its Latin form) if the complex is an anion, as in the name hexacyanoferrate(II) for [Fe(CN)6]4. The number of a particular type of ligand in a complex is indicated by the prefixes mono-, di-, tri-, and tetra-. The same prefixes are used to state the number of metal atoms if more than one is present in a complex, as in octachloridodirhenate(III), [Re2Cl8]2 (11). Where confusion with the names of ligands is likely, perhaps because the name already includes a prefix, as with ethylenediamine, the alternative prefixes bis-, tris-, and tetrakisare used, with the ligand name in parentheses. For example, dichlorido- is unambiguous but tris(ethylenediamine) shows more clearly that there are three ethylenediamine ligands, as in tris(ethylenediamine)cobalt(II), [Co(en)3]2. Ligands that bridge two metal centres are denoted by a prefix μ (mu) added to the name of the relevant ligand, as in Constitution and geometry μ-oxido-bis(pentamminecobalt(III)) (12). If the number of centres bridged is greater than two, a subscript is used to indicate the number; for instance a hydride ligand bridging three metal atoms is denoted μ3-H. 203 Bite angle A note on good practice The letter is also used to indicate the number of points of attachment: thus a bidentate ethylenediamine ligand bound through both N atoms is indicated as 2N. The letter (eta) is used to indicate bonding modes of certain organometallic ligands (Section 22.4). Square brackets are used to indicate which groups are bound to a metal atom, and should be used whether the complex is charged or not; however, in casual usage, neutral complexes and oxoanions are often written without brackets, as in Ni(CO)4 for tetracarbonylnickel(0)1 and MnO4 for tetraoxidomanganate(VII) (‘permanganate’). The metal symbol is given first, then the ligands in alphabetical order (the earlier rule that anionic ligands precede neutral ligands has been superseded), as in [Co(Cl)2(NH3)4] for tetraamminedichloridocobalt(III). This order is sometimes varied to clarify which ligand is involved in a reaction. Polyatomic ligand formulas are sometimes written in an unfamiliar sequence (as for OH2 in [Fe(OH2)6]2 for hexaaquairon(II)) to place the donor atom adjacent to the metal atom and so help to make the structure of the complex clear. The donor atom of an ambidentate ligand is sometimes indicated by underlining it, for example [Fe(OH2)5(NCS)]2. Note that, somewhat confusingly, the ligands in the formula are in alphabetical order of binding element, and thus the formula and name of the complex may differ in the order in which the ligands appear. 10 2– 104° Re Cl E X A MPL E 7.1 Naming complexes 11 [Re [R 2Cl8]22– Name the complexes (a) [Pt(Cl)2(NH3)4]2; (b) [Ni(CO)3(py)]; (c) [Cr(edta)]; (d) [Co(Cl)2(en)2]; (e) [Rh(CO)2I2]. Answer To name a complex, we start by working out the oxidation number of the central metal atom and then add the names of the ligands in alphabetical order. (a) The complex has two anionic ligands (Cl), four neutral ligands (NH3) and an overall charge of 2; hence the oxidation number of platinum must be 4. According to the alphabetical order rules, the name of the complex is tetraamminedichloridoplatinum(IV). (b) The ligands CO and py (pyridine) are neutral, so the oxidation number of nickel must be 0. It follows that the name of the complex is tricarbonylpyridinenickel(0). (c) This complex contains the hexadentate edta4 ion as the sole ligand. The four negative charges of the ligand result in a complex with a single negative charge if the central metal ion is Cr3. The complex is therefore ethylenediaminetetraacetatochromate(III). (d) This complex contains two anionic chloride ligands and two neutral en ligands. The overall charge of 1 must be the result of the cobalt having oxidation number 3. The complex is therefore dichloridobis(ethylenediamine)cobalt(III). (e) This complex contains two anionic I (iodido) ligands and two neutral CO ligands. The overall charge of 1 must be the result of the rhodium having oxidation number 1. The complex is therefore dicarbonyldiiodidorhodate(I). Self-test 7.1 Write the formulas of the following complexes: (a) diaquadichlorido-platinum(II); (b) diamminetetra(thiocyanato-N)chromate(III); (c) tris(ethylenediamine)rhodium(III); (d) bromidopentacarbonylmanganese(I); (e) chloridotris(triphenylphosphine)rhodium(I). Constitution and geometry Key points: The number of ligands in a complex depends on the size of the metal atom, the identity of the ligands, and the electronic interactions. The coordination number of a metal atom or ion is not always evident from the composition of the solid, as solvent molecules and species that are potentially ligands may simply fill spaces within the structure and not have any direct bonds to the metal ion. For example, X-ray diffraction shows that CoCl2.6H2O contains the neutral complex [Co(Cl)2(OH2)4] and two uncoordinated H2O molecules occupying well-defined positions in the crystal. Such additional solvent molecules are called solvent of crystallization. 1 When assigning oxidation numbers in carbonyl complexes, CO is ascribed a net oxidation number of 0. 4+ NH3 Co O 12 [(H3N)5CoOCo(NH3)5]4+ 204 7 An introduction to coordination compounds Three factors govern the coordination number of a complex: 1. The size of the central atom or ion. 2. The steric interactions between the ligands. 3. Electronic interactions between the central atom or ion and the ligands. In general, the large radii of atoms and ions lower down the periodic table favour higher coordination numbers. For similar steric reasons, bulky ligands often result in low coordination numbers, especially if the ligands are also charged (when unfavourable electrostatic interactions also come into play). High coordination numbers are also most common on the left of a period, where the ions have larger radii. They are especially common when the metal ion has only a few electrons because a small number of valence electrons means that the metal ion can accept more electrons from Lewis bases; one example is [Mo(CN)8]4. Lower coordination numbers are found on the right of the d block, particularly if the ions are rich in electrons; an example is [PtCl4]2. Such atoms are less able to accept electrons from any Lewis bases that are potential ligands. Low coordination numbers occur if the ligands can form multiple bonds with the central metal, as in MnO4 and CrO42, as now the electrons provided by each ligand tend to exclude the attachment of more ligands. We consider these coordination number preferences in more detail in Chapter 20. 7.3 Low coordination numbers Key points: Two-coordinate complexes are found for Cu and Ag; these complexes often accommodate more ligands if they are available. Complexes may have coordination numbers higher than their empirical formulas suggest. P Pt P P 13 [Pt(PCy3)3], Cy= cycloC6H11 The best known complexes of metals with coordination number 2 that are formed in solution under ordinary laboratory conditions are linear species of the Group 11 ions. Linear two-coordinate complexes with two identical symmetric ligands have D∞h symmetry. The complex [AgCl2], which is responsible for the dissolution of solid silver chloride in aqueous solutions containing excess Cl ions, is one example, dimethyl mercury, MeHgMe, is another. A series of linear Au(I) complexes of formula LAuX, where X is a halogen and L is a neutral Lewis base such as a substituted phosphine, R3P, or thioether, R2S, are also known. Two-coordinate complexes often gain additional ligands to form three- or fourcoordinate complexes. A formula that suggests a certain coordination number in a solid compound might conceal a polymeric chain with a higher coordination number. For example, CuCN appears to have coordination number 1, but it in fact exists as linear CuCNCuCN chains in which the coordination number of copper is 2. Three-coordination is rare among metal complexes, but is found with bulky ligands such as tricyclohexylphosphine, as in [Pt(PCy3)3] (13), where Cy denotes cyclohexyl (C6H11), with its trigonal arrangement of the ligands. MX3 compounds, where X is a halogen, are usually chains or networks with a higher coordination number and shared ligands. Three-coordinate complexes with three identical symmetric ligands normally have D3h symmetry. 7.4 Intermediate coordination numbers Complexes of metal ions with the intermediate coordination numbers four, five, and six are the most important class of complex. They include the vast majority of complexes that exist in solution and almost all the biologically important complexes. (a) Four-coordination Key points: Tetrahedral complexes are favoured over higher coordinate complexes if the central atom is small or the ligands large; square-planar complexes are typically observed for metals with d8 configurations. 14 Tetrahedral complex (Td) Four-coordination is found in an enormous number of compounds. Tetrahedral complexes of approximately Td symmetry (14) are favoured over higher coordination numbers when Constitution and geometry the central atom is small and the ligands are large (such as Cl, Br, and I) because then ligandligand repulsions override the energy advantage of forming more metalligand bonds. Four-coordinate s- and p-block complexes with no lone pair on the central atom, such as [BeCl4]2, [BF4], and [SnCl4], are almost always tetrahedral, and tetrahedral complexes are common for oxoanions of metal atoms on the left of the d block in high oxidation states, such as [MoO4]2. Examples of tetrahedral complexes from Groups 511 are: [VO4]3, [CrO4]2, [MnO4], [FeCl4]2, [CoCl4]2, [NiBr4]2, and [CuBr4]2. Another type of four-coordinate complex is also found: those where the four ligands surround the central metal in a square-planar arrangement (15). Complexes of this type were originally identified because they can lead to different isomers when the complex has the formula MX2L2. We discuss this isomerism in Section 7.7. Square-planar complexes with four identical symmetric ligands have D4h symmetry. Square-planar complexes are rarely found for s- and p-block complexes but are abundant for d8 complexes of the elements belonging to the 4d- and 5d-series metals such as Rh, Ir, Pd2, Pt2, and Au3, which are almost invariably square planar. For 3d-metals with d8 configurations (for example, Ni2), square-planar geometry is favoured by ligands that can form π bonds by accepting electrons from the metal atom, as in [Ni(CN)4]2. Examples of square-planar complexes from Groups 9, 10, and 11 are [RhCl(PPh3)3], trans-[Ir(CO) Cl(PMe3)2], [Ni(CN)4]2, [PdCl4]2, [Pt(NH3)4]2, and [AuCl4]. Square-planar geometry can also be forced on a central atom by complexation with a ligand that contains a rigid ring of four donor atoms, much as in the formation of a porphyrin complex (16). Section 20.1f gives a detailed explanation of the factors that help to stabilize square-planar complexes. 15 Square-planar complex (D4h) N N Zn N N 16 HN N N N (b) Five-coordination Fe N N 17 Key points: In the absence of polydentate ligands that enforce the geometry, the energies of the various geometries of five-coordinate complexes differ little from one another and such complexes are often fluxional. Five-coordinate complexes, which are less common than four- or six-coordinate complexes, are normally either square pyramidal or trigonal bipyramidal. A square-pyramidal complex would have C4v symmetry if all the ligands were identical and the trigonalbipyramidal complex would have D3h symmetry with identical ligands. Distortions from these ideal geometries are common, and structures are known at all points between the two ideal geometries. A trigonal-bipyramidal shape minimizes ligandligand repulsions, but steric constraints on ligands that can bond through more than one site to a metal atom can favour a square-pyramidal structure. For instance, square-pyramidal five-coordination is found among the biologically important porphyrins, where the ligand ring enforces a square-planar structure and a fifth ligand attaches above the plane. Structure (17) shows part of the active centre of myoglobin, the oxygen transport protein; the location of the Fe atom above the plane of the ring is important to its function (Section 27.7). In some cases, five-coordination is induced by a polydentate ligand containing a donor atom that can bind to an axial location of a trigonal bipyramid, with its remaining donor atoms reaching down to the three equatorial positions (18). Ligands that force a trigonal-bipyramidal structure in this fashion are called tripodal. 205 2+ N Me Co Br 18 [CoBrN(CH2CH2NMe2)3]2+ (c) Six-coordination Key point: The overwhelming majority of six-coordinate complexes are octahedral or have shapes that are small distortions of octahedral. Six-coordination is the most common arrangement for metal complexes and is found in s-, p-, d-, and f-metal coordination compounds. Almost all six-coordinate complexes are octahedral (19), at least if we consider the ligands as represented by structureless points. A regular octahedral (Oh) arrangement of ligands is highly symmetric (Fig. 7.1). It is especially important, not only because it is found for many complexes of formula ML6 but also because it is the starting point for discussions of complexes of lower symmetry, such as those shown in Fig. 7.2. The simplest deviation from Oh symmetry is tetragonal (D4h), and occurs when two ligands along one axis differ from the other four; these two ligands, which are trans to each other, might be closer in than the other four or, more commonly, further away. For the d9 configuration (particularly for Cu2 complexes), a tetragonal distortion may occur even when all ligands are identical because of an inherent effect known 19 Octahedral complex (Oh) 206 7 An introduction to coordination compounds C4 C2 C3 C3 C2 C3 C2 σd C3 σh C2 C2 i C4 C4 Figure 7.1 The highly symmetric octahedral arrangement of six ligands around a central metal atom and the corresponding symmetry elements of an octahedron. Note that not all the d are shown. (a) (b) (c) (d) Figure 7.2 Distortions of a regular octahedron: (a) and (b) tetragonal distortions, (c) rhombic distortion, (d) trigonal distortion. C2 as the JahnTeller distortion (Section 20.1g). Rhombic (D2h) distortions, in which a trans pair of ligands are close in and another trans pair are further out, can occur. Trigonal (D3d) distortions occur when two opposite faces of the octahedron move away and give rise to a large family of structures that are intermediate between regular octahedral and trigonal prismatic (20); such structures are sometimes referred to as rhombohedral. Trigonal-prismatic (D3h) complexes are rare, but have been found in solid MoS2 and WS2; the trigonal prism is also the shape of several complexes of formula [M(S2C2R2)3] (21). Trigonal-prismatic d0 complexes such as [Zr(CH3)6]2 have also been isolated. Such structures require either very small -donor ligands, ligands that bind by forming a bond to the central atom, or favourable ligandligand interactions that can constrain the complex into a trigonal prism; such ligandligand interactions are often provided by ligands that contain sulfur atoms, which can form long, weak covalent bonds to each other. A chelating ligand that permits only a small bite angle can cause distortion from octahedral towards trigonal-prismatic geometry in six-coordinate complexes (Fig. 7.3). 7.5 Higher coordination numbers Key points: Larger atoms and ions, particularly those of the f block, tend to form complexes with high coordination numbers; nine-coordination is particularly important in the f block. Seven-coordination is encountered for a few 3d complexes and many more 4d and 5d complexes, where the larger central atom can accommodate more than six ligands. Seven-coordination resembles five-coordination in the similarity in energy of its various geometries. These limiting ‘ideal’ geometries include the pentagonal bipyramid (22), a capped octahedron (23), and a capped trigonal prism (24); in each of the latter two, the seventh capping ligand occupies one face. There are a number of intermediate structures, and interconversion between them is often rapid at room temperature. Examples include [Mo(CNR)7]2, [ZrF7]3, [TaCl4(PR3)3], and [ReCl6O]2 from the d block and [UO2(OH2)5]2 from the f block. A method to force seven- rather than six-coordination on the lighter elements is to synthesize a ring of five donor atoms (25) that then occupy the equatorial positions, leaving the axial positions free to accommodate two more ligands. Stereochemical non-rigidity is also shown in eight-coordination; such complexes may be square antiprismatic (26) in one crystal but dodecahedral (27) in another. Two examples of 20 Trigonal prism (D3h) Figure 7.3 A chelating ligand that permits only a small bite angle can distort an octahedral complex into trigonal-prismatic geometry. Constitution and geometry CF3 207 C S Re 23 Capped octahedron 21 [Re(S(CF3)C=C(CF3)S)3] 22 Pentagonal bipyramid (D5h) H3C CH3 N N N N N 25 24 Capped trigonal prism complexes with these geometries are shown as (28) and (29), respectively. Cubic geometry (30) is rare. Nine-coordination is important in the structures of f-block elements; their relatively large ions can act as host to a large number of ligands. A simple example of a nine-coordinate lanthanoid complex is [Nd(OH2)9]3. More complex examples arise with the MCl3 solids, with M ranging from La to Gd, where a coordination number of nine is achieved through metalhalidemetal bridges (Section 23.5). An example of nine-coordination in the d block is [ReH9]2 (31), which has small enough ligands for this coordination number to be feasible; the geometry can be thought of as a tricapped trigonal prism. 26 Square antiprism (D4d) 3– Mo 30 Cube (Oh) CN 2– 3– 28 [Mo(CN)8] (D4) H Re 4– Zr 27 Dodecahedron; triangulated decahedron (D2d) ox 29 [Zr(ox)4]4− 31 [ReH9]2– (D3h) 208 7 An introduction to coordination compounds 2− Ce O N O O Ten- and twelve-coordination are encountered in complexes of the f-block M3 ions. Examples include [Ce(NO3)6]2 (32), which is formed in the reaction of Ce(IV) salts with nitric acid. Each NO3 ligand is bonded to the metal atom by two O atoms. An example of a ten-coordinate complex is [Th(ox)4(OH2)2]4, in which each oxalate ion ligand (ox, C2O42) provides two O donor atoms. These high coordination numbers are rare with s-, p-, and d- block ions. 7.6 Polymetallic complexes − 32 [Ce(NO3)6]2 Key point: Polymetallic complexes are classified as metal clusters if they contain MM bonds or as cage complexes if they contain ligand-bridged metal atoms. H2O Cu CH3CO2 33 [(H2O)Cu(µ-CH3CO2)4Cu(OH2)] SR 2– Fe S 34 [Fe4S4(SR)4]2– 35 Hg2Cl2 , D∞h CO Isomerism and chirality Key points: A molecular formula may not be sufficient to identify a coordination compound: linkage, ionization, hydrate, and coordination isomerism are all possible for coordination compounds. g H Cl Polymetallic complexes are complexes that contain more than one metal atom. In some cases, the metal atoms are held together by bridging ligands; in others there are direct metalmetal bonds; in yet others there are both types of link. The term metal cluster is usually reserved for polymetallic complexes in which there are direct metalmetal bonds that form triangular or larger closed structures. This rigorous definition, however, would exclude linear MM compounds, and is normally relaxed. We shall consider any MM bonded system to be a cluster. When no metalmetal bond is present, polymetallic complexes are referred to as cage complexes (or ‘cage compounds’).2 Polymetallic complexes are known for metals in every coordination number and geometry. Cage complexes may be formed with a wide variety of anionic ligands. For example, two Cu2 ions can be held together with acetate-ion bridges (33). Structure (34) is an example of a cubic structure formed from four Fe atoms bridged by RS ligands. This type of structure is of great biological importance as it is involved in a number of biochemical redox reactions (Section 27.8). With the advent of modern structural techniques, such as automated X-ray diffractometers and multinuclear NMR, many polymetallic clusters containing metalmetal bonds have been discovered and have given rise to an active area of research. A simple example is the mercury(I) cation Hg22, and complexes derived from it, such as [Hg2(Cl)2] (35), which is commonly written simply Hg2Cl2. A metal cluster containing nine CO ligands and two Mn atoms is shown as (36). Mn 36 [(CO)5MnMn(CO)5] A molecular formula often does not give enough information to identify a compound unambiguously. We have already noted that the existence of ambidentate ligands gives rise to the possibility of linkage isomerism, in which the same ligand may link through different atoms. This type of isomerism accounts for the red and yellow isomers of the formula [Co(NH3)5(NO2)]2. The red compound has a nitrito-O CoO link (5); the yellow isomer, which forms from the unstable red form over time, has a nitro nitrito-N CoN link (4). We will consider three further types of isomerism briefly before looking at geometric and optical isomerism in more depth. Ionization isomerism occurs when a ligand and a counterion in one compound exchange places. An example is [PtCl2(NH3)4]Br2 and [PtBr2(NH3)4]Cl2. If the compounds are soluble, the two isomers exist as different ionic species in solution (in this example, with free Br and Cl ions, respectively). Very similar to ionization isomerism is hydrate isomerism, which arises when one of the ligands is water, for example there are three differently coloured hydration isomers of a compound with molecular formula CrCl3.6H2O: the violet [Cr(OH2)6]Cl3, the pale green [CrCl(OH2)5] Cl2.H2O, and the dark green [CrCl2(OH2)4]Cl.2H2O. Coordination isomerism arises when there are different complex ions that can form from the same molecular formula, as in [Co(NH3)6][Cr(CN)6] and [Cr(NH3)6][Co(CN)6]. 2 The term ‘cage compound’ has a variety of meanings in inorganic chemistry and it is important to keep them distinct. For example, another use of the term is as a synonym of a clathrate compound (an inclusion compound, in which a species is trapped in a cage formed by molecules of another species). Isomerism and chirality Once we have established which ligands bind to which metals, and through which donor atoms, we can consider how to arrange these ligands in space. The three-dimensional character of metal complexes can result in a multitude of possible arrangements of the ligands. We now explore these varieties of isomerism by considering the permutations of ligand arrangement for each of the common complex geometries: this type of isomerism is known as geometric isomerism. 209 NH3 Cl Pt E X A MPL E 7. 2 Isomerism in metal complexes What types of isomerism are possible for complexes with the following molecular formulas: (a) [Pt(PEt3)3SCN], (b) CoBr(NH3)5SO4, (c) FeCl2.6H2O? Answer (a) The complex contains the ambidentate thiocyanate ligand, SCN, which can bind through either the S or the N atom to give rise to two linkage isomers: [Pt(SCN)(PEt3)3] and [Pt(NCS)(PEt3)3]. (b) With an octahedral geometry and five coordinated ammonia ligands, it is possible to have two ionization isomers: [Co(NH3)5SO4]Br and [CoBr(NH3)5]SO4. (c) Hydrate isomerism occurs as complexes of formula [Fe(OH2)6]Cl2, [FeCl(OH2)5]Cl.H2O, and [FeCl2(OH2)4].2H2O are possible. 37 cis-[PtCl2(NH3)2] Cl NH3 Pt Self-test 7.2 Two types of isomerism are possible for the six-coordinate complex Cr(NO2)2.6H2O. Identify all isomers. 7.7 Square-planar complexes 38 trans-[PtCl2(NH3)2] Key point: The only simple isomers of square-planar complexes are cistrans isomers. Werner studied a series of four-coordinate Pt(II) complexes formed by the reactions of PtCl2 with NH3 and HCl. For a complex of formula MX2L2, only one isomer is expected if the species is tetrahedral, but two isomers are expected if the species is square planar, (37) and (38). Because Werner was able to isolate two nonelectrolytes of formula [PtCl2(NH3)2], he concluded that they could not be tetrahedral and were, in fact, square planar. The complex with like ligands on adjacent corners of the square is called a cis isomer (37, point group C2v) and the complex with like ligands opposite is the trans isomer (38, D2h). Geometric isomerism is far from being of only academic interest: platinum complexes are used in cancer chemotherapy, and it is found that only cis-Pt(II) complexes can bind to the bases of DNA for long enough to be effective. In the simple case of two sets of two different monodentate ligands, as in [MA2B2], there is only the case of cis/trans isomerism to consider, (39) and (40). With three different ligands, as in [MA2BC], the locations of the two A ligands also allow us to distinguish the geometric isomers as cis and trans, (41) and (42). When there are four different ligands, as in [MABCD], there are three different isomers and we have to specify the geometry more explicitly, as in (43), (44), and (45). Bidentate ligands with different endgroups, as in [M(AB)2], can also give rise to geometrical isomers that can be classified as cis (46) and trans (47). A B M 39 cis-[MA2B2] B A M E X A MPL E 7. 3 Identifying isomers from chemical evidence 40 trans-[MA2B2] Use the reactions indicated in Fig. 7.4 to show how the cis and trans geometries of a pair of platinum complexes may be assigned. Answer The cis diamminedichlorido isomer reacts with Ag2O to lose Cl, and the product adds one oxalato dianion (C2O2 ) at adjacent positions. The trans isomer loses Cl, but the product cannot displace the 4 two OH ligands with only one C2O2 anion. A reasonable explanation is that the C2O42 anion cannot 4 reach across the square plane to bridge two trans positions. This conclusion is supported by X-ray crystallography. Self-test 7.3 The two square-planar isomers of [PtBrCl(PR3)2] (where PR3 is a trialkylphosphine) have different 31P-NMR spectra (Fig. 7.5). For the sake of this exercise, we ignore coupling to 195Pt (I 21 at 33 per cent abundance). One isomer (A) shows a single 31P resonance; the other (B) shows two 31P resonances, each of which is split into a doublet by the second 31P nucleus. Which isomer is cis and which is trans? A C M B 41 cis-[MA2BC] 210 7 An introduction to coordination compounds 2– Cl Cl Pt Cl NH3 Cl H3 N Pt H3 N Cl Pt Ag2O HCl HCl OH H3N Cl Pt OH H3 N Pt NH3 O HCl O O O O– OH O Pt NH3 Cl Ag2O C2 O42– H3 N NH3 NH3 NH3 Cl Figure 7.4 The preparation of cis- and trans-diamminedichloridoplatinum(II) and a chemical method for distinguishing the isomers. 2+ NH3 NH3 H3N O Pt NH3 OH O C2O4 2– H3N Pt NH3 OH NH3 A A A B M M D C C M D B 42 trans-[MA2BC] B C 43 [MABCD], A trans to B 44 [MABCD] A trans to C A A C M B B M A M B D 45 [MABCD] A trans to D 46 cis-[M(AB)2] 47 trans-[M(AB)2] 7.8 Tetrahedral complexes Key point: The only simple isomers of tetrahedral complexes are optical isomers. (a) A isomer (b) B isomer Figure 7.5 Idealized 31P-NMR spectra of two isomers of [PtBrCl(PR3)2]. The fine structure due to Pt is not shown. The only isomers of tetrahedral complexes normally encountered are those where either all four ligands are different or where there are two unsymmetrical bidentate chelating ligands. In both cases, (48) and (49), the molecules are chiral, not superimposable on their mirror image (Section 6.4). Two mirror-image isomers jointly make up an enantiomeric pair. The existence of a pair of chiral complexes that are each other’s mirror image (like a right hand and a left hand), and that have lifetimes that are long enough for them to be separable, is called optical isomerism. Optical isomers are so-called because they are optically active, in the sense that one enantiomer rotates the plane of polarized light in one direction and the other rotates it through an equal angle in the opposite direction. 7.9 Trigonal-bipyramidal and square-pyramidal complexes Key points: Five-coordinate complexes are not stereochemically rigid; two chemically distinct coordination sites exist within both trigonal-bipyramidal and square-pyramidal complexes. Isomerism and chirality The energies of the various geometries of five-coordinate complexes often differ little from one another. The delicacy of this balance is underlined by the fact that [Ni(CN)5]3 can exist as both square-pyramidal (50) and trigonal-bipyramidal (51) conformations in the same crystal. In solution, trigonal-bipyramidal complexes with monodentate ligands are often highly fluxional (that is, able to twist into different shapes), so a ligand that is axial at one moment becomes equatorial at the next moment: the conversion from one stereochemistry to another may occur by a Berry pseudorotation (Fig. 7.6). Thus, although isomers of five-coordinate complexes do exist, they are commonly not separable. It is important to be aware that both trigonalbipyramidal and square-pyramidal complexes have two chemically distinct sites: axial (a) and equatorial (e) for the trigonal bipyramid (52) and axial (a) and basal (b) for the square pyramid (53). Certain ligands will have preferences for the different sites because of their steric and electronic requirements, but we do not go into detail here. B A There are huge numbers of complexes with nominally octahedral geometry, where in this context the nominal structure ‘ML6’ is taken to mean a central metal atom surrounded by six ligands, not all of which are necessarily the same. M D C 48 [MABCD] enantiomers B 7.10 Octahedral complexes 211 M A 49 [M(AB)2] enantiomers (a) Geometrical isomerism Key points: Cis and trans isomers exist for octahedral complexes of formula [MA4B2], and mer and fac isomers are possible for complexes of formula [MA3B3]. More complicated ligand sets lead to further isomers. Whereas there is only one way of arranging the ligands in octahedral complexes of general formula [MA6] or [MA5B], the two B ligands of an [MA4B2] complex may be placed on adjacent octahedral positions to give a cis isomer (54) or on diametrically opposite positions to give a trans isomer (55). Provided we treat the ligands as structureless points, the trans isomer has D4h symmetry and the cis isomer has C2v symmetry. There are two ways of arranging the ligands in [MA3B3] complexes. In one isomer, three A ligands lie in one plane and three B ligands lie in a perpendicular plane (56). This complex is designated the mer isomer (for meridional) because each set of ligands can be regarded as lying on a meridian of a sphere. In the second isomer, all three A (and B) ligands are adjacent and occupy the corners of one triangular face of the octahedron (57); this complex is designated the fac isomer (for facial). Provided we treat the ligands as structureless points, the mer isomer has C2v symmetry and the fac isomer has C3v symmetry. For a complex of composition [MA2B2C2], there are five different geometrical isomers: an all-trans isomer (58); three different isomers where one pair of ligands are trans with the other two cis, as in (59), (60), and (61); and an enantiomeric pair of all-cis isomers (62). More complicated compositions, such as [MA2B2CD] or [MA3B2C], result in more extensive geometrical isomerism. For instance the rhodium compound [RhH(C≡CR)2(PMe3)3] exists as three different isomers: fac (63), mer-trans (64), and mer-cis (65). Although octahedral complexes are normally stereochemically rigid, isomerization reactions do sometimes occur (Section 21.9). (a) (b) (c) 3– CN Ni 50 [Ni(CN)5]3–, square pyramidal (C4v) 3– CN Ni 51 [Ni(CN)5]3–, trigonal bipyramidal (D3h) Figure 7.6 A Berry pseudorotation in which (a) a trigonal-bipyramidal Fe(CO)5 complex distorts into (b) a square-pyramidal isomer and then (c) becomes trigonal bipyramidal again, but with the two initially axial ligands now equatorial. 212 7 An introduction to coordination compounds A A a e e e b b 52 Trigonal bipyramid (D3h) M B b b a M B a 53 Square pyramid (C4v) 55 trans-[MA4B2] 54 cis-[MA4B2] A C A A M B B M A M 56 mer-[MA3B3] M B B C 57 fac-[MA3B3] 58 [MA2B2C2] 59 [MA2B2C2] A A B M M C A B C C B M 60 [MA2B2C2] 61 [MA2B2C2] 62 [MA2B2C2] enantiomers R CC PMe3 Rh PMe3 H CC Rh R H H 63 fac-[RhH(C≡CR)2(PMe3)3] R H 64 mer-trans-[RhH(C≡CR)2(PMe3)3] Rh CC PMe3 65 mer-cis-[RhH(C≡CR)2(PMe3)3] (b) Chirality and optical isomerism Key points: A number of ligand arrangements at an octahedral centre give rise to chiral compounds; isomers are designated or depending on their configuration. In addition to the many examples of geometrical isomerism shown by octahedral compounds, many are also chiral. A very simple example is [Mn(acac)3] (66), where three bidentate Isomerism and chirality 213 acac Cl l C + o C Co en 66 [Mn(acac)3] enantiomers en 67 cis-[CoCl2(en)2]+ enantiomers acetylacetonato (acac) ligands result in the existence of enantiomers. One way of looking at the optical isomers that arise in complexes of this nature is to imagine looking down one of the threefold axes and see the ligand arrangement as a propellor or screw thread. Chirality can also exist for complexes of formula [MA2B2C2] when the ligands of each pair are cis to each other (62). In fact, many examples of optical isomerism are known for octahedral complexes with both monodentate and polydentate ligands, and we must always be alert to the possibility of optical isomerism. As a further example of optical isomerism, consider the products of the reaction of cobalt(III) chloride and ethylenediamine in a 1:2 mole ratio. The product includes a pair of dichlorido complexes, one of which is violet (67) and the other green (68); they are, respectively, the cis and trans isomers of dichloridobis(ethylenediamine)cobalt(III), [CoCl2(en)2]. As can be seen from their structures, the cis isomer cannot be superimposed on its mirror image. It is therefore chiral and hence (because the complexes are long-lived) optically active. The trans isomer has a mirror plane and can be superimposed on its mirror image; it is achiral and optically inactive. The absolute configuration of a chiral octahedral complex is described by imagining a view along a threefold rotation axis of the regular octahedron and noting the handedness of the helix formed by the ligands (Fig. 7.7). Clockwise rotation of the helix is then designated ∆ (delta) whereas the anticlockwise rotation is designated (lambda). The designation of the absolute configuration must be distinguished from the experimentally determined direction in which an isomer rotates polarized light: some compounds rotate in one direction, others rotate in the opposite direction, and the direction may change with wavelength. The isomer that rotates the plane of polarization clockwise (when viewed into the oncoming beam) at a specified wavelength is designated the d-isomer, or the ()-isomer; the one rotating the plane anticlockwise is designated the l-isomer, or the ()-isomer. Box 7.1 describes how the specific isomers of a complex might be synthesized and Box 7.2 describes how enantiomers of metal complexes may be separated. Complexes with coordination numbers of greater than six have the potential for a great number of isomers, both geometrical and optical. As these complexes are often stereochemically nonrigid, the isomers are usually not separable and we do not consider them further. 68 trans-[CoCl2(en)2]+ ∆ Λ N N N N N N N ∆-[Co(en)3]3+ N N N N Λ-[Co(en)3]3+ O O N O O O O O ∆-[Co(ox)3]3– O O O O O Λ-[Co(ox)3]3– Figure 7.7 Absolute configurations of M(LL)3 complexes. ∆ is used to indicate clockwise rotation of the helix and . to indicate anticlockwise rotation. B OX 7.1 The synthesis of specific isomers The synthesis of specific isomers often requires subtle changes in synthetic conditions. For example, the most stable Co(II) complex in ammoniacal solutions of Co(II) salts, [Co(NH3)6]2, is only slowly oxidized. As a result, a variety of complexes containing other ligands as well as NH3 can be prepared by bubbling air through a solution containing ammonia and a Co(II) salt. Starting with ammonium carbonate yields [Co(CO3) (NH3)4], in which CO2 is a bidentate ligand that occupies two adjacent 3 coordination positions. The complex cis-[CoL2(NH3)4] can be prepared by displacement of the CO2 ligand in acidic solution. When concentrated 3 hydrochloric acid is used, the violet cis-[CoCl2(NH3)4]Cl compound (B1) can be isolated: Co(CO3)(NH3)4 2 H(aq) 3 Cl(aq) → cis-[CoCl2(NH3)4]Cl(s) H2CO3(aq) Cl Cl NH3 NH3 Co C B1 Co B2 By contrast, reaction of [Co(NH3)6]3 directly with a mixture of HCl and H2SO4 in air gives the bright green trans-[CoCl2(NH3)4]Cl isomer (B2). 214 7 An introduction to coordination compounds B OX 7. 2 The resolution of enantiomers Optical activity is the only physical manifestation of chirality for a compound with a single chiral centre. However, as soon as more than one chiral centre is present, other physical properties, such as solubility and melting points, are affected because they depend on the strengths of intermolecular forces, which are different between different isomers (just as there are different forces between a given nut and bolts with left- and right-handed threads). One method of separating a pair of enantiomers into the individual isomers is therefore to prepare diastereomers. As far as we need be concerned, diastereomers are isomeric compounds that contain two chiral centres, one being of the same absolute configuration in both components and the other being enantiomeric between the two components. An example of diastereomers is provided by the two salts of an enantiomeric pair of cations, A, with an optically pure anion, B, and hence of composition [-A] [-B] and [ -A][-B]. Because diastereomers differ in physical properties (such as solubility), they are separable by conventional techniques. A classical chiral resolution procedure begins with the isolation of a naturally optically active species from a biochemical source (many naturally occurring compounds are chiral). A convenient compound is d-tartaric acid (B3), a carboxylic acid obtained from grapes. This molecule is a chelating ligand for complexation of antimony, so a convenient resolving agent is the potassium salt of the singly charged antimony d-tartrate anion. This anion is used for the resolution of [Co(en)2(NO2)2] as follows: The enantionmeric mixture of the cobalt(III) complex is dissolved in warm water and a solution of potassium antimony d-tartrate is added. The mixture B3 is cooled immediately to induce crystallization. The less soluble diastereomer {l-[Co(en)2(NO2)2]} {d-[SbOC4H4O6]} separates as fine yellow crystals. The filtrate is reserved for isolation of the d-enantiomer. The solid diastereomer is ground with water and sodium iodide. The sparingly soluble compound l-[Co(en)2(NO2)2]I separates, leaving sodium antimony tartrate in the solution. The d-isomer is obtained from the filtrate by precipitation of the bromide salt. Further reading A. von Zelewsky, Stereochemistry of coordination compounds. Wiley, Chichester (1996). W.L. Jolly, The synthesis and characterization of inorganic compounds. Waveland Press, Prospect Heights (1991). E X A M PL E 7. 4 Identifying types of isomerism When the four-coordinate square-planar complex [IrCl(PMe3)3] (where PMe3 is trimethylphosphine) reacts with Cl2, two six-coordinate products of formula [Ir(Cl)3(PMe3)3] are formed. 31P-NMR spectra indicate one P environment in one of these isomers and two in the other. What isomers are possible? Cl Ir Answer Because the complexes have the formula [MA3B3], we expect meridional and facial isomers. Structures (69) and (70) show the arrangement of the three Cl ions in the fac and mer isomers, respectively. All P atoms are equivalent in the fac isomer and two environments exist in the mer isomer. Self-test 7.4 When the anion of the amino acid glycine, H2NCH2CO2 (gly), reacts with cobalt(III) oxide, both the N and an O atom of gly coordinate and two Co(III) nonelectrolyte mer and fac isomers of [Co(gly)3] are formed. Sketch the two isomers. PMe3 7.11 Ligand chirality 69 fac-[IrCl3(PMe3)3] Cl Ir PMe3 70 mer-[IrCl3(PMe3)3] Key point: Coordination to a metal can stop a ligand inverting and hence lock it into a chiral configuration. In certain cases, achiral ligands can become chiral on coordination to a metal, leading to a complex that is chiral. Usually the nonchiral ligand contains a donor that rapidly inverts as a free ligand, but becomes locked in one configuration on coordination. An example is MeNHCH2CH2NHMe, where the two N atoms become chiral centres on coordination to a metal atom. For a square-planar complex, this imposed chirality results in four isomers, one pair of chiral enantiomers (71) and two complexes that are not chiral (72) and (73). N N N N N M N N M N N M N N M N N N N N 72 73 71 The thermodynamics of complex formation 215 – E X A MPL E 7. 5 Recognizing chirality – Cr Which of the complexes (a) [Cr(edta)], (b) [Ru(en)3]2, (c) [Pt(dien)Cl] are chiral? Answer If a complex has either a mirror plane or centre of inversion, it cannot be chiral. If we look at the schematic complexes in (74), (75), and (76), we can see that neither (74) nor (75) has a mirror plane or a centre of inversion; so both are chiral (they also have no higher Sn axis). Conversely, (76) has a plane of symmetry and hence is achiral. (Although the CH2 groups in a dien ligand are not in the mirror plane, they oscillate rapidly above and below it.) edta 74 [Cr(edta)]– enantiomers Self-test 7.5 Which of the complexes (a) cis-[Cr(Cl)2(ox)2]3, (b) trans-[Cr(Cl)2(ox)2]3, (c) cis-[RhH(CO)(PR3)2] are chiral? 2+ 2+ Ru The thermodynamics of complex formation When assessing chemical reactions we need to consider both thermodynamic and kinetic aspects because although a reaction may be thermodynamically feasible, there might be kinetic constraints. en 75 [Ru(en)3]2+ enantiomers 7.12 Formation constants Cl Key points: A formation constant expresses the interaction strength of a ligand relative to the interaction strength of the solvent molecules (usually H2O) as a ligand; a stepwise formation constant is the formation constant for each individual solvent replacement in the synthesis of the complex; an overall formation constant is the product of the stepwise formation constants. Consider the reaction of Fe(III) with SCN to give [Fe(SCN)(OH2)5]2, a red complex used to detect either iron(III) or the thiocyanate ion: [Fe(OH ) ]3(aq) SCN(aq) [Fe(SCN)(OH ) ]2(aq) H O(l) 2 6 Kf = 2 5 2 2 2 5 [Fe(SCN)(OH ) ] [Fe(OH 2 )63 ][SCN ] The equilibrium constant, Kf, of this reaction is called the formation constant of the complex. The concentration of solvent (normally H2O) does not appear in the expression because it is taken to be constant in dilute solution and ascribed unit activity. The value of Kf indicates the strength of binding of the ligand relative to H2O: if Kf is large, the incoming ligand binds more strongly than the solvent, H2O; if Kf is small, the incoming ligand binds more weakly than H2O. Because the values of Kf can vary over a huge range (Table 7.3), they are often expressed as their logarithms, log Kf. A note on good practice In expressions for equilibrium constants and rate equations, we omit the brackets that are part of the chemical formula of the complex; the surviving square brackets denote molar concentration of a species (with the units mol dm3 removed). The discussion of stabilities is more involved when more than one ligand may be replaced. For instance, in the reaction of [Ni(OH2)6]2 to give [Ni(NH3)6]2, [Ni(OH2)6]2(aq) 6NH3(aq) ➝ [Ni(NH3)6]2(aq) 6 H2O(l) there are at least six steps, even if cistrans isomerization is ignored. For the general case of the complex MLn, for which the overall reaction is M n L ➝ MLn, the stepwise formation constants are [ML] Kf1 M L ML [M][L] ML L ML2 Kf2 [ML 2 ] [ML][L] Pt dien 76 [PtCl(dien)]+ 216 7 An introduction to coordination compounds Table 7.3 Formation constants for the ction [M(H2O)n ]m L [M (L ) (OH2 )n1]m + H2O Ion Ligand Kf log Kf Ion Ligand Kf log Kf Mg2 NH3 1.7 0.23 Pd2 Cl 1.25 105 5.1 Ca2 NH3 0.64 0.2 Na SCN 1.2 104 4.08 3 2 NH3 525 2.72 Cr SCN 1.2 10 3.08 Cu NH3 8.50 105 5.93 Fe3 SCN 234 2.37 Cu2 NH3 2.0 104 4.31 Co2 SCN 11.5 1.06 Hg2 NH3 6.3 108 8.8 Fe2 pyridine 5.13 0.71 Cl 0.17 0.77 Zn 2 pyridine 8.91 0.95 Cl 4.17 0.62 Cu2 pyridine 331 2.52 pyridine 93 1.97 Ni Rb Mg2 Cr Cl 7.24 0.86 Co2 Cl 4.90 0.69 3 Ag 3 and so on, and in general MLn1 L MLn Kfn M n L MLn n [ML n ] [ML n1 ][L] These stepwise constants are the ones to consider when seeking to understand the relationships between structure and reactivity. When we want to calculate the concentration of the final product (the complex MLn) we use the overall formation constant, n: [ML n ] [M][L]n As may be verified by multiplying together the individual stepwise constants, the overall formation constant is the product of the stepwise constants: n Kf1Kf 2 ...Kfn The inverse of each Kf, the dissociation constant, Kd, is also sometimes useful, and is often preferred when we are interested in the concentration of ligand that is required to give a certain concentration of complex: ML M L Kd1 [M][L] 1 [ML] Kf1 For a 1:1 reaction, like the one above, when half the metal ions are complexed and half are not, so that [M] [ML], then Kd1 [L]. In practice, if initially [L] [M], so that there is an insignificant change in the concentration of L when M is added and undergoes complexation, Kd is the ligand concentration required to obtain 50 per cent complexation. Because Kd has the same form as Ka for acids, with L taking the place of H, its use facilitates comparisons between metal complexes and Brønsted acids. The values of Kd and Ka can be tabulated together if the proton is considered to be simply another cation. For instance, HF can be considered as the complex formed from the Lewis acid H with the Lewis base F playing the role of a ligand. 7.13 Trends in successive formation constants Key points: Stepwise formation constants typically lie in the order Kfn > Kfn+1, as expected statistically; deviations from this order indicate a major change in structure. The magnitude of the formation constant is a direct reflection of the sign and magnitude of the standard Gibbs energy of formation (because rG ° RT ln Kf). It is commonly observed that stepwise formation constants lie in the order Kf1 > Kf2 > … > Kfn. This general trend can be explained quite simply by considering the decrease in the number of the ligand H2O molecules available for replacement in the formation step, as in M(OH2)5L L(aq) M(OH2)4L2 H2O(l) The thermodynamics of complex formation Table 7.4 Formation constants of Ni(II) ammines, [Ni(NH3)n(OH2)6n]2 n Kf log Kf Kn /Kn1 Experimental Statistical 0.28 0.42 1 525 2.72 2 148 2.17 3 45.7 1.66 0.31 0.53 4 13.2 1.12 0.29 0.56 5 4.7 0.63 0.35 0.53 6 1.1 0.04 0.23 0.42 Based on ratios of numbers of ligands available for replacement, with the reaction enthalpy assumed constant. compared with M(OH2)4L2 L(aq) M(OH2)3L3 H2O(l) The decrease in the stepwise formation constants reflects the diminishing statistical factor as successive ligands are replaced, coupled with the fact that an increase in the number of bound ligands increases the likelihood of the reverse reaction. That such a simple explanation is more or less correct is illustrated by data for the successive complexes in the series from [Ni(OH2)6]2 to [Ni(NH3)6]2 (Table 7.4). The reaction enthalpies for the six successive steps are known to vary by less than 2 kJ mol1. A reversal of the relation Kfn > Kfn+1 is usually an indication of a major change in the electronic structure of the complex as more ligands are added. An example is the observation that the tris(bipyridine) complex of Fe(II), [Fe(bpy)3]2, is strikingly stable compared with the bis complex, [Fe(bpy)2(OH2)2]2. This observation can be correlated with the change in electronic configuration from a high-spin (weak-field) t24geg2 configuration in the bis complex (note the presence of weak-field H2O ligands) to a low-spin (strong-field) t26g configuration in the tris complex, where there is a considerable increase in the LFSE (see Sections 20.1 and 20.2). [Fe(OH2)6]2(aq) bpy(aq) [Fe(bpy)(OH2)4]2(aq) 2 H2O(l) log Kf1 4.2 [Fe(bpy)(OH2)4]2(aq) bpy(aq) [Fe(bpy)2(OH2)2]2(aq) 2 H2O(l) log Kf2 3.7 [Fe(bpy)2(OH2)2]2(aq) bpy(aq) [Fe(bpy)3]2(aq) 2 H2O(l) log Kf3 9.3 A contrasting example is the halogeno complexes of Hg(II), where Kf3 is anomalously low compared with Kf2: [Hg(OH2)6]2(aq) Cl(aq) HgCl(OH2)5 H2O(l) log Kf1 6.74 HgCl(OH2)5 Cl(aq) [HgCl2(OH2)4] (aq) H2O(l) log Kf2 6.48 HgCl2(OH2)4 Cl(aq) HgCl3(OH2) 3 H2O(l) log Kf3 0.95 The decrease between the second and third values is too large to be explained statistically and suggests a major change in the nature of the complex, such as the onset of four-coordination: OH2 Cl Hg H2O OH2 Cl OH2 OH2 + Cl⫺ Hg⫺ Cl Cl Cl 3 H2O E X A MPL E 7.6 Interpreting irregular successive formation constants The successive formation constants for complexes of cadmium with Br are Kf1 36.3, Kf2 3.47, Kf3 1.15, Kf4 2.34. Suggest an explanation of why Kf4 > Kf3. 217 218 7 An introduction to coordination compounds Answer The anomaly suggests a structural change, so we need to consider what it might be. Aqua complexes are usually six-coordinate whereas halogeno complexes of M2 ions are commonly tetrahedral. The reaction of the complex with three Br groups to add the fourth is CdBr3(OH2)3 Br(aq) ➝ [CdBr4]2(aq) 3 H2O(l) This step is favoured by the release of three H2O molecules from the relatively restricted coordination sphere environment. The result is an increase in Kf. Self-test 7.6 Assuming the displacement of a water by a ligand were so favoured that the back reaction could be ignored, calculate all the stepwise formation constants you would expect in the formation of [ML6]2 from [M(OH2)6]2, and the overall formation constant, given that Kf1 1 105. 7.14 The chelate and macrocyclic effects Key points: The chelate and macrocyclic effects are the greater stability of complexes containing coordinated polydentate ligands compared with a complex containing the equivalent number of analogous monodentate ligands; the chelate effect is largely an entropic effect; the macrocyclic effect has an additional enthalpic contribution. When Kf1 for the formation of a complex with a bidentate chelate ligand, such as ethylenediamine (en), is compared with the value of 2 for the corresponding bis(ammine) complex, it is found that the former is generally larger: [Cd(OH2)6]2(aq) en(aq) [Cd(en)(OH2)4]2(aq) 2 H2O(l) log Kf1 5.84 ∆rH O 29.4 kJ mol1 ∆rS O 13.0 J K1 mol1 [Cd(OH2)6]2(aq) 2 NH3(aq) [Cd(NH3)2(OH2)4]2(aq) 2 H2O(l) log 2 4.95 N N HN N N N NH N 77 ∆rH O 29.8 kJ mol1 ∆rS O 5.2 J K1 mol1 Two similar CdN bonds are formed in each case, yet the formation of the chelatecontaining complex is distinctly more favourable. This greater stability of chelated complexes compared with their nonchelated analogues is called the chelate effect. The chelate effect can be traced primarily to differences in reaction entropy between chelated and nonchelated complexes in dilute solutions. The chelation reaction results in an increase in the number of independent molecules in solution. By contrast, the nonchelating reaction produces no net change (compare the two chemical equations above). The former therefore has the more positive reaction entropy and hence is the more favourable process. The reaction entropies measured in dilute solution support this interpretation. The entropy advantage of chelation extends beyond bidentate ligands, and applies, in principle, to any polydentate ligand. In fact, the greater the number of donor sites the multidentate ligand has, the greater is the entropic advantage of displacing monodentate ligands. Macrocyclic ligands, where multiple donor atoms are held in a cyclic array, such as crown ethers or phthalocyanin (77), give complexes of even greater stability than might otherwise be expected. This so-called macrocyclic effect is thought to be a combination of the entropic effect seen in the chelate effect, together with an additional energetic contribution that comes from the preorganized nature of the ligating groups (that is, no additional strains are introduced to the ligand on coordination). The chelate and macrocyclic effects are of great practical importance. The majority of reagents used in complexometric titrations in analytical chemistry are polydentate chelates like edta4, and most biochemical metal binding sites are chelating or macrocylic ligands. A formation constant as high as 1012 to 1025 is generally a sign that the chelate or macrocyclic effect is in operation. In addition to the thermodynamic rationalization for the chelate effect we have described, there is an additional role in the chelate effect for kinetics. Once one ligating group of a polydentate ligand has bound to a metal ion, it becomes more likely that its other 219 The thermodynamics of complex formation ligating groups will bind, as they are now constrained to be in close proximity to the metal ion; thus chelate complexes are favoured kinetically too. N N M 78 7.15 Steric effects and electron delocalization 79 Key point: The stability of chelate complexes of d metals involving diimine ligands is a result of the chelate effect in conjunction with the ability of the ligands to act as π acceptors as well as donors. N Steric effects have an important influence on formation constants. They are particularly important in chelate formation because ring completion may be difficult geometrically. Chelate rings with five members are generally very stable because their bond angles are near ideal in the sense of there being no ring strain. Six-membered rings are reasonably stable and may be favoured if their formation results in electron delocalization. Three-, four-, and seven-membered (and larger) chelate rings are found only rarely because they normally result in distortions of bond angles and unfavourable steric interactions. Complexes containing chelating ligands with delocalized electronic structures may be stabilized by electronic effects in addition to the entropy advantages of chelation. For example, diimine ligands (78), such as bipyridine (79) and phenanthroline (80), are constrained to form five-membered rings with the metal atom. The great stability of their complexes with d metals is probably a result of their ability to act as π acceptors as well as donors and to form π bonds by overlap of the full metal d orbitals and the empty ring π orbitals (Section 20.2). This bond formation is favoured by electron population in the metal t2g orbitals, which allows the metal atom to act as a π donor and transfer electron density to the ligand rings. An example is the complex [Ru(bpy)3]2 (81). In some cases the chelate ring that forms can have appreciable aromatic character, which stabilizes the chelate ring even more. Box 7.3 describes how complicated chelating and macrocyclic ligands might be synthesized. N N N 80 2+ Ru bpy 81 [Ru(bpy)3]2+ B OX 7. 3 Making rings and knots A metal ion such as Ni(II) can be used to assemble a group of ligands that then undergo a reaction among themselves to form a macrocyclic ligand, a cyclic molecule with several donor atoms. A simple example is N H 2+ NH NH2 O H H NH NH2 O Zn 2+ N N Ni2+ N N Zn Ni H H N N N H This phenomenon, which is called the template effect, can be applied to produce a surprising variety of macrocyclic ligands. The reaction shown above is an example of a condensation reaction, a reaction in which a bond is formed between two molecules, and a small molecule (in this case H2O) is eliminated. If the metal ion had not been present, the condensation reaction of the component ligands would have been an illdefined polymeric mixture, not a macrocycle. Once the macrocycle has been formed, it is normally stable on its own, and the metal ion may be removed to leave a multidentate ligand that can be used to complex other metal ions. A wide variety of macrocyclic ligands can be synthesized by the template approach. Two more complicated ligands are shown below. N O N N N C u C N C N 2+ u C N N N N N (Continued) 220 7 An introduction to coordination compounds B OX 7. 3 (Continued) The origin of the template effect may be either kinetic or thermodynamic. For example, the condensation may stem either from the increase in the rate of the reaction between coordinated ligands (on account of their proximity or electronic effects) or from the added stability of the chelated ring product. OH HO OH More complicated template syntheses can be used to construct topologically complex molecules, such as the chain-like catenanes, molecules that consist of interlinked rings. An example of the synthesis of a catenane containing two rings is shown below. O O 2 O O N C u N N 2IC H C u N H 2(C H 2)4C H 2I 2OC O N O N e s a b N N N O N O C u O HO OH O HO 2+ N N O N u C N OH 2+ O N u C N N O N N O 2C u 2IC H H 2(C H 2OC H 2)5C e s a b OH N O O OH N O Even more complicated systems, equivalent to knots and links,1 can be constructed with multiple metals. The following synthesis gives rise to a single molecular strand tied in a trefoil knot: Here, two bipyridine-based ligands are coordinated to a copper ion, and then the ends of each ligand are joined by a flexible linkage. The metal ion can then be removed to give a catenand (catenane ligand), which can be used to complex other metal ions. 2 O N 2I O O O N u C u C N O N N N N O N N O HO OH O O 1 Knotted and linked systems are far from being purely of academic interest and many proteins exist in these forms: see C. Liang and K. Mislow, J. Am. Chem. Soc., 1994, 116, 3588 and 1995, 117, 4201. FURTHER READING G.B. Kauffman, Inorganic coordination compounds. Wiley, New York (1981). A fascinating account of the history of structural coordination chemistry. G.B. Kauffman, Classics in coordination chemistry: I. Selected papers of Alfred Werner. Dover, New York (1968). Provides translations of Werner’s key papers. G.J. Leigh and N. Winterbottom (ed.), Modern coordination chemistry: the legacy of Joseph Chatt. Royal Society of Chemistry, Cambridge (2002). A readable historical discussion of this area. A. von Zelewsky, Stereochemistry of coordination compounds. Wiley, Chichester (1996). A readable book that covers chirality in detail. J.A. McCleverty and T.J. Meyer (ed.), Comprehensive coordination chemistry II. Elsevier (2004). N.G. Connelly, T. Damhus, R.M. Hartshorn, and A.T. Hutton, Nomenclature of inorganic chemistry, IUPAC recommendations 2005. Royal Society of Chemistry, Cambridge (2005). Also known as ‘The IUPAC red book’, the definitive guide to naming inorganic compounds. R.A. Marusak, K. Doan, and S.D. Cummings, Integrated approach to coordination chemistry—an inorganic laboratory guide. Wiley (2007). This unusual textbook describes the concepts of coordination chemistry and illustrates these concepts through well-explained experimental projects. J.-M. Lehn (ed.), Transition metals in supramolecular chemistry, Volume 5 of Perspectives in Supramolecular Chemistry. Wiley (2007). Inspiring accounts of new developments and applications in coordination chemistry. Problems 221 EXERCISES 7.13 For which of the following square-planar complexes are isomers possible? Draw all the isomers. [Pt(NH3)2(ox)], [PdBrCl(PEt3)2], [IrH(CO)(PR3)2], [Pd(gly)2]. 7.1 Name and draw structures of the following complexes: (a) [Ni(CO)4], (b) [Ni(CN)4]2, (c) [CoCl4]2, (d) [Mn(NH3)6]2. 7.2 Give formulas for (a) chloridopentaamminecobalt(III) chloride, (b) hexaaquairon(3) nitrate, (c) cis-dichloridobis(ethylenediamine)ruthenium(II), (d) μ-hydroxidobis(penta-amminechromium(III)) chloride. 7.14 For which of the following octahedral complexes are isomers possible? Draw all the isomers. [FeCl(OH2)5]2, [Ir(Cl)3(PEt3)3], [Ru(bpy)3]2, [Co(Cl)2(en)(NH3)2], [W(CO)4(py)2]. 7.3 Name the octahedral complex ions (a) cis-[CrCl2(NH3)4], (b) trans-[Cr(NH3)2(NCS)4], (c) [Co(C2O4)(en)2]. 7.15 Ignoring optical isomers, how many isomers are possible for an octahedral complex of general formula [MA2BCDE]? How many isomers are possible (include optical isomers)? 7.4 (a) Sketch the two structures that describe most four-coordinate complexes. (b) In which structure are isomers possible for complexes of formula MA2B2? 7.16 Which of the following complexes are chiral? (a) [Cr(ox)3]3, (b) cis-[Pt(Cl)2(en)], (c) cis-[Rh(Cl)2(NH3)4], (d) [Ru(bpy)3]2, (e) fac-[Co(NO2)3(dien)], (f) mer-[Co(NO2)3(dien)]. Draw the enantiomers of the complexes identified as chiral and identify the plane of symmetry in the structures of the achiral complexes. 7.5 Sketch the two structures that describe most five-coordinate complexes. Label the two different sites in each structure. 7.6 (a) Sketch the two structures that describe most six-coordinate complexes. (b) Which one of these is rare? 7.17 Which isomer is the following tris(acac) complex? 7.7 Explain the meaning of the terms monodentate, bidentate, and tetradentate. O 7.8 What type of isomerism can arise with ambidentate ligands? Give an example. O O (a) P(OPh)3 (b) P Me2 (c) N N N N 7.10 Draw the structures of representative complexes that contain the ligands (a) en, (b) ox2, (c) phen, (d) edta4. 7.12 For which of the following tetrahedral complexes are isomers possible? Draw all the isomers. [CoBr2Cl2], [CoBrCl2(OH2)], [CoBrClI(OH2)]. O 7.18 Draw both and ∆ isomers of the [Ru(en)3]2 cation. (d) 7.11 The two compounds [RuBr(NH3)5]Cl and [RuCl(NH3)5]Br are what types of isomers? O O 7.9 Which of the following molecules could act as bidentate ligands? Which could act as chelating ligands? Me2P Mn 7.19 The stepwise formation constants for complexes of NH3 with [Cu(OH2)6]2(aq) are log Kf1 4.15, log Kf2 3.50, log Kf3 2.89, log Kf4 2.13, and log Kf5 0.52. Suggest a reason why Kf5 is so different. 7.20 The stepwise formation constants for complexes of NH2CH2CH2NH2 (en) with [Cu(OH2)6]2(aq) are log Kf1 10.72 and log Kf2 9.31. Compare these values with those of ammonia given in Exercise 7.19 and suggest why they are different. PROBLEMS 7.1 The compound Na2IrCl6 reacts with triphenylphosphine in diethyleneglycol in an atmosphere of CO to give trans-[IrCl(CO) (PPh3)2], known as ‘Vaska’s compound’. Excess CO gives a fivecoordinate species and treatment with NaBH4 in ethanol gives [IrH(CO)2(PPh3)2]. Derive a formal name for ‘Vaska’s compound’. Draw and name all isomers of the two five-coordinate complexes. 7.2 A pink solid has the formula CoCl3.5NH3.H2O. A solution of this salt is also pink and rapidly gives three 3 mol AgCl on titration with silver nitrate solution. When the pink solid is heated, it loses 1 mol H2O to give a purple solid with the same ratio of NH3:Cl:Co. The purple solid, on dissolution and titration with AgNO3, releases two of its chlorides rapidly. Deduce the structures of the two octahedral complexes and draw and name them. 7.3 The hydrated chromium chloride that is available commercially has the overall composition CrCl3.6H2O. On boiling a solution, it becomes violet and has a molar electrical conductivity similar to that of [Co(NH3)6]Cl3. By contrast, CrCl3.5H2O is green and has a lower molar conductivity in solution. If a dilute acidified solution of the green complex is allowed to stand for several hours, it turns violet. Interpret these observations with structural diagrams. 7.4 The complex first denoted -[PtCl2(NH3)2] was identified as the trans isomer. (The cis isomer was denoted α.) It reacts slowly with solid Ag2O to produce [Pt(NH3)2(OH2)2]2. This complex does not react with ethylenediamine to give a chelated complex. Name and draw the structure of the diaqua complex. A third isomer of composition PtCl2.2NH3 is an insoluble solid that, when ground with AgNO3, gives a mixture containing Pt(NH3)42 and a new solid phase of composition Ag2[PtCl4]. Give the structures and names of each of the three Pt(II) compounds. 7.5 Phosphane (phosphine) and arsane (arsine) analogues of [PtCl2(NH3)2] were prepared in 1934 by Jensen. He reported zero dipole moments for the isomers, where the designation represents the product of a synthetic route analogous to that of the ammines. Give the structures of the complexes. 7.6 Air oxidation of Co(II) carbonate and aqueous ammonium chloride gives a pink chloride salt with a ratio of 4NH3:Co. On addition of HCl to a solution of this salt, a gas is rapidly evolved and the solution slowly turns violet on heating. Complete evaporation of the violet solution yields CoCl3.4NH3. When this is heated in concentrated HCl, a green salt can be isolated 222 7 An introduction to coordination compounds with composition CoCl3.4NH3.HCl. Write balanced equations for all the transformations occurring after the air oxidation. Give as much information as possible concerning the isomerism occurring and the basis of your reasoning. Is it helpful to know that the form of [Co(Cl)2(en)2] that is resolvable into enantiomers is violet? 7.7 When cobalt(II) salts are oxidized by air in a solution containing ammonia and sodium nitrite, a yellow solid, [Co(NO2)3(NH3)3], can be isolated. In solution it is nonconducting; treatment with HCl gives a complex that, after a series of further reactions, can be identified as trans-[Co(Cl)2(NH3)3(OH2)]. It requires an entirely different route to prepare cis-[Co(Cl)2(NH3)3(OH2)]. Is the yellow substance fac or mer? What assumption must you make to arrive at a conclusion? 7.8 The reaction of [ZrCl4(dppe)] (dppe is a bidentate phosphine ligand) with Mg(CH3)2 gives [Zr(CH3)4(dppe)]. NMR spectra indicate that all methyl groups are equivalent. Draw octahedral and trigonal prism structures for the complex and show how the conclusion from NMR supports the trigonal prism assignment. (P.M. Morse and G.S. Girolami, J. Am. Chem. Soc., 1989, 111, 4114.) 7.9 The resolving agent d-cis[Co(NO2)2(en)2]Br can be converted to the soluble nitrate by grinding in water with AgNO3. Outline the use of this species for resolving a racemic mixture of the d and l enantiomers of K[Co(edta)]. (The l-[Co(edta)] enantiomer forms the less soluble diastereomer. See F.P. Dwyer and F.L. Garvan, Inorg. Synth., 1965, 6, 192.) 7.10 Show how the coordination of two MeHNCH2CH2NH2 ligands to a metal atom in a square-planar complex results in not only cis and trans but also optical isomers. Identify the mirror planes in the isomers that are not chiral. 7.11 The equilibrium constants for the successive reactions of ethylenediamine with Co2, Ni2, and Cu2 are as follows. [M(OH2)6]2 en [M(en)(OH2)4]2 2 H2O [M(en)(OH2)4]2 en [M(en)2(OH2)2]2 2 H2O [M(en)2(OH2)2] en [M(en)3]2 2 H2O Ion log K1 log K2 log K3 Co2 5.89 4.83 3.10 Ni2 7.52 6.28 4.26 10.72 9.31 1.0 Cu K2 K3 2 2 K1 Discuss whether these data support the generalizations in the text about successive formation constants. How do you account for the very low value of K3 for Cu2? 7.12 How may the aromatic character of a chelate ring provide additional stabilization of a complex? See A. Crispini and M. Ghedini, J. Chem. Soc., Dalton Trans. 1997, 75. 7.13 Use the internet to find out what rotaxanes are. Discuss how coordination chemistry might be used to synthesize such molecules. 8 Physical techniques in inorganic chemistry All the structures of the molecules and materials to be covered in this book have been determined by applying one or more kinds of physical technique. The techniques and instruments available vary greatly in complexity and cost, as well as in their suitability for meeting particular challenges. All the methods produce data that help to determine a compound’s structure, its composition, or its properties. Many of the physical techniques used in contemporary inorganic research rely on the interaction of electromagnetic radiation with matter and there is hardly a section of the electromagnetic spectrum that is not used. In this chapter, we introduce the most important physical techniques that are used to investigate the atomic and electronic structures of inorganic compounds and study their reactions. Diffraction methods 8.1 X-ray diffraction 8.2 Neutron diffraction Absorption spectroscopy 8.3 Ultraviolet–visible spectroscopy 8.4 Infrared and Raman spectroscopy Resonance techniques 8.5 Nuclear magnetic resonance 8.6 Electron paramagnetic resonance 8.7 Mössbauer spectroscopy Diffraction methods Ionization-based techniques Diffraction techniques, particularly those using X-rays, are the most important methods available to the inorganic chemist for the determination of structures. X-ray diffraction has been used to determine the structures a quarter of a million different substances, including tens of thousands of purely inorganic compounds and many organometallic compounds. The method is used to determine the positions of the atoms and ions that make up a solid compound and hence provides a description of structures in terms of features such as bond lengths, bond angles, and the relative positions of ions and molecules in a unit cell. This structural information has been interpreted in terms of atomic and ionic radii, which then allow chemists to predict structure and explain trends in many properties. Diffraction methods are nondestructive in the sense that the sample remains unchanged and may be analysed further by using a different technique. 8.8 Photoelectron spectroscopy 8.9 X-ray absorption spectroscopy 8.10 Mass spectrometry Chemical analysis 8.11 Atomic absorption spectroscopy 8.12 CHN analysis 8.13 X-ray fluorescence elemental analysis 8.14 Thermal analysis 8.1 X-ray diffraction Magnetometry Electrochemical techniques Computational techniques Key points: The scattering of radiation with wavelengths of about 100 pm from crystals gives rise to diffraction; the interpretation of the diffraction patterns gives quantitative structural information and in many cases the complete molecular or ionic structure. FURTHER READING EXERCISES PROBLEMS Diffraction is the interference between waves that occurs as a result of an object in their path. X-rays are scattered elastically (with no change in energy) by the electrons in atoms, and diffraction can occur for a periodic array of scattering centres separated by distances similar to the wavelength of the radiation (about 100 pm), such as exist in a crystal. If we think of scattering as equivalent to reflection from two adjacent parallel planes of atoms separated by a distance d (Fig. 8.1), then the angle at which constructive interference occurs (to produce a diffraction intensity maximum) between waves of wavelength is given by Bragg’s equation: 2d sin n (8.1) where n is an integer. Thus an X-ray beam impinging on a crystalline compound with an ordered array of atoms will produce a set of diffraction maxima, termed a diffraction pattern, with each maximum, or reflection, occurring at an angle corresponding to a different separation of planes of atoms, d, in the crystal. d d sin Figure 8.1 Bragg’s equation is derived by treating layers of atoms as reflecting planes. X-rays interfere constructively when the additional path length 2d sin is equal to an integral multiple of the wavelength . 224 8 Physical techniques in inorganic chemistry An atom or ion scatters X-rays in proportion to the number of electrons it possesses and the intensities of the measured diffraction maxima are proportional to the square of that number. Thus the diffraction pattern produced is characteristic of the positions and types (in terms of their number of electrons) of atom present in the crystalline compound and the measurement of X-ray diffraction angles and intensities provides structural information. Because of its dependence on the number of electrons, X-ray diffraction is particularly sensitive to any electron-rich atoms in a compound. Thus, X-ray diffraction by NaNO3 displays all three nearly isoelectronic atoms similarly, but for Pb(OH)2 the scattering and structural information is dominated by the Pb atom. There are two principal X-ray techniques: the powder method, in which the materials being studied are in polycrystalline form, and single-crystal diffraction, in which the sample is a single crystal of dimensions of several tens of micrometres or larger. Sample 2 (a) Powder X-ray diffraction X-rays Figure 8.2 A cone of diffraction that results from X-ray scattering by a powdered sample. The cone consists of thousands of individual diffraction spots from individual crystallites that merge together. X-ray tube Tube focus Detector Sample plate 2 Measuring circle Diffraction intensity (a) 8 (b) 16 24 32 Diffraction angle, 2 /° 40 Figure 8.3 (a) Schematic diagram of a powder diffractometer operating in reflection mode in which the X-ray scattering occurs from a sample mounted as a flat plate. For weakly absorbing compounds the samples may be mounted in a capillary and the diffraction data collected in transmission mode. (b) The form of a typical powder diffraction pattern showing a series of reflections as a function of angle. Key point: Powder X-ray diffraction is used mainly for phase identification and the determination of lattice parameters and lattice type. A powdered (polycrystalline) sample contains an enormous number of very small crystallites, typically 0.1 to 10 μm in dimension and orientated at random. An X-ray beam striking a polycrystalline sample is scattered in all directions; at some angles, those given by Bragg’s equation, constructive interference occurs. As a result, each set of planes of atoms with lattice spacing d gives rise to a cone of diffraction intensity. Each cone consists of a set of closely spaced diffracted rays, each one of which represents diffraction from a single crystallite within the powder sample (Fig. 8.2). With a very large number of crystallites these rays merge together to form the diffraction cone. A powder diffractometer (Fig. 8.3a) uses an electronic detector to measure the angles of the diffracted beams. Scanning the detector around the sample along the circumference of a circle cuts through the diffraction cones at the various diffraction maxima and the intensity of the X-rays detected is recorded as a function of the detector angle (Fig. 8.3b). The number and positions of the reflections depend on the cell parameters, crystal system, lattice type, and wavelength used to collect the data; the peak intensities depend on the types of atoms present and their positions. Nearly all crystalline solids have a unique powder X-ray diffraction pattern in terms of the angles of the reflections and their intensities. In mixtures of compounds, each crystalline phase present contributes to the powder diffraction pattern its own unique set of reflection angles and intensities. Typically, the method is sensitive enough to detect a small level (5 to 10 per cent by mass) of a particular crystalline component in a mixture. The effectiveness of powder X-ray diffraction has led to it becoming the major technique for the characterization of polycrystalline inorganic materials (Table 8.1). Many of the powder diffraction data sets collected from inorganic, organometallic, and organic compounds have been compiled into a database by the Joint Committee on Powder Diffraction Standards (JCPDS). This database, which contains over 50 000 unique powder X-ray diffraction patterns, can be used like a fingerprint library to identify an unknown material from its powder pattern alone. Powder X-ray diffraction is used routinely in the investigation of phase formation and changes in structures of solids. The synthesis of a metal oxide can be verified by collecting a powder diffraction pattern and demonstrating that the data are consistent with a single pure phase of that material. Indeed, the progress of a chemical reaction is often monitored by observing the formation of the product phase at the expense of the reactants. Basic crystallographic information, such as lattice parameters, can normally be extracted easily from powder X-ray diffraction data, usually with high precision. The presence or absence of certain reflections in the diffraction pattern permits the determination of the lattice type. In recent years the technique of fitting the intensities of the peaks in the diffraction pattern has become a popular method of extracting structural information such as atomic positions. The analysis, which is known as the Rietveld method, involves fitting a calculated diffraction pattern to the experimental trace. The technique is not as powerful as the single-crystal methods, for it gives less accurate atomic positions, but has the advantage of not requiring the growth of a single crystal. Diffraction methods 225 Table 8.1 Application of powder X-ray diffraction Application Typical use and information extracted Identification of unknown materials Rapid identification of most crystalline phases Determination of sample purity Monitoring the progress of a chemical reaction occurring in the solid state Determination and refinement of lattice parameters Phase identification and monitoring structure as a function of composition Investigation of phase diagrams/ new materials Mapping out composition and structure Determination of crystallite size/stress Particle size measurement and uses in metallurgy Structure refinement Extraction of crystallographic data from a known structure type Ab initio structure determination Structure determination (often at high precision) is possible in some cases without initial knowledge of the crystal structure Phase changes/expansion coefficients Studies as a function of temperature (cooling or heating typically in the range 100–1200 K). Observation of structural transitions E X A M PL E 8 .1 Using powder X-ray diffraction Titanium dioxide exists as several polymorphs, the most common of which are anatase, rutile, and brookite. The experimental diffraction angles for the six strongest reflections collected from each of these different polymorphs are summarized in the table in the margin. The powder X-ray diffraction pattern collected using 154 pm X-radiation from a sample of white paint, known to contain TiO2 in one or more of these polymorphic forms, showed the diffraction pattern in Fig. 8.4. Identify the TiO2 polymorphs present. Answer We need to identify the polymorph that has a diffraction pattern that matches the one observed. The lines closely match those of rutile (strongest reflections) and anatase (a few weak reflections), so the paint contains these phases with rutile as the major TiO2 phase. Self-test 8.1 Chromium(IV) oxide also adopts the rutile structure. By consideration of Bragg’s equation and the ionic radii of Ti4 and Cr4 (Resource section 1) predict the main features of the CrO2 powder X-ray diffraction pattern. (b) Single-crystal X-ray diffraction Key point: The analysis of the diffraction patterns obtained from single crystals allows the full determination of the structure. 20 30 40 2 /° 53.97 44.14 25.36 37.01 37.85 39.28 41.32 Counts 36.15 54.44 27.50 Analysis of the diffraction data obtained from single crystals is the most important method of obtaining the structures of inorganic solids. Provided a compound can be grown as a 50 60 Figure 8.4 A powder diffraction pattern obtained from a mixture of TiO2 polymorphs (see Example 8.1). Rutile Anatase Brookite 27.50 25.36 19.34 36.15 37.01 25.36 39.28 37.85 25.71 41.32 38.64 30.83 44.14 48.15 32.85 54.44 53.97 34.90 226 8 Physical techniques in inorganic chemistry Sample 2 X-ray beam Detector Figure 8.5 The layout of a four-circle diffractometer. A computer controls the location of the detector as the four angles are changed systematically. crystal of sufficient size and quality, the data provide definitive information about molecular and extended lattice structures. The collection of diffraction data from a single crystal is normally carried out by using a four-circle or area-detector diffractometer (Fig. 8.5). A four-circle diffractometer uses a scintillation detector to measure the diffracted X-ray beam intensity as a function of the angles shown in the illustration. An area-detector diffractometer uses an image plate that is sensitive to X-rays and so can measure a large number of diffraction maxima simultaneously; many new systems use this technology because the data can typically be collected in just a few hours (Fig. 8.6). Analysis of the diffraction data from single crystals is formally a complex process involving the locations and intensities of many thousands of reflections, but with increasing advances in computation power a skilled crystallographer can complete the structure determination of a small inorganic molecule in under an hour. Single-crystal X-ray diffraction can be used to determine the structures of the vast majority of inorganic compounds when they can be obtained as crystals with dimensions of about 50 50 50 μm or larger. Positions for most atoms, including C, N, O, and metals, in most inorganic compounds can be determined with sufficient accuracy that bond lengths can be defined to within a fraction of a picometre. As an example, the SS bond length in monoclinic sulfur has been reported as 204.7 0.3 pm. A note on good (or at least conventional) practice Crystallographers still generally use the ångström (1 Å 10 −10 m 10 −8 cm 10 –2 pm) as a unit of measurement. This unit is convenient because bond lengths typically lie between 1 and 3 Å. The SS bond length in monoclinic sulfur would be reported as 2.047 ± 0.003 Å. Figure 8.6 Part of a single-crystal X-ray diffraction pattern. Individual spots arise by diffraction of X-rays scattered from different planes of atoms within the crystal. H O Cs C O The positions of H atoms can be determined for inorganic compounds that contain only light atoms (Z less than about 18, Ar), but their locations in many inorganic compounds that also contain heavy atoms, such as the 4d- and 5d-series elements, can be difficult or impossible. The problem lies with the small number of electrons on an H atom (just 1), which is often reduced even further when H forms bonds to other atoms. Other techniques, such as neutron diffraction (Section 8.2), can often be applied to determine the positions of H in inorganic compounds. Molecular structures obtained by the analysis of single-crystal X-ray diffraction data are often represented in ORTEP diagrams (Fig. 8.7; the acronym stands for Oak Ridge Thermal Ellipsoid Program). In an ORTEP diagram an ellipsoid is used to represent the volume within which the atomic nucleus most probably lies, taking into account its thermal motion. The size of the ellipsoid increases with temperature and, as a result, so does the imprecision of the bond lengths extracted from the data. (c) X-ray diffraction at synchrotron sources Key point: High-intensity X-ray beams generated by synchrotron sources allow the structures of very complex molecules to be determined. Figure 8.7 An ORTEP diagram of caesium oxalate monohydrate, Cs2C2O4.H2O. The ellipsoids correspond to a 90 per cent probability of locating the atoms. Much more intense X-ray beams than are available from laboratory sources can be obtained by using synchrotron radiation. Synchrotron radiation is produced by electrons circulating close to the speed of light in a storage ring and is typically several orders of magnitude more intense than laboratory sources. Because of their size, synchrotron X-ray sources are normally national or international facilities. Diffraction equipment located at such an X-ray source permits the study of much smaller samples and crystals as small as 10 10 10 μm can be used. Furthermore, data collection can be undertaken much more rapidly and more complex structures, such as those of enzymes, can be determined more easily. 8.2 Neutron diffraction Key point: The scattering of neutrons by crystals yields diffraction data that give additional information on structure, particularly the positions of light atoms. Diffraction occurs from crystals for any particle with a velocity such that its associated wavelength (through the de Broglie relation, h/mv) is comparable to the separations Absorption spectroscopy of the atoms or ions in the crystal. Neutrons and electrons travelling at suitable velocities have wavelengths of the order of 100–200 pm and thus undergo diffraction by crystalline inorganic compounds. Neutron beams of the appropriate wavelength are generated by ‘moderating’ (slowing down) neutrons generated in nuclear reactors or through a process known as spallation, in which neutrons are chipped off the nuclei of heavy elements by accelerated beams of protons. The instrumentation used for collecting data and analysing single-crystal or powder neutron diffraction patterns is often similar to that used for X-ray diffraction. The scale is much larger, however, because neutron beam fluxes are much lower than laboratory X-ray sources. Furthermore whereas many chemistry laboratories have X-ray diffraction equipment for structure characterization, neutron diffraction can be undertaken only at a few specialist sources worldwide. The investigation of an inorganic compound with this technique is therefore much less routine and its application is essentially limited to systems where X-ray diffraction fails. The advantages of neutron diffraction stem from the fact that neutrons are scattered by nuclei rather than by the surrounding electrons. As a result, neutrons are sensitive to structural parameters that often complement those for X-rays. In particular, the scattering is not dominated by the heavy elements, which can be a problem with X-ray diffraction for most inorganic compounds. For example, locating the position of a light element such as H and Li in a material that also contains Pb can be impossible with X-ray diffraction, as almost all the electron density is associated with the Pb atoms. With neutrons, in contrast, the scattering from light atoms is often similar to that of heavy elements, so the light atoms contribute significantly to the intensities in the diffraction pattern. Thus neutron diffraction is frequently used in conjunction with X-ray diffraction techniques to define an inorganic structure more accurately in terms of atoms such as H, Li, and O when they are in the presence of heavier, electronrich metal atoms. Typical applications include studies of the complex metal oxides, such as the high-temperature superconductors (where accurate oxide ion positions are required in the presence of metals such as Ba and Tl) and systems where H atom positions are of interest. Another use for neutron diffraction is to distinguish nearly isoelectronic species. In X-ray scattering, pairs of neighbouring elements in a period of the periodic table, such as O and N or Cl and S, are nearly isoelectronic and scatter X-rays to about the same extent, therefore they are hard to tell apart in a crystal structure that contains them both. However, the atoms of these pairs do scatter neutrons to very different extents, N 50 per cent more strongly than O, and Cl about four times better than S, so the identification of the atoms is much easier than by X-ray diffraction. Absorption spectroscopy The majority of physical techniques used to investigate inorganic compounds involve the absorption and sometimes the re-emission of electromagnetic radiation. The frequency of the radiation absorbed provides useful information on the energy levels of an inorganic compound and the intensity of the absorption can often be used to provide quantitative analytical information. Absorption spectroscopy techniques are normally nondestructive as after the measurement the sample can be recovered for further analysis. The spectrum of electromagnetic radiation used in chemistry ranges from the short wavelengths associated with - and X-rays (about 1 nm), to radiowaves with wavelengths of several metres (Fig. 8.8). This spectrum covers the full range of atomic and molecular energies associated with characteristic phenomena such as ionization, vibration, rotation, and nuclear reorientation. Thus, X- and ultraviolet (UV) radiation can be used to determine the electronic structures of atoms and molecules and infrared (IR) radiation can be used to examine their vibrational behaviour. Radiofrequency (RF) radiation, in nuclear magnetic resonance (NMR), can be used to explore the energies associated with reorientations of the nucleus in a magnetic field, and those energies are sensitive to the chemical environment of the nucleus. In general, absorption 227 Radio Figure 8.8 The electromagnetic spectrum with wavelengths and techniques that make use of the different regions. Table 8.2 Typical timescales of some common characterization methods X-ray diffraction 10–18 s Mössbauer 10–18 s Electronic spectroscopy UV–visible 10–15 s Vibrational spectroscopy IR/Raman 10–12 s NMR c.10–3–10–6 s EPR 10–6 s Microwave 10–5 Wavelength, λ/m 10–6 10–7 10–8 Far infrared NMR EPR Rotational spectroscopy Near infrared 10–9 10–10 10–11 10–12 10–13 10–14 1 pm 10–4 1 nm 10–3 1 µm 10–2 700 nm 420 nm 10–1 1 dm 1m 1 1 mm 8 Physical techniques in inorganic chemistry 1 cm 228 Vacuum ultraviolet X-ray Visible Ultraviolet Vibrational UV/Visible Photoelectron spectroscopy spectroscopy spectroscopy γ-ray Cosmic rays Mössbauer spectroscopy spectroscopic methods make use of the absorption of electromagnetic radiation by a molecule or material at a characteristic frequency corresponding to the energy of a transition between the relevant energy levels. The intensity is related to the probability of the transition, which in turn is determined in part by symmetry rules, such as those described in Chapter 6 for vibrational spectroscopy. The various spectroscopic techniques involving electromagnetic radiation have different associated timescales. This variation can influence the structural information that is extracted. When a photon interacts with an atom or molecule we need to consider factors such as the lifetime of any excited state and a how a molecule may change during that interval. Table 8.2 summarizes the timescales associated with various spectroscopic techniques discussed in this section. Thus IR spectroscopy takes a much faster snapshot of the molecular structure than NMR, for a molecule may have time to reorientate or change shape in a nanosecond. The temperature at which data are collected should also be taken into account as molecular reorientation rates increase with increasing temperature. ■ A brief illustration. Iron pentacarbonyl, Fe(CO)5, illustrates why consideration of such timescales is important when analysing the spectra to obtain structural information. Infrared spectroscopy suggests that Fe(CO)5 has D3h symmetry with distinct axial and equatorial carbonyl groups whereas NMR suggests that all the carbonyl groups are equivalent. ■ 8.3 Ultraviolet–visible spectroscopy Key points: The energies and intensities of electronic transitions provide information on electronic structure and chemical environment; changes in spectral properties are used to monitor the progress of reactions. Ultraviolet–visible spectroscopy (UV–visible spectroscopy) is the observation of the absorption of electromagnetic radiation in the UV and visible regions of the spectrum. It is sometimes known as electronic spectroscopy because the energy is used to excite electrons to higher energy levels. UV–visible spectroscopy is among the most widely used techniques for studying inorganic compounds and their reactions, and most laboratories possess a UV–visible spectrophotometer (Fig. 8.9). This section describes only basic principles; they are elaborated in later chapters, particularly Chapter 20. Detector Sample Source Reference Grating Beam combiner Figure 8.9 The layout of a typical UV–visible absorption spectrometer. Absorption spectroscopy 229 ⎛I ⎞ A log10 ⎜ 0 ⎟ ⎝ I⎠ 589 nm The sample for a UV–visible spectrum determination is usually a solution but may also be a gas or a solid. A gas or liquid is contained in a cell (a ‘cuvette’) constructed of an optically transparent material such as glass or, for UV spectra at wavelengths below 320 nm, pure silica. Usually, the beam of incident radiation is split into two, one passing through the sample and the other passing through a cell that is identical except for the absence of the sample. The emerging beams are compared at the detector (a photodiode) and the absorption is obtained as a function of wavelength. Conventional spectrometers sweep the wavelength of the incident beam by changing the angle of a diffraction grating, but it is now more common for the entire spectrum to be recorded at once using a diode array detector. For solid samples, the intensity of UV–visible radiation reflected from the sample is more easily measured than that transmitted through a solid and an absorption spectrum is obtained by subtraction of the reflected intensity from the intensity of the incident radiation (Fig. 8.10). The intensity of absorption is measured as the absorbance, A, defined as Absorbance, A (a) Measuring a spectrum (8.2) where I0 is the incident intensity and I is the measured intensity after passing through the sample. The detector is the limiting factor for strongly absorbing species because the measurement of low photon flux is unreliable. ■ A brief illustration. A sample that attenuates the light intensity by 10 per cent (so I0 /I 100/90) has an absorbance of 0.05, one that attenuates it by 90 per cent (so I0 /I 100/10) has an absorbance of 1.0, and one that attenuates it by 99 per cent (so I0 /I 100/1) an absorbance of 2.0, and so on. ■ 750 675 600 500 450 375 Wavelength, λ/nm Figure 8.10 The UV–visible absorption spectrum of the solid ultramarine blue Na7[SiAlO4]6(S3). The empirical Beer–Lambert law is used to relate the absorbance to the molar concentration [J] of the absorbing species J and optical pathlength L: (8.3) E X A MPL E 8 . 2 Relating UV–visible spectra and colour Answer We need to be aware that the removal of light of a particular wavelength from incident white light results in the remaining light being perceived as having its complementary colour. Complementary colours are diametrically opposite each other on an artist’s colour wheel (Fig. 8.13). The only absorption from 10 000 15 000 25 000 30 000 Figure 8.12 shows the UV–visible absorption spectra of PbCrO4 and TiO2. What colour would you expect PbCrO4 to be? Visible 20 000 where ε (epsilon) is the molar absorption coefficient (still commonly referred to as the ‘extinction coefficient’ and sometimes the ‘molar absorptivity’). Values of ε range from above 105 dm3 mol–1 cm–1 for fully allowed transitions, for example an electron transferring from the 3d to 4p energy levels in an atom (∆l 1), to less than 1 dm3 mol–1 cm–1 for ‘forbidden’ transitions, such as those with ∆l 0. Selection rules also exist for transitions between molecular orbitals, although in complex molecules they are frequently broken (Chapter 20). For small molar absorption coefficients, the absorbing species may be difficult to observe unless the concentration or pathlength is increased accordingly. Figure 8.11 shows a typical solution UV–visible spectrum obtained from a d-metal compound, in this case Ti(III), which has a d1 configuration. From the wavelength of the radiation absorbed, the energy levels of the compound, including the effect of the ligand environment on d-metal atoms, can be inferred. The type of transition involved can often be inferred from the value of ε. The proportionality between absorbance and concentration provides a way to measure properties that depend on concentration, such as equilibrium compositions and the rates of reaction. Absorbance, A A ε[J]L Figure 8.11 The UV–visible spectrum of [Ti(OH2)6]3(aq). Absorbance is given as a function of wavenumber. 230 8 Physical techniques in inorganic chemistry PbCrO4 in the visible region is in the blue region of the spectrum. The rest of the light is scattered into the eye, which, according to Fig. 8.13, perceives the complementary colour yellow. Absorbance Self-test 8.2 Explain why TiO2 is widely used in sunscreens to protect against harmful UVA radiation (UV radiation with wavelengths in the range 320–360 nm). 300 (b) Spectroscopic monitoring of titrations and kinetics 400 500 600 Wavelength, l /nm 700 Figure 8.12 The UV–visible spectra of PbCrO4 (red) and TiO2 (blue). Absorbance is given as a function of wavelength. 800 400 620 430 580 490 560 l /nm Figure 8.13 An artist’s colour wheel: complementary colours lie opposite each other across a diameter. When the emphasis is on measurement of the intensities rather than the energies of transitions, the spectroscopic investigation is usually called spectrophotometry. Provided at least one of the species involved has a suitable absorption band, it is usually straightforward to carry out a ‘spectrophotometric titration’ in which the extent of reaction is monitored by measuring the concentrations of the components present in the mixture. The measurement of UV–visible absorption spectra of species in solution also provides a method for monitoring the progress of reactions and determining rate constants. Techniques that use UV–visible spectral monitoring range from those measuring reactions with half-lives of picoseconds (photochemically initiated by an ultrafast laser pulse) to the monitoring of slow reactions with half-lives of hours and even days. The stoppedflow technique (Fig. 8.14) is commonly used to study reactions with half-lives of between 5 ms and 10 s and which can be initiated by mixing. Two solutions, each containing one of the reactants, are mixed rapidly by a pneumatic impulse, then the flowing, reacting solution is brought to an abrupt stop by filling a ‘stop-syringe’ chamber, and triggering the monitoring of absorbance. The reaction can be monitored at a single wavelength or successive spectra can be measured very rapidly by using a diode array detector. The spectral changes incurred during a titration or the course of a reaction also provide information about the number of species that form during its progress. An important case is the appearance of one or more isosbestic points, wavelengths at which two species have equal values for their molar absorption coefficients (Fig. 8.15; the name comes from the Greek for ‘equal extinguishing’). The detection of isosbestic points in a titration or during the course of a reaction is evidence for there being only two dominant species (reactant and product) in the solution. 8.4 Infrared and Raman spectroscopy Key points: Infrared and Raman spectroscopy are often complementary in that a particular type of vibration may be observed in one method but not the other; the information is used in many ways, ranging from structural determination to the investigation of reaction kinetics. Vibrational spectroscopy is used to characterize compounds in terms of the strength, stiffness, and number of bonds that are present. It is also used to detect the presence of known compounds (fingerprinting), to monitor changes in the concentration of a species during a reaction, to determine the components of an unknown compound (such as the presence of CO ligands), to determine a likely structure for a compound, and to measure properties of bonds (their force constants). Reservoir syringes Drive syringes Figure 8.14 The structure of a stopped-flow instrument for studying fast reactions in solution. Detector Mixing chamber Computer Observation chamber Stop syringe Drive Light source Absorption spectroscopy 231 A bond in a molecule behaves like a spring: stretching it through a distance x produces a restoring force F. For small displacements, the restoring force is proportional to the displacement and F –kx, where k is the force constant of the bond: the stiffer the bond, the greater the force constant. Such a system is known as a harmonic oscillator, and solution of the Schrödinger equation gives the energies Eν (v 12 ) (8.4a) where ω (k/) , v 0, 1, 2,…, and μ is the effective mass of the oscillator. For a diatomic molecule composed of atoms of masses mA and mB, 1/2 mA mB mA mB Absorption (a) The energies of molecular vibrations (8.4b) This effective mass is different for isotopologues (molecules composed of different isotopes of an element), which in turn leads to changes in Eν .If mA >> mB, then μ ≈ mB and only atom B moves appreciably during the vibration: in this case the vibrational energy levels are determined largely by mB, the mass of the lighter atom. The frequency is therefore high when the force constant is large (a stiff bond) and the effective mass of the oscillator is low (only light atoms are moved during the vibration). Vibrational energies are usually expressed in terms of the wavenumber /2πc; typical values of lie in the range 300–3800 cm–1 (Table 8.3). A note on good practice You will commonly see µ referred to as the ‘reduced mass’, for the same term appears in the separation of internal motion from translational motion. However, in polyatomic molecules each vibrational mode corresponds to the motion of different quantities of mass that depends on the individual masses in a much more complicated way, and the ‘effective mass’ is the more general term for use when discussing vibrational modes. A molecule consisting of N atoms can vibrate in 3N – 6 different, independent ways if it is nonlinear and 3N – 5 different ways if it is linear. These different, independent vibrations are called normal modes. For instance, a CO2 molecule has four normal modes of vibration (as was shown in Fig. 6.15), two corresponding to stretching the bonds and two corresponding to bending the molecule in two perpendicular planes. Bending modes typically occur at lower frequencies than stretching modes and their effective masses, and therefore their frequencies, depend on the masses of the atoms in a complicated way that reflects the extents to which the various atoms move in each mode. The modes are labelled 1, 2, etc. and are sometimes given evocative names, such as ‘symmetric stretch’ and ‘antisymmetric stretch’. Only normal modes that correspond to a changing electric dipole moment can absorb infrared radiation so only these modes are IR active and contribute to an IR spectrum. A normal mode is Raman active if it corresponds to a change in polarizability. As we saw in Chapter 6, group theory is a powerful tool for predicting the IR and Raman activities of molecular vibrations. The lowest level (v 0) of any normal mode corresponds to E0 12 , the so-called zero-point energy, which is the lowest vibrational energy that a bond can possess. In addition to the fundamental transitions with ∆v 1, vibrational spectra may also show bands arising from double quanta (∆v 2) at 2 , known as overtones, and combinations of two different vibrational modes (for example, 1 2). These special transitions may be helpful as they often arise even when the fundamental transition is not allowed by the selection rules. (b) The techniques The IR vibrational spectrum of a compound is obtained by exposing the sample to infrared radiation and recording the variation of the absorbance with frequency, wavenumber, or wavelength. A note on good practice An absorption is often stated as occurring at, say, ‘a frequency of 1000 wavenumbers’. This usage is doubly wrong. First, ‘wavenumber’ is the name of a physical observable related to frequency by = / c . Second, wavenumber is not a unit: the dimensions of wavenumber are 1/length and it is commonly reported in inverse centimetres (cm1). 500 600 700 Wavelength, l /nm Figure 8.15 Isosbestic points observed in the absorption spectral changes during reaction of HgTPP (TPP tetraphenylporphyrin) with Zn2 in which Zn replaces Hg in the macrocycle. The initial and final spectra are those of the reactant and product, which indicates that free TPP does not reach a detectable concentration during the reaction. (Adapted from C. Grant and P. Hambright, J. Am. Chem. Soc. 1969, 91, 4195.) Table 8.3 Characteristic fundamental stretching wavenumbers of some common molecular species as free ions or coordinated to metal centres Species Range/cm1 OH 3400–3600 NH 3200–3400 CH 2900–3200 BH 2600–2800 CN 2000–2200 CO (terminal) 1900–2100 CO (bridging) ∕ C=0 ⁄ NO 1600–1760 O2 920–1120 1800–1900 1675–1870 2 2 800–900 SiO 900–1100 MetalCl 250–500 Metalmetal bonds 120–400 O 232 Transmittance 8 Physical techniques in inorganic chemistry 4000 3000 2000 1000 ~ –1 Wavenumber, /cm Figure 8.16 The IR spectrum of nickel acetate tetrahydrate showing characteristic absorptions due to water and the carbonyl ∕ group (OH stretch at 3600 cm1 and C = 0 ⁄ stretch at c.1700 cm1). Rayleigh Intensity Anti-Stokes In early spectrometers, the transmission was measured as the frequency of the radiation was swept between two limits. Now the spectrum is extracted from an interferogram by Fourier transformation, which converts information in the time domain (based on the interference of waves travelling along paths of different lengths) to the frequency domain. The sample must be contained in a material that does not absorb IR radiation, which means that glass cannot be used and aqueous solutions are unsuitable unless the spectral bands of interest occur at frequencies not absorbed by water. Optical windows are typically constructed from CsI. Traditional procedures of sample preparation include KBr pellets (where the sample is diluted with dried KBr and then pressed into a translucent disc) and paraffin mulls (where the sample is produced as a suspension that is then placed as a droplet between the optical windows). These methods are still widely used, although becoming more popular are total internal reflectance devices in which the sample is simply placed in position. A typical range of an IR spectrum is between 4000 and 250 cm–1, which corresponds to a wavelength of electromagnetic radiation of between 2.5 and 40 μm; this range covers many important vibrational modes or inorganic bonds. Figure 8.16 shows a typical spectrum. In Raman spectroscopy the sample is exposed to intense laser radiation in the visible region of the spectrum. Most of the photons are scattered elastically (with no change of frequency) but some are scattered inelastically, having given up some of their energy to excite vibrations. The latter photons have frequencies different from that of the incident radiation (0) by amounts equivalent to vibrational frequencies (i) of the molecule. An advantage of Raman spectroscopy over IR spectroscopy is that aqueous solutions can be used, but a disadvantage is that linewidths are usually much greater. Conventional Raman spectroscopy involves the photon causing a transition to a ‘virtual’ excited state which then collapses back to a real lower state, emitting the detected photon in the process. The technique is not very sensitive but great enhancement is achieved if the species under investigation is coloured and the excitation laser is tuned to a real electronic transition. The latter technique is known as resonance Raman spectroscopy and is particularly valuable for studying the environment of d-metal atoms in enzymes because only vibrations close to the electronic chromophore (the group primarily responsible for the electronic excitation) are excited and the many thousands of bonds in the rest of the molecule are silent. Raman spectra may be obtained over a similar range to IR spectra (200–4000 cm–1) and Fig. 8.17 shows a typical spectrum. Note that energy can be transferred to the sample, leading to Stokes lines, lines in the spectrum at energies lower (at a lower wavenumber and greater wavelength) than the excitation energy, or transferred from the sample to the photon, leading to anti-Stokes lines, which appear at energies higher than the excitation energy. Raman spectroscopy is often complementary to IR spectroscopy as the two techniques probe vibrational modes with different activities: one mode might correspond to a change in dipole moment (IR active and observed in the IR spectrum) and another to a change in polarizability (a change in the electron distribution in a molecule caused by an applied electric field) and be seen in the Raman spectrum. We saw in Sections 6.5 and 6.6 that for group-theoretical reasons no mode can be both IR and Raman active in a molecule with a centre of inversion (the exclusion rule). Stokes Raman shift, δ~/cm–1 Figure 8.17 A typical Raman spectrum showing Rayleigh scattering (scattering of the laser light with no change in wavelength) and Stokes and anti-Stokes lines. 3000 2000 1000 0 –1000 –2000 –3000 (c) Applications of IR and Raman spectroscopy One important application of vibration spectroscopy is in the determination of the shape of an inorganic molecule. A five-coordinate structure, AX5, for instance, may adopt a square-pyramidal (C4v) or trigonal-bipyramidal (D3h) geometry. A normal mode analysis of these geometries of the kind explained in Section 6.6 reveals that a trigonal-bipyramidal AX5 molecule has five stretching modes (of symmetry species 2A A2 E, the last is a pair of doubly degenerate vibrations) of which three are IR active (A2 E, corresponding to two absorption bands) and four are Raman active (2A E, corresponding to three bands when the degeneracy of the E modes are taken into account). A similar analysis of square-pyramidal geometry shows it has four IR active stretching modes (2A1 E, three bands) and five Raman active stretching modes (2A1 B1 E, four bands). The spectra of BrF5 show three IR and four Raman BrF stretching bands, showing that this molecule is square-pyramidal, just as expected from VSEPR theory (Section 2.3). Resonance techniques 233 E X A MPL E 8 . 3 Identifying molecular geometry IR Figure 8.18 shows the IR and Raman spectra obtained from XeF4 over the same range of wavenumbers. Does XeF4 adopt a square-planar or tetrahedral geometry? Self-test 8.3 Use VSEPR theory to predict a molecular shape for XeF2 and hence determine the total number of vibrational modes expected to be observed in its IR and Raman spectra. Would any of these absorptions occur at the same frequency in both the Raman and IR spectra? A major use of IR and Raman spectroscopy is the study of numerous compounds of the d block that contain carbonyl ligands, which give rise to intense vibrational absorption bands in a region where few other molecules produce absorptions. Free CO absorbs at 2143 cm–1, but when coordinated to a metal atom in a compound the stretching frequency (and correspondingly the wavenumber) is lowered by an amount that depends on the extent to which electron density is transferred into the 2π antibonding orbital (the LUMO) by back donation from the metal atom (Section 22.5). The CO stretching absorption also allows terminal and bridging ligands to be distinguished, with bridging ligands occurring at lower frequencies. Isotopically labelled compounds show a shift in the absorption bands (by about 40 cm−1 to lower wavenumbers when 13C replaces 12C in a CO group) through a change in effective mass (eqn 8.4b), and this effect can be used to assign spectra and probe reaction mechanisms involving this ligand. The speed of data acquisition possible with Fourier-transform IR (FTIR) spectroscopy has meant that it can be incorporated into rapid kinetic techniques, including ultrafast laser photolysis and stopped-flow methods. Raman and IR spectroscopy are excellent methods for studying molecules that are formed and trapped in inert matrices, the technique known as matrix isolation. The principle of matrix isolation is that highly unstable species that would not normally exist can be generated in an inert matrix such as solid xenon. Resonance techniques Several techniques of structural investigation depend on bringing energy level separations into resonance with electromagnetic radiation, with the separations in some cases controlled by the application of a magnetic field. Two of these techniques involve magnetic resonance: in one, the energy levels are those of magnetic nuclei (nuclei with non-zero spin, I > 0); in the other, they are the energy levels of unpaired electrons. 8.5 Nuclear magnetic resonance Key points: Nuclear magnetic resonance is suitable for studying compounds containing elements with magnetic nuclei, especially hydrogen. The technique gives information on molecular structure, including chemical environment, connectivity, and internuclear separations. It also probes molecular dynamics and is an important tool for investigating rearrangement reactions occurring on a millisecond timescale. Nuclear magnetic resonance (NMR) is the most powerful and widely used spectroscopic method for the determination of molecular structures in solution and pure liquids. In many cases it provides information about shape and symmetry with greater certainty than is possible with other spectroscopic techniques, such as IR and Raman spectroscopy. It also provides information about the rate and nature of the interchange of ligands in fluxional molecules and can be used to follow reactions, in many cases providing exquisite mechanistic detail. The technique has been used to obtain the structures of protein molecules with molar masses of up to 30 kg mol–1 (corresponding to a molecular mass of 30 kDa) and complements the more static descriptions obtained with X-ray single-crystal diffraction. However, unlike X-ray diffraction, NMR studies of molecules in solution generally cannot Signal Answer An AB4 molecule may be tetrahedral (Td), square-planar (D4h), square-pyramidal (C4v), or see-saw (C2v). The spectra have no absorption energy in common and therefore it is likely that the molecule has a centre of symmetry. Of the symmetry groups available, only square-planar geometry, D4h, has a centre of symmetry. 580 1136 1105 Raman 1000 500 Wavenumber, ~/cm–1 Figure 8.18 The IR and Raman spectra of XeF4. 234 8 Physical techniques in inorganic chemistry provide precise bond distances and angles, although it can provide some information on internuclear separations. It is a nondestructive technique because the sample can be recovered from solution after the resonance spectrum has been collected. The sensitivity of NMR depends on several factors, including the abundance of the isotope and the size of its nuclear magnetic moment. For example, 1H, with 99.98 per cent natural abundance and a large magnetic moment, is easier to observe than 13C, which has a smaller magnetic moment and only 1.1 per cent natural abundance. With modern multinuclear NMR techniques it is particularly easy to observe spectra for 1H, 19F, and 31P, and useful spectra can also be obtained for many other elements. Table 8.4 lists a selection of nuclei and their sensitivities. A common limitation for exotic nuclei is the presence of a nuclear quadrupole moment, a nonuniform distribution of electric charge (which is present for all nuclei with nuclear spin quantum number I > 12 ), which broadens signals and degrades spectra. Nuclei with even atomic numbers and even mass numbers (such as 12 C and 16O) have zero spin and are invisible in NMR. mI = – 12 (β) B=0 B>0 ∆E = hγB (a) Observation of the spectrum mI = + 12 (α) Figure 8.19 When a nucleus with spin I > 0 is in a magnetic field, its 2I 1 orientations (designated mI) have different energies. This diagram shows the energy levels of a nucleus with I 21 (as for 1H, 13C, 31P). A nucleus of spin I can take up 2I 1 orientations relative to the direction of an applied magnetic field. Each orientation has a different energy (Fig. 8.19), with the lowest level the most highly populated. The energy separation of the two states mI 12 and mI − 12 of a spin- 12 nucleus (such as 1H or 13C) is ∆E = B0 (8.5) where B0 is the magnitude of the applied magnetic field (more precisely, the magnetic induction in tesla, 1 T 1 kg s–2 A–1) and is the magnetogyric ratio of the nucleus, that is the ratio of its magnetic moment to its spin angular momentum. With modern superconducting magnets producing a field of 5–20 T, resonance is achieved with electromagnetic radiation in the range 200–900 MHz. Because the difference in energy of the mI 12 and mI − 12 states in the applied magnetic field is small, the population of the lower level is only marginally greater than that of the higher level. Consequently, Table 8.4 Nuclear spin characteristics of common nuclei Natural abundance/% Sensitivity 1 99.98 5680 1 2 2 0.015 0.00821 1 15.351 7 92.58 1540 3 2 38.863 80.42 754 3 2 32.072 1.11 1.00 1 2 25.145 10.137 H H Li 11 B 13 C 100.000 N 0.37 0.0219 1 2 O 0.037 0.0611 3 2 13.556 100 4730 1 2 94.094 100 525 3 2 26.452 19.867 15 17 Spin NMR frequency/MHz† Nucleus 19 F 23 Na 29 Si 4.7 2.09 1 2 31 P 100 377 1 2 40.481 100 0.668 1 2 4.900 100 0.177 1 2 3.185 4.654 89 Y 103 Rh Ag 48.18 0.276 1 2 Sn 8.58 28.7 1 2 37.272 14.4 0.0589 1 2 4.166 33.8 19.1 1 2 21.462 5.42 1 2 17.911 109 119 183 W 195 Pt 199 Hg 16.84 Sensitivity is relative to C 1 and is the product of the relative sensitivity of the isotope and the natural abundance. † At 2.349 T (a ‘100 MHz spectrometer’). 13 Resonance techniques Superconducting magnet 235 Computer Probe Preamplifier Receiver Detector Figure 8.20 The layout of a typical NMR spectrometer. The link between transmitter and detector is arranged so that only lowfrequency signals are processed. Transmitter the sensitivity of NMR is low but can be increased by using a stronger magnetic field, which increases the energy difference and therefore the population difference and the signal intensity. Spectra were originally obtained in a continuous wave (CW) mode in which the sample is subjected to a constant radiofrequency and the resonances encountered as the field is increased are recorded as the spectrum, or the field is held constant and the radiofrequency is swept. In contemporary spectrometers, the energy separations are identified by exciting nuclei in the sample with a sequence of radiofrequency pulses and then observing the return of the nuclear magnetization back to equilibrium. Fourier transformation then converts the time-domain data to the frequency domain with peaks at frequencies corresponding to transitions between the different nuclear energy levels. Figure 8.20 shows the experimental arrangement of an NMR spectrometer. (b) Chemical shifts The frequency of an NMR transition depends on the local magnetic field experienced by the nucleus and is expressed in terms of the chemical shift, , the difference between the resonance frequency of nuclei () in the sample and that of a reference compound (o): ␯ ␯ 106 ␯ 100 Hz (8.6) A note on good practice The chemical shift is dimensionless. However, common practice is to report is as ‘parts per million’ (ppm) in acknowledgement of the factor of 106 in the definition. This practice is unnecessary. A common standard for 1H, 13C, or 29Si spectra is tetramethylsilane Si(CH3)4 (TMS). When < 0 the nucleus is said to be shielded (with a resonance that is said to occur to ‘high field’) relative to the standard; > 0 corresponds to a nucleus that is deshielded (with a resonance that is said to occur to ‘low field’) with respect to the reference. An H atom bound to a closed-shell, low-oxidation-state, d-block element from Groups 6 to 10 (such as [HCo(CO)4]) is generally found to be highly shielded whereas in an oxoacid (such as H2SO4) it is deshielded. From these examples it might be supposed that the higher the electron density around a nucleus, the greater its shielding. However, as several factors contribute to the shielding, a simple physical interpretation of chemical shifts in terms of electron density is generally not possible. The chemical shifts of 1H and other nuclei in various chemical environments are tabulated, so empirical correlations can often be used to identify compounds or the element to which the resonant nucleus is bound. For example the H chemical shift in CH4 is only 0.1 because the H nuclei are in an environment similar to that in tetramethylsilane, but the H chemical shift in GeH4 is 3.1 (Fig. 8.21). Chemical shifts are different for the same element in inequivalent positions within a molecule. For instance, in ClF3 the chemical shift of the unique 19F nucleus is separated by ∆ 120 from that of the other two F nuclei (Fig. 8.22). δ 3.1 Figure 8.21 The 1H-NMR spectrum of GeH4. 236 8 Physical techniques in inorganic chemistry F(ax) (c) Spin–spin coupling F(ax) F(ax Cll F(eq) F(eq) F(ax) F(ax δ Figure 8.22 The 19F-NMR spectrum of ClF3. Structural assignment is often helped by the observation of the spin–spin coupling, which gives rise to multiplets in the spectrum due to interactions between nuclear spins. Spin– spin coupling arises when the orientation of the spin of a nearby nucleus affects the energy of another nucleus and causes small changes in the location of the latter’s resonance. The strength of spin–spin coupling, which is reported as the spin–spin coupling constant, J (in hertz, Hz), decreases rapidly with distance through chemical bonds, and in many cases is greatest when the two atoms are directly bonded to each other. In first-order spectra, which are being considered here, the coupling constant is equal to the separation of adjacent lines in a multiplet. As can be seen in Fig. 8.21, J(1H–73Ge) ≈ 100 Hz. The resonances of chemically equivalent nuclei do not display the effects of their mutual spin–spin coupling. Thus, a single 1H signal is observed for the CH3I molecule even though there is coupling between the H nuclei. A multiplet of 2I 1 lines is obtained when a spin- 12 nucleus (or a set of symmetry-related spin- 12 nuclei) is coupled to a nucleus of spin I. In the spectrum of GeH4 shown in Fig. 8.21, the single central line arises from the four equivalent H nuclei in GeH4 molecules that contain Ge isotopes with I 0. This central line is flanked by ten evenly spaced but less intense lines that arise from a small fraction of GeH4 that contains 73Ge, for which I 92 , the four 1 H nuclei are coupled to the 73Ge nucleus to yield a ten-line multiplet (2 92 1 10). The coupling of the nuclear spins of different elements is called heteronuclear coupling; the GeH coupling just discussed is an example. Homonuclear coupling between nuclei of the same element is detectable when the nuclei are in chemically inequivalent locations. ■ A brief illustration. The 19F NMR spectrum of ClF3 is shown in Fig. 8.22. The signal ascribed to the two axial F nuclei (each with I 21 ) is split into a doublet by the single equatorial 19F nucleus, and the latter is split into a triplet by the two axial 19F nuclei (19F is in 100 per cent abundance). Thus, the pattern of 19F resonances readily distinguishes this unsymmetrical structure from trigonal-planar and trigonal-pyramidal structures, both of which would have equivalent F nuclei and hence a single 19 F resonance. ■ The sizes of 1H1H homonuclear coupling constants in organic molecules are typically 18 Hz or less. By contrast, 1HX heteronuclear coupling constants can be several hundred hertz. Homonuclear and heteronuclear coupling between nuclei other than 1H can lead to coupling constants of many kilohertz. The sizes of coupling constants are often related to the geometry of a molecule by noting empirical trends. In square-planar Pt(II) complexes, J(PtP) is sensitive to the group trans to a phosphine ligand and the value of J(PtP) increases in the following order of trans ligands: R– H– PR3 NH3 Br– Cl– For example, cis-[PtCl2(PEt3)2], where Cl is trans to P, has J(Pt–P) 3.5 kHz, whereas trans-[PtCl2(PEt3)2], with P trans to P, has J(PtP) 2.4 kHz. These systematic variations allow cis and trans isomers to be distinguished quite readily. The rationalization for the variation in the sizes of the coupling constants above stems from the fact that a ligand that exerts a large trans influence (Section 21.4) substantially weakens the bond trans to itself, causing a reduction in the NMR coupling between the nuclei. (d) Intensities The integrated intensity of a signal arising from a group of chemically equivalent nuclei is proportional to the number of nuclei in the group. Provided sufficient time is allowed during spectrum acquisition for the full relaxation of the observed nucleus, integrated intensities can be used to aid spectral assignment for most nuclei. However, for nuclei with low sensitivities (such as 13C), allowing sufficient time for full relaxation may not be realistic, so quantitative information from signal intensities is difficult to obtain. For instance, in the spectrum of ClF3 the integrated 19F intensities are in the ratio 2:1 (for the doublet and triplet, respectively). This pattern is consistent with the structure and the splitting pattern because it indicates the presence of two equivalent F nuclei and one inequivalent F nucleus. Thus, the pattern of 19F resonances distinguishes this less symmetrical structure from a trigonal-planar structure, D3h, which would have a single resonance with all F environments equivalent. The relative intensities of the 2N 1 lines in a multiplet that arises from coupling to N equivalent spin- 12 nuclei are given by Pascal’s triangle (1); thus three equivalent protons 237 Resonance techniques give a 1:3:3:1 quartet. Groups of nuclei with higher spin quantum numbers give different patterns. The 1H-NMR spectrum of HD, for instance, consists of three lines of equal intensity as a result of coupling to the 2H nucleus (I 1, with 2I 1 3 orientations). 1 1 1 3 4 1 5 1 2 3 1 E X A MPL E 8 . 4 Interpreting an NMR spectrum Explain why the 19F-NMR spectrum of SF4 consists of two 1:2:1 triplets of equal intensity. 1 1 6 0 1 1 4 0 1 1 5 1 Pascal’s triangle Answer We need to recall that an SF4 molecule (2) has a distorted tetrahedral or ‘see-saw’ structure based on a trigonal-bipyramidal arrangement of electron pairs with a lone pair occupying one of the equatorial sites. The two axial F nuclei are chemically different from the two equatorial F nuclei and give two signals of equal intensity. The signals are in fact 1:2:1 triplets, as each 19F nucleus couples to the two chemically distinct 19F nuclei. F S Self-test 8.4 (a) Explain why the 77Se-NMR spectrum of SeF4 consists of a triplet of triplets. (b) Use the isotope information in Table 8.4 to explain why the 1H resonance of the hydrido (H–) ligand in cis-[Rh(CO) H(PMe3)2] consists of eight lines of equal intensity (77Se I = 12 ). (e) Fluxionality The timescale of NMR is slow in the sense that structures can be resolved provided their lifetime is not less than a few milliseconds. For example, Fe(CO)5 shows just one 13C resonance, indicating that, on the NMR timescale, all five CO groups are equivalent. However, the IR spectrum (of timescale about 1 ps) shows distinct axial and equatorial CO groups, and by implication a trigonal-bipyramidal structure. The observed 13C-NMR spectrum of Fe(CO)5 is the weighted average of these separate resonances. Because the temperature at which NMR spectra are recorded can easily be changed, samples can often be cooled down to a temperature at which the rate of interconversion becomes slow enough for separate resonances to be observed. Figure 8.23, for instance, shows the idealized 31P-NMR spectra of [RhMe(PMe3)4] at room temperature and at –80ºC. At low temperatures the spectrum consists of a doublet of doublets of relative intensity 3 near −24, which arises from the equatorial P atoms (coupled to 103Rh and the single axial 31P), and a quartet of doublets of intensity 1, derived from the equatorial P atom (coupled to 103Rh and the three equatorial 31P atoms). At room temperature the scrambling of the PMe3 groups makes them all equivalent and a doublet is observed (from coupling to 103Rh). Careful control allows determination of the temperature at which the spectrum changes from the high-temperature form to the low-temperature form (‘the coalescence temperature’) and thence to determine the barrier to interconversion. 2 SF4 PMe3 CH3 Rh 3 [RhMe(PMe3)4] (f) Solid-state NMR The NMR spectra of solids rarely show the high resolution that can be obtained from solution NMR. This difference is mainly due to anisotropic interactions such as dipolar magnetic couplings between nuclei, which are averaged in solution due to molecular tumbling, and Room temperature –80°C 0 –4 –8 –12 –16 –20 δ –24 Figure 8.23 The 31P-NMR spectra of [RhMe(PMe3)4] (3) at room temperature and at –80°C. 1 238 8 Physical techniques in inorganic chemistry B=0 B>0 hν = g B B ms = + 12 (α) ms = – 12 (β) Signal Figure 8.24 When an unpaired electron is in a magnetic field, its two orientations (, ms 21 and , ms – 21 ) have different energies. Resonance is achieved when the energy separation matches the energy of the incident microwave photons. Microwave source Magnetic field Detector Electromagnet Sample cavity Phase Modulation sensitive input detector Figure 8.25 The layout of a typical continuous wave EPR spectrometer. long-range magnetic interactions arising from the fixed atomic positions. These effects mean that in the solid state chemically equivalent nuclei might be in different magnetic environments and so have different resonance frequencies. A typical result of these additional couplings is to produce very broad resonances, often more than 10 kHz wide. To average out anisotropic interactions, samples are spun at very high speeds (typically 10–25 kHz) at the ‘magic angle’ (54.7º) with respect to the field axis. At this angle the interaction of parallel magnetic dipoles and quadrupole interactions, which typically vary as 1 – 3 cos2 θ, is zero. This so-called magic-angle spinning (MAS) reduces the effect of anisotropy substantially but often leaves signals that are still significantly broadened compared to those from solution. The broadening of signals can sometimes be so great that signal widths are comparable to the chemical shift range for some nuclei. This broadening is a particular problem for 1H, which typically has a chemical shift range of ∆ 10. Broad signals are less of a problem for nuclei such as 195Pt, for which the range in chemical shifts is ∆ 16 000, although this large range can be reflected in large anisotropic linewidths. Quadrupolar nuclei (those with I > 12 ) present additional problems as the peak position becomes field dependent and can no longer be identified with the chemical shift. Despite these difficulties, developments in the technique have made possible the observation of high-resolution NMR spectra for solids and are of far-reaching importance in many areas of chemistry. An example is the use of 29Si-MAS-NMR to determine the environments of Si atoms in natural and synthetic aluminosilicates such as zeolites. The techniques of homonuclear and heteronuclear ‘decoupling’ enhance the resolution of spectra and the use of multiple pulse sequences has allowed the observation of spectra with some difficult samples. The high-resolution technique CPMAS-NMR, a combination of MAS with cross-polarization (CP), usually with heteronuclear decoupling, has been used to study many compounds containing 13C, 31P, and 29Si. The technique is also used to study molecular compounds in the solid state. For example, the 13C-CPMAS spectrum of [Fe2(C8H8)(CO)5] at –160ºC indicates that all C atoms in the C8 ring are equivalent on the timescale of the experiment. The interpretation of this observation is that the molecule is fluxional even in the solid state. 8.6 Electron paramagnetic resonance Key points: Electron paramagnetic resonance spectroscopy is used to study compounds possessing unpaired electrons, particularly those containing a d-block element; it is often the technique of choice for identifying and studying metals such as Fe and Cu at the active sites of metalloenzymes. Electron paramagnetic resonance (EPR; or electron spin resonance, ESR) spectroscopy, the observation of resonant absorption by unpaired electrons in a magnetic field, is a technique for studying paramagnetic species such as organic and main-group radicals. Its principal importance in inorganic chemistry is for characterizing compounds containing the d- and f-block elements. The simplest case is for a species having one unpaired electron (s 12 ): by analogy with NMR, the application of an external magnetic field B0 produces a difference in energy between the ms 12 and ms – 12 states of the electron, with E gB B0 (8.7) where µB is the Bohr magneton and g is a numerical factor known simply as the g value (Fig. 8.24). The conventional method of recording EPR spectra is to use a continuous wave (CW) spectrometer (Fig. 8.25), in which the sample is irradiated with a constant microwave frequency and the applied magnetic field is varied. The resonance frequency for most spectrometers is approximately 9 GHz, and the instrument is then known as an ‘X-band spectrometer’. The magnetic field in an X-band spectrometer is about 0.3 T. Laboratories specializing in EPR spectroscopy often have a range of instruments, each operating at different fields. Thus an S-band (resonant frequency 3 GHz) spectrometer and particularly those operating at high fields, Q-band (35 GHz) and W-band (95 GHz) spectrometers, are used to complement the information gained with an X-band spectrometer. The use of high fields improves resolution and can simplify spectra by reducing interactions between paramagnetic centres. Pulsed EPR spectrometers are becoming commercially available and offering new opportunities analogous to the way that pulsed Fourier-transform techniques have revolutionized Resonance techniques 239 NMR. Pulsed EPR techniques provide time resolution, making it possible to measure the dynamic properties of paramagnetic systems. (a) The g value For a free electron g 2.0023 but in compounds this value is altered by spin–orbit coupling which changes the local magnetic field experienced by the electron. For many species, particularly d-metal complexes, g values may be highly anisotropic so that the resonance condition depends on the angle that the paramagnetic species makes to the applied field (Fig. 8.26). The illustration shows the EPR spectra of frozen solutions or ‘glasses’, expected for isotropic (all three g values the same along perpendicular axes), axial (two the same), and rhombic (all three different) spin systems. The sample (which is usually contained in a quartz tube) comprises the paramagnetic species in dilute form, either in the solid state (doped crystals or powders) or in solution. Relaxation is so efficient for d-metal ions that the spectra are often too broad to detect; consequently, liquid nitrogen and sometimes liquid helium are used to cool the sample. Frozen solutions behave as amorphous powders, so resonances are observed at all the g values of the compound, analogous to powder X-ray diffraction. More detailed studies can be made with oriented single crystals. Provided relaxation is slow, EPR spectra can also be observed at room temperature in liquids. Spectra can also be obtained for systems having more than one unpaired electron, such as triplet states, but the theoretical background is much more complicated. Whereas species having an odd number of electrons are usually detectable, it can be difficult to observe spectra for systems having an even number of electrons. Table 8.5 shows the suitability of common paramagnetic species for EPR detection. g (a) g\|\| g⊥ (b) (b) Hyperfine coupling The hyperfine structure of an EPR spectrum, the multiplet structure of the resonance lines, is due to the coupling of the electron spin to any magnetic nuclei present. A nucleus with spin I splits an EPR line into 2I 1 lines of the same intensity (Fig. 8.27). A distinction is sometimes made between the hyperfine structure due to coupling to the nucleus of the atom on which the unpaired electron is primarily located and the ‘superhyperfine coupling’, the coupling to ligand nuclei. Superhyperfine timescoupling to ligand nuclei is used to measure the extent of electron delocalization and covalence in metal complexes (Fig. 8.28). Table 8.5 EPR detectability of common d-metal ions Usually easy to study Species Usually difficult to study or diamagnetic S Species S Ti(III) 1 2 Ti(II) 1 Cr(III) 3 2 Ti(IV) 0 V(IV) 1 2 Cr(II) 2 Fe(III) 1 2 , 52 V(III) 1 Co(II) 3 2 , 1 2 V(V) 0 Ni(III) 3 2 , 1 2 Fe(II) 2, 1, 0 Ni(I) 1 2 Co(III) 0 Cu(II) 1 2 Co(I) 0 Mo(V) 1 2 Ni(II) 1 W(V) 1 2 Cu(I) 0 Mo(VI) 0 Mo(IV) 1, 0 W(VI) 0 gxx (c) gzz gyy Figure 8.26 The forms of EPR powder (frozen solution) spectra expected for different types of g value anisotropy. The green line is the absorption and the red line the first derivative of the absorption (its slope). For technical reasons related to the detection technique, the first derivative is normally observed in EPR spectrometers. mI = + mI = – 1 2 1 2 mI = – mI = + 1 2 1 2 ms = + 12 (α) ms = – 12 (β) Figure 8.27 When a magnetic nucleus is present, its 2I 1 orientations give rise to a local magnetic field that splits each spin state of an electron into 2I 1 levels. The allowed transitions (∆ms 1, ∆mI 0) give rise to the hyperfine structure of an EPR spectrum. 240 8 Physical techniques in inorganic chemistry A\|\| E X A M PL E 8 . 5 Interpreting superhyperfine coupling Use the data provided in Table 8.4 to suggest how the EPR spectrum of a Co(II) complex might change when a single OH ligand is replaced by an F ligand. Answer We need to note that 19F (100 per cent abundance) has nuclear spin I 21 whereas 16O (close to 100 per cent) has I 0. Consequently, any part of the EPR spectrum might be split into two lines. g\|\| = 2.30 g⊥ = 2.08 Self-test 8.5 Predict how you might show that an EPR signal that is characteristic of a new material arises from tungsten sites. Figure 8.28 The EPR spectrum of Cu2 (d9, one unpaired electron) in frozen aqueous solution. The tetragonally distorted Cu2 ion shows an axially symmetrical spectrum in which hyperfine coupling to Cu (I 32 , four hyperfine lines) is clearly evident for the g\|\| component. Cobalt source 57 Co 136.32 eV 9% 14.41 eV 0 57 Fe 91% 57 57 Fe Fe γ-Ray photon (14.41 eV) Detector Sample Figure 8.29 The layout of a Mössbauer spectrometer. The speed of the carriage is adjusted until the Doppler shifted frequency of the emitted -ray matches the corresponding nuclear transition in the sample. The inset shows the nuclear transitions responsible for the emission of the -ray. 8.7 Mössbauer spectroscopy Key point: Mössbauer spectroscopy is based on the resonant absorption of -radiation by nuclei and exploits the fact that nuclear energies are sensitive to the electronic and magnetic environment. The Mössbauer effect makes use of recoilless absorption and emission of -radiation by a nucleus. To understand what is involved, consider a radioactive 57Co nucleus that decays by electron capture to produce an excited state of 57Fe, denoted 57Fe (Fig. 8.29). This nuclide decays to another excited state, denoted 57Fe, that lies 14.41 eV above the ground state and which emits a -ray of energy 14.41 eV as it decays. If the emitting nucleus (the source) is pinned down in a rigid lattice the nucleus does not recoil and the radiation produced is highly monochromatic. If a sample containing 57Fe (which occurs with 2 per cent natural abundance) is placed close to the source, the monochromatic -ray emitted by 57Fe can be expected to be absorbed resonantly. However, because changes in the exact electronic and magnetic environment affect the nuclear energy levels to a small degree, resonant absorption occurs only if the environment of the sample 57Fe is chemically identical to that of the emitting 57Fe nucleus. It might seem that the energy of the -ray cannot easily be varied, but the Doppler effect can be exploited for moving the transmitting nucleus at a speed v relative to the sample and a frequency shift is induced of magnitude ∆ (v/c). This shift is sufficient even for velocities of a few millimetres per second, to match the absorption frequency to the transmitted frequency. A Mössbauer spectrum is a portrayal of the resonant absorption peaks that occur as the velocity of the source is changed. The Mössbauer spectrum of a sample containing iron in a single chemical environment can be expected to consist of a single line due to absorption of radiation at the energy required (∆E) to excite the nucleus from its ground state to the excited state. The difference between ∆E of the sample and that of metallic 57Fe is called the isomer shift and is expressed in terms of the velocity that is required to achieve resonance by the Doppler shift. The value of ∆E depends on the magnitude of the electron density at the nucleus and although this effect is primarily due to s electrons (because their wavefunctions are nonzero at the nucleus), shielding effects cause ∆E to be sensitive to the number of p and d electrons too. As a result, different oxidation states, such as Fe(II), Fe(III), and Fe(IV), can be distinguished as well as ionic and covalent bonding. The element most suited for study by Mössbauer spectroscopy is iron. This element has great importance and is common in minerals, oxides, alloys, and biological samples, for which other many other characterisation techniques, for example NMR, are less effective. Mössbauer spectroscopy is also used to study some other nuclei, including 119Sn, 129I, and 197Au which have suitable nuclear energy levels and -emission half-lives to get useful spectra. Because a 57Fe nucleus has I 32 , it possesses an electric quadrupole moment (a nonspherical distribution of electric charge) which interacts with electric field gradients. As a result, and providing the electronic environment of the nucleus is not isotropic, the Mössbauer spectrum splits into two lines of separation ∆EQ (Fig. 8.30). The splitting is a good guide as to the state of Fe in proteins and minerals as it depends on the oxidation state and the distribution of d-electron density. Magnetic fields produced by a large magnet or internally in some ferromagnetic materials also cause changes in the energies of the various spin orientation states of an 57Fe nucleus. As a result, the Mössbauer spectrum splits into six lines. Ionization-based techniques 3 ± 2 3 2 1 – 2 1 2 1 2 3 + 2 241 – ± 3 2 1 2 1 – 2 1 2 Figure 8.30 The effects of an electric field gradient and a magnetic field on the energy levels involved in the Mössbauer technique for Fe samples. The derived spectra show, from left to right, the origins of the isomer shift, quadrupole coupling, and magnetic hyperfine coupling (but no quadrupole splitting). The spectrum of (a) K4Fe(CN)6.3H2O, octahedral low spin d6, is representative of a highly symmetric environment and shows a single peak with an isomer shift. The spectrum of (b) FeSO4.7H2O, d6 in a nonsymmetric environment, shows quadrupolar splitting. E X A MPL E 8 .6 Interpreting a Mössbauer spectrum The isomer shifts of Fe(II) compounds relative to metallic iron, Fe(0), are generally in the range 1 to 1.5 mm s–1 whereas isomer shifts for Fe(III) compounds lie in the range 0.2 to 0.5 mm s1. Explain these values in terms of the electronic configurations of Fe(0), Fe(II), and Fe(III). Answer The outermost electron configurations of Fe(0), Fe(II), and Fe(III) are 4s23d6, 3d6, and 3d5, respectively. The s-electron density at the nucleus is reduced in Fe(II) compared with Fe(0), producing a large positive isomer shift. When a 3d electron is removed to produce Fe(III) from Fe(II), there is a small increase in s-electron density at the nucleus (as the 3d electrons partly screen the nucleus from the inner s electrons) and the isomer shift becomes less positive. Self-test 8.6 Predict a likely isomer shift for iron in Sr2FeO4. UV X-ray Ionization-based techniques Ionization-based techniques measure the energies of products, electrons, or molecular fragments generated when a sample is ionized by bombardment with high-energy radiation or particles. 8.8 Photoelectron spectroscopy Key point: Photoelectron spectroscopy is used to determine the energies and order of orbitals in molecules and solids by analysing the kinetic energies of photoejected electrons. The basis of photoelectron spectroscopy (PES) is the measurement of the kinetic energies of electrons (photoelectrons) emitted by ionization of a sample that is irradiated with highenergy monochromatic radiation (Fig. 8.31). It follows from the conservation of energy Kinetic energy, Ek Valence orbitals Ionization energy Core orbitals Figure 8.31 In photoelectron spectroscopy, high-energy electromagnetic radiation (UV for the ejection of valence electrons, X-ray for core electrons) expels an electron from its orbital, and the kinetic energy of the photoelectron is equal to the difference between the photon energy and the ionization energy of the electron. 242 8 Physical techniques in inorganic chemistry that the kinetic energy of the ejected photoelectrons, Ek, is related to their ionization energies, Ei, from their orbitals by the relation Ek h – Ei πg πu σg 12 14 16 18 20 Ionization energy, I/eV Figure 8.32 The UV photoelectron spectra of O2. Loss of an electron from 2g (see the MO energy-level diagram, Fig. 2.12) gives rise to two bands because the unpaired electron that remains can be parallel or antiparallel to the two unpaired electrons in the 1πg orbitals. (8.8) where is the frequency of the incident radiation. Koopmans’ theorem states that the ionization energy is equal to the negative of the orbital energy, so the determination of the kinetic energies of photoelectrons can be used to determine orbital energies. The theorem assumes that the energy involved in electron reorganization after ionization is offset by the increase in electron–electron repulsion energy as the orbital contracts. This approximation is usually taken as reasonably valid. There are two major types of photoionization technique, X-ray photoelectron spectroscopy (XPS) and ultraviolet photoelectron spectroscopy (UPS). Although much more intense sources can be obtained using synchrotron beam lines, the standard laboratory source for XPS is usually a magnesium or aluminium anode that is bombarded by a highenergy electron beam. This bombardment results in radiation at 1.254 and 1.486 keV, respectively, due to the transition of a 2p electron into a vacancy in the 1s orbital caused by ejection of an electron. These energetic photons cause ionizations from core orbitals in other elements that are present in the sample; the ionization energies are characteristic of the element and its oxidation state. Because the linewidth is high (usually 1–2 eV), XPS is not suitable for probing fine details of valence orbitals but can be used to study the band structures of solids. The mean free path of electrons in a solid is only about 1 nm, so XPS is suitable for surface elemental analysis, and in this application it is commonly known as electron spectroscopy for chemical analysis (ESCA). The source for UPS is typically a helium discharge lamp that emits He(I) radiation (21.22 eV) or He(II) radiation (40.8 eV). Linewidths are much smaller than in XPS, so the resolution is far greater. The technique is used to study valence-shell energy levels and the vibrational fine structure often provides important information on the bonding or antibonding character of the orbitals from which electrons are ejected (Fig. 8.32). When the electron is removed from a nonbonding orbital the product is formed in its vibrational ground state, and a narrow line is observed. However, when the electron is removed from a bonding or antibonding orbital the resulting ion is formed in several different vibrational states and extensive fine structure is observed. Bonding and antibonding orbitals can be distinguished by determining whether the vibrational frequencies in the resulting ion are higher or lower than for the original molecule. Another useful aid is the comparison of photoelectron intensities for a sample irradiated with He(I) and He(II). The higher energy source preferentially ejects electrons from d or f orbitals, allowing these contributions to be distinguished from s and p orbitals, for which He(I) causes higher intensities. The origin of this effect lies in differences in absorption cross-sections (see Further reading). 8.9 X-ray absorption spectroscopy Ei + 50 eV EXAFS NEXAFS XANES Ei Ei+ 10 eV Pre-edge Signal Key point: X-ray absorption spectra can be used to determine the oxidation state of an element in a compound and to investigate its local environment. Energy, E Figure 8.33 A typical X-ray absorption edge spectrum defining the various regions discussed in the text. As mentioned in the previous section, the intense X-ray radiation from synchrotron sources may be used to eject electrons from the cores of elements present in a compound. X-ray absorption spectra (XAS) are obtained by varying the photon energy across a range of energies at which electrons in the various atoms present in a compound can be excited and ionized (typically between 0.1 and 100 keV). The characteristic absorption energies correspond to the binding energies of different inner-shell electrons of the various elements present. Thus the frequency of an X-ray beam may be swept across an absorption edge of a selected element and information on the oxidation state and neighbourhood of this chosen chemical element obtained. Figure 8.33 shows a typical X-ray absorption spectrum. Each region of the spectrum can provide different useful information on the chemical environment of the element under investigation: 1. Just prior to the absorption edge is the ‘pre-edge’ where core electrons are excited to higher empty orbitals but not ejected. This ‘pre-edge structure’ can provide information on the energies of excited electronic states and also on the local symmetry of the atom. Ionization-based techniques 243 In the edge region, where the photon energy, E is between Ei and Ei 10 eV, where Ei is the ionization energy, the ‘X-ray absorption near-edge structure’ (XANES) is observed. Information that can be extracted from the XANES region includes oxidation state and the coordination environment, including any subtle geometrical distortions. The near-edge structure can also be used as a ‘fingerprint’, as it is characteristic of a specific environment and valence state. The presence and quantity of a compound in a mixture may be determined from analysis of this region of the spectrum. 3. The‘near-edge X-ray absorption fine structure’ (NEXAFS) region lies between Ei 10 eV and Ei 50 eV. It has particular application to chemisorbed molecules on surfaces because it is possible to infer information about the orientation of the adsorbed molecule. 4. The ‘extended X-ray absorption fine structure’ (EXAFS) region lies at energies greater than Ei 50 eV. The photoelectrons ejected from a particular atom by absorption of X-ray photons with energies in this region may be backscattered by any adjacent atoms. This effect can result in an interference pattern that is detected as periodic variations in intensity at energies just above the absorption edge. In EXAFS these variations are analysed to reveal the nature (in terms of their electron density) and number of nearby atoms and the distance between the absorbing atom and the scattering atom. An advantage of this method is that it can provide bond lengths in amorphous samples and for species in solutions. 8.10 Mass spectrometry Key point: Mass spectrometry is a technique for determining the mass of a molecule and of its fragments. Mass spectrometry measures the mass-to-charge ratio of gaseous ions. The ions can be either positively or negatively charged, and it is normally trivial to infer the actual charge on an ion and hence the mass of a species. It is a destructive analytical technique because the sample cannot be recovered for further analysis. The precision of measurement of the mass of the ions varies according to the use being made of the spectrometer (Fig. 8.34). If all that is required is a crude measure of the mass, for instance to within ±mu (where mu is the atomic mass constant, 1.66054 1027 kg), then the resolution of the mass spectrometer need be of the order of only 1 part in 104. In contrast, to determine the mass of individual atoms so that the mass defect can be determined, the precision must approach 1 part in 1010. With a mass spectrometer of this precision, molecules of nominally the same mass such as 12C16O (of mass 27.9949mu) can be distinguished from 14N2 (of mass 28.0061mu) and the elemental and isotopic composition of ions of nominal mass less than 1000mu may be determined unambiguously. (a) Ionization and detection methods The major practical challenge with mass spectrometry is the conversion of a sample into gaseous ions. Typically, less than a milligram of compound is used. Many different experimental arrangements have been devised to produce gas-phase ions but all suffer from a tendency to fragment the compound of interest. Electron impact ionization (EI) relies on bombarding a sample with high-energy electrons to cause both vaporization and ionization. A disadvantage is that EI tends to induce considerable decomposition in larger molecules. Fast atom bombardment (FAB) is similar to EI, but bombardment of the sample with Sector magnet Ion path Ion source Accelerating plate Detector Figure 8.34 A magnetic sector mass spectrometer. The molecular fragments are deflected according to their mass-to-charge ratio, allowing for separation at the detector. 244 8 Physical techniques in inorganic chemistry Accelerating plate Figure 8.35 A time-of-flight (TOF) mass spectrometer. The molecular fragments are accelerated to different speeds by the potential difference and arrive at different times at the detector. Ion source High vacuum TOF tube Detector fast neutral atoms is used to vaporize and ionize the sample; it induces less fragmentation than EI. Matrix-assisted laser desorption/ionization (MALDI) is similar to EI, but a short laser pulse is used to the same effect; this technique is particularly effective with polymeric samples. In electrospray ionization (ESI), charged droplets of solution are sprayed into a vacuum chamber where solvent evaporation results in generation of individually charged ions; ESI mass spectrometry is becoming more widely used and is often the method of choice for ionic compounds in solution. The traditional method of ion separation relies on the acceleration of ions with an electric field and then using a magnetic field to deflect the moving ions: ions with a lower mass-to-charge ratio are deflected more than heavier ions. As the magnetic field is changed, ions with different mass-to-charge ratio are directed on to the detector (Fig. 8.34). In a time-of-flight (TOF) mass spectrometer, the ions from a sample are accelerated by an electric field for a fixed time and then allowed to fly freely (Fig. 8.35). Because the force on all the ions of the same charge is the same, the lighter ions are accelerated to higher speeds than the heavier ions and strike a detector sooner. In an ion cyclotron resonance (ICR) mass spectrometer (often denoted FTICR, for Fourier transform-ICR) ions are collected in a small cyclotron cell inside a strong magnetic field. The ions circle round in the magnetic field, effectively behaving as an electric current. Because an accelerated current generates electromagnetic radiation, the signal generated by the ions can be detected and used to establish their mass-to-charge ratio. Mass spectrometry is most widely used in organic chemistry but is also very useful for the analysis of inorganic compounds. However, many inorganic compounds, such as those with ionic structures or covalently bonded networks (for example SiO2), are not volatile and do not fragment into molecular ion units (even with the MALDI technique) so cannot be analysed by this method. Conversely, the weaker bonding in some inorganic coordination compounds means that they fragment much more easily than organic compounds in the mass spectrometer. (b) Interpretation Figure 8.36 shows a typical mass spectrum. To interpret a spectrum, it is helpful to detect a peak corresponding to the singly charged, intact molecular ion. Sometimes a peak occurs at half the molecular mass and is then ascribed to a doubly charged ion. Peaks from multiply charged ions are usually easy to identify because the separation between the peaks 1 Signal 0.80 0.60 0.40 0.2 0 Figure 8.36 The mass spectrum of [Mo(η6-C6H6)(CO)2PMe3]. 160 180 200 220 240 m /z 260 280 300 320 Chemical analysis 100 Relative abundance/% from the different isotopomers is no longer mu but fractions of that mass. For instance, in a doubly charged ion, isotopic peaks are 12 mu apart, in a triply charged ion they are 13 mu apart, and so on. In addition to indicating the mass of the molecule or ion that is being studied (and hence its molar mass), a mass spectrum also provides information about fragmentation pathways of molecules. This information can be used to confirm structural assignments. For example, complex ions often lose ligands and peaks are observed that correspond to the complete ion less one or more ligands. Multiple peaks are observed when an element is present as a number of isotopes (for instance, chlorine is 75.5 per cent 35Cl and 24.5 per cent 37Cl). Thus, for a molecule containing chlorine, the mass spectrum will show two peaks 2mu apart in an intensity ratio of about 3:1. Different patterns of peaks are obtained for elements with a more complex isotopic composition and can be used to identify the presence of an element in compounds of unknown composition. An Hg atom, for instance, has six isotopes in significant abundance (Fig. 8.37). The actual proportion of isotopes of an element varies according to its geographic source, and this subtle aspect is easily identified with high-resolution mass spectrometers. Thus, the precise determination of the proportions of isotopes can be used to determine the source of a sample. 245 75 50 25 0 182 192 202 m/z 212 Figure 8.37 The mass spectrum of a sample containing mercury showing the isotopic composition of the atoms. E X A MPL E 8 .7 Interpreting a mass spectrum Figure 8.36 shows part of the mass spectrum of [Mo(6-C6H6)(CO)2PMe3]. Assign the main peaks. Self-test 8.7 Explain why the mass spectrum of ClBr3 (Fig. 8.38) consists of five peaks separated by 2mu. Chemical analysis A classic application of physical techniques is to the determination of the elemental composition of compounds. The techniques now available are highly sophisticated and in many cases can be automated to achieve rapid, reliable results. In this section we include thermal techniques that can be used to follow the phase changes of substances without change of composition, as well as processes that result in changes of composition. In each of these techniques the compound is usually destroyed during the analysis. 100 Relative abundance/% Answer The complex has an average molecular mass of 306mu, but a simple molecular ion is not seen because Mo has a large number of isotopes. Ten peaks centred on 306mu are detected. The most abundant isotope of Mo is 98Mo (24 per cent) and the ion that contains this isotope has the highest intensity of the peaks of the molecular ion. In addition to the peaks representing the molecular ion, peaks at M 28, M 56, M 76, M 104, and M 132 are seen. These peaks represent the loss of one CO, two CO, PMe3, PMe3 CO, and PMe3 2CO ligands, respectively, from the parent compound. 75 50 25 0 258 268 278 m/z 288 Figure 8.38 The mass spectrum of ClBr3. 8.11 Atomic absorption spectroscopy Key point: Almost every metallic element can be determined quantitatively by using the spectral absorption characteristics of atoms. The principles of atomic absorption spectroscopy are similar to those of UV–visible spectroscopy except that the absorbing species are free atoms or ions. Unlike molecules, atoms and ions do not have rotational or vibrational energy levels and the only transitions that occur are between electronic energy levels. Consequently, atomic absorption spectra consist of sharply defined lines rather than the broad bands typical of molecular spectroscopy. Figure 8.39 shows the basic components of an atomic absorption spectrophotometer. The gaseous sample is exposed to radiation of a specific wavelength from a ‘hollow cathode’ lamp, which consists of a cathode constructed of a particular element and a tungsten anode in a sealed tube filled with neon. If a particular element is present in the sample, the radiation emitted by the lamp for that element is reduced in intensity because it stimulates absorption. By determining the level of absorption relative to standard materials a quantitative measurement can be made of the amount of the element. A different lamp is required for each element that is to be analysed. Output Sample Lamp Atomizer Detector Monochromator Figure 8.39 The layout of a typical atomic absorption spectrophotometer. 246 8 Physical techniques in inorganic chemistry The major differences in instrumentation arise from the different methods used to convert the analyte (the substance being analysed) to free, unbound atoms or ions. In flame atomization the analyte solution is mixed with the fuel in a ‘nebulizer’, which creates an aerosol. The aerosol enters the burner where it passes into a fuel–oxidant flame. Typical fuel– oxidant mixtures are acetylene–air, which produces flame temperatures of up to 2500 K, and acetylene–nitrous oxide, which generates temperatures of up to 3000 K. A common type of ‘electrothermal atomizer’ is the graphite furnace. The temperatures reached in the furnace are comparable to those attained in a flame atomizer but detection limits can be 1000 times better. The increased sensitivity is due to the ability to generate atoms quickly and keep them in the optical path for longer. Another advantage of the graphite furnace is that solid samples may be used. Because the ionization process may produce spectral lines from other components of the analyte, a monochromator is placed after the atomizer to isolate the desired wavelength for passage to the detector. Almost every metallic element can be analysed using atomic absorption spectroscopy, although not all with high sensitivity or a usefully low detection limit. For example, the detection limit for Cd in a flame ionizer is 1 part per billion (1 ppb 1 in 109) whereas that for Hg is only 500 ppb. Limits of detection using a graphite furnace can be as low as 1 part in 1015. Direct determination is possible for any element for which hollow cathode lamp sources are available. Other species can be determined by indirect procedures. For example, PO43 reacts with MoO42 in acid conditions to form H3PMo12O40, which can be extracted into an organic solvent and analysed for molybdenum. To analyse for a particular element, a set of calibration standards is prepared in a similar matrix to the sample, and the standards and the sample are analysed under the same conditions. 8.12 CHN analysis Key point: The carbon, hydrogen, nitrogen, oxygen, and sulfur content of a sample can be determined by high-temperature decomposition. Instruments are available that allow automated analysis of C, H, N, O, and S. Figure 8.40 shows the arrangement for an instrument that analyses for C, H, and N, sometimes referred to as CHN analysis. The sample is heated to 900ºC in oxygen and a mixture of carbon dioxide, carbon monoxide, water, nitrogen, and nitrogen oxides is produced. A stream of helium sweeps the products into a tube furnace at 750ºC, where copper reduces nitrogen oxides to nitrogen and removes oxygen. Copper oxide converts carbon monoxide to carbon dioxide. The resulting mixture is analysed by passing it through a series of three thermal conductivity detectors. The first detector measures hydrogen and then water is removed in a trap. At the second detector the carbon is measured, and carbon dioxide is removed in a second trap. The remaining nitrogen is measured at the third detector. The data obtained from this technique are reported as mass percentage C, H, and N. Oxygen may be analysed if the reaction tube is replaced with a quartz tube filled with carbon that has been coated with catalytic platinum. When the gaseous products are swept through this tube, the oxygen is converted to carbon monoxide, which is then converted to carbon dioxide by passage over hot copper oxide. The rest of the procedure is the same as described above. Sulfur can be measured if the sample is oxidized in a tube filled with copper oxide. Water is removed by trapping in a cool tube and the sulfur dioxide is determined at what is normally the hydrogen detector. Detector 1 Sample Detector 2 Reduction Combustion Figure 8.40 The layout of the apparatus used for CHN analysis. H2O trap CO2 trap Detector 3 Chemical analysis 247 E X A MPL E 8 . 8 Interpreting CHN analytical data A CHN analysis of a compound of iron gave the following mass percentages for the elements present C: 64.54, N: 0, and H: 5.42, with the residual mass being iron. Determine the empirical formula of the compound. Answer The molar masses of C, H, and Fe are 12.01, 1.008, and 55.85 g mol1, respectively. The mass of each element in exactly 100 g of the sample is 64.54 g C, 5.42 g H, and (the difference from 100 g) 30.04 g Fe. The amounts present are therefore n (C ) 64.54 g 5.37 mol 12.01 g mol1 n (H) 5.42 g 1.008g mol n (eF ) 1 5.38 mol 30.04 g 0.538 mol 55.85 g mol1 These amounts are in the ratio 5.37:5.38:0.538 ≈ 10:10:1. The empirical formula of the compound is therefore C10H10Fe. Self-test 8.8 Why are the percentages of hydrogen in 5d-series compounds as determined by CHN methods less accurate than those determined for the analogous 3d-series compounds? 8.13 X-ray fluorescence elemental analysis Key points: Qualitative and quantitative information on the elements present in a compound may be obtained by exciting and analysing X-ray emission spectra. S Signal As discussed in Section 8.9, ionization of core electrons can occur when a material is exposed to short-wavelength X-rays. When an electron is ejected in this way, an electron from a higher energy orbital can take its place and the difference in energy is released in the form of a photon, which is also typically in the X-ray region with an energy characteristic of the atoms present. This fluorescent radiation can be analysed either by energydispersive analysis or by wavelength-dispersive analysis. By matching the peaks in the spectrum with the characteristic values of the elements it is possible to identify the presence of a particular element. This is the basis of the X-ray fluorescence (XRF) technique. The intensity of the characteristic radiation is also directly related to the amount of each element in the material. Once the instrument is calibrated with appropriate standards it can be used to determine quantitatively most elements with Z > 8 (oxygen). Figure 8.41 shows a typical XRF energy-dispersive spectrum. A technique similar to XRF is used in electron microscopes where the method is known as energy-dispersive analysis of X-rays (EDAX) or energy-dispersive spectroscopy (EDS). Here the X-rays generated by bombardment of a sample by energetic electrons result in the ejection of core electrons and X-ray emission occurs as the outer electrons fall into the vacancies in the core levels. These X-rays are characteristic of the elements present and their intensities are representative of the amounts present. The spectrum can be analysed to determine qualitatively and quantitatively the presence and amount of most elements (generally those with Z > 8) in the material. 8.14 Thermal analysis Key points: Thermal methods include thermogravimetric analysis, differential thermal analysis, and differential scanning calorimetry. Thermal analysis is the analysis of a change in a property of a sample induced by heating. The sample is usually a solid and the changes that occur include melting, phase transition, sublimation, and decomposition. The analysis of the change in the mass of a sample on heating is known as thermogravimetric analysis (TGA). The measurements are carried out using a thermobalance, C 0 O Na Zn K AlSi 1 2 3 Energy, E/keV 4 Figure 8.41 An XRF spectrum obtained from a metal silicate sample showing the presence of various elements by their characteristic X-ray emission lines. 248 8 Physical techniques in inorganic chemistry Carrier gas Balance Output and controller Programmer Mass, m CuSO4 CuSO4.H2O CuSO4.5H2O CuSO4.3H2O Figure 8.42 A thermogravimetric analyser: the mass of the sample is monitored as the temperature is raised. 275 225 175 125 75 25 dm/dθ which consists of an electronic microbalance, a temperature-programmable furnace, and a controller, which enables the sample to be simultaneously heated and weighed (Fig. 8.42). The sample is weighed into a sample holder and then suspended from the balance within the furnace. The temperature of the furnace is usually increased linearly, but more complex heating schemes, isothermal heating (heating that maintains constant temperature at a phase transition), and cooling protocols can also be used. The balance and furnace are situated within an enclosed system so that the atmosphere can be controlled. That atmosphere may be inert or reactive, depending on the nature of the investigation, and can be static or flowing. A flowing atmosphere has the advantage of carrying away any volatile or corrosive species and prevents the condensation of reaction products. In addition, any species produced can be fed into a mass spectrometer for identification. Thermogravimetric analysis is most useful for desorption, decomposition, dehydration, and oxidation processes. For example, the thermogravimetric curve for CuSO4.5H2O from room temperature to 300ºC shows three stepwise mass losses (Fig. 8.43), corresponding to the three stages in the dehydration to form first CuSO4.3H2O, then CuSO4.H2O, and finally CuSO4. The most widely used thermal method of analysis is differential thermal analysis (DTA). In this technique the temperature of the sample is compared to that of a reference material while they are both subjected to the same heating procedure. In a DTA instrument, the sample and reference are placed in low thermal conductivity sample holders that are then located within cavities in a block in the furnace. Common reference samples for the analysis of inorganic compounds are alumina, Al2O3, and carborundum, SiC. The temperature of the furnace is increased linearly and the difference in temperature between the sample and the reference is plotted against the furnace temperature. If an endothermic event takes place within the sample, the temperature of the sample lags behind that of the reference and a minimum is observed on the DTA curve. If an exothermal event takes place, the temperature of the sample rises above that of the reference and a maximum is observed on the curve. The area under the endotherm or exotherm (the resulting curve in each case) is related to the enthalpy change accompanying the thermal event. A technique closely related to DTA is differential scanning calorimetry (DSC). In DSC, the sample and the reference are maintained at the same temperature throughout the heating procedure by using separate power supplies to the sample and reference holders. Any difference between the power supplied to the sample and reference is recorded against the furnace temperature. Thermal events appear as deviations from the DSC baseline as either endotherms or exotherms, depending on whether more or less power has to be supplied to the sample relative to the reference. In DSC, endothermic reactions are usually represented as positive deviations from the baseline, corresponding to increased power supplied to the sample. Exothermic events are represented as negative deviations from the baseline. The information obtained from DTA and DSC is very similar. The former can be used up to higher temperatures although the quantitative data, such as the enthalpy of a phase change, obtained from DSC are more reliable. Both DTA and DSC are used for ‘fingerprint’ comparison of the results obtained from a sample with those of a reference material. Information about the temperatures and enthalpy changes of transitions, such as a change in structure or melting, can be extracted. E X A M PL E 8 . 9 Interpreting thermal analysis data Temperature, /°C Figure 8.43 The thermogravimetric curve obtained for CuSO4.5H2O as the temperature is raised from 20°C to 500°C. The red line is the mass of the sample and the green line is its first derivative (the slope of the red line). When a sample of bismuth nitrate hydrate, Bi(NO3)3.nH2O, of mass 100 mg was heated to 500°C and dryness, the loss in mass observed was 18.56 mg. Determine n. Answer We need to make a stoichiometric analysis of the decomposition, Bi(NO3)3.nH2O → Bi(NO3)3 nH2O to determine the value of n. The molar mass of Bi(NO3)3.nH2O is 395.01 18.02n g mol–1, so the initial amount of Bi(NO3)3.nH2O present is (100 mg)/(395.01 18.02n g mol–1). As each formula unit of Bi(NO3)3.nH2O contains n mol H2O, the amount of H2O present in the solid is n times this amount, or n(100 mg)/(395.01 18.02n g mol1) 100n/(395.01 18.02n) mmol. The mass loss is 18.56 mg. This loss is entirely due to the loss of water, so the amount of H2O lost is (18.56 mg)/(18.02 g mol–1) 1.030 mmol. We equate this amount to the amount of H2O in the solid initially: 100n 1.030 395.0118.02n Electrochemical techniques 249 (The units mmol have cancelled.) It follows that n 5, and that the solid is Bi(NO3)3.5H2O. Self-test 8.9 Reduction of a 10.000 mg sample of an oxide of tin in hydrogen at 600°C resulted in the formation of 7.673 mg of tin metal. Determine the stoichiometry of the tin oxide. Magnetometry Vibrator Key point: Magnetometry is used to determine the characteristic response of a sample to an applied magnetic field. The classic way of monitoring the magnetic properties of a sample is to measure the attraction into or repulsion out of an inhomogeneous magnetic field by monitoring the change in the apparent weight of a sample when the magnetic field is applied by using a Gouy balance (Fig. 8.44a). The sample is hung on one side of a balance by a fine thread so that one end of it lies in the field of a powerful electromagnet and the other end of the sample is in just the Earth’s magnetic field. The sample is weighed with the electromagnet field applied and then with it turned off. From the change in apparent weight, the force acting on the sample as a result of the application of the field can be determined and from this, with a knowledge of various instrumental constants, sample volume and the molar mass, the molar susceptibility. The effective magnetic moment of a d-metal ion present in a material may be deduced from the magnetic susceptibility and used to infer the number of unpaired electrons and the spin state (Chapter 20). In a Faraday balance a magnetic field gradient is generated between two curved magnets; this technique yields precise susceptibility measurements and also allows collection of magnetization data as a function of the magnitude and direction of the applied field. The more modern vibrating sample magnetometer (VSM, Fig. 8.44b) measures the magnetic properties of a material by using a modified version of a Gouy balance. The sample is placed in a uniform magnetic field, which induces a net magnetization. As the sample is vibrated, an electrical signal is induced in suitably placed pick-up coils. The signal has the same fre quency of vibration and its amplitude is proportional to the induced magnetization. The vibrating sample may be cooled or heated, allowing a study of magnetic properties as a function of temperature. Measurements of magnetic properties are now made more routinely using a superconducting quantum interference device (SQUID, Fig. 8.45). A SQUID makes use of the quantization of magnetic flux and the property of current loops in superconductors that are part of the circuit. The current that flows in the loop in a magnetic field is determined by the value of the magnetic flux and hence the magnetic susceptibility of the sample. Electrochemical techniques (a) Receiving coils (b) Figure 8.44 A schematic diagram of (a) a Gouy balance and (b, insert) a schematic diagram of a vibrating sample magnetometer modification. Mgnetic a field rrent u C a Sp lm e Key point: Cyclic voltammetry measures the electrical currents due to reduction and oxidation of electroactive species in solution. In cyclic voltammetry the current flowing between two electrodes immersed in a solution is measured as the potential difference is changed cyclically. It provides direct information on reduction potentials and the stabilities of different products of oxidation or reduction. The technique gives rapid qualitative insight into the redox properties of an electroactive compound and reliable quantitative information on its thermodynamic and kinetic properties. The ‘working electrode’ at which the electrochemical reaction of interest occurs is usually constructed from platinum, silver, gold, or graphite. The reference electrode is normally a silver/silver-chloride electrode and the counter electrode is normally platinum. To understand what is involved, consider the redox couple [Fe(CN)6]3–/[Fe(CN)6]4–, where, initially, only the reduced form (the Fe(II) complex) is present (Fig. 8.46). The concentration of electroactive species is usually quite low (less than 0.001 m) and the solution contains a relatively high concentration of inert ‘supporting’ electrolyte (at concentrations greater than about 0.1 m) to provide conductivity. The potential difference is applied between the working electrode and the reference electrode and is scanned back and forth between two limits, tracing out a triangular waveform. Sre p u cond cting u ire w ID U Q S Figure 8.45 The magnetic susceptibility of a sample is measured by using a SQUID: the sample, which is exposed to a magnetic field, is moved through the loops in small increments and the potential difference across the SQUID is monitored. 250 8 Physical techniques in inorganic chemistry Relative current Epa 0 Epc –0.2 0 +0.2 (E – E°)/V Figure 8.46 The cyclic voltammogram for an electroactive species present in solution as the reduced form and displaying a reversible one-electron reaction at an electrode. The peak potentials Epa and Epc for oxidation and reduction, respectively, are separated by 0.06 V. The reduction potential is the mean of Epa and Epc. No current flows while the potential is low. As it approaches the reduction potential of the Fe(III)/Fe(II) couple, the Fe(II) is oxidized at the working electrode and a current starts to flow. This current rises to a peak then decreases steadily because Fe(II) becomes depleted close to the electrode (the solution is unstirred) and must be supplemented by species diffusing from increasingly distant regions of the solution. Once the upper potential limit is reached, the potential sweep is reversed. Initially, Fe(II) diffusing to the electrode continues to be oxidized but eventually the potential difference becomes sufficiently negative to reduce the Fe(III) that had been formed; the current reaches a peak then decreases gradually to zero as the lower potential limit is reached. The average of the two peak potentials is a good approximation to the reduction potential E under the prevailing conditions (E is not in general the standard potential because nonstandard conditions are usually adopted). In the ideal case, the oxidation and reduction peaks are similar in magnitude and separated by a small potential increment, which is usually (59 mV)/e at 25ºC, where e is the number of electrons transferred during the reaction at the electrode. This case is an example of a reversible redox reaction, with electron transfer at the electrode sufficiently fast that equilibrium is always maintained throughout the potential sweep. In such a case the current is usually limited by diffusion of electroactive species to the electrode. Slow kinetic processes at the electrode result in a large separation of reduction and oxidation peaks that increases with increasing scan rate. This separation arises because an overpotential (Section 5.18, effectively a driving force) is required to overcome barriers to electron transfer in each direction. Moreover, the peak due to a reduction or an oxidation in the initial part of the cycle process is often not matched by a corresponding peak in the reverse direction. This absence occurs because the species that is initially generated undergoes a further chemical reaction during the cycle and produces either a species with a different reduction potential or one that is not electroactive within the range of potential scanned. Inorganic chemists often refer to this behaviour as ‘irreversible’. An electrochemical reaction followed by a chemical reaction is known as an EC process. By analogy, a CE process is a reaction in which the species able to undergo the electrochemical (E) reaction must first be generated by a chemical reaction. Thus, for a molecule that is suspected of decomposing upon oxidation, it may be possible to observe the initial unstable species formed by the E process provided the scan rate is sufficiently fast to re-reduce it before it undergoes further reaction. Consequently, by varying the scan rate, the kinetics of the chemical reaction can be determined. Computational techniques Key point: Computational procedures use either ab initio methods or parametrized semi-empirical methods to calculate the properties of molecules and solids. Graphical techniques can be used to display the results. Computation has proved to be one of the most important techniques in chemistry. Computer modelling is the use of numerical models for exploring the structures and properties of individual molecules and materials. The methods used range from rigorous, and therefore computationally very time-consuming, treatments, known as ab initio methods, based on the numerical solution of the Schrödinger equation for the system, to the more rapid and necessarily less detailed ‘semi-empirical techniques’, which use approximate or ‘effective functions’ to describe the forces between particles. There are two principal approaches to solving the Schrödinger equation numerically for many-electron polyatomic molecules. In the more fundamental ab initio methods, an attempt is made to calculate structures from first principles, using only the atomic numbers of the atoms present and their general arrangement in space. Such an approach is intrinsically reliable but computationally very demanding. For complex problems involving molecules and materials with numerous atoms, such methods are so computationally time-consuming that alternative methods involving experimental data are needed. In these semi-empirical methods, integrals that occur in the formal solution of the Schrödinger equation are set equal to parameters that have been chosen to lead to the best fit to experimental quantities, such as enthalpies of formation. Semi-empirical methods are applicable to a wide range of molecules with an almost limitless number of atoms, and are widely popular. Further Reading Both types of procedure typically adopt a self-consistent field (SCF) procedure, in which an initial guess about the composition of the linear combinations of atomic orbitals (LCAO) used to model molecular orbitals is successively refined until the composition and the corresponding energy remains unchanged in a cycle of calculation. The most common type of ab initio calculation is based on the Hartree–Fock method in which the primary approximation is applied to the electron–electron repulsion. Various methods of correcting for the explicit electron–electron repulsion, referred to as the correlation problem, are the Møller–Plesset perturbation theory (MPn, where n is the order of correction), the generalized valence bond (GVB) method, multi-configurations self-consistent field (MCSCF), configuration interaction (CI), and coupled cluster theory (CC). A currently popular alternative to the ab initio method is density functional theory (DFT), in which the total energy is expressed in terms of the total electron density \|\|2 rather than the wavefunction itself. When the Schrödinger equation is expressed in terms of , it becomes a set of equations called the Kohn–Sham equations, which are solved iteratively starting from an initial estimate and continuing until they are self-consistent. The advantage of the DFT approach is that it is less demanding computationally, requires less computer time, and—in some cases, particularly d-metal complexes—gives better agreement with experimental values than is obtained from other procedures. Semi-empirical methods are set up in the same general way as Hartree–Fock calculations but within this framework certain pieces of information, such as integrals representing the interaction between two electrons, are approximated by importing empirical data or simply ignored. To soften the effect of these approximations, parameters representing other integrals are adjusted so as to give the best agreement with experimental data. Semi-empirical calculations are much faster than the ab initio calculations but the quality of results is very dependent on using a reasonable set of experimental parameters that can be transferred from structure to structure. Thus semi-empirical calculations have been very successful in organic chemistry with just a few types of element and molecular geometries. Semi-empirical methods have also been devised specifically for the description of inorganic species. The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each molecular orbital and the energies of these orbitals. The graphical representation of a molecular orbital uses stylized shapes to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colours. The total electron density at any point (the sum of the squares of the wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density (Fig. 8.47). An important aspect of a molecule other than its geometrical shape is the distribution of charge over its surface. A common procedure begins with calculation of the net electric potential at each point on an isodensity surface by subtracting the potential due to the electron density at that point from the potential due to the nuclei. The result is an electrostatic potential surface (an ‘elpot surface’) in which net positive potential is shown in one colour and net negative potential is shown in another, with intermediate gradations of colour. Computer modelling is applied to solids as well as to individual molecules and is useful for predicting the behaviour of a material, for example for indicating which crystal structure of a compound is energetically most favourable, for predicting phase changes, for calculating thermal expansion coefficients, identifying preferred sites for dopant ions, and calculating a diffusion pathway through a lattice. However, it is worth noting that very few aspects of inorganic chemistry can be computed exactly. Although modelling computation can give a very useful insight into materials chemistry, it is not yet at a stage where it can be used reliably to predict the exact structure or properties of any complex compound. FURTHER READING Although this chapter has introduced many of the methods used by chemists to characterize inorganic compounds, it is not exhaustive. Other techniques used for investigating the structures and properties of solids and solutions include electron microscopy (Chapter 25) and inelastic neutron scattering to name two. The following references are a source of information on these techniques and a greater depth of coverage on the major techniques that have been introduced here. 251 Figure 8.47 The output of computations of the electronic structure of a molecule is conveyed in a variety of ways. Here we show the electric potential surface of SF5CF3, a molecule that has been found to act as a very powerful greenhouse gas but of uncertain origin in the atmosphere. Red areas indicate regions of negative potential and green regions of positive potential. 252 8 Physical techniques in inorganic chemistry A.K. Brisdon, Inorganic spectroscopic methods. Oxford Science Publications (1998). J.R. Ferraro and K. Nakamoto, Introductory Raman spectroscopy. Academic Press (1994). R.P. Wayne, Chemical instrumentation. Oxford Science Publications (1994). K. Nakamoto, Infrared and Raman spectra of inorganic and coordination compounds. Wiley–Interscience (1997). D.A. Skoog, F.J. Holler, and T.A. Nieman, Principles of instrumental analysis. Brooks Cole (1997). R.S. Drago, Physical methods for chemists. Saunders (1992). J.K.M. Saunders and B.K. Hunter, Modern NMR spectroscopy, a guide for chemists. Oxford University Press (1993). S.K. Chatterjee, X-ray diffraction: Its theory and applications. PrenticeHall of India (2004). J.A. Iggo, NMR spectroscopy in inorganic chemistry. Oxford University Press (1999). B.D. Cullity and S.R. Stock, Elements of X-ray diffraction. Prentice Hall (2003). J.W. Akitt and B.E. Mann, NMR and chemistry. Stanley Thornes, Cheltenham (2000). B. Henderson and G. F. Imbusch, Optical spectroscopy of inorganic solids (Monographs on the Physics & Chemistry of Materials). Oxford University Press (2006). K.J.D. MacKenzie and M.E. Smith, Multinuclear solid-state nuclear magnetic resonance of inorganic materials. Pergamon (2004). E.I. Solomon and A.B.P. Lever, Inorganic electronic structure and spectroscopy: Methodology: Volume 1. Wiley (2006). E.I. Solomon and A.B.P. Lever, Inorganic electronic structure and spectroscopy: applications and case studies: Volume 2. Wiley (2006). J.S. Ogden, Introduction to molecular symmetry. Oxford University Press (2001). F. Siebert and P. Hildebrandt, Vibrational spectroscopy in life science. Wiley VCH (2007). D.P.E. Dickson and F.J. Berry, Mössbauer spectroscopy. Cambridge University Press; (2005). M.E. Brown, Introduction to thermal analysis. Kluwer Academic Press (2001). P.J. Haines, Principles of thermal analysis and calorimetry. Royal Society of Chemistry (2002). A.J. Bard and L.R. Faulkner, Electrochemical methods: fundamentals and applicatons. 2nd edn. Wiley (2001). O. Kahn, Molecular magnetism. VCH, New York (1993). 8.1 How might you determine what crystalline components are present in a natural mineral sample? 8.2 The reaction of sodium carbonate, boron oxide, and silicon dioxide gives a borosilicate glass. Explain why the powder diffraction pattern of this product shows no diffraction maxima. 8.3 The minimum size of crystal that can typically be studied using a laboratory single crystal diffractometer is 50 50 50 μm. The X-ray flux from a synchrotron source is expected to be 106 times the intensity of a laboratory source. Calculate the minimum size of a cubic crystal that could be studied on a diffractometer by using this source. A neutron flux is 103 times weaker. Calculate the minimum size of crystal that could be studied by single crystal neutron diffraction. Photoelectron flux EXERCISES 28 27 26 8.4 Calculate the wavelength associated with a neutron moving at 2.20 km s–1. Is this wavelength suitable for diffraction studies (mn 1.675 10–27 kg)? Figure 8.48 8.5 Suggest reasons for the order of stretching frequencies observed with diatomic species: CN > CO > NO. 77 8.6 Use the data in Table 8.3 to estimate the OO stretching wavenumber expected for a compound believed to contain the oxygenyl species O2. Would you expect to observe this stretching vibration in (i) the IR spectrum or (ii) the Raman spectrum? 8.7 Figure 8.48 shows the UV photoelectron spectrum of NH3. Explain why the band at approximately 11 eV shows such a long and sharply resolved progression. 18 17 16 15 14 13 12 Ionization energy, I/eV 11 10 8.10 Predict the form of the 19F-NMR and the 77Se-NMR spectra of SeF4. For 77Se, I 12 . 8.11 Explain the observation that the 19F-NMR spectrum of XeF5 consists of a central peak symmetrically flanked by two peaks, each of which is roughly one-sixth of the intensity of the central peak. 8.12 Determine the g values of the EPR spectrum shown in Fig. 8.49, measured for a frozen sample using a microwave frequency of 9.43 GHz. 8.13 Which technique is sensitive to the slowest processes, NMR or EPR? 8.8 Explain why a Raman band assigned to the symmetric NC stretching mode in N(CH3)3 shows a shift to lower frequency when 14 N is substituted by 15N but no such shift is observed for the NSi symmetric stretch in N(SiH3)3. 8.14 For a paramagnetic compound of a d-metal compound having one unpaired electron, outline the main difference you would expect to see between an EPR spectrum measured in aqueous solution at room temperature and that recorded for a frozen solution. 8.9 Explain why the 13C-NMR spectrum of Co2(CO)9 shows only a single peak at room temperature. 8.15 Predict a value for the isomer shift for iron in BaFe(VI)O4. Problems 253 0.720 V Current, I/µA 10 0.240 V Start 0 0.180 V 300 340 380 420 460 500 –10 B/mT 0 Figure 8.49 8.16 How would you determine whether the compound Fe4[Fe(CN)6]3 contains discrete Fe(II) and Fe(III) sites? 8.17 Suggest a reason why no resolved quadrupole splitting is observed in the 121Sb (I 5/2) Mössbauer spectrum of solid SbF5. 8.18 Explain why, even though the average atomic mass of silver is 107.9mu, no peak at 108mu is observed in the mass spectrum of pure silver. What effect does this absence have on the mass spectra of silver compounds? 8.19 What peaks should you expect in the mass spectrum of [Mo(C6H6)(CO)3]? 8.20 Interpret the cyclic voltammogram shown in Fig. 8.50, which has been recorded for an Fe(III) complex in aqueous solution. 0.5 Potential difference, E/V 1.0 Figure 8.50 8.21 Thermogravimetric analysis of a zeolite of composition CaAl2Si6O16.nH2O shows a mass loss of 20 per cent on heating to dryness. Determine n. 8.22 A cobalt(II) salt was dissolved in water and reacted with excess acetylacetone (1,2-pentanedione, CH3COCHCOCH3) and hydrogen peroxide. A green solid was formed that gave the following results for elemental analysis: C, 50.4 per cent; H, 6.2 per cent; Co, 16.5 per cent (all by mass). Determine the ratio of cobalt to acetylacetonate ion in the product. PROBLEMS 8.1 Discuss the importance of X-ray crystallography in inorganic chemistry. See for example: The history of molecular structure determination viewed through the Nobel Prizes. W.P. Jensen, G.J. Palenik, and I.-H. Suh, J. Chem. Educ., 2003, 80, 753. 8.2 Discuss why the length of an OH bond obtained from X-ray diffraction experiments averages 85 pm whereas that obtained in neutron diffraction experiments averages 96 pm. Would you expect to see similar effects with CH bond lengths measured by these techniques? 8.3 How is single crystal neutron diffraction of use in inorganic chemistry? 8.4 Refer to Fig. 8.51. Discuss the likely reason for the shifts in max for spectra of I2 recorded in different solvent media: (a) heptane, (b) benzene, (c) diethylether/CCl4, (d) pyridine/heptane, (e) triethylamine/heptane. ε/(103 mol–1 dm3 cm–1) 2 (e) (d) (c) 1 (b) (a) 0 400 Figure 8.51 450 500 550 Wavelength, l /nm 8.5 Discuss how you would carry out the following analyses: (a) calcium levels in breakfast cereal, (b) mercury in shell fish, (c) the geometry of BrF5, (d) number of organic ligands in a d-metal complex, (e) water of crystallization in an inorganic salt. 8.6 Write a review of the applications of thermal methods of analysis in inorganic chemistry. 8.7 Discuss the challenges involved in carrying out each of the following scenarios: (a) identification of inorganic pigments in a valuable painting, (b) analysis of evolved inorganic gases from a volcano, (c) determination of pollutants at the sea bed. 8.8 How may a consideration of 31P chemical shifts and 1H coupling constants be used to distinguish the isomers of the octahedral complex [Rh(CCR)2H(PMe3)3]? (See J.P. Rourke, G. Stringer, D.S. Yufit, J.A.K. Howard, and T.B. Marder, Organometallics, 2002, 21, 429.) 8.9 Ionization techniques in mass spectrometry frequently induce fragmentation and other undesirable reactions. However, under some circumstances these reactions can be helpful. Using fullerenes as an example, discuss this phenomenon. (See M.M. Boorum, Y.V. Vasil’ev, T. Drewello, and L.T. Scott, Science, 2001, 294, 828.) 8.10 The iron content of a dietary supplement tablet was determined using atomic absorption spectrometry. A tablet (0.4878 g) was ground to a fine powder and 0.1123 g was dissolved in dilute sulfuric acid and transferred to a 50 cm3 volumetric flask. A 10 cm3 sample of this solution was taken and made up to 100 cm3 in another volumetric flask. A series of standards was prepared that contained 1.00, 3.00, 5.00, 7.00, and 10.0 ppm iron. The absorptions of 254 8 Physical techniques in inorganic chemistry the standards and the sample solution were measured at the iron absorption wavelength: Concentration/ppm Absorbance Concentration/ppm Absorbance 100 0.152 200 0.388 1.00 0.095 300 0.590 3.00 0.265 400 0.718 5.00 0.450 500 0.865 7.00 0.632 Sample 0.751 10.0 0.910 Sample 0.545 Calculate the mass of iron in the tablet. 8.11 A sample of water from a reservoir was analysed for copper content. The sample was filtered and diluted tenfold with deionized water. A set of standards was prepared with copper concentrations between 100 and 500 ppm. The standards and the sample were aspirated into an atomic absorption spectrometer and the absorbance was measured at the copper absorption wavelength. The results obtained are given below. Calculate the concentration of copper in the reservoir. 8.12 A sample from an effluent stream was analysed for phosphate levels. Dilute hydrochloric acid and excess sodium molybdate were added to a 50 cm3 sample of the effluent. The molybdophosphoric acid, H3PMo12O40, that was formed was extracted into two 10 cm3 portions of an organic solvent. A molybdenum standard with a concentration of 10 ppm was prepared in the same solvent. The combined extract and the standard were aspirated into an atomic absorption spectrometer that was set up to measure molybdenum. The extracts gave an absorbance of 0.573 and the standard gave an absorbance of 0.222. Calculate the concentration of phosphate in the effluent. PART 2 The elements and their compounds This part of the book describes the physical and chemical properties of the elements as they are set out in the periodic table. This ‘descriptive chemistry’ of the elements reveals a rich tapestry of patterns and trends, many of which can be rationalized and explained by application of the concepts developed in Part 1. The first chapter of this part, Chapter 9, summarizes the trends and patterns in the context of the periodic table and the principles described in Part 1. The trends described in this chapter are illustrated throughout the following chapters. Chapter 10 deals with the chemistry of the unique element hydrogen. The following eight chapters (Chapters 1118) proceed systematically across the main groups of the periodic table. The elements of these groups demonstrate the diversity, intricacy, and fascinating nature of inorganic chemistry. The chemical properties of the d-block elements are so diverse and extensive that the next four chapters of the part are devoted to them. Chapter 19 reviews the descriptive chemistry of the three series of d-block elements. Chapter 20 describes how electronic structure affects the chemical and physical properties of the d-metal complexes and Chapter 21 describes their reactions in solution. Chapter 22 deals with the industrially vital d-metal organometallic compounds. Our tour of the periodic table draws to a close in Chapter 23 with a description of the remarkably uniform f-block elements. This page intentionally left blank 9 Periodic trends The periodic table provides an organizing principle that coordinates and rationalizes the diverse physical and chemical properties of the elements. Periodicity is the regular manner in which the physical and chemical properties of the elements vary with atomic number. This chapter reviews the material in Chapter 1 and summarizes these variations in a manner that should be kept in mind throughout the chapters of this part of the text. Although the chemical properties of the elements can seem bewilderingly diverse, the periodic table helps to show that they vary reasonably systematically with atomic number. Once these trends and patterns are recognized and understood, much of the detailed properties of the elements no longer seem like a random collection of unrelated facts and reactions. In this chapter we summarize some of the trends in the physical and chemical properties of the elements and interpret them in terms of the underlying principles presented in Chapter 1. Periodic properties of the elements 9.1 Valence electron configurations 9.2 Atomic parameters 9.3 Occurrence 9.4 Metallic character 9.5 Oxidation states Periodic characteristics of compounds 9.6 Coordination numbers 9.7 Bond enthalpy trends 9.8 Anomalies 9.9 Binary compounds 9.10 Wider aspects of periodicity Periodic properties of the elements The general structure of the modern periodic table was discussed in Section 1.8. Almost all trends in the properties of the elements can be traced to the electronic configuration of the atoms and atomic radii, and their variation with atomic number. 9.1 Valence electron configurations The valence electron configuration of the ground state of an atom of an element can be inferred from its group number. For example, in Group 1 all the elements have an ns1 valence configuration, where n is the period number. As we saw in Chapter 1, the valence electron configurations vary with group number as follows: 1 ns 2 1 13 2 14 2 ns 1 2 ns np 15 2 16 2 ns np 3 17 2 ns np 4 ns np 18 2 5 ns2np6 ns np Electron configurations in the d block are slightly less systematic, but involve the filling of the (n 1)d orbitals. In Period 4 they are as follows: 3 4 1 2 3d 4s 5 2 2 3d 4s 6 3 2 3d 4s 7 5 1 3d 4s 8 5 2 3d 4s 9 6 2 3d 4s 10 7 2 3d 4s 11 8 2 3d 4s 12 10 1 3d 4s 3d104s2 Note the manner in which half-filled and full d subshells are favoured. 9.2 Atomic parameters Although this part of the text deals with the chemical properties of the elements and their compounds, we need to keep in mind that these chemical properties spring from the physical characteristics of atoms. As we saw in Chapter 1, these physical characteristics—the radii of atoms and ions, and the energy changes associated with the formation of ions— vary periodically. Here we review these variations. FURTHER READING EXERCISES PROBLEMS 258 9 Periodic trends 1 2 i L 1 2 5 2 3 1 e B B 3 1 3 8 Na M g Al 1 0 8 0 6 1 1 3 4 1 7 4 K 7 2 5 6 Figure 9.1 The variation of atomic radii (in picometres) through the main groups of the periodic table. Rb 8 4 2 sC 5 6 2 C a 2 1 rS 5 1 2 a B 2 7 1 5 1 C 5 7 7 3 7 9 1 4 1 N 7 3 i S 4 0 1 5 2 1 7 1 7 1 8 1 O F Ne 1 7 P 5 1 S C l Ar 9 e G a G 3 2 1 6 1 As e S r B b S eT I o P At 4 1 In 1 3 6 1 4 S n 1 4 3 4 1 l T 1 0 7 5 7 1 P b 5 1 7 6 1 3 1 i B Kr Xe Rn (a) Atomic radii Key points: Atomic radii increase down a group and, within the s and p blocks, decrease from left to right across a period. Cross Reference: Section 19.2 Cross Reference: Table 14.1 As we saw in Section 1.9, atomic radii increase down a group and decrease from left to right across a period. Across a period, as a result of the joint effects of penetration and shielding, there is an increase in effective nuclear charge. This increase draws in the electrons and results in a smaller atom. On descending a group, electrons occupy successive shells outside a completed core and the radii increase (Fig. 9.1). Atomic radii in the 5d series of the d block are very similar to their congeners in the 4d series even though the atoms have a greater number of electrons. For example, the radii of Mo and W in Group 6 are 140 and 141 pm, respectively. This reduction of radius below that expected, the lanthanide contraction, is due to the presence of 4f electrons in the intervening lanthanoids: the poor shielding properties of f electrons results in a higher effective nuclear charge than expected on the basis of a simple extrapolation from other atoms. A similar contraction is found in the elements that follow the d block. For example, although there is a substantial increase in atomic radius between C and Si (77 and 118 pm, respectively), the atomic radius of Ge (122 pm) is only slightly greater than that of Si. (b) Ionization energies and electron affinities Key point: Ionization energy increases across a period and decreases down a group. Electron affinities are highest for elements near fluorine, particularly the halogens. We need to be aware of the energies needed to form cations and anions of the elements. Ionization energies are relevant to the formation of cations; electron affinities are relevant to the formation of anions. The ionization energy of an element is the energy required to remove an electron from a gas-phase atom (Section 1.9). Ionization energies correlate strongly with atomic radii, and elements that have small atomic radii generally have high ionization energies. Therefore, as the atomic radius increases down a group, the ionization energy decreases. Likewise, the decrease in radius across a period is accompanied by an increase in ionization energy (Fig. 9.2). As discussed in Section 1.9, there are variations in this trend: in particular, high ionization energies occur when electrons are removed from half-filled or full shells or subshells. Thus the first ionization energy of nitrogen ([He]2s22p3) is 1402 kJ mol1, which is higher than the value for oxygen ([He]2s22p4, 1314 kJ mol1). Similarly, the ionization energy of phosphorus (1011 kJ mol1) is higher than that of sulfur (1000 kJ mol1). The energies of electrons in hydrogenic atoms are proportional to Z2/n2. A first approximation to the energies of electrons in many-electron atoms is that they are proportional to Zeff2/n2, where Zeff is the effective nuclear charge (Section 1.6), although this proportionality should not be taken too seriously. Plots of first ionization energies against Zeff2/n2 for the outermost electrons for the elements Li to Ne (n = 2) and Hf to Hg (n = 6) are shown 259 Periodic properties of the elements 2 13 14 15 16 17 2 Li 513 Be 899 B 801 C 1086 N 1402 O 1314 F 1681 Ne 2080 3 Na 495 Mg 737 Al 577 Si 786 P 1011 S 1000 Cl 1251 Ar 1520 4 K 419 Ca 589 Ga 579 Ge 762 As 947 Se 941 Br 1139 Kr 1351 5 Rb 403 Sr 549 In 558 Sn 708 Sb 834 Te 869 I 1008 Xe 1170 6 Cs 375 Ba 502 Tl 590 Pb 716 Bi 704 Po 812 At 926 Rn 1036 in Figs 9.3 and 9.4, respectively. The graphs confirm that the proportionality is broadly followed, especially at high values of n when the outermost electron experiences an interaction with an almost point-like core. The electron affinity plays a role in assessing the energy required to form an anion. As we saw in Section 1.9c, an element has a high electron affinity if the additional electron can enter a shell where it experiences a strong effective nuclear charge. Therefore, elements close to F (other than the noble gases) have the highest electron affinities as Zeff is then large. The addition of an electron to a singly charged ion (as in the formation of O2 from O) is invariably negative (that is, the process is endothermic) because it takes energy to push an electron on to a negatively charged species. However, that does not mean it cannot happen, for it is important to assess the overall consequences of ion formation, and it is often the case that the interaction between highly charged ions in the solid overcomes the additional energy needed to form them. When assessing the energetics of compound formation, it is essential to think globally and not to rule out an overall process simply because an individual step is endothermic. (c) Electronegativity Key points: Electronegativity increases across a period and decreases down a group. We saw in Section 1.9 that the electronegativity, , is the power of an atom of the element to attract electrons to itself when it is part of a compound. As we saw in Section 1.9b, trends in electronegativity can be correlated with trends in atomic radii. This correlation is most readily understood in terms of the Mulliken definition of electronegativity as the mean of the ionization energy and electron affinity of an element. If an atom has a high ionization energy (so it is unlikely to give up electrons) and a high electron affinity (so there are energetic advantages in its gaining electrons), then it is more likely to attract an electron to itself. Consequently, the electronegativities of the elements, tracking the trends in ionization energies and electron affinities, which in turn track atomic radii, typically increase left to right across a period and decrease down a group. However, the Pauling values of electronegativity are commonly used (Fig. 9.5). There are some exceptions to this general trend as can be seen from the following electronegativities: Al Si 1.61 1.90 Ga Ge 1.81 2.01 In Sn 1.78 1.96 Figure 9.2 The variation of first ionization energy (in kilojoules per mole) through the main groups of the periodic table. 2500 Ionization energy, I/(kJ mol–1) 1 18 He 2373 18/VIII 2000 Ne F N 1500 O 1000 Be C B 500 Li 0 0 2 4 6 Zeff2/n2 8 10 Figure 9.3 Plot of first ionization energies against Zeff2/n2 for the outermost electrons for the elements lithium to neon (n = 2). 1200 Ionization energy, I/(kJ mol–1) H 1312 Hg 1000 Pt 800 Ir Au Os W Re 600 Ta Hf 400 200 0 2 2.5 3 3.5 Zeff2/n2 4 Figure 9.4 Plot of first ionization energies against Zeff2/n2 for the outermost electrons for the elements hafnium to mercury (n = 6). 260 9 Periodic trends H 2.20 Figure 9.5 The variation of Pauling electronegativity through the main groups of the periodic table. Cross Reference: Sections 13.1, 14.5, and 15.11b 2 3 1 2 13 14 Li 157 Be 112 B 88 C Na Mg 160 Al 143 Si 118 Li 0.98 Be 1.57 B 2.04 C Na Mg 1.31 Al 1.61 Si 1.90 (a) 2 3 Figure 9.6 The diagonal relationship between (a) atomic radii (in picometres) and (b) Pauling electronegativity in Periods 2 and 3. Cross Reference: Section 12.3 Cross Reference: Section 13.1 1 2 13 14 15 16 17 2 Li 0.98 Be 1.57 B 2.04 C 2.55 N 3.04 O 3.44 F 3.98 Ne 3 Na 0.93 Mg 1.31 Al 1.61 Si 1.90 P 2.19 S 2.58 Cl 3.16 Ar 4 K 0.82 Ca 1.00 Ga 1.81 Ge 2.01 As 2.18 Se 2.55 Br 2.96 Kr 5 Rb 0.82 Sr 0.95 In 1.78 Sn 1.96 Sb 2.05 Te 2.10 I 2.66 Xe 6 Cs 0.79 Ba 0.89 Tl 2.04 Pb 2.33 Bi 2.02 Po 2.0 At Rn This departure from a smooth decrease down the group is called the alternation effect and is due to the intervention of the 3d subshell earlier in Period 4. The alternation effect also appears in a more chemically direct manner, where it summarizes (but does not explain) the nonexistence of various compounds in Groups 13 to 15, as in the following examples from Group 15, where the grey denotes unknown compounds (AsCl5 is unstable above 50°C): NF5 NCl5 NBr5 PF5 PCl5 PBr5 AsF5 AsCl5 AsBr5 SbF5 SbCl5 SbBr5 BiF5 BiCl5 BiBr5 Although electronic factors such as electronegativity no doubt play a role in these examples, steric effects are important too, especially for N. The term ‘electropositive’, which describes an element’s ability to lose electrons, needs to be used with caution. In some applications it means that the element has a low electronegativity; in others it means that the redox couple Mn/M has a strongly negative standard potential (so that M is a reducing metal). We use the term only in the latter sense in this book. (d) Diagonal relationships (b) Cross Reference: Section 11.3 18 He 18/VIII Key points: The atomic radius, and hence some chemical properties, of some Period 2 elements is similar to that of the element to their lower right in the periodic table. Most of the trends in the chemical properties of the elements within the periodic table are best discussed in terms of vertical trends within groups or horizontal trends across periods. The element at the head of each group also commonly possesses a diagonal relationship with the element to its lower right. Diagonal relationships arise because the atomic radii, charge densities, electronegativities, and hence many of the chemical properties, of the two elements are similar (Fig. 9.6). The most striking diagonal relationships are those between Li and Mg. For example, whereas the Group 1 elements form compounds that are essentially ionic in nature, Li and Mg salts have some degree of covalent character in their bonding. There is a strong diagonal relationship between Be and Al: both elements form covalent hydrides, halides, and oxides; the analogous compounds of Group 2 are predominantly ionic. The diagonal relationship between B and Si is illustrated by the fact that both elements form flammable, gaseous hydrides whereas aluminium hydride is a solid. (e) Enthalpies of atomization Key point: The enthalpy of atomization increases with increasing number of valence electrons. The enthalpy of atomization of an element, ∆aH O , is a measure of the energy required to form gaseous atoms. For solids, the enthalpy of atomization is the enthalpy change Periodic properties of the elements 261 Table 9.1 Enthalpies of atomization, ∆aH O /(kJ mol–1) Li Be B C N O F 161 321 590 715 473 248 79 Na Mg Al Si P S Cl 109 150 314 439 315 223 121 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br 90 193 340 469 515 398 279 418 427 431 339 130 289 377 290 202 112 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I 86 164 431 611 724 651 648 640 556 390 289 113 244 301 254 199 107 Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po 176 427 669 774 844 791 782 665 565 369 61 186 196 208 144 9.3 Occurrence Key point: Hardhard and softsoft interactions help to systematize the terrestrial distribution of the elements. Some elements occur in their elemental state in nature, for example the gases nitrogen and oxygen, the nonmetal sulfur, and the metals silver and gold. Most elements however, occur naturally in ores as compounds with other elements. The concept of hardness and softness (Section 4.12) helps to rationalize a great deal of inorganic chemistry, including the type of compound that the element forms in Nature. Thus, soft acids tend to bond to soft bases and hard acids tend to bond to hard bases. These tendencies explain certain aspects of the Goldschmidt classification of the elements into four types (Fig 9.9), a scheme widely used in geochemistry: Lithophiles are found primarily in the Earth’s crust (the lithosphere) in silicate minerals, and include Li, Mg, Ti, Al, and Cr (as their cations). These cations are hard, and are found in association with the hard base O2. 800 Period 2 600 400 Period 3 200 0 1 2 13 14 15 16 17 Group Figure 9.7 Variation of the enthalpy of atomization in the s- and p-block elements. Cross Reference: Section 19.3b Enthlapy of atomization, ΔaH °/(kJ mol–1) associated with the atomization of the solid; for molecular species, it is the enthalpy of dissociation of the molecules. As can be seen in Table 9.1, enthalpies of atomization first increase and then decrease across Periods 2 and 3, reaching a maximum at C in Period 2 and Si in Period 3. The values decrease between C and N, and Si and P: even though N and P each have five valence electrons, two of these electrons form a lone pair and only three are involved in bonding. A similar effect is seen between N and O, where O has six valence electrons of which four form lone pairs and only two are involved in bonding. These trends are shown in Fig. 9.7. The enthalpies of atomization of the d-block elements are higher than those of the s- and p-block elements, in line with their greater number of valence electrons and consequently stronger bonding. The values reach a maximum at Groups 5 and 6 (Fig. 9.8), where there is a maximum number of unpaired electrons available to form bonds. The middle of each row shows an irregularity due to spin correlation (Section 1.7a), which favours a halffilled d shell for the free atom. This effect is particularly evident for the 3d series, in which Cr (3d54s1) and Mn (3d54s2) have significantly lower atomization energies than expected from a simple consideration of their number of valence electrons. The enthalpy of atomization decreases down a group in the s and p blocks but increases down a group in the d block. Thus s and p orbitals become less effective at forming bonds as the period number increases, whereas d orbitals become more effective. These trends are attributed to the expansion of p orbitals on descending a group from optimal for overlap to too diffuse for extensive overlap and, in contrast, d orbitals expanding in size from too contracted to optimal for overlap. The same trends can be seen in the melting points (Table 9.2) of the elements, where a greater number of valence electrons leads to greater binding energy and a higher melting temperature. Enthalpy of atomization, ΔaH °/(kJ mol–1) Cs 79 5d-Series 800 4d-Series 600 400 200 0 3d-Series 3 4 5 6 7 8 9 10 11 12 Group Figure 9.8 Variation of enthalpy of atomization in the d-block elements. 262 9 Periodic trends Table 9.2 Normal melting points of the elements, θmp /°C Li Be B C N O F 180 1280 2300 3730 –210 –218 –220 Na Mg Al Si P S Cl 97.8 650 660 1410 113 –110 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge 63.7 850 1540 1675 1900 1890 1240 1535 1492 1453 1083 420 29.8 937 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn 38.9 768 1500 1850 2470 2610 2200 2500 1970 1550 961 321 2000 232 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po 28.7 714 920 2220 3000 3410 3180 3000 2440 1769 1063 13.6 304 327 271 254 As Se Br 217 –7.2 Sb Te I 630 450 114 Chalcophiles are often found in combination with sulfide (and selenide and telluride) minerals, and include Cd, Pb, Sb, and Bi. These elements (as their cations) are soft, and are found in association with the soft base S2 (or Se2 and Te2). Zinc cations are borderline hard, but softer than Al3 and Cr3, and Zn is also often found as its sulfide. Siderophiles are intermediate in terms of hardness and softness and show an affinity for both oxygen and sulfur. They occur mainly in their elemental state and include Pt, Pd, Ru, Rh, and Os. Atmophiles are gases such as H, N, and Group 18 elements (the noble gases). E X A M PL E 9.1 Explaining the Goldschmidt classification The common ores of Ni and Cu are sulfides. By contrast Al is obtained from the oxide and Ca from the carbonate. Can these observations be explained in terms of hardness? Answer We need to assess whether the hard–hard and soft–soft rule applies. From Table 4.5 we know that O2– and CO32– are hard bases; S2– is a soft base. The table also shows that the cations Ni2+ and Cu2+ are considerably softer acids than Al3+ or Ca2+. Hence the hard–hard and soft–soft rule accounts for the sorting observed. Self-test 9.1 Of the metals Cd, Rb, Cr, Pb, Sr, and Pd, which might be expected to be found in aluminosilicate minerals and which in sulfides? 18 1 1 2 i 2 L e B a N g M 3 sL le p o ith ls h p ro e id S s ie p lco a h C s ile h p o tm A 3 4 5 a K C 4 H e H 13 5 6 7 8 9 10 11 14 15 16 17 B C N O F e N l A i S P S l C r A 12 c S iT V r C n M e F o C i N u C n Z a G e G s A e S r B r K b R r S Y r Z b N o M cT u R h R d P g A d C In n S b S eT I e X s C a B a L f H aT W e R s O Ir t P u A g H l T b P i B o P t A n R r F a R c A f R b D g S h B s H t M s D g R 6 7 e P C r N dm PS mE uG dT bD yH o rE T mY b Figure 9.9 The Goldschmidt classification of the elements. h P T a UN pP uA mC mB k C f E sF mM dN oL r u L Periodic properties of the elements 263 9.4 Metallic character Key point: The metallic character of the elements decreases across a period and increases down a group. Many elements in the p block exist as allotropes. The chemical properties of the metallic elements can be considered as arising from the ability of the elements to lose electrons to form the electron sea that binds together the cations and accounts for metallic bonding (Section 3.19). Consequently, elements with low ionization energies are likely to be metals and those with high ionization energies are likely to be nonmetals. Thus, as ionization energies decrease down a group the elements become more metallic, and as the ionization energies increase across a row the elements become less metallic (Fig. 9.10). These trends can also be directly related to the trends in atomic radii as large atoms typically have low ionization energies and are more metallic in character. This trend is most noticeable within Groups 13 to 16, where the elements at the head of the group are nonmetals and those at the foot of the group are metals. Within this general trend there are allotropic variations in the sense that some elements exist as both metals and nonmetals. An example is Group 15: N and P are nonmetals, As exists as nonmetal, metalloid, and metallic allotropes, and Sb and Bi are metals. Elements in the p block typically form several allotropes (Table 9.3). Cross Reference: Sections 13.1, 14.1, 15.1, and 16.1 Cross Reference: Section 15.1 Metallic character decreases Metallic character increases 18 1 H 1 2 Li 3 Be Na Mg 3 4 5 4 5 6 Cr 7 8 Mn Fe K Ca Sc Ti V Rb Sr Y Zr Nb Mo Tc Ru Cs Ba La Hf Ta Ra Ac Rf Db Sg Bh 6 7 Fr He Metals Metalloids Nonmetals 2 Ce Pr Th Pa W Re Os 9 10 Co Ni Rh Pd Ir Hs Mt Pt 11 14 15 16 17 B C N O F Ne Al Si P S Cl Ar Se Br Kr I Xe Po At Rn Er Tm Yb Lu 12 Cu Zn Ga Ge As Ag Cd In Sn Sb Te Tl Pb Bi Au Hg Ds Rg Nd Pm Sm Eu Gd Tb U 13 Dy Ho Np Pu Am Cm Bk Cf Es Fm Md No Lr Table 9.3 Some allotropes of the p-block elements C Diamond, graphite, amorphous, fullerenes Sn Grey, white O Dioxygen, ozone P White, red, black S Many catenated rings, chains, amorphous As Yellow, metallic/grey, black Se Red (α, β, γ), grey, black Sb Blue, yellow, black Bi Amorphous, crystalline Figure 9.10 The variation of metallic character through the periodic table. 264 9 Periodic trends 9.5 Oxidation states Key points: The group oxidation number can be predicted from the electron configuration of an element. The inert pair effect leads to an increasing stability of an oxidation state that is 2 less than the group oxidation number for the heavier elements. d-Block elements exhibit a variety of oxidation states. Cross Reference: Sections 13.14, 14.8, and 15.13 Cross Reference: Section 19.3a Cross Reference: Section 19.4 The trends in stable oxidation states within the periodic table can be understood to some extent by considering electron configurations. Related factors such as ionization energies and spin correlation also play a role. A complete or half-full valence shell imparts greater stability than a partially filled shell. Therefore, there is a tendency for atoms to gain or lose electrons until they acquire that configuration. A noble-gas configuration is achieved in the s and p blocks when eight electrons occupy the s and p subshells of the valence shell. In Groups 1, 2, and 13 the loss of electrons to leave the inner complete shell can be achieved with a relatively small input of energy. Thus, for these elements the oxidation numbers typical of the groups are 1, 2, and 3, respectively. From Group 14 to Group 17 it becomes increasingly energetically favourable—provided we consider the overall contributions to the energy, such as the interaction between oppositely charged ions—for the atoms to accept electrons in order to complete the valence shell. Consequently, the group oxidation numbers are 4, 3, 2, 1 with more electronegative elements. Group 18 elements already have a complete octet of electrons and are neither readily oxidized nor reduced. The heavier elements of the p block also form compounds with the element with an oxidation number 2 less than the group oxidation number. The relative stability of an oxidation state in which the oxidation number is 2 less than the group oxidation number is an example of the inert pair effect and it is a recurring theme within the p block. For example, in Group 13 whereas the group oxidation number is 3, the 1 oxidation state increases in stability down the group. In fact, the most common oxidation state of thallium is Tl(I). There is no simple explanation for this effect: it is often ascribed to the large energy that is needed to remove the ns2 electrons after the np1 electron has been removed. However, the sum of the first three ionization energies of Tl (5438 kJ mol1) is not higher than the value for Ga (5521 kJ mol1) and only slightly higher than the value for In (5083 kJ mol1). Another contribution to the effect may be the low MX bond enthalpies for the heavier p-block elements and the decreasing lattice energy as the atomic radii increase down a group. The group oxidation number is achieved in the d block only for Groups 3 to 8 and even then highly oxidizing F or O are required to bring it out. For Group 7 and 8, only O can produce anions or neutral oxides with an element with oxidation numbers 7 and 8, as in the permanganate anion, MnO4 and osmium tetraoxide, OsO4. The range of observed oxidation states is shown in Table 9.4. As can be seen, up to Mn, all the 3d and 4s electrons can participate in bonding and the maximum oxidation state corresponds to the group number. Once the d5 electron configuration is exceeded the tendency for the d electrons to participate in bonding decreases and high oxidation states are not observed. Similar trends exist for the 4d and 5d series. However, the stability of high oxidation states increases down a group in Groups 4 to 10 as the atomic radius increases. The relative stabilities of oxidation states of the 4d and 5d series members of each group are similar as their Table 9.4 The range of observed positive oxidation states for 3d-series elements Sc Ti V Cr Mn Fe Co Ni Cu Zn d1s2 d2s2 d3s2 d5s1 d5s2 d6s2 d7s2 d8s2 d10s1 d10s2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +3 +1 +3 +1 +3 +3 +3 +3 +3 +3 +3 +4 +4 +4 +4 +4 +4 +4 +5 +5 +5 +5 +5 +6 +6 +6 +7 Periodic characteristics of compounds atomic radii are very similar (due to the lanthanide contraction). As already remarked, half-filled shells of electrons with parallel spins are particularly stable due to spin correlation (Section 1.7a). This additional stability has important consequences for the chemistry of the d-block elements with exactly half-filled shells. Manganese has the configuration 3d54s2; as a result, Mn(II) is particularly stable and Mn(III) compounds are uncommon. The importance of spin correlation diminishes as the orbitals become larger, for example Tc and Re, which are the 4d and 5d counterparts of Mn, do not form M(II) compounds. d-Block elements also form stable compounds with the metal in its zero oxidation state. These complexes are typically stabilized by a ligand that acts as a π-acid. Cross Reference: Section 22.12 Periodic characteristics of compounds The number and type of bonds that elements form depend to a large extent on the relative strength of the bonds and the relative sizes of the atoms. 9.6 Coordination numbers Key points: Low coordination numbers generally dominate for small atoms; high coordination numbers are possible as a group is descended. The coordination number of an atom in a compound depends very much on the relative sizes of the central atom and the surrounding atoms. In the p block, low coordination numbers are most common for the compounds of Period 2 elements but higher coordination numbers are observed as each group is descended and the radius of the central atom increases. For example, in Group 15 N forms three-coordinate molecules such as NCl3 and four-coordinate ions such as NH4, whereas its congener P forms molecules with coordination numbers of 3 and 5, as in PCl3 and PCl5, and six-coordinate ionic species such as PCl6. This higher coordination number in Period 3 elements is an example of hypervalence and is sometimes attributed to the participation of d orbitals in bonding. However, as remarked in Section 2.3b, it is more likely to be due to the possibility of arranging a greater number of atoms or molecules around the larger central atom. In the d block the 4d and 5d series of elements tend to exhibit higher coordination numbers than the 3d series elements due to their larger radii. For example, in Group 3 Sc forms the 6-coordinate ScF63− ion whereas La forms the fluoride-bridged 9-coordinate LaF96− ion. Very high coordination numbers have been observed for some large atoms, for example Th forms the 10-coordinate [Th(C2O4)4(OH2)2]4 ion. 9.7 Bond enthalpy trends Key points: Bond enthalpies EH for p-block elements decrease down a group whereas in the d block they increase. For an atom E that has no lone pairs, the EX bond enthalpy decreases down the group; for an atom that has lone pairs, it typically increases between Periods 2 and 3, and then decreases down the group. As explained in Section 2.10, the mean bond enthalpy is the average bond dissociation enthalpy taken over a series of compounds. In the p block, EH bonds get weaker on descending a group whereas in the d block they get stronger. These trends are attributed to the same orbital contraction and expansion effects as affect atomization enthalpies (Section 9.2f). Mean EX bond enthalpies for atoms that have lone pairs typically decrease down a group. However, the bond enthalpy for an element at the head of a group in Period 2 is anomalous and smaller than that of an element in Period 3: B/(kJ mol–1) B/(kJ mol–1) NN 163 NCl 200 PP 201 PCl 319 AsAs 180 AsCl 317 Cross Reference: Section 15.1 Cross Reference: Section 19.7 265 266 9 Periodic trends The relative weakness of single bonds between atoms of Period 2 elements is often ascribed to the proximity of the lone pairs on neighbouring atoms and the repulsion between them. For a p-block element E that has no lone pairs, the EX bond enthalpy decreases down a group: B/(kJ mol–1) CC 348 B/(kJ mol–1) CCl 338 SiSi 226 SiCl 391 GeGe 188 GeCl 342 Smaller atoms form stronger bonds because the shared electrons are closer to each of the atomic nuclei. The strength of the SiCl bond is attributed to the fact that the atomic orbitals of the two elements have similar energies and efficient overlap. High values are also sometimes attributed to a contribution from π-bonding involving d orbitals. E X A M PL E 9. 2 Using bond enthalpies to rationalize structures Explain why elemental sulfur forms rings or chains with SS single bonds, whereas oxygen exists as diatomic molecules. Answer We need to consider the relative magnitude of the bond enthalpies for the single and double bonds: B/(kJ mol1) OO 142 SS 263 B/(kJ mol1) OO 498 SS 431 Because an OO bond is more than three times as strong as an OO bond, there is a much stronger tendency for oxygen to form OO bonds than OO bonds, as in dioxygen, O2. An SS bond is less than twice as strong as an SS bond, so the tendency to form SS bonds is not as strong as in oxygen and the formation of SS bonds is more likely. Self-test 9.2 Why does sulfur form catenated polysulfides of formula [SSS]2 and [SSSSS]2 whereas polyoxygen anions beyond O3 are unknown? One application of bond enthalpy arguments concerns the existence of subvalent compounds, compounds in which fewer bonds are formed than valence rules suggest, such as PH2. Although this compound is thermodynamically stable with respect to dissociation into the constituent atoms, it is unstable with respect to the disproportionation: 3 PH2(g) → 2 PH3(g) 1 4 P4(s) The origin of the spontaneity of this reaction is the strength of the PP bonds in molecular phosphorus, P4. There are the same number (six) of PH bonds in the reactants as there are in the products, but the reactants have no PP bonds. In the d block, bond enthalpies generally increase down a group, a trend that is the opposite of the general trend for the p block: B/(kJ mol–1) CoH 226 RhH 247 B/(kJ mol–1) FeC 390 RuC 528 OsC 598 As we saw in Section 9.3e, d orbitals appear to become more effective at forming bonds down a group as they expand in size from contracted to optimal for overlap, and unpaired electrons become much more common. 9.8 Anomalies Key point: The first member of each group within the p block shows differences from the rest of the group that are attributed to smaller atomic radii and a lack of low-lying d orbitals. The 3d metals form Periodic characteristics of compounds 267 compounds with lower coordination numbers and oxidation states than the 4d and 5d elements. The oxidation state Ln(III) dominates in compounds of the lanthanoids, whereas the actinoids exhibit a variety of oxidation states. The chemical properties of the first member of each group in the p block are significantly different from its congeners. These anomalies are attributable to the small atomic radius and its correlates, high ionization energies, high electronegativities, and low coordination numbers. For example, in Group 14 carbon forms an enormous number of catenated hydrocarbons with strong CC bonds. Carbon also forms strong multiple bonds in the alkenes and alkynes. This tendency to catenation is much reduced for its congeners and the longest silane formed contains just four Si atoms. Nitrogen shows distinct differences from phosphorus and the rest of Group 15. Thus, nitrogen commonly exhibits a coordination number of 3, as in NF3, and a coordination number of 4 in species such as NH4 and NF4 whereas phosphorus can form 3- and 5-coordinate compounds, such as PF3 and PF5, and 6-coordinate species such as PF6. The extent of hydrogen bonding is much greater in the compounds of the first member of each group. For example, the boiling point of ammonia is −33C, which is higher than that of the other Group 15 hydrides. Likewise, water and hydrogen fluoride are liquids at room temperature whereas H2S and HCl are gases. The striking differences between the Period 2 elements and their congeners is also reflected, but less markedly, in the d block. Thus, the properties of the 3d-series metals differ from those of the 4d and 5d series. The lower oxidation states are more stable in the 3d series, with the stability of higher oxidation states increasing down each group. For example, the most stable oxidation state of chromium is Cr(III) whereas it is M(VI) for Mo and W. The degree of covalence and the coordination numbers increase between the compounds of 3d and the 4d and 5d elements. For instance, the dihalides of the 3d-series elements are mostly ionic solids, as in CrF2, whereas the 4d- and 5d-series elements form higher halides such as MoF6 and WF6, which are liquids at room temperature. These differences are attributable to the smaller ionic radii of the 3d-series elements and the fact that the radii of the 4d- and 5d-series elements are quite similar (due to lanthanide contraction). The anomalies of the heading element are also found in the f block. The elements Ce through to Lu, generically Ln, are all highly electropositive, with the Ln3/Ln standard potential lying between those of Li and Mg. The elements favour the oxidation state Ln(III) with a uniformity that is unprecedented in the periodic table. Other properties of the elements vary significantly. For example, the radii of Ln3 ions contract steadily across the series. This decrease is attributed in part to the increase in Zeff as electrons are added to the 4f subshell, but relativistic effects also make a substantial contribution (Section 1.9a). This decrease in radius leads to an increase in the hydration enthalpy across the series. The reduction potentials of the lanthanoids are all similar, with values ranging from 1.99 V for Eu3/Eu to 2.38 V for La3/La (an honorary member of the f block). The elements from thorium (Th, Z = 90) to lawrencium (Lr, Z = 103) have ground-state electron configurations that involve the filling of the 5f subshell, and in this sense are analogues of the lanthanoids. However, the actinoids do not exhibit the chemical uniformity of the lanthanoids and occur in a rich variety of oxidation states. The f orbitals of the actinoids extend beyond the Rn core and participate in bonding. Like the lanthanoids, the actinoids have large atomic and ionic radii and as a result often have high coordination numbers. For example, U in solid UCl4 is 8-coordinate and in solid UBr4 it is 7-coordinate in a pentagonal-bipyramidal array. Solid-state structures with coordination numbers up to 12 have been observed (Section 7.5). In addition to the differences between the first member of each group and its congeners there are also similarities between elements with atomic numbers Z and Z 8; for the d block, the similarities are between elements with atomic numbers Z and Z 22. For example, Al (Z = 13) shows similarities to Sc (Z = 21). The similarities are understandable in terms of the electron configurations, for Al in Group 13 and Sc in Group 3 both have three valence electrons. Their atomic radii are fairly similar: Al is 143 pm and Sc is 160 pm and the standard potential of Al3/Al (1.66 V) is closer to that of Sc3/Sc (1.88 V) than that of Ga3/Ga (0.53 V). Similarities also exist between the following pairs: Z Z8 14 Si 22 15 P 23 16 S 24 17 Cl 25 Ti V Cr Mn Cross Reference: Section 14.7a Cross Reference: Section 14.7b Cross Reference: Section 15.1 Cross Reference: Section 15.10 Cross Reference: Sections 16.8a and 17.8 Cross Reference: Section 19.4 Cross Reference: Section 19.7 Cross Reference: Section 23.3 Cross Reference: Section 23.9 268 9 Periodic trends For example, S and Cr form anions of the type SO42, S2O72, CrO42, and Cr2O72, and both Cl and Mn form the oxidizing peroxoanions, ClO4 and MnO4. These similarities are observed when the elements are in their highest oxidation states and the d-block element has a d0 configuration. For d-block elements the similarity is between the Z and Z22 elements. E X A M PL E 9. 3 Predicting the chemical properties of a Z + 8 element The perchlorate ion, ClO4, is a very powerful oxidizing agent and its compounds can detonate on contact or with heat. Predict whether the analogous compound of the Z 8 element would be a suitable replacement for a perchlorate compound in a reaction. Answer We need to identify the Z 8 element. Because the atomic number of Cl is 17, the Z 8 element is Mn (Z 25). The compound of Mn analogous to ClO4− is the permanganate ion, MnO4−, which is in fact an oxidizing agent, but less hazardous. Permanganate is likely to be a suitable replacement for perchlorate. Self-test 9.3 Xenon is very unreactive but forms a few compounds with oxygen and fluorine such as XeO4. Predict the shape of XeO4 and identify the Z 22 compound with the same structure. 9.9 Binary compounds The simple binary compounds of the elements exhibit interesting trends in their structure and properties. Hydrogen, oxygen, and the halogens form compounds with most elements and the hydrides, oxides, and halides are reviewed here to give some insight into trends in bonding and properties. (a) Hydrides of the elements Key point: The hydrides of the elements are classified as molecular, saline, or metallic. Cross Reference: Sections 13.6, 14.7, 15.10, 16.8, and 17.2 Cross Reference: Section 11.6 and 12.6 Cross Reference: Section X.X Hydrogen reacts with most elements to form hydrides that can be described as molecular, saline, or metallic, although some cannot be easily classified and are termed intermediate (Fig. 9.11). Molecular compounds of hydrogen are common for the nonmetallic, electronegative elements of Groups 13 to 17; some examples are B2H6, CH4, NH3, H2O, and HF. These covalent hydrides are gases, with the exception of water (due to extensive hydrogen bonding). The saline hydrides are formed by the electropositive elements of Group 1 and Group 2 (with the exception of Be). The saline hydrides are ionic solids with high melting points. Nonstoichiometric metallic hydrides are formed by all the d-block metals of Groups 3, 4, and 5, and by the f-block elements. (b) Oxides of the elements Key point: Metals form basic oxides and nonmetals form acidic oxides. The elements form normal oxides, peroxides, superoxides, suboxides, and nonstoichiometric oxides. The high reactivity of oxygen and its high electronegativity leads to a large number of binary oxygen compounds, many of which bring out high oxidation states in the second element. The range of possible oxides is shown in Table 9.5. 18 1 1 2 Li 3 2 Be Na Mg 3 4 5 Figure 9.11 Classification of the binary hydrides of the s-, p-, and d-block elements. 7 4 5 6 Cr 7 8 Mn Fe K Ca Sc Ti V Rb Sr Y Zr Nb Mo Tc Ru Hf Ta Cs Ba 6 Unknown Saline Intermediate Metallic Molecular W Re Os 9 10 Co Ni Rh Pd Ir Pt 11 He 13 14 15 16 17 B C N O F Ne Al Si P S Cl Ar Se Br Kr I Xe At Rn 12 Cu Zn Ga Ge As Ag Cd In Sn Sb Te Tl Pb Bi Au Hg Po Periodic characteristics of compounds 269 Table 9.5 Possible oxides of the elements H2O H2O2 Li2O BeO B2O3 network solids CO CO2 C3O2 N2O NO N2O3 NO2 N2O4 N2O5 O2 O3 OF2 O2F2 SiO2 P4O6 P4O10 SO2 SO3 Cl2O Cl2O3 ClO2 Cl2O4 Cl2O6 Cl2O7 glasses Na2O Na2O2 Al2O3 MgO MgO2 glasses minerals K2O K2O2 KO2 KO3 CaO CaO2 Sc2O3 TiO TiO2 Ti2O3 VO VO2 V2O3 V2O5 V3O5 CrO2 CrO3 Cr2O3 Cr3O4 MnO MnO2 Mn2O3 Mn2O7 Mn3O4 FeO Fe2O3 Fe3O4 CoO Co3O4 NiO Ni2O3 CuO Cu2O ZnO Ga2O3 GeO GeO2 As2O3 As2O5 SeO2 SeO3 Br2O Br2O3 BrO2 Rb2O Rb2O2 RbO2 RbO3 Rb9O2 SrO SrO2 Y2O3 ZrO2 NbO NbO2 Nb2O5 MoO MoO2 MoO3 Mo2O3 Mo2O5 TcO2 Tc2O7 RuO2 RuO3 RhO2 Rh2O3 PdO PdO2 AgO Ag2O CdO In2O3 SnO SnO2 Sb2O3 Sb2O5 TeO2 TeO3 I2O4 I2O4 I2O5 I4O9 Cs2O Cs2O2 CsO2 CsO3 BaO BaO2 La2O3 HfO2 TaO TaO2 Ta2O3 Ta2O5 WO2 WO3 ReO2 ReO3 Re2O3 Re2O7 OsO2 OsO4 IrO2 Ir2O3 PtO PtO2 PtO3 Au2O3 HgO Hg2O Tl2O Tl2O2 PbO PbO2 Pb3O4 Bi2O3 Bi2O5 Metals typically form basic oxides. The electropositive metal forms a cation readily and the oxide anion abstracts a proton from water. For example, OHions are produced when barium oxide reacts with water: BaO(s) H2O(l) → Ba2(aq) 2 OH(aq) Nonmetals form acidic oxides. The electronegative element pulls in electrons from coordinated H2O molecules, liberating H. For example, sulfur trioxide reacts with water to produce hydronium ions (represented here as simply H(aq)): XeO3 XeO4 Cross Reference: Section 11.8 and 12.8 Cross Reference: Sections 15.13, 16.12, and 17.2 SO3(g) H2O(l) → 2 H(aq) SO42(aq) (c) Halides of the elements Key point: The s-block halides are predominantly ionic and the p-block halides are predominantly covalent. In the d block, low oxidation state halides tend to be ionic and high oxidation state halides tend to be covalent. The halogens form compounds with most elements, but not always directly. The range of chlorides that are formed is illustrated in Table 9.6. With the exception of Li and Be, the s-block halides are ionic and the p-block fluorides are predominantly covalent. Fluorine and Cl bring out the group oxidation number in most of the elements as well as an oxidation number 2 lower, as expected from the inert pair effect. 18 Acidity increases He 2 13 14 15 16 17 Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Ga Ge As Se Br Kr Rb Sr In Sn Sb Te I Xe Cs Ba Tl Pb Bi At Rn 1 2 Li Acidity decreases The acidic nature of the oxides increases across a row and decreases down a group for a given oxidation state (Fig. 9.12). In Group 13 the element at the head of the group, B, is a nonmetal and forms the acidic oxide B2O3. At the bottom of the group the metallic character has increased and the inert pair effect has reduced the stable oxidation state from 3 to 1 and the oxide of thallium is the basic Tl2O. 3 4 5 Po 6 Figure 9.12 The general variation of acidic nature of the oxides of the elements through the periodic table. 270 9 Periodic trends Table 9.6 Simple chlorides of the elements HCl LiCl BeCl2 BCl3 CCl4 NCl3 OCl2 ClF ClF3 ClF5 NaCl MgCl2 AlCl3 SiCl4 PCl3 PCl5 S2Cl2 SCl2 Cl2 KCl CaCl2 ScCl3 TiCl2 TiCl3 TiCl4 VCl2 VCl3 VCl4 CrCl2 CrCl3 CrCl4 MnCl2 MnCl3 FeCl2 FeCl3 CoCl2 CoCl3 NiCl2 CuCl2 CuCl3 ZnCl2 GaCl3 GeCl4 AsCl3 AsCl5 SeCl4 BrCl RbCl SrCl2 YCl3 ZrCl2 ZrCl4 NbCl3 NbCl4 NbCl5 TcCl4 MoCl6 RuCl2 RuCl3 RhCl3 PdCl2 AgCl CdCl2 InCl InCl2 InCl3 SnCl2 SnCl4 SbCl3 SbCl5 TeCl4 ICl ICl3 I2Cl6 CsCl BaCl2 LaCl3 HfCl4 TaCl3 TaCl4 TaCl5 MoCl2 MoCl3 MoCl4 MoCl5 MoCl6 WCl2 WCl4 WCl6 ReCl4 ReCl5 ReCl6 OsCl4 OsCl5 OsCl6 IrCl2 IrCl3 IrCl4 PtCl3 PtCl4 AuCl HgCl2 Hg2Cl2 TlCl TlCl2 TlCl3 PbCl2 PbCl4 BiCl3 BiCl5 Cross Reference: Section 19.7 At Exceptions are N, O, and S, where only lower oxidation state halides are formed. The d-block elements form halides with a range of oxidation states. The higher oxidation state halides are formed with F and Cl. The lower oxidation state halides are ionic solids. The higher oxidation state chlorides formed predominantly by the 4d- and 5d-series elements are covalent and there is an increased tendency to form cluster compounds with metalmetal bonds. 9.10 Wider aspects of periodicity The differences in chemical properties of elements and compounds result from a complex interplay of periodic trends. In this final section we illustrate how these trends compensate, conflict with, and enhance each other. (a) Ionic chlorides Key point: For ionic compounds, trends in lattice enthalpies, ionization energies, and enthalpies of atomization have significant effects on the enthalpy of formation of ionic halides. Cross Reference: Section 12.7 and 11.7 As can be seen from Table 9.7, the values of ∆fH O for the Group 1 halides are reasonably constant on descending a group. The ionization energy and enthalpy of atomization both become less positive down the group as the atomic radii increase but these trends are largely offset by changes in the lattice enthalpy (Section 3.11). The values of ∆fH O for Group 2 halides are up to twice the values for the Group 1 halides. The ionization energy and enthalpy of atomization both become more positive between Groups 1 and 2 as the radii decrease. However, the more significant factor is the very large increase in lattice enthalpy as the radii decrease and the charges on the ions increase from 1 and 1 to 2 and 2 1. As a result, Group 2 halides are more stable than those of Group 1. (b) Covalent halides Key point: Bond enthalpy and entropy effects are the most important factors in determining whether or not Group 16 halides exist. Table 9.7 Standard enthalpies of formation of Group 1 and 2 chlorides, ∆fH O /(kJ mol–1) LiCl −409 BeCl2 −512 NaCl −411 MgCl2 −642 KCl −436 CaCl2 −795 RbCl −431 SrCl2 −828 CsCl −433 BaCl2 −860 Periodic characteristics of compounds Compounds formed between sulfur and the halogens can provide some insight into the factors that influence the values of ∆fH O for covalent halides. Sulfur forms several different compounds with F, most of which are gases. Sulfur hexafluoride, SF6, sulfur difluoride, SF2, and sulfur dichloride, SCl2, exist whereas SCl6 is not known. The values of ∆fH O calculated from bond enthalpy data are: ∆fH O /(kJ mol1) SF2 SF6 SCl2 SCl6 298 1220 49 74 Cross Reference: Section 16.9 Thus, although the formation of SCl6 is more exothermic than SCl2, other factors ensure that SCl6 cannot be prepared under standard conditions. The explanation can be found by considering the bond enthalpies of sulfurhalogen bonds: B/(kJ mol1) FSF FSF5 ClSCl 367 329 271 There is a decrease in bond enthalpy between SF2 and SF6 due possibly to steric crowding around the S atom and repulsion between crowded F atoms. A similar decrease would be expected between SCl2 and SCl6. This weak bond is one factor in the nonexistence of SCl6. In addition, there would be a higher entropic cost when a SCl6 molecule is formed from three dihalogen molecules compared to just one when SCl2 is formed. In contrast, compounds containing the PCl6− ion are known. Bonding between P and Cl should be stronger than that between S and Cl because P is less electronegative than S. E X A M PL E 9. 4 Assessing the factors affecting the formation of a compound Estimate ∆fH O (SH6, g) by assuming that B(HS) is the same for HSH5 as for HSH (375 kJ mol1). The value of B(H–H) is 436 kJ mol1 and B(SS) is 263 kJ mol1. Suggest a way of reconciling your answer with observation. Answer We estimate ∆fH O (SH6 ,g) from the difference between the enthalpies of bonds broken and bonds formed in the reaction 1 8 S8(s) + 3 H2(g) → SH6(g) The enthalpy change accompanying bond breaking is 263 kJ mol1 3(436 kJ mol1) 1571 kJ mol1. The enthalpy change accompanying bond making is 6 (375 kJ mol1) 2250 kJ mol1. Therefore, ∆fH O (SH6, g) 1571 kJ mol1 2250 kJ mol1 679 kJ mol1 indicating that the compound is exothermic and, on the basis of this calculation, can be expected to exist. However, SH6 does not exist. The reason must lie in the SH bond being much weaker than that used in the calculation. A further contribution to the Gibbs energy of formation is the unfavourable entropy change due to forming the molecule from three H2 molecules instead of only one when SH2 is formed. Self-test 9.4 Comment on the following ∆fH O values (kJ mol−1): S(g) Se(g) Te(g) SF4 SeF4 TeF4 SF6 SeF6 TeF6 223 202 199 762 850 1036 1220 1030 1319 (c) Oxides in the d block Key point: The d-block oxides deviate from the ionic model as metalmetal bonding becomes more important. The contrasting enthalpies of formation of various metal oxides of formula MO (Table 9.8) give some interesting insights into different aspects of periodicity. The high exothermocity of the Group 2 oxides is a result of the relatively low ionization energy and low enthalpy of atomization of the s-block metals. The experimental lattice enthalpies are very close to those calculated from the Kapustinskii equation, indicating that the compounds conform well to the ionic model. Values of ∆fH O (MO) for the 3d-series elements become less negative across Cross Reference: Section 12.8 271 272 9 Periodic trends Table 9.8 Some thermodynamic data (in kJ mol–1) for d-metal oxides, MO ∆ion(1+2)H O ∆aH O ∆fH O ∆HLO (calc) ∆HLO (exp) Ca 1735 177 CaO −636 3464 3390 V 2064 514 VO −431 3728 4037 Ni 2490 430 NiO −240 3860 4436 Nb 2046 726 NbO −406 3760 4206 As calculated from the Kapustinskii equation. Cross Reference: Section 19.8 the period. There are opposing trends across this series because, although the ionization energy increases, the atomization enthalpy decreases. The experimental lattice enthalpies of the oxides on the right of the series deviate from those calculated from the Kapustinskii equation, indicating that the ionic model is no longer adequate. Although the ionization energy of a 4d-series element is lower than that of its 3d-series congener, its atomization enthalpy is much higher, reflecting the stronger metallic bonding due to better overlap between 4d orbitals than between 3d orbitals. E X A M PL E 9. 5 Predicting the thermal stabilities of d-block oxides Compare the stabilities of V2O5 and Nb2O5 towards thermal decomposition in the reaction M2O5(s) → 2 MO(s) + 32 O2(g) Use the data in Table 9.8 and the enthalpies of formation of Nb2O5 and V2O5 of 1901 and 1552 kJ mol–1, respectively. Answer We need to consider the reaction enthalpy for each oxide. The enthalpies of reaction can be calculated from the difference between the enthalpies of formation of products and those of the reactants: For Nb2O5: ∆rH O = 2(406 kJ mol1) (1901 kJ mol1) = +1089 kJ mol1 For V2O5: ∆rH O = 2(431 kJ mol1) (1552 kJ mol1) = +690 kJ mol1 The enthalpy of the reaction for V2O5 is less endothermic than that of Nb2O5. Therefore, V2O5 is thermally more stable. Self-test 9.5 Given that ∆fH O (P4O10, s) = 3012 kJ mol–1, what further data would be useful when drawing comparisons with the value for V2O5? FURTHER READING P. Enghag, Encyclopedia of the elements. John Wiley & Sons (2004). D.M.P. Mingos, Essential trends in inorganic chemistry. Oxford University Press (1998). An overview of inorganic chemistry from the perspective of structure and bonding. N.C. Norman, Periodicity and the s- and p-block elements. Oxford University Press (1997). Includes coverage of essential trends and features of s-block chemistry. E.R. Scerri, The periodic table: Its story and its significance. Oxford University Press (2007). EXERCISES 9.1 Give the expected maximum stable oxidation state for (a) Ba, (b) As, (c) P, (d) Cl. 9.2 With the exception of one member of the group, the elements form saline hydrides. They form oxides and peroxides and all the carbides react with water to liberate a hydrocarbon. Identify this group of elements. 9.3 The elements vary from metals through metalloids to nonmetals. They form halides in oxidation states 5 and 3 and the hydrides are all toxic gases. Identify this group of elements. 9.4 Draw a BornHaber cycle for the formation of the hypothetical compound NaCl2. State which thermochemical step is responsible for the fact that NaCl2 does not exist. 9.5 Predict how the inert pair effect would manifest itself beyond Group 15 and compare your predictions with the chemical properties of the elements involved. 9.6 Summarize the relationship between ionic radii, ionization energy, and metallic character. 9.7 Give the names of the ores from which (a) Mg, (b) Al, and (c) Pb are extracted. 9.8 Identify the Z 8 element for P. Briefly summarize any similarities between the elements. 9.9 Use the following data to calculate average values of B(SeF) in SeF4 and SeF6. Comment on your answers in view Problems of the corresponding values for B(SF) in SF4 (340 kJ mol1) and SF6 (329 kJ mol1): ∆aH O (Se) = 227 kJ mol1, 273 ∆aH O (F) = 159 kJ mol1, ∆fH O l(SeF6, g) = 1030 kJ mol1, ∆fH O l(SeF4, g) = 850 kJ mol1. PROBLEMS 9.1 In the paper ‘What and how physics contributes to understanding the periodic law’ (Foundations of Chemistry, 2001, 3, 145) V. Ostrovsky describes the philosophical and methodological approaches taken by physicists to explain periodicity. Compare and contrast the approaches of physics and chemistry to explaining chemical periodicity. 9.2 P. Christiansen et al. describe ‘Relativistic effects in chemical systems’ in their 1985 paper (Ann. Rev. Phys. Chem., 2001, 36, 407). How did they define relativistic effects? Briefly summarize the most important consequences of relativistic effects in chemistry. 9.3 Many models of the periodic table have been proposed since the version devised by Mendeleev. Review the more recent versions and discuss the theoretical basis for each one. 10 Hydrogen Part A: The essentials 18 H 10.1 The element He 10.2 Simple compounds 1 2 Li Be Part B: The detail 13 14 15 16 17 B C N O F Ne 10.3 Nuclear properties 10.4 Production of dihydrogen 10.5 Reactions of dihydrogen 10.6 Compounds of hydrogen 10.7 General methods for synthesis FURTHER READING EXERCISES PROBLEMS Hydrogen has a very rich chemistry despite its simple atomic structure. In this chapter we discuss reactions of hydrogen species that are particularly interesting in terms of fundamental chemistry as well as important applications, including energy. We describe how H2 is produced in the laboratory, on an industrial scale from fossil fuels, and by using renewable resources. We summarize the properties of binary compounds that range from volatile, molecular compounds to salt-like and metallic solids. We shall see that many of the properties of these compounds are understandable in terms of their ability to supply H or H ions, and that it is often possible to anticipate which one is likely to dominate. We explain how H-bonding stabilizes the structures of H2O and DNA and how H2 can behave as a ligand to d-block metal atoms and ions. PART A: THE ESSENTIALS Hydrogen is the most abundant element in the universe and the tenth most abundant by mass on Earth, where it is found in the oceans, minerals, and in all forms of life. The partial depletion of elemental hydrogen from Earth reflects its volatility during formation of the planet. The stable form of elemental hydrogen under normal conditions is dihydrogen, H2, which occurs at trace levels in the Earth’s lower atmosphere (0.5 ppm) and is essentially the only component of the extremely thin outer atmosphere. Dihydrogen has many uses (Fig. 10.1). It is produced naturally, as a product of fermentation and as a byproduct of ammonia biosynthesis (Box 10.1). It is often cited as the ‘fuel of the future’ on account of its availability from fully renewable resources (water and sunlight) and its clean and highly exothermic reaction with O2, but the volatility and low energy density of H2 pose challenges for storage. Mta e l ctio u d ro p n Feed ck -sto CH3OH 10.1 The element M H2 –C–C– iM rg a n e Fu el NH3 Fertiilzers stics la p Figure 10.1 The major uses of H2. The hydrogen atom, with ground-state configuration 1s1, has only one electron so it might be thought that the element’s chemical properties will be limited, but this is far from the case. Hydrogen has richly varied chemical properties and forms compounds with nearly every other element. It ranges in character from being a strong Lewis base (the hydride ion, H) to being a strong Lewis acid (as the hydrogen cation, H, the proton; Section 4.1). Under certain circumstances H atoms can form bonds to more than one other atom simultaneously. The ‘hydrogen bond’ formed when an H atom bridges two electronegative atoms is fundamental to life: it is because of hydrogen bonding that water occurs as a liquid rather than a gas, and proteins and nucleic acids fold into the complex, highly organized three-dimensional structures that determine their functions. The essentials 275 B OX 10.1 The biological hydrogen cycle Hydrogen is cycled by microbial organisms using metalloenzymes (Section 27.14). Although H2 is present only to the extent of about 0.5 ppm at the Earth’s surface, its levels are hundreds of times higher in anaerobic environments, such as wetland soils and sediments at the bottoms of deep lakes and hot springs. Hydrogen is produced in these O2-free zones as a waste product by strict anaerobes (fermentative bacteria) which break down organic material (biomass) using H as an oxidant acting as a terminal electron acceptor. It is also produced by thermophilic organisms that derive their carbon and energy entirely from CO, and by nitrogen fixing bacteria, which yield H2 as a byproduct of ammonia formation. Other microorganisms, many of them aerobic, use H2 instead as a ‘food’ (a fuel) and are responsible for the formation of the familiar gases CH4 (methanogens) and H2S (Desulfovibrio) as well as nitrite and other products. Figure B10.1 summarizes some of the overall processes occurring in a freshwater environment. In animals, including humans, the anaerobic environment of the large intestine is host to bacteria that form H2 by the breakdown of carbohydrates. The mucus layer of the mouse intestine has been found to contain H2 at levels above 0.04 mmol dm3, equivalent to a partial atmosphere of 5 per cent H2. In turn, this H2 is utilized by methanogens, such as those found in ruminating mammals, to produce CH4 and by other bacteria, including dangerous pathogens such as those of the Salmonella genus and Helicobacter pylori, which is responsible for gastric ulcers. High levels of H2 in the breath have been used to diagnose conditions related to carbohydrate-intolerance and these levels may reach 70 ppm following lactose ingestion by lactose-intolerant patients. H2O Organic matter NO2– O2 Fe(II) NO3– H2S Fe(III) CH4 SO42– Fermentative bacteria H2 CO2 Figure B10.1 Some of the processes that contribute to the biological hydrogen cycle in a freshwater environment. Industrial production of H2 by microorganisms (biohydrogen) is an important area for research and development. There are two different approaches, each of which uses renewable energy. The first of these approaches is to use anaerobic organisms to ferment biomass from sources ranging from cultivated biomass (including seaweed) to domestic waste. The second involves manipulation of photosynthetic organisms such as green algae and cyanobacteria to produce H2 as well as biomass. In either case H2 can be extracted continuously by gas filters without the interruptions that would be required were harvesting required. (a) The atom and its ions Key points: The proton, H, is always found in combination with a Lewis base and is highly polarizing; the hydride ion, H, is highly polarizable. There are three isotopes of hydrogen: hydrogen itself (1H), deuterium (D, 2H), and tritium (T, 3H); tritium is radioactive. The lightest isotope, 1H (very occasionally called protium), is by far the most abundant. Deuterium has variable natural abundance with an average value of about 16 atoms in 100 000. Tritium occurs to the extent of only 1 in 1021. The different names and symbols for the three isotopes reflect the significant differences in their masses and the chemical properties that stem from mass, such as the rates of diffusion and bond-cleavage reactions. The nuclear spin of 1H (I = 12) is exploited in NMR spectroscopy (Section 8.5) for identifying hydrogen-containing molecules and determining their structures. The free hydrogen cation (H, the proton) has a very high charge/radius ratio and it is not surprising to find that it is a very strong Lewis acid. In the gas phase it readily attaches to other molecules and atoms; it even attaches to He to form HeH. In the condensed phase, H is always found in combination with a Lewis base, and its ability to transfer between Lewis bases gives it the special role in chemistry explored in detail in Chapter 4. The molecular cations H2 and H3 have only a transitory existence in the gas phase and are unknown in solution. In contrast to H, which is highly polarizing; the hydride ion, H, is highly polarizable because two electrons are bound by just one proton. The radius of H varies considerably depending on the atom to which it is attached. The lack of core electrons to scatter X-rays means that bond distances and angles involving a H atom in a compound are difficult to measure by X-ray diffraction: for this reason, neutron diffraction is used when it is crucial to determine the precise positions of H atoms. (b) Properties and reactions Key points: Hydrogen has unique atomic properties that place it in a special position in the periodic table. Dihydrogen is an inert molecule and its reactions require a catalyst or initiation by radicals. Hydrogen’s unique properties distinguish it from all other elements in the periodic table. It is often placed at the head of Group 1 because, like the alkali metals, it has only one electron in its valence shell. That position, however, does not truly reflect the chemical or 276 10 Hydrogen physical properties of the element. In particular, its ionization energy is far higher than those of the other Group 1 elements, so hydrogen is not a metal, although it may be found naturally in a metallic state where extreme pressures exist, such as the core of Jupiter. In some versions of the periodic table hydrogen is placed at the head of Group 17 because, like the halogens, it requires only one electron to complete its valence shell. But the electron affinity of hydrogen is far lower than that of any of the elements of Group 17 and the discrete hydride ion, H−, is encountered only in certain compounds. To reflect its unique characteristics, we place H in its own special position at the head of the entire table. Because H2 has so few electrons, the intermolecular forces between H2 molecules are weak, and at 1 atm the gas condenses to a liquid only when cooled to 20 K. If an electric discharge is passed through H2 gas at low pressure, the molecules dissociate, ionize, and recombine, forming a plasma containing, in addition to H2, spectroscopically observable amounts of H, H, H2, and H3. The H2 molecule has a high bond enthalpy (436 kJ mol1) and a short bond length (74 pm). The high bond strength results in H2 being an inert molecule. However, it is still much easier to dissociate H2 homolytically than heterolytically,1 because in heterolytic dissociation energy is required to separate opposite charges: H2(g) → H(g) H(g) ∆rH O = 436 kJ mol1 H2(g) → H(g) H(g) ∆rH O = 1675 kJ mol1 Because of its inherently inert nature, reactions of H2 do not occur readily unless a special activation pathway has been provided. These pathways include homolytic or heterolytic dissociation catalysed by molecules or active surfaces, and initiation of radical chain reactions. The explosive reaction of H2 with O2, for instance 2 H2(g) O2(g) → 2 H2O(g) ∆rH O = 242 kJ mol1 proceeds by a complex radical chain mechanism. Hydrogen is an excellent fuel for large rockets on account of its high specific enthalpy (the standard enthalpy of combustion divided by the mass), which is approximately three times that of a typical hydrocarbon (Box 10.2). In addition to reactions that result in dissociation of the HH bond, H2 6n also react reversibly without cleavage to form dihydrogen d-metal complexes (Section 10.6 and Section 22.7). 10.2 Simple compounds H C 109.5° 1 Methane, CH4 (a) Classification of binary compounds Key points: Compounds formed between hydrogen and other elements vary in their nature and stability. In combination with metals, hydrogen is often regarded as a hydride; hydrogen compounds with elements of similar electronegativity have low polarity. N H The nature of the bonding in binary compounds of hydrogen (EHn) is largely rationalized by noting that an H atom has a high ionization energy (1312 kJ mol1) and a low but positive electron affinity (73 kJ mol1). Although binary hydrogen compounds are often known as ‘hydrides’, very few actually contain a discrete H anion. The (Pauling) electronegativity of 2.2 (Section 1.9) is intermediate in value, so it is normally assigned the oxidation number 1 when in combination with metals (as in NaH and AlH3) and 1 when in combination with nonmetals (as in H2O and HCl). 107° 2 Ammonia, NH3 The binary compounds of hydrogen fall into three classes, although there is a range of structural types, and some elements form compounds with hydrogen that do not fall strictly into any one category: 1. Molecular hydrides exist as individual, discrete molecules; they are usually formed with p-block elements of similar or higher electronegativity than H. Their EH bonds are best regarded as covalent. O 104.5° H 3 Water, H2O Familiar examples of molecular hydrides include methane, CH4 (1), ammonia, NH3 (2), and water, H2O (3). 2. Saline hydrides, also known as ionic hydrides, are formed with the most electropositive elements. 1 In homolytic dissociation the bond breaks symmetrically to give one product. In heterolytic dissociation the bond breaks unsymmetrically to give two different products. The essentials 277 B OX 10. 2 Dihydrogen as a fuel The use of H2 as a fuel (an energy carrier) has been investigated seriously since the 1970s when oil prices first rose dramatically; interest has increased greatly in more recent times owing to environmental pressures on further use of fossil fuels. Hydrogen is clean burning, nontoxic, and its production from fully renewable resources is slowly but inevitably replacing its production from fossil carbon feedstocks. Table B10.1 compares the performance data for H2 and other energy carriers, including hydrocarbon fuels and a lithium ion battery. Among all fuels, H2 has the highest specific enthalpy (its standard enthalpy of combustion divided by its mass) but has a very low energy density (its standard enthalpy of combustion divided by its volume). It is obvious that H2 is an excellent fuel for vehicles provided the problems of on-board containment are solved (see Box 10.4). In addition to its choice as a rocket fuel (due to its high specific enthalpy), H2 can be used in conventional internal combustion engines with little if any modification Table B10.1 Specific enthalpies and energy densities of common energy carriers (1 MJ = 0.278 kWh) Fuel Specific enthalpy /(MJ kg−1) Energy density /(MJ dm−3) Liquid H2 120 8.5 H2 at 200 bar 120 1.9 Liquid natural gas 50 20.2 Natural gas at 200 bar 50 8.3 Petrol (gasoline) 46 34.2 Diesel 45 38.2 Coal 30 27.4 Ethanol 27 22.0 Methanol 20 15.8 Wood 15 14.4 Lithium battery 2.0 6.1 to their design or specifications. However, the most important way of utilizing H2 in a vehicle is to react it in a fuel cell to produce electricity directly (Box 5.1). The efficient and reliable power output of H2 fuel cells makes it viable to produce H2 ‘on board’ by steam reforming of methanol, a transportable and energy-dense fuel. (Direct methanol fuel cells, discussed in Box 5.1, produce less power than H2 fuel cells and are therefore less attractive for vehicles.) An automotive steam reformer (the principle of which is shown in Fig. B10.2) mixes methanol vapour with H2O (steam) and with O2 (air) to produce H2 by the following reactions ⎯⎯⎯ → CO2(g) 3 H2(g) CH3OH(g) H2O(g) ⎯Cu/ZnO ⎯→ CO2(g) 2 H2(g) CH3OH(g) 21 O2(g) ⎯Pd ∆rH O = 49 kJ mol1 ∆rH O = −155 kJ mol1 The reactions, which occur over the temperature range 200350°C, are controlled to ensure that the heat produced by the exothermic oxidation reaction just offsets that required for the reaction with steam and the vaporization of all components. Excessive heat results in production of CO, which poisons the Pt catalyst of the PEM fuel cell. The CO2 and H2 products are separated with a Pd membrane. CH3OH + H2O Pd membrane tube CO2 H2 O2 (as air) Cu/ZnO and Pd catalysts (Li1xCoO2, see Box 11.1) Denotes an energy carrier that is easily derived or recharged from renewable resources. Figure B10.2 Schematic cross-sectional view of an on-board methanol reformer. Saline hydrides, such as LiH and CaH2, are nonvolatile, electrically nonconducting, crystalline solids although only those in Group 1 and the heavier elements of Group 2 should be regarded as hydride ‘salts’ containing H ions. 3. Metallic hydrides are non-stoichiometric, electrically conducting solids with a metallic lustre. Metallic hydrides are formed with many d- and f-block elements (Table 3.12). The H atoms are often regarded as occupying interstitial sites within the metal structure although this occupation rarely occurs without expansion or phase change. Figure 10.2 (which reproduces Fig. 9.11) summarizes this classification and the distribution of the different classes through the periodic table. It also identifies ‘intermediate’ hydrides that do not fall strictly into any of these categories, and elements for which binary hydrides have not been characterized. In addition to binary compounds, hydrogen is found in complex anions of some p-block elements, examples being the BH4 ion (tetrahydridoborate, also known in older texts as ‘borohydride’) in NaBH4 or the AlH4 ion (tetrahydridoaluminate, also known in older texts as ‘aluminiumhydride’) in LiAlH4. (b) Thermodynamic considerations Key points: In the s and p blocks, strengths of EH bonds decrease down each group. In the d block, strengths of EH bonds increase down each group. To fuel cell 278 10 Hydrogen 18 1 1 2 Li 3 Saline Intermediate Metallic Molecular 2 Be Na Mg 3 4 Figure 10.2 Classification of the binary hydrogen compounds of the s-, p-, and d-block elements. Although some d-block elements such as iron and ruthenium do not form binary hydrides they do form metal complexes containing the hydride ligand. 5 6 4 5 6 Cr 7 8 Mn Fe K Ca Sc Ti V Rb Sr Y Zr Nb Mo Tc Ru Cs Ba La Hf Ta W Re Os Unknown 9 10 Co Ni 11 Ir Pt 13 14 15 16 17 B C N O F Ne Al Si P S Cl Ar Se Br Kr I Xe At Rn 12 Cu Zn Rh Pd He Ga Ge As Ag Cd In Sn Sb Te Tl Pb Bi Au Hg Po 7 The standard Gibbs energies of formation of the hydrogen compounds of s- and p-block elements reveal a regular variation in stability (Table 10.1). With the possible exception of BeH2 (for which good data are not available) all the s-block hydrides are exergonic (∆fG O < 0) and therefore thermodynamically stable with respect to their elements at room temperature. The trend is erratic in Group 13, in that only AlH3 is exergonic at room temperature. In all the other groups of the p block, the simple hydrogen compounds of the first members of the groups (CH4, NH3, H2O, and HF) are exergonic but the analogous compounds of their congeners become progressively less stable down the group, a trend that is illustrated by decreasing EH bond energies (Fig. 10.3). The heavier hydrides become more stable on going from Group 14 across to the halogens. For example, SnH4 is highly endergonic (∆fG O > 0) whereas HI is barely so. These thermodynamic trends can be traced to the variation in atomic properties. The HH bond is the strongest single homonuclear bond known (apart from DD or TT bonds) and in order for a compound to be exergonic and stable with respect to its elements, it needs to have EH bonds that are even stronger than HH. For molecular hydrides of the p-block elements, bonding is strongest with the Period 2 elements and becomes progressively weaker down each group. The weak bonds formed by the heavier p-block elements are due to the poor overlap between the relatively compact H1s orbital and the more diffuse s and p orbitals of their atoms. Although d-block elements do not form binary molecular compounds, many complexes contain one or more hydride ligands. Metalhydrogen bond strengths in the d block increase down a group because the 3d orbitals are too contracted to overlap well with the H1s orbital and better overlap is afforded by 4d and 5d orbitals. (c) Reactions of binary compounds Key point: The reactions of binary compounds of hydrogen fall into three classes depending on the polarity of the EH bond. 600 568 Bond enthalpy/kJ mol–1 Period 2 463 3 368 4 5 312 298 402 400 200 323 391 321 289 297 253 257 432 366 267 In compounds where E and H have similar electronegativities, cleavage of the EH bond tends to be homolytic, producing, initially, an H atom and a radical, each of which can go on to combine with other available radicals. Table 10.1 Standard Gibbs energy of formation, ∆ fG O /(kJ mol1) of binary s- and p-block hydrogen compounds at 25C. Values in parentheses are estimates. Group Period 1 2 13 14 15 16 17 2 LiH(s) BeH2(s) B2H6(g) CH4(g) NH3(g) H2O(I) HF(g) 68.4 (20) 86.7 50.7 16.5 237.1 273.2 NaH(s) MgH2(s) AIH3(s) SiH4 PH3(g) H2S(g) HCl(g) 33.5 35.9 (91) 56.9 13.4 33.6 95.3 KH(s) CaH2(s) Ga2H6(s) GeH4(g) AsH3(g) H2Se(g) HBr(g) 0 3 0 14 4 15 16 Group Figure 10.3 Average bond energies (kJ mol−1) for binary molecular hydrides of p-block elements. 17 (36) 147.2 113.4 68.9 15.9 53.5 5 RbH(s) SrH2(s) SnH4(g) SbH3(g) H2Te(g) HI(g) (30) (141) 188.3 147.8 0 1.7 6 CsH(s) BaH2(s) (32) (140) The detail For (E) ≈ (H): 279 EH → E· H· Common examples of homonuclear cleavage include the thermolysis and combustion of hydrocarbons. In compounds where E is more electronegative than H, heterolytic cleavage occurs, releasing a proton. For (E) (H): EH → E H The compound behaves as a Brønsted acid and is able to transfer H to a base. In such compounds the H atom is termed protonic. Heterolytic bond cleavage also occurs in compounds where E is less electronegative than H, including saline hydrides For (E) (H): EH → E H In this case the H atom is hydridic and an H ion is transferred to a Lewis acid. The reducing agents NaBH4 and LiAlH4 used in organic synthesis are examples of hydride-transfer reagents. PART B: THE DETAIL In this part of the chapter we present a more detailed discussion of the chemical properties of hydrogen, identifying and interpreting trends. We explain how dihydrogen is prepared on a small scale in the laboratory and how it is produced industrially from fossil fuels, then we outline methods for its production from water using renewable energy. We describe the reactions that dihydrogen undergoes with other elements and classify the different types of compounds formed. Finally we present the strategies for synthesizing various hydrogen-containing compounds. 10.3 Nuclear properties Key point: The three hydrogen isotopes H, D, and T have large differences in their atomic masses and different nuclear spins, which give rise to easily observed changes in IR, Raman, and NMR spectra of molecules containing these isotopes. Neither 1H nor 2H (deuterium, D) is radioactive, but 3H (tritium, T) decays by the loss of a particle to yield a rare but stable isotope of helium: 3 1 H → 23 He + − The half-life for this decay is 12.4 years. Tritium’s abundance of 1 in 1021 hydrogen atoms in surface water reflects a steady state between its production by bombardment of cosmic rays on the upper atmosphere and its loss by radioactive decay. Tritium can be synthesized by neutron bombardment of 6Li or 7Li. 1 0 3 n + 63 Li → H + 42 He + 4.78 MeV 1 1 0 n + 73 Li → 13 H + 42 He + 01 n − 2.87 MeV Continuous production of tritium from lithium is a key step in the projected future generation of energy from nuclear fusion rather than nuclear fission. In a fusion reactor tritium and deuterium are heated to over 100 MK to give a plasma in which the nuclei react to produce 4He and a neutron. 2 1 4 H + 31 H → He + 01 n + 17.6 MeV 2 (∆H = −1698 MJ mol1 ) The neutron is used to bombard a lithium blanket that has been enriched in 6Li to generate further tritium. This process carries far fewer environmental risks than fission of 235U and is essentially renewable: of the two primary fuels required, deuterium is readily available from water, although the natural abundance of lithium is low (Chapter 11). The physical and chemical properties of isotopologues, isotopically substituted molecules, are usually very similar, but not when D is substituted for H, as the mass of the substituted atom is doubled. Table 10.2 shows that the differences in boiling points and bond enthalpies are easily measurable for H2 and D2. The difference in boiling point between H2O and D2O reflects the greater strength of the O ... DO hydrogen bond (Section 10.6) compared with that of the O ... HO bond because the zero-point energy (Section 8.4) of the former is lower. The compound D2O is known as ‘heavy water’ and is used as a moderator in the nuclear power industry; it slows down emitted neutrons and increases the rate of induced fission. A note on good practice An isotopologue is a molecular entity that differs only in isotopic composition. An isotopomer is an isomer having the same number of each isotopic atom but differing in their positions. Reaction rates are often measurably different for processes in which EH and ED bonds, where E is another element, are broken, made, or rearranged. The detection of this kinetic isotope effect can often help to support a proposed reaction mechanism. Kinetic isotope effects are frequently observed when an H atom is transferred from one atom to another in an activated complex. For example, the electrochemical reduction of H(aq) to H2(g) occurs with a substantial isotope effect, with H2 being liberated much more rapidly. A practical consequence of the difference in rates of formation of H2 and D2 is that D2O may be concentrated electrolytically, thus facilitating separation of Table 10.2 The effect of deuteration on physical properties H2 Normal boiling point/°C –1 Mean bond enthalpy/(kJ mol ) D2 H 2O D 2O 252.8 249.7 100.0 101.4 436.0 443.3 463.5 470.9 280 10 Hydrogen the two isotopes: the pure D2O that accumulates is then used to produce pure HD (by reaction with LiAlH4) or D2 (by electrolysis). In general, reactions involving D2O occur more slowly than those involving H2O and, not surprisingly, D2O and Dsubstituted foods ingested in large quantities are poisonous for higher organisms. Because the frequencies of molecular vibrations depend on the masses of atoms, they are strongly influenced by substitution of D for H. The heavier isotope results in the lower frequency (Section 8.4). The isotope effect can be exploited by observing the IR spectra of isotopologues to determine whether a particular infrared absorption involves significant motion of a hydrogen atom in the molecule. The distinct properties of the isotopes make them useful as tracers. The involvement of H and D through a series of reactions can be followed by infrared (IR, Section 8.4) and mass spectrometry (Section 8.10) as well as by NMR spectroscopy (Section 8.5). Tritium can be detected by its radioactivity, which can be a more sensitive probe than spectroscopy. Another important property of the hydrogen nucleus is its spin. The nucleus of hydrogen, a proton, has I = 12 ; the nuclear spins of D and T are 1 and 12 , respectively. As explained in Chapter 8, proton NMR detects the presence of H nuclei in a compound and is a powerful method for determining structures of molecules, even proteins with molecular masses as high as 20 kDa. Figure 10.4 shows some typical 1H-NMR chemical shifts for some compounds of p- and d-block elements. Hydrogen atoms bonded to electronegative elements (protonic H atoms) display deshielded NMR signals (at more positive values of chemical shift), whereas hydrogens coordinated to metal ions with incomplete d subshells typically display more negative chemical shifts. Molecular hydrogen, H2, exists in two forms that differ in the relative orientations of the two nuclear spins: in ortho-hydrogen the spins are parallel (I = 1), in para-hydrogen the spins are antiparallel (I = 0). At T = 0 hydrogen is 100 per cent para. As the temperature is raised, the proportion of the ortho form in a mixture at equilibrium increases until at room temperature there is approximately 75 per cent ortho and 25 per cent para. Most physical properties of the two forms are the same, but the melting and boiling points of para-hydrogen are about 0.1˚C lower than those of normal hydrogen, and the thermal conductivity of para-hydrogen is about 50 per cent greater than that of the ortho form. The heat capacities also differ. d-Block 14 Group SnH4 SiH4 GeH4 CH4 15 PH3 NH3 16 H2O H2S 17 +16 +12 +8 –4 HBr HCl 0 δ –4 –8 Hydrogen is important both as a raw material for the chemical industry and, increasingly, as a fuel. Although it is not present in significant quantities in the Earth’s atmosphere or in underground gas deposits, there is a high biological turnover because various microorganisms use H as an oxidant or H2 as a fuel (Box 10.1). Industrially, most H2 is produced from natural gas by using steam reforming (in the USA, about 95 per cent is produced in this way). Increasingly, H2 is being produced by other methods, notably coal gasification (with carbon dioxide capture, Box 14.4) and thermally assisted electrolysis. In 2007, world production of H2 was approximately 50 Mt. Most H2 is used close to its site of production for the synthesis of ammonia (the Haber process), hydrogenation of unsaturated fats and large-scale manufacture of organic chemicals. In the future, H2 may be produced from entirely renewable sources such as water and capturing the energy of sunlight. (a) Small-scale preparation Key points: In the laboratory, H2 is easily produced by the reactions of electropositive elements with aqueous acid or alkali, or by hydrolysis of saline hydrides. It is also produced by electrolysis. There are many straightforward procedures for preparing small quantities of pure H2. In the laboratory, H2 is produced by reaction of Al or Si with hot alkali solution. 2 Al(s) 2 OH(aq) 6 H2O(l) → 2 Al(OH)4(aq) 3 H2(g) Si(s) 2 OH(aq) H2O(l) → SiO32(aq) 2 H2(g) or, at room temperature, by reaction of Zn with mineral acids: Zn(s) 2 H3O(aq) → Zn2(aq) H2(g) 2 H2O(l) The reaction of metal hydrides with water provides a convenient way to obtain small amounts of H2 outside the laboratory. Calcium hydride is particularly suited for this purpose as it is commercially available and inexpensive and it reacts with H2O at room temperature: CaH2(s) 2 H2O(l) → Ca(OH)2(s) 2 H2(g) Calcium hydride is used as a portable H2 generator and has important applications in remote places, such as filling meteorological balloons. Pure H2 is also produced in small amounts using a simple electrolysis cell; electrolysis of heavy water is also a convenient way to prepare pure D2. (b) Production from fossil sources MnH(CO)5 CoH(CO)4 PtClH(PEt3)2 HF 10.4 Production of dihydrogen HI –12 –16 –20 Figure 10.4 Typical 1H-NMR chemical shifts. The tinted boxes show families of elements together. Key point: Most H2 for industry is produced by high-temperature reaction of H2O with CH4 or a similar reaction with coke. Hydrogen is produced in huge quantities to satisfy the needs of industry, in fact production is often integrated directly (without transport) into chemical processes that require H2 as a feedstock. The main commercial process for the production of H2 is currently steam reforming, the catalysed reaction of H2O (as steam) and hydrocarbons (typically methane from natural gas) at high temperatures: CH4(g) H2O(g) → CO (g) 3H2(g) ∆rH O = 206.2 kJ mol1 281 The detail O2 Increasingly, coal or coke is used. This reaction, which occurs at 1000oC, is C(s) H2O(g) → CO(g) H2(g) H2 ∆rH O = 131.4 kJ mol1 The mixture of CO and H2 is known as water gas and further reaction with water (the water gas shift reaction) produces more H2: CO(g) H2O(g) → CO2(g) H2(g) ∆rH O = 41.2 kJ mol1 – Overall, coal gasification (and hydrocarbon reforming) result in production of CO2 and H2. C(s) 2H2O(g) → CO2(g) 2H2(g) ∆rH O = 90.2 kJ mol1 By implementing a system for capturing CO2 from the mixture (Box 14.4), it is possible to use fossil fuels and minimize release of the greenhouse gas CO2 into the atmosphere. However, this process is not a renewable route for H2 production as it is based on the use of fossil fuels. Dihydrogen for immediate consumption by on-board fuel cells in vehicles is produced from methanol by using an automotive steam reformer (Box 10.2). (c) Production from renewable sources Key points: Production of H2 by electrolysis of water is costly and viable only in areas where electricity is cheap or if it is a byproduct of an economically important process. Environmental pressures are driving technologies to produce H2 more efficiently from surplus or renewable energy, including solar and biological sources. Electrolysis is used to produce H2 that is free from contaminants. H2O(l) → H2(g) 12 O2(g) E cell ° 1.23 V, ∆rGO 237 kJ mol1 Ni Fe Ni Fe Ni Fe Ni Membrane Figure 10.5 An industrial electrolysis cell for H2 production using Ni anodes and Fe cathodes connected in series. electron-chemistry) or biological processes (modified photosynthesis). The physical methods under investigation are outlined in Box 10.3. Biological production could take place in ‘hydrogen farms’ by nurturing photosynthetic microorganisms that have been modified to produce H2 as well as organic molecules. 10.5 Reactions of dihydrogen Key points: Molecular hydrogen is activated by homolytic or heterolytic dissociation on a metal or metal oxide surface or by coordination to a d-block metal. Reactions of hydrogen with O2 and halogens involve a radical chain mechanism. Although H2 is quite an inert molecule, it reacts very rapidly under special conditions. Conditions for activating H2 include: 1. Homolytic dissociation into H atoms, induced by adsorption at certain metal surfaces: ⫹ ⫺ Pt ⫺ Pt ⫺ Pt ⫺ Pt ⫺ H ⫺ H2 H ⫺ ⫺ Pt ⫺ Pt ⫺ Pt ⫺ Pt ⫺ Heterolytic dissociation into H and H ions induced by adsorption on a heteroatom surface, such as a metal oxide: H ⫹ H⫺ H2 ⫹ ⫺ Zn⫺ O ⫺ Zn⫺ O ⫺ ⫺ ⫺ To drive this reaction, a large overpotential is required to offset the sluggish electrode kinetics, particularly for the production of O2. The best catalysts are based on platinum, but it is too expensive to justify its use in large-scale plants. As a consequence, electrolysis of water is economical and environmentally benign only if the electrical power stems from cheap, renewable resources or if it is surplus to demand. These conditions are found in countries that have plenty of hydroelectric or nuclear energy. There is also scope for off-shore wind farms that can employ resources far from the electricity grids and are far from the population areas that are otherwise necessary for conventional power generation. Electrolysis is carried out using hundreds of cells arranged in series, each operating at 2 V with iron or nickel electrodes and aqueous NaOH (or an ion-selective membrane) as electrolyte (Fig. 10.5). Temperatures of 8085ºC are used to increase the electrolytic current and to lower the overpotential. The most important electrolytic H2 production method is the chlor-alkali process (Box 11.2), in which H2 is produced as a byproduct of NaOH manufacture. In this process the other gaseous product is Cl2, which requires a lower overpotential than O2. As yet, however, only about 0.1 per cent of the global H2 demand is produced by electrolysis, including that produced in the chlor-alkali process. This percentage could be improved by the development of cheap and efficient electrocatalysts to reduce the economically wasteful overpotential. Hydrogen can be produced by fermentation, using anaerobic bacteria that use cultivated biomass or biological waste as their energy source (Box 10.1). Research is also under way to establish how best to produce H2 by exploiting solar energy directly either through physical methods (solar-powered thermolysis or photo- Fe ⫺ Zn⫺ O ⫺ Zn⫺ O ⫺ Initiation of a radical chain reaction: X H2 XH⭈ ⫹ H⭈ XH⭈ O2 2OH⭈ HOO⭈ H2 H2O ⫹ H⭈ ... etc. (a) Homolytic dissociation High temperatures are required to dissociate H2 into atoms. An important example of homolytic dissociation at normal temperatures is the reaction of H2 at finely divided Pt or Ni metal (Section 26.11). 282 10 Hydrogen B OX 10. 3 Hydrogen from solar energy The Earth receives about 100 000 TW from the Sun, which is approximately 7000 times greater than the present global rate of energy consumption (15 TW). Solar energy is already harnessed in several familiar ways, such as wind turbines, photosynthesis (biomass), and photovoltaic cells but ultimately the use of solar energy to generate H2 from water (water splitting) provides the greatest opportunity to end the world’s dependence on fossil fuels and help curb global climate change. Two technologies under development are high-temperature solar H2 production and solar photoelectrochemical H2 production. The so-called ‘sunbelt’ regions, which include Australia, southern Europe, the Sahara desert, and southwestern states of the USA, receive about 1 kW m2 of solar power. These regions are suitable sites for high-temperature solar H2 production using solar concentrating systems that reflect and focus solar radiation onto a receiver furnace, producing temperatures in excess of 1500C. The intense heat, which is also available in the mantel surrounding a nuclear reactor, can be used to drive a turbine for generating electricity or to split water into H2 and O2, so producing a fuel. Direct, single-step thermolysis of water requires temperatures in excess of 4000C, which is well above the threshold readily attainable in a solar concentrator or compatible with containment materials and engineering. By using a multi-step process, however, it is possible to produce H2 at much lower temperatures. Many systems are under investigation and development, the simplest of which are two-stage processes involving metal oxides, such as the sequence Fe3O4(s) → 3 FeO(s) 21 O2(g) ∆rH O = 319.5 kJ mol1 H2O(l) 3 FeO(s) → Fe3O4(s) H2(g) ∆rH O = −33.6 kJ mol1 Dihydrogen production by this route still requires temperatures in excess of 2200°C. Water splitting at lower temperatures has been achieved with hybrid processes that combine thermochemical and electrochemical reactions, such as 425 C ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → H2(g) 2 CuCl(s) 2 Cu(s) 2 HCl(g) ⎯ electrolysis ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 2 Cu(s) 2 CuCl2(s) 4 CuCl(s) ⎯ 325 C ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → Cu2OCl2(s) 2 HCl(g) 2 CuCl2(s) H2O ⎯ Cu2OCl2(s) 550 C ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 2 CuCl(s) 21 O2(g) This reaction, in which H2 is dissociatively chemisorbed as H atoms, is used to catalyse the hydrogenation of alkenes and the reduction of aldehydes to alcohols. Platinum is also used as the electrocatalyst for H2 oxidation in proton-exchange membrane fuel cells (Box 5.1). The facile chemisorption of H2 at Pt anodes results in a minimal overpotential for H2 oxidation and optimal performance. Another example of homolytic cleavage involves the initial coordination of molecular H2 as an η2-H2 species in discrete metal complexes, which is described briefly in Section 10.6d and in more detail in Section 22.7. Dihydrogen complexes provide examples of species intermediate between molecular H2 and a dihydrido complex. No dihydrogen complexes are known for the early d-block (Groups 3, 4, and 5), f-block, or p-block metals. If the metal is sufficiently electron rich, back donation of d electrons into the 1σu orbital splits the HH bond, resulting in the formation of a cis-dihydrido complex in which the formal oxidation number of the metal has increased by 2: H⫺H Mn⫹ ⫹ H2 Mn⫹ H⫺ H⫺ M(n⫹2)⫹ Solar photoelectrochemical H2 production operates on a principle similar to that used by plants for photosynthesis. To split water electrochemically a cell potential greater than 1.23 V is required, which in principle could be provided by light with a wavelength below 1000 nm. The principle of a photoelectrochemical water-splitting system based on light-sensitive particles is shown in Fig. B10.3, which represents water splitting in terms of two separate half-reactions. The essentials are (a) a mechanism for generating an excited electronic state by photon capture, (b) efficient transfer of electrons between the site of excitation and catalytic centres, (c) sites for the H2 production halfreaction, and (d) sites for the O2 production half-reaction. The excitation is produced at a photosensitive centre P, an organic molecule or metal complex attached to the particle. The sites for H2 and O2 production must be catalysts in order for these reactions to occur on timescales that are short compared to the lifetimes of the excited and oxidized states of P. For H2, the catalyst can be Pt, although much cheaper alternatives are necessary in order to achieve a feasible industrial-scale system. The major challenge for photoelectrochemical water splitting is to achieve rapid and efficient production of O2, and there are intense efforts to find substances that mimic the Mn catalyst used in plant photosynthesis (Section 27.10). 1 2 P hn H2O – e 1 2 H2 P e– 1 4 O2 1 2 P+ H2O Figure B10.3 Schematic cycle for photoelectrochemical production of H2 and O2 by catalysts attached to particles of a conducting material such as doped TiO2, to which a photosensitiser P is also attached. Excitation by visible light produces P, a powerful reducing agent for producing H2. Electron transfer from P produces P, a powerful oxidizing agent for producing O2. (b) Heterolytic dissociation Heterolytic dissociation of H2 depends on a metal ion (for hydride coordination) and a Brønsted base being in close proximity. Reaction of H2 with a ZnO surface appears to produce a Zn(II)-bound hydride and an O-bound proton. This reaction is involved in the production of methanol by catalytic hydrogenation of carbon monoxide over Cu/ZnO/Al2O3. CO(g) 2 H2(g) → CH3OH(g) Another example in which H2 is dissociated into a hydride and a proton is during its oxidation at the active site of metalloenzymes known as hydrogenases (Section 27.14). (c) Radical chain reactions Radical chain mechanisms account for the thermally or photochemically initiated reactions between H2 and the halogens in which atoms are generated that act as radical chain carriers in the propagation reaction: Initiation, by heat or light: Br2 → Br· Br· The detail Propagation: Br· H2 → HBr H· H· Br2 → HBr Br· The activation energy for radical attack is low because a new bond is formed as one bond is lost, so once initiated the formation and consumption of radicals is self-sustaining and the production of HBr is very rapid. Chain termination occurs when the radicals recombine: Termination: H· H· → H2 Table 10.3 Some common molecular hydrogen compounds Group Formula Traditional name IUPAC name 13 B2H6 Diborane Diborane(6) 14 15 Br· Br· → Br2 H· + Br· → HBr Termination becomes more important towards the end of the reaction when the concentrations of H2 and Br2 are low. The highly exothermic reaction of H2 with O2 also occurs by a radical chain mechanism. Certain mixtures explode violently when detonated: 2 H2(g) O2(g) → 2 H2O(g) ∆rH O = 242 kJ mol1 283 16 CH4 Methane Methane SiH4 Silane Silane GeH4 Germane Germane SnH4 Stannane Stannane NH3 Ammonia Azane PH3 Phosphine Phosphane AsH3 Arsine Arsane SbH3 Stibine Stibane H2O Water Oxidane H2S Hydrogen sulfide Sulfane H2Se Hydrogen selenide Sellane H2Te Hydrogen telluride Tellane 10.6 Compounds of hydrogen Hydrogen forms compounds with most of the elements. These compounds are classified into molecular hydrides, saline hydrides (salts of the hydride anion), metallic hydrides (interstitial compounds of d-block elements), and discrete complexes of d-block elements in which hydride or dihydrogen are ligands. (a) Molecular hydrides Molecular hydrides are formed with p-block elements and Be. The bonding is covalent but variations in bond polarity (depending on the electronegativity of the atoms to which hydrogen is attached) result in a range of reaction types in which hydrogen is formally transferred as H, H−, or H·. (i) Nomenclature and classification Key point: Molecular compounds of hydrogen are classified as electronrich, electron-precise, or electron-deficient. Electron-deficient hydrides provide some of the most intriguing examples of molecular structure and bonding as their simplest units tend to associate via bridging hydrogen atoms to form dimers and higher polymers. The systematic names of the molecular hydrogen compounds are formed from the name of the element and the suffix -ane, as in phosphane for PH3. The more traditional names, however, such as phosphine and hydrogen sulfide (H2S, sulfane) are still widely used (Table 10.3). The common names ammonia and water are universally used rather than their systematic names azane and oxidane. Molecular compounds of hydrogen are divided further into three subcategories: Electron-precise, in which all valence electrons of the central atom are engaged in bonds. Electron-rich, in which there are more electron pairs on the central atom than are needed for bond formation (that is, there are lone pairs on the central atom). Electron-deficient, in which there are too few electrons to be able to write a Lewis structure for the molecule. Electron-precise molecular hydrogen compounds include hydrocarbons such as methane and ethane, and their heavier analogues silane, SiH4, and germane, GeH4 (Chapter 14). All these molecules are characterized by the presence of two-centre, two-electron bonds (2c,2e bonds) and the absence of lone pairs on the central atom. Electron-rich compounds are formed by the elements in Groups 15 to 17. Important examples include ammonia, water, and the hydrogen halides. Electron-deficient hydrogen compounds are common for boron and aluminium. The analogous simple hydride of boron, BH3, is not found. instead it occurs as a dimer, B2H6 (diborane, 4) in which the two B atoms are bridged by a pair of H atoms in two three-centre, two-electron bonds (3c,2e bonds). B H 4 Diborane, B2H6 The shapes of the molecules of the electron-precise and electron-rich compounds can all be predicted by the VSEPR rules (Section 2.3). Thus, CH4 is tetrahedral (1), NH3 is trigonal pyramidal (2), H2O is angular (3), and HF is (necessarily) linear. Electron-deficient compounds provide some of the most interesting and unusual examples of structure and bonding. A Lewis structure for diborane, B2H6, would require at least 14 valence electrons to bind the eight atoms together, but the molecule has only 12 valence electrons. The simple explanation of its structure is the presence of BHB three-centre, two-electron bonds (3c,2e; Section 2.11e) acting as bridges between the two B atoms, so two electrons contribute to binding three atoms. These bridging BH bonds are longer and weaker than the terminal BH bonds. Another way of viewing this structure is that each BH3 284 10 Hydrogen moiety is a strong Lewis acid and gains the share of an electron pair from a BH bond in the other BH3 moiety. Being so small the H atoms pose little or no steric hindrance to dimer formation. The structures of boron hydrides are described more fully in Chapter 13. As expected, aluminium shows related behaviour that is modified by the larger atomic radius of this Period 3 element. The compound AlH3 does not exist as a monomer but forms a polymer in which each relatively large Al atom is surrounded octahedrally by six H atoms. Beryllium, unlike its congeners, exhibits a diagonal relationship with Al and also forms a polymeric covalent hydride BeH2. Although BH3 and AlH3 do not exist as monomers they do form important complex anions in combination with the hydride anion. The common reagents sodium tetrahydridoborate (NaBH4) and lithium tetrahydridoaluminate (LiAlH4) are examples of adduct formation between BH3 or AlH3, each a Lewis acid, and the Lewis base H. (ii) Reactions of molecular hydrides Key points: Homolytic dissociation of an EH bond to produce a radical E· and hydrogen atom H occurs most readily for the hydrides of the heavy p-block elements. Hydrogen attached to an electronegative element has protic character and the compound is typically a Brønsted acid. Hydrogen attached to an electropositive element can be transferred to an acceptor as a hydride ion. As was summarized briefly in Section 10.2 of Part A, the reactions of binary molecular hydrides are discussed in terms of their ability to undergo homolytic dissociation and, when the dissociation is heterolytic, in terms of their protic or hydridic character. Homolytic dissociation occurs readily for the hydrogen compounds of some p-block elements, especially the heavier elements. For example, the use of a radical initiator greatly facilitates the reaction of trialkylstannanes, R3SnH, with haloalkanes, RX, as a result of the formation of R3Sn· radicals: R3 SnH + R ′ X → R ′ H + R3 SnX The tendency towards radical reactions increases towards the heavier elements in each group, and SnH compounds are in general more prone to radical reactions than are SiH compounds. The ease of homolytic EH bond cleavage correlates with the decrease in EH bond strength down a group. The order of reactivity for haloalkanes with trialkylstannanes is In contrast, water (OH bond enthalpy 464 kJ mol1), which is a highly exothermic hydride, is only 4 per cent dissociated into H2 and O2 at 2200°C (Box 10.3): H2O(g) → 12 O2(g) H2(g) ∆rH O = 242 kJ mol1 Direct thermolysis of water is therefore not a practical solution for H2 production. As we saw in Section 10.2, compounds reacting by proton donation are said to show protic behaviour: in other words, they are Brønsted acids. We saw in Section 4.1 that Brønsted acid strength increases from left to right across a period in the p block (in the order of increasing electron affinity) and down a group (in the order of decreasing bond energy). One striking example of this trend is the increase in acidity from CH4 to HF and then from HF to HI. Binary hydrogen compounds of elements on the right of the periodic table typically undergo these reactions. Molecules in which hydrogen is bound to a more electropositive element can act as hydride ion donors. Important examples are the complex hydrido anions such as BH4 and AlH4, which are used to hydrogenate compounds containing a multiple bond. 4H O 2 AlH 4− + 4 RCHO → Al(OCH 2 R)4− ⎯⎯⎯ → Al(OH)4− + 4 RCH 2OH E X A MPL E 10.1 Determining which hydrogen atoms in a molecule are the most acidic Phosphorous acid, H3PO3, is a diprotic acid and is more helpfully written as OP(H)(OH)2. Explain why the H atom bound to P is much less protonic than the two H atoms bound to O. Answer We approach this problem by adapting the principles used to explain the Brønsted acidity of simple molecules. In Section 4.1 we saw that the Brønsted acidity of an acid EH depends upon the E−H bond enthalpy and electron affinity of B. In OP(H)(OH)2 the P−H bond (bond enthalpy in PH3 = 321 kJ mol1) is considerably weaker than an O−H bond (bond enthalpy in H2O = 464 kJ mol1) and on this basis we would expect a P−H bond to be more protonic. But the determining factor is that O is much more electronegative than P and therefore better able to accommodate the negative charge left by the departing H. Formic acid, HCO(OH), is another example of a molecule containing H atoms with very different protonic character. Self-test 10.1 Which of the following, CH4, SiH4, or GeH4, would you expect to be (a) the strongest Brønsted acid and (b) the strongest hydride donor? RF RCl RBr RI Thus fluoroalkanes do not react with R3SnH, chloroalkanes require heat, photolysis, or chemical radical initiators, and bromoalkanes and iodoalkanes react spontaneously at room temperature. This trend indicates that the initiation step is halogen abstraction. Thermal decomposition reactions of molecular hydrides yielding H2 and the element occur by homolytic dissociation. Decomposition temperatures usually correlate with EH bond energies and inversely with enthalpies of formation. For example, AsH3 (AsH bond enthalpy 297 kJ mol1), which is an endothermic hydride, decomposes quantitatively at 250300C: AsH3(g) → As(s) 32 H2(g) ∆rH O = 66.4 kJ mol1 (iii) Hydrogen bonding Key points: Compounds and functionalities containing H atoms attached to electronegative elements with at least one lone pair often associate through hydrogen bonds. An EH bond between an electronegative element E and hydrogen is highly polar, EH, and the partially positively charged H atom can interact with a lone pair on the E atom of another molecule, forming a bridge that is known as a hydrogen bond. Striking evidence for hydrogen bonding is provided by the trends in normal boiling points, which are unusually high for the strongly hydrogen-bonded molecules water (in which there are OH O bonds), ammonia (containing NH N bonds), and The detail hydrogen fluoride (containing FH F bonds, Fig. 10.6). The relatively low boiling points of PH3, H2S, HCl, and the heavier p-block molecular hydrides indicate that these molecules do not form strong hydrogen bonds. Although hydrogen bonds are usually much weaker than conventional bonds (Table 10.4), their collective action is responsible for stabilizing complex structures such as the open network structure of ice (Fig. 10.7). Collective hydrogen bonding interactions play a large part in maintaining the structure of protein molecules (Section 27.2). They are also responsible for the recognition between specific DNA bases, adenine/thymine and guanine/cytosine, that underlies gene replication (Fig. 10.8) Similarly, solid HF consists of chain structures that survive partially even in the vapour (5). Boiling point/°C 100 H2S HCl PH3 SiH4 H2O HF H2Se HBr AsH3 GeH4 285 Table 10.4 Comparison of hydrogen bond enthalpies with the corresponding EH covalent bond enthalpies (kJ mol1) Hydrogen bond SH 363 17 NH 386 22 OH 464 29 FH 574 55 ClH 428 HSH...SH2 H NH...NH 7 HOH...OH2 FH...FH 2 3 HOH...Cl FH...F Covalent bond 163 H2Te HI SbH3 SnH4 0 5 (HF)5 NH3 –100 CH4 –200 2 3 4 5 Period Figure 10.6 Normal boiling points of p-block binary hydrogen compounds. Hydrogen bonding may be symmetrical or unsymmetrical. In unsymmetrical hydrogen bonding, the H atom is not midway between the two nuclei, even when the heavier linked atoms are identical. For example, the [ClHCl] ion is linear but the H atom is not midway between the Cl atoms (Fig. 10.9). By contrast, in the bifluoride ion, [FHF], the H atom lies midway between the F atoms; the FF separation (226 pm) is significantly less than twice the van der Waals radius of the F atom (2 × 135 pm). Hydrogen bonding is readily detected by the shift to lower frequency and broadening of EH stretching bands in infrared spectra (Fig. 10.10) and by unusual proton chemical shifts in 1 H-NMR. The structures of hydrogen-bonded complexes have been observed in the gas phase by microwave spectroscopy. The lone-pair orientation of electron-rich compounds implied by VSEPR theory (Section 2.3) shows good agreement with the HF orientation (Fig. 10.11). For example, HF is oriented along the threefold axis of NH3, collinear with HCN, out of the H2O plane in its complex with H2O, and off the HF axis in the HF dimer. X-ray single crystal structure determinations often show the same patterns, as, for example, in the structure of ice and in Figure 10.7 The structure of ice. The structure shows all possible atom positions, but only half are actually occupied. (b) Potential energy (a) Cl 137 F Cyt Gua Figure 10.8 Base pairing in DNA. Cytosine recognizes guanine through the formation of three hydrogen bonds. 185 113 113 Cl– F– Bond lengths Figure 10.9 The variation of the potential energy with the position of the proton between two atoms in a hydrogen bond. (a) The single minimum potential characteristic of a strong hydrogen bond. (b) The double minimum potential characteristic of a weak hydrogen bond. 286 10 Hydrogen CCl4 H2O i Transmittance PrOH i 3700 Xe H PrO–H...OiPr 3300 ~ ν/cm–1 Figure 10.10 Infrared spectra of 2-propanol. In the upper curve, 2-propanol is present as unassociated molecules in dilute solution. In the lower curve, the pure alcohol is associated through hydrogen bonds. The association lowers the frequency and broadens the OH stretching absorption band. (from N.B. Colthrup, L.H. Daly, and S.E. Wiberley, Introduction to infrared and Raman spectroscopy. Academic Press, New York (1975).) H N H H O 46° H H H C N H F 70° Figure 10.11 The orientation of lone pairs as indicated by VSEPR theory compared with the orientation of HF in the gas-phase hydrogen-bonded complex. The HF molecule is oriented along the threefold axis of NH3, collinear with HCN, out of the H2O plane in its complex with H2O, and off the HF axis in the HF dimer. Figure 10.12 The cages of water molecules in clathrate hydrates; in this case Xe4(CCl4)8(H2O)68. Water can also form clathrate hydrates, consisting of hydrogenbonded cages of water molecules surrounding foreign molecules or ions. One example is the clathrate hydrate of composition Xe4(CCl4)8(H2O)68 (Fig. 10.12). In this structure, the cages with O atoms defining their corners consist of 14-faced and 12-faced polyhedra in the ratio 3:2. These O atoms are held together by hydrogen bonds and guest molecules occupy the interiors of the polyhedra. Aside from their interesting structures, which illustrate the organization that can be enforced by hydrogen bonding, clathrate hydrates are often used as models for the way in which water appears to become organized around nonpolar groups, such as those in proteins. Methane clathrate hydrates occur in the Earth at high pressures, and it is estimated that huge quantities of CH4 are trapped in these formations (see Box 14.2). Some ionic compounds form clathrate hydrates in which the anion is incorporated into the framework by hydrogen bonding. This type of clathrate is particularly common with the very strong hydrogen bond acceptors F and OH. One such example is N(CH3)4F.4H2O (6). H2O F– N(CH3)4+ 6 N(CH3)4F·4H2O solid HF, but packing forces in solids may have a strong influence on the orientation of the relatively weak hydrogen bond. One of the most interesting manifestations of hydrogen bonding is the structure of ice. There are at least ten different phases of ice but only one is stable under ordinary conditions. The familiar low-pressure phase of ice, ice-I, crystallizes in a hexagonal unit cell with each O atom surrounded tetrahedrally by four others (as shown in Fig. 10.7). These O atoms are held together by hydrogen bonds with OH O and O HO bonds largely randomly distributed through the solid. The resulting structure is quite open, which accounts for the density of ice being lower than that of water. When ice melts, the network of hydrogen bonds partially collapses. (b) Saline hydrides Key points: Hydrogen compounds of the most electropositive metals may be regarded as ionic hydrides; they liberate H2 in contact with Brønsted acids and transfer H to electrophiles. As direct hydride donors they react with halide compounds to form anionic hydride complexes. The saline hydrides are ionic solids containing discrete H ions and are analogous to corresponding halide salts. The ionic radius of H varies from 126 pm in LiH to 154 pm in CsH. This wide variability reflects the poor control that the single charge of the proton has on its two surrounding electrons and the resulting high compressibility and polarizability of H. Hydrides The detail of Group 1 and 2 elements, with the exception of Be, are ionic compounds. All Group 1 hydrides adopt the rock-salt structure. With the exception of MgH2, which has the rutile structure, the Group 2 hydrides adopt the PbCl2 structure (Table 10.5). Table 10.5 Structures of s-block hydrides Compound Crystal structure LiH, NaH, RbH, CsH Rock salt MgH2 Rutile CaH2, SrH2, BaH2 Distorted PbCl2 The saline hydrides are insoluble in common nonaqueous solvents but they do dissolve in molten alkali halides. Electrolysis of these molten-salt solutions produces hydrogen gas at the anode (the site of oxidation): 2 H(melt) → H2(g) 2 e This reaction provides chemical evidence for the existence of H ions. The reaction of saline hydrides with water, as in NaH(s) H2O(l) → NaOH(aq) H2(g) is dangerously vigorous. Alkali metal hydrides are convenient reagents for making other hydride compounds because they are direct providers of H ions for the following synthetically useful reactions: 1. Metathesis with a halide, such as the reaction of finely divided lithium hydride with silicon tetrachloride in dry diethyl ether (et): 4 LiH(s) SiCl4(et) → 4 LiCl(s) SiH4(g) 2. Addition to a Lewis acid, for example reaction with a trialkylboron compound yields a hydride complex that is a useful reducing agent and source of hydride ions in organic solvents: NaH(s) B(C2H5)3(et) → NaHB(C2H5)3 where ‘et’ denotes solution in diethyl ether 3. Reaction with a proton source, to produce H2: NaH(s) CH3OH(et) → NaOCH3(s) H2(g) The absence of convenient solvents limits the use of saline hydrides as reagents, but this problem is partially overcome by the availability of commercial dispersions of finely divided NaH in oil. Even more finely divided and reactive alkali metal hydrides can be prepared from the metal alkyl and hydrogen. Saline hydrides are pyrophoric; indeed, finely divided sodium hydride can ignite simply if it is left exposed to humid air. Such fires are difficult to extinguish because even carbon dioxide is reduced when it comes into contact with hot metal hydrides (water, of course, forms even more flammable hydrogen); they may, however, be blanketed with an inert solid, such as sand. Magnesium dihydride, MgH2, is under investigation as a hydrogen storage medium for transport purposes, where lightness is important (Box 10.4). The amount of H atoms in a given 287 volume of MgH2 is about 50 per cent higher than in the same volume of liquid H2. Calcium dihydride is used as a portable H2 generator. (c) Metallic hydrides Key points: No stable binary metal hydrides are known for the metals in Groups 7 to 9; metallic hydrides have metallic conductivity and in many the hydrogen is very mobile. Many of the d- and f-block elements react with H2 to produce metallic hydrides. Most of these compounds (and the hydrides of alloys) have a metallic lustre and are electrically conducting (hence their name). They are less dense than the parent metal and are brittle. Most metallic hydrides have variable composition (they are nonstoichiometric). For example, at 550°C zirconium hydride exists over a composition range from ZrH1.30 to ZrH1.75; it has the fluorite structure (Fig. 3.38) with a variable number of anion sites unoccupied. The variable stoichiometry and metallic conductivity of these hydrides can be understood in terms of a model in which the band of delocalized orbitals responsible for the conductivity accommodates the electrons supplied by arriving H atoms. In this model, the H atoms as well as the metal atoms take up equilibrium positions in the electron sea. The conductivities of metallic hydrides typically vary with hydrogen content, and this variation can be correlated with the extent to which the conduction band is filled or emptied as hydrogen is added or removed. Thus, whereas CeH2x is a metallic conductor, CeH3 (which has a full conduction band) is an insulator and is more like a saline hydride. Metallic hydrides are formed by all the d-block elements of Groups 3, 4, and 5 and by almost all f-block elements (Fig. 10.13). However, the only hydride in Group 6 is CrH, and no hydrides are known for the unalloyed metals of Groups 7, 8, and 9. The region of the periodic table covered by Groups 7 through to 9 is sometimes referred to as the hydride gap because few, if any, stable binary metalhydrogen compounds are formed by these elements. However, these metals are important as hydrogenation catalysts because they can activate hydrogen. Claims have been made, however, that hydrogen dissolves in iron at very high pressure and that iron hydride is abundant at the centre of the Earth. The Group 10 metals, especially Ni and Pt, are often used as hydrogenation catalysts in which surface hydride formation is thought to be involved (Chapter 26). However, somewhat surprisingly, at moderate pressures only Pd forms a stable bulk phase; its composition is PdHx, with x 1. Nickel forms hydride phases at very high pressures but Pt does not form any at all. Apparently the PtH bond enthalpy is sufficiently great to disrupt the HH bond but not strong enough to offset the loss of PtPt bonding, which would occur on formation of a bulk platinum hydride. In agreement with this interpretation, the enthalpies of sublimation, which reflect MM bond enthalpies, increase in the order Pd (378 kJ mol1) Ni (430 kJ mol1) Pt (565 kJ mol1). The MH bond enthalpy is a crucial factor in the design of metal hydride batteries, which are described in Box 10.5. Another striking property of many metallic hydrides is the high mobility of hydrogen within the material at slightly elevated temperatures. This mobility is used in the ultrapurification of H2 by diffusion through a palladium/silver alloy tube (Fig. 10.14). 288 10 Hydrogen B OX 10. 4 The quest for reversible H2 storage materials The need to develop practical systems for on-board hydrogen storage is considered to be a major obstacle to the future use of H2 as an energy carrier in vehicles. The problem is only partly resolved by compression and liquefaction. Compression to high-pressure gaseous H2 at 200 bar (energy density 0.53 kWh dm3) then refrigeration to form liquid H2 (energy density 2.37 kWh dm3) requires considerable energy and containment costs, and these are particularly prohibitive for small private vehicles for which space and cost are also of prime concern. The challenge, therefore, is to identify and develop materials that can store H2 in a fully reversible manner at high rates under reasonable temperatures and pressure conditions. One such material is LaNi5H6, which stores H2 reversibly with a density of 2 per cent by mass, but for transport these materials also have to be lightweight. Materials under investigation include hydrides, borohydrides, and amides of the lightest metals. Examples of these compounds and their H2 storage densities (as a percentage of hydrogen by mass) are MgH2 (8 per cent), LiBH4 (20 per cent), LiNH2 (10 per cent), and Al(BH4)3 (17 per cent). The last is a liquid that melts at 65C. Some of the principles are illustrated by the LiNH2 system, for which reversible hydrogen storage takes place in two reactions: of Li2NH (antifluorite) is closely related to that of LiNH2 (defect antifluorite structure with half the Li sites occupied in an ordered manner). The small Li ions can migrate within such a structure by a hopping mechanism involving transitory defect sites, so allowing H2 uptake by protonation of NH2 with corresponding expansion of the unit cell (Fig. B10.4) and coupled formation of a LiH phase. A problem that must be overcome with amides and other complex hydrides is their tendency to decompose to undesirable products, such as NH3. Liquid organic nitrogen-containing heterocyclic compounds that can take up H2 reversibly are also under investigation. Compounds based on imidazole are easily handled and the discharged (dehydrogenated) product could simply be exchanged for hydrogenated fuel at a filling station. Another direction for reversible H2 storage is as molecular clathrates, such as H2(H2O)2. The H2 molecule is held by weak van der Waals forces and is released by decreasing the pressure or increasing the temperature. N Li 3 bar H , 210° C ⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯ → Li NH(s) LiH(s) Li3N(s) H2(g) ← 2 Vacuum, >320 C 2 (∆rH O = 148 kJ mol1 at 298 K) 3 bar H , 255 C ⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯ → LiNH (s) LiH(s) Li2NH(s) H2(g) ← 2 Vacuum, GaCl3 The detail By contrast, towards a soft Lewis base (such as dimethylsulfane, Me2S, which is soft because of its S atom), the Lewis acidities strengthen as the softness of the acceptor element increases: GaX3 > AlX3 > BX3 ( X = Cl or Br ) Aluminium trichloride is a useful starting material for the synthesis of other Al compounds: AlCl3 (sol) + 3LiR(sol) → AlR3 (sol) + 3LiCl(s) This reaction is an example of transmetallation, which is important in the preparation of main-group organometallic compounds. In transmetallation reactions, the halide formed is that of the more electronegative element and the high lattice enthalpy of that compound can be regarded as the ‘driving force’ of the reaction (as we saw in Example 13.7). The main industrial application of AlCl3 is as a FriedelCrafts catalyst in organic synthesis. The thallium trihalides are much less stable than those of its lighter congeners. A trap for the unwary is that thallium triiodide is a compound of Tl(I) rather than Tl(III), as it contains the I3 ion, not I. This is confirmed by considering the standard potentials, which indicate that Tl(III) is rapidly reduced to Tl(I) by iodide: Tl3+ (aq) + 2e− → Tl + (aq) E O = +1.25 V I3− (aq) + 2e− → 3I− (aq) E O = +0.55 V However, in excess iodide Tl(III) is stabilized by the formation of a complex: TlI3 (s) + I− (aq) → [ TlI4 ]− (aq) In keeping with the general tendency towards higher coordination numbers for the larger atoms of later p-block elements, halides of Al and its heavier congeners may take on more than one Lewis base: AlCl 3 + N(CH3 )3 → Cl3 AlN(CH3 )3 Cl3 Al(N(CH3 )3 )2 Cl3 AlN(CH3 )3 + N(CH3 )3 → 345 The formula GaCl2 is deceiving, as this solid and most other apparently divalent salts do not contain Ga(II); instead they are mixed-oxidation-state compounds containing Ga(I) and Ga(III). Mixed-oxidation-state halogen compounds are also known for the heavier metals, such as InCl2 and TlBr2. The presence of M3 ions is indicated by the existence of MX4 complexes in these salts with short MX distances, and the presence of M ions is indicated by longer and less regular separation from the halide ions. There is in fact only a fine line between the formation of a mixed-oxidation-state ionic compound and the formation of a compound that contains MM bonds. For example, mixing GaCl2 with a solution of [N(CH3)4]Cl in a nonaqueous solvent yields the compound [N(CH3)4]2[Cl3GaGaCl3], in which the anion has an ethane-like structure with a GaGa bond. Indium monohalides are prepared by direct interaction of the elements or by heating the metal with HgX2. The stability increases from the chloride to the iodide and is enhanced by the formation of complexes such as In[AlX4]. Gallium(I) and In(I) halides both disproportionate when dissolved in water: 3MX(s) → 2 M(s) + M3+ (aq) + 3X − (aq) (M = Ga, In; X = Cl,Br, I) Thallium(I) is stable with respect to disproportionation in water because Tl3 is difficult to achieve. Thallium(I) halides are prepared by the action of HX on an acidified solution of a soluble Tl(I) salt. Thallium(I) fluoride has a distorted rock-salt structure whereas TlCl and TlBr have the caesium-chloride structure (Section 3.9). Yellow TlI has an orthorhombic layer structure in which the inert pair manifests itself structurally but, when pressure is applied, is converted to red TlI with a caesium-chloride structure. Thallium(I) iodide is used in photomultiplier tubes to detect ionizing radiation. Other low-oxidation-state halides of indium and thallium are known: TlX2 is actually TlI[Tl 3IIIX4] and Tl2X3 is Tl3I [TlIIIX6] and In4Br6 is In6I [IIIIBr6]2. E X A MPL E 13 . 8 Proposing reactions of Group 13 halides 13.15 Low-oxidation-state halides of aluminium, gallium, indium, and thallium Propose chemical equations (or indicate no reaction) for reactions between (a) AlCl3 and (C2H5)3NGaCl3 in toluene, (b) (C2H5)3NGaCl3 and GaF3 in toluene, (c) TlCl and NaI in water. Key point: The 1 oxidation state becomes progressively more stable from aluminium to thallium. Answer (a) We need to note that the trichlorides are excellent Lewis acids and that Al(III) is a stronger and harder Lewis acid than Ga(III). Therefore, the following reaction can be expected: All the AlX compounds, GaF, and InF are unstable, gaseous species that disproportionate in the solid phase: 3 AlX(s) → 2 Al(s) + AlX3 (s) The other monohalides of Ga, In, and Tl are more stable. Gallium monohalides are formed by reacting GaX3 with the metal in a 1:2 ratio: GaX3 (s) + 2Ga(s) → 3GaX(s) (X = Cl,Br,or I) The stability increases from the chloride to the iodide. The stability of the 1 oxidation state is increased by the formation of complexes such as Ga[AlX4]. The apparently divalent GaCl2 can be prepared by heating GaX3 with gallium metal in a 2:1 ratio: ∆ → 3GaX (s) 2GaX3 (s) + Ga(s) ⎯⎯ 2 (X = Cl,Br,or I) AlCl3 + (C 2H5 )3 NGaCl3 → (C 2H5 )3 NAlCl3 + GaCl3 (b) In this case we need to note that the fluorides are ionic, so GaF3 has a very high lattice enthalpy and is not a good Lewis acid. There is no reaction. (c) Now we note that Tl(I) is a chemically borderline soft Lewis acid, so it combines with the softer I ion rather than Cl: TlCl(s) + Nal(aq) → Tll(s) + NaCl(aq) Like silver halides, Tl(I) halides have low solubility in water, so the reaction will probably proceed very slowly. Self-test 13.8 Propose, with reasons, the chemical equation (or indicate no reaction) for reactions between (a) (CH3)2SAlCl3 and GaBr3, (b) TlCl and formaldehyde (HCHO) in acidic aqueous solution. (Hint: Formaldehyde is easily oxidized to CO2 and H.) 346 13 The Group 13 elements 13.16 Oxo compounds of aluminium, gallium, indium, and thallium Key points: Aluminium and gallium form and forms of the oxide in which the elements are in their 3 oxidation state; thallium forms an oxide, in which it is in its 1 oxidation state, and a peroxide. The most stable form of Al2O3, -alumina, is a very hard and refractory material. In its mineral form it is known as corundum and as a gemstone it is sapphire or ruby, depending on the metal ion impurities. The blue of sapphire arises from a charge-transfer transition from Fe2 to Ti4 ion impurities (Section 20.5). Ruby is -alumina in which a small fraction of the Al3 ions are replaced by Cr3. The structure of -alumina and gallia, Ga2O3, consists of an hcp array of O2 ions with the metal ions occupying two-thirds of the octahedral holes in an ordered array. Dehydration of aluminium hydroxide at temperatures below 900C leads to the formation of -alumina, which is a metastable polycrystalline form with a defect spinel structure (Section 3.9b) and a very high surface area. Partly because of its surface acid and base sites, this material is used as a solid phase in chromatography and as a heterogeneous catalyst and catalyst support (Section 26.10) The and forms of Ga2O3 have the same structures as their Al analogues. The metastable form is -Ga2O3, which has a ccp structure with Ga(III) in distorted octahedral and tetrahedral sites. Half the Ga(III) ions are therefore four-coordinate despite its large radius (compared to Al(III)). This coordination may be due to the effect of the filled 3d10 shell of electrons, as remarked previously. Indium and Tl form In2O3 and Tl2O3, respectively. Thallium also forms the Tl(I) oxide and peroxide, Tl2O and Tl2O2. Indium tin oxide (ITO) is In2O3 doped with 10 per cent by mass SnO2 to form an n-type semiconductor. The material is transparent in the visible region, and is electrically conducting. It is deposited in thin films on surfaces by a variety of methods such as physical vapour deposition and ion beam sputtering. The main uses of these films are as transparent conducting coatings for liquid crystal and plasma displays, touch panels, solar cells, and organic light-emitting diodes. They are also used as infrared reflecting mirrors and as an anti-reflective coating on binoculars, telescopes, and spectacles. A coating of ITO on aircraft renders them invisible to radar. The melting point of ITO is 1900°C, which makes ITO thin film strain gauges very useful in harsh environments such as jet engines and gas turbines. 13.17 Sulfides of gallium, indium, and thallium Key point: Gallium, indium, and thallium form many sulfides with a wide range of structures. The only sulfide of Al is Al2S3, which is prepared by direct reaction of the elements at elevated temperatures: ∆ 2 Al(s) + 3S(s) ⎯⎯→ Al2 S3 (s) It is rapidly hydrolysed in aqueous solution: Al2 S3 (s) + 6 H 2O(l) → 2 Al(OH)3 (s) + 3H 2 S(g) Table 13.4 Selected sulfides of gallium, indium, and thallium Sulfide Structure GaS Layer structure with GaGa bonds -Ga2S3 Defect wurtzite structure (hexagonal) -Ga2S3 Defect sphalerite structure (cubic) InS Layer structure with InIn bonds -In2S3 Defect spinel (as -Al2O3) TlS Chains of edge-shared TlIIIS4 tetrahedra Tl4S3 Chains of [TlIIIS4] and TlI[TlIIIS3] tetrahedra Aluminium sulfide exists in , , and forms. The structures of the and forms are based on the wurtzite structure (Section 3.9): in -Al2S3 the S2 ions are hcp and the Al3 ions occupy two-thirds of the tetrahedral sites in an ordered fashion; in -Al2S3 the Al3 ions occupy two-thirds of the tetrahedral sites randomly. The form adopts the same structure as -Al2O3. The sulfides of Ga, In, and Tl are more numerous and varied than those of Al and adopt many different structural types. Some examples are given in Table 13.4. Many of the sulfides are semiconductors, photoconductors, or light emitters and are used in electronic devices. 13.18 Compounds with Group 15 elements Key point: Aluminium, gallium, and indium react with phosphorus, arsenic, and antimony to form materials that act as semiconductors. The compounds formed between Group 13 and Group 15 elements (the nitrogen group) are important commercially and technologically as they are isolectronic with Si and Ge and act as semiconductors (Sections 14.1 and 24.19). The nitrides adopt the wurtzite structure and the phosphides, arsenides, and stibnides all adopt the zinc blende (sphalerite) structure (Section 3.9). All the Group 13/15 (still commonly ‘Group III/V’) binary compounds can be prepared by direct reaction of the elements at high temperature and pressure. Ga(s) + As(s) → GaAs(s) The most widely used Group 13/15 semiconductor is gallium arsenide, GaAs, which is used to make devices such as integrated circuits, light-emitting diodes, and laser diodes. Its band gap is similar to that of Si and larger than those of other Group 13/15 compounds (Table 13.5). Gallium arsenide is superior to Si for such applications because it has higher electron mobility, allowing it to function at frequencies in excess of 250 GHz. Gallium arsenide devices also generate less electronic noise than silicon devices. Table 13.5 Band gap at 298 K Eg /eV GaAs 1.35 GaSb 0.67 InAs 0.36 InSb 0.16 Si 1.11 The detail One disadvantage of Group 13/15 semiconductors is that the compounds decompose in moist air and must be kept under an inert atmosphere, usually nitrogen, or be completely encapsulated. 347 tetrahydridoboranate ion, BH4 (Section 13.6). The sodium salt can be obtained by a simple addition reaction: BPh3 + NaPh → Na+ [BPh4 ]− 13.19 Zintl phases Key point: Group 13 elements form Zintl phases with Group 1 and Group 2 elements which are poor conductors and diamagnetic. The Group 13 elements form Zintl phases (Section 3.8c) with Group 1 or Group 2 metals. Zintl phases are compounds of two metals which are brittle, diamagnetic, and poor conductors. They are therefore quite different from alloys. Zintl phases are formed between a very electropositive Group 1 or 2 element and a moderately electronegative p-block metal or metalloid. They are ionic, with electrons being transferred from the Group 1 or 2 metal to the more electronegative element. The anion, referred to as the ‘Zintl ion’, has a complete octet of valence electrons and is polymeric; the cations are located within the anionic lattice. The structure of NaTl consists of the polymeric anion in a covalent diamond structure with Na ions fitted into the anionic lattice. In Na2Tl the polymeric anion is tetrahedral Tl48−. The Zintl anions can be isolated by reacting with salts containing the tetraalkylammonium ion, which substitutes for the Group 1 or 2 metal ion, or by encapsulation within a cryptand. Some compounds appear to be Zintl phases but are conducting and paramagnetic. For example, K8In11 8− anion, which has one delocalized electron contains the In11 per formula unit. The Na salt is soluble in water but the salts of most large, monopositive ions are insoluble. Consequently, the anion is useful as a precipitating agent and can be used in gravimetric analysis. (b) Organoaluminium compounds Key points: Methyl- and ethylaluminium are dimers; bulky alkyl groups result in monomeric species. Alkylaluminium compounds can be prepared on a laboratory scale by transmetallation of a mercury compound: 2 Al + 3Hg(CH3 )2 → Al2 (CH3 )6 + 3Hg Trimethylaluminium is prepared commercially by the reaction of Al metal with chloromethane to give Al2Cl2(CH3)4. This intermediate is then reduced with Na and the Al2(CH3)6 (39) is removed by fractional distillation. CH3 Al 39 Al2(CH3)6 13.20 Organometallic compounds The most important organometallic compounds of the Group 13 elements are those of B and Al. Organoboron compounds are commonly treated as organometallic compounds even though B is not a metal. (a) Organoboron compounds Key points: Organoboron compounds are electron deficient and act as Lewis acids; tetraphenylborate is an important ion. Organoboranes of the type BR3 can be prepared by hydroboration of an alkene with diborane. B2 H6 + 6CH 2C H 2 → 2 B(CH 2CH3 )3 Alternatively, they can be produced from a Grignard reagent (Section 12.13): (C2 H 5 )2 O:BF3 + 3RMgX → BR3 + 3MgXF + (C2 H 5 )2 O Alkylboranes are not hydrolysed but are pyrophoric. The aryl species are more stable. They are all monomeric and planar. Like other B compounds, the organoboron species are electron deficient and consequently act as Lewis acids and form adducts easily. An important anion is the tetraphenylborate ion, [B(C6H5)4], more commonly written BPh4, analogous to the Alkylaluminium dimers are similar in structure to the analogous dimeric halides (Section 13.6) but the bonding is different. In the halides, the bridging AlClAl bonds are 2c,2e bonds, that is each AlCl bond involves an electron pair. In the alkylaluminium dimers the AlCAl bonds are longer than the terminal AlC bonds, which suggests that they are 3c,2e bonds, with one bonding pair shared across the AlCAl unit, somewhat analogous to the bonding in diborane, B2H6 (Section 13.6). Triethylaluminium and higher alkyl compounds are prepared from the metal, an appropriate alkene, and hydrogen gas at elevated temperatures and pressures. 60 110ºC,10 20MPa 2 Al + 3 H 2 + 6CH 2 CH 2 ⎯⎯⎯⎯⎯⎯⎯ → Al2 (CH 2CH3 )6 This route is relatively cost effective and, as a result, alkylaluminium compounds have found many commercial applications. Triethylaluminium, often written as the monomer Al(C2H5)3, is an organometallic complex of Al of major industrial importance. It is used in the ZieglerNatta polymerization catalyst (Section 26.16). Steric factors have a powerful effect on the structures of alkylaluminiums. Where dimers are formed, the long weak bridging bonds are easily broken. This tendency increases with the bulkiness of the ligand. So, for example, triphenylaluminium is a dimer but trimesitylaluminium (where trimesityl is 2,4,6-(CH3)3C6H2) is a monomer. 348 13 The Group 13 elements FURTHER READING R.B. King, Inorganic chemistry of the main group elements. John Wiley & Sons (1994). R.B. King (ed.), Encyclopedia of inorganic chemistry. John Wiley & Sons (2005). D.M.P. Mingos, Essential trends in inorganic chemistry. Oxford University Press (1998). A survey of inorganic chemistry from the perspective of structure and bonding. C.E. Housecroft, Boranes and metalloboranes. Ellis Horwood, Chichester (2005). An introduction to borane chemistry. N.C. Norman, Periodicity and the s- and p-block elements. Oxford University Press (1997). Includes coverage of essential trends and features of s-block chemistry. EXERCISES 13.1 Give a balanced chemical equation and conditions for the recovery of boron. 13.12 Draw the B12 unit that is a common motif of boron structures; take a viewpoint along a C2 axis. 13.2 Describe the bonding in (a) BF3, (b) AlCl3, (c) B2H6. 13.13 Which boron hydride would you expect to be more thermally stable, B6H10 or B6H12? Give a generalization by which the thermal stability of a borane can be judged. 13.3 Arrange the following in order of increasing Lewis acidity: BF3, BCl3, AlCl3. In the light of this order, write balanced chemical reactions (or no reaction) for (a) BF3N(CH3)3 BCl3 →, (b) BH3CO BBr3→. 13.4 Thallium tribromide (1.11 g) reacts quantitatively with 0.257 g of NaBr to form a product A. Deduce the formula of A. Identify the cation and anion. 13.5 Identify compounds A, B, and C. BF3 LiAlH4 A H 2O CaF2 C heat B 13.6 Does B2H6 survive in air? If not, write the equation for the reaction. 13.7 Predict how many different boron environments would be present in the proton-decoupled 11B-NMR of a) B5H11, b) B4H10. 13.8 Predict the products from the hydroboration of (a) (CH3)2CCH2, (b) CH⬅CH. 13.9 Diborane has been used as a rocket propellant. Calculate the energy released from 1.00 kg of diborane given the following values O of ∆fH /kJ mol1: B2H6 = 31, H2O = 242, B2O3 = 1264. The combustion reaction is B2H6 (g) 3 O2(g) → 3 H2O (g) B2O3 (s). What would be the problem with diborane as a fuel? 13.10 Using BCl3 as a starting material and other reagents of your choice, devise a synthesis for the Lewis acid chelating agent, F2BC2H4BF2. 13.14 How many skeletal electrons are present in B5H9? 13.15 (a) Give a balanced chemical equation (including the state of each reactant and product) for the air oxidation of pentaborane(9). (b) Describe the probable disadvantages, other than cost, for the use of pentaborane as a fuel for an internal combustion engine. 13.16 (a) From its formula, classify B10H14 as closo, nido, or arachno. (b) Use Wade’s rules to determine the number of framework electron pairs for decaborane(14). (c) Verify by detailed accounting of valence electrons that the number of cluster valence electrons of B10H14 is the same as that determined in (b). 13.17 Starting with B10H14 and other reagents of your choice, give the equations for the synthesis of [Fe(nido-B9C2H11)2]2, and sketch the structure of this species. 13.18 (a) What are the similarities and differences in structure of layered BN and graphite (Section 14.5)? (b) Contrast their reactivity with Na and Br2. (c) Suggest a rationalization for the differences in structure and reactivity. 13.19 Devise a synthesis for the borazines (a) Ph3N3B3Cl3 and (b) Me3N3B3H3, starting with BCl3 and other reagents of your choice. Draw the structures of the products. 13.20 Give the structures and names of B4H10, B5H9, and 1,2-B10C2H12. 13.21 Arrange the following boron hydrides in order of increasing Brønsted acidity, and draw a structure for the probable structure of the deprotonated form of one of them: B2H6, B10H14, B5H9. 13.11 Given NaBH4, a hydrocarbon of your choice, and appropriate ancillary reagents and solvents, give formulas and conditions for the synthesis of (a) B(C2H5)3, (b) Et3NBH3. PROBLEMS 13.1 Borane exists as the molecule B2H6 and trimethylborane exists as a monomer B(CH3)3. In addition, the molecular formulas of the compounds of intermediate compositions are observed to be B2H5(CH3), B2H4(CH3)2, B2H3(CH3)3, and B2H2(CH3)4. Based on these facts, describe the probable structures and bonding in the latter series. 13.2 11B-NMR is an excellent spectroscopic tool for inferring the structures of boron compounds. With 11B11B coupling ignored, it is possible to determine the number of attached H atoms by the multiplicity of a resonance: BH gives a doublet, BH2 a triplet, and BH3 a quartet. B atoms on the closed side of nido and arachno clusters are generally more shielded than those on the open face. Assuming no BB or BHB coupling, predict the general pattern of the 11B-NMR spectra of (a) BH3CO and (b) [B12H12]2. 13.3 Identify the incorrect statements in the following description of Group 13 chemistry and provide corrections along with explanations of the principle or chemical generalization that applies. (a) All the elements in Group 13 are nonmetals. (b) The increase in chemical hardness on going down the group is illustrated by greater oxophilicity and fluorophilicity for the heavier elements. (c) The Lewis acidity increases for BX3 from X = F to Br and this may be explained Problems by stronger BrB π bonding. (d) Arachno-boron hydrides have a 2(n 3) skeletal electron count and they are more stable than nidoboron hydrides. (e) In a series of nido-boron hydrides acidity increases with increasing size. (f) Layered boron nitride is similar in structure to graphite and because it has a small separation between HOMO and LUMO, it is a good electrical conductor. 13.4 Use suitable molecular orbital software to calculate the wavefunctions and energy levels for closo-[B6H6]2. From that output, draw a molecular orbital energy diagram for the orbitals primarily involved in BB bonding and sketch the form of the orbitals. How do these orbitals compare qualitatively with the qualitative description for this anion in this chapter? Is BH bonding neatly separated from the BB bonding in the computed wavefunctions? 13.5 In his paper ‘Covalent and ionic molecules: why are BeF2 and AlF3 high melting point solids whereas BF3 and SiF4 are gases?’ (J. Chem. Educ., 1998, 75, 923), R.J. Gillespie makes a case for the classification of the bonding in BF3 and SiF4 as predominantly ionic. Summarize his arguments and describe how these differ from the conventional view of bonding in gaseous molecules. 13.6 Nanotubes of C and BN have been synthesized by C. Colliex et al. (Science, 1997, 278, 653). (a) What are the advantageous properties 349 of these nanotubes over carbon analogues? (b) Outline the method used for the preparation of these compounds. (c) What was the main structural feature of the nanotubes and how could this be exploited in applications? 13.7 M. Montiverde discusses ‘Pressure dependence of the superconducting temperature of MgB2’ (Science, 2001, 292, 75). (a) Describe the theoretical bases of the two theories that have been postulated to explain the superconductivity of MgB2. (b) How does the Tc of MgB2 vary with pressure? What insight does this provide into the superconductivity? 13.8 Use the references in Z.W. Pan, Z.R. Dai, and Z.L. Wang (Science, 2001, 291, 1947) as a starting point to write a review of wire-like nanomaterials of Group 13 elements. Indicate how In2O3 nanobelts were prepared and give the dimensions of a typical nanobelt. 13.9 In their paper ‘New structural motifs in metallaborane chemistry: synthesis, characterization, and solid-state structures of (CpW)3( -H)B8H8, (CpW)2B7H9, and (CpRe)2B7H7 (Cp = 5-C5Me5)’ (Organometallics, 1999, 18, 853). A.S. Weller, M. Shang, and T.P. Fehlner discuss the synthesis and characterization of some novel boron-rich metalloboranes. Sketch and explain the 11 B- and 1H-NMR spectra of (CpW)2B7H9. 14 The Group 14 elements Part A: The essentials 14.1 The elements 14.2 Simple compounds 14.3 Extended siliconoxygen compounds Part B: The detail 18 H He 1 2 Li Be 13 14 15 16 17 B C N O F Ne Al Si P 14.4 Occurrence and recovery 14.5 Diamond and graphite Ga Ge As 14.6 Other forms of carbon 14.7 Hydrides 14.8 Compounds with halogens 14.9 Compounds of carbon with oxygen and sulfur 14.10 Simple compounds of silicon with oxygen 14.11 Oxides of germanium, tin, and lead 14.12 Compounds with nitrogen 14.13 Carbides In Sn Sb The elements of Group 14 are arguably the most important of all, carbon providing the basis for life on Earth and silicon being vital for the physical structure of the natural environment in the form of crustal rocks. The elements of this group exhibit great diversity in their properties, ranging as they do from the nonmetallic carbon to the well-known metals tin and lead. All the elements form binary compounds with other elements. In addition, silicon forms a diverse range of network solids. Many of the organocompounds of the Group 14 elements are commercially important. The elements of Group 14, carbon, silicon, germanium, tin, and lead, show considerable diversity in their chemical and physical properties. Carbon, of course, is the building block of life and central to organic chemistry. In this chapter, our focus with carbon is on its inorganic chemistry. Silicon is widely distributed in the natural environment, and tin and lead find widespread applications in industry and manufacturing. Tl Pb Bi 14.14 Silicides 14.15 Extended siliconoxygen compounds 14.16 Organosilicon compounds 14.17 Organometallic compounds FURTHER READING EXERCISES PROBLEMS PART A: THE ESSENTIALS The elements of Group 14 (the carbon group) are of fundamental importance in industry and nature. We discuss carbon in many contexts throughout this text, including organometallic compounds in Chapter 22 and catalysis in Chapter 26. The focus of this section is on the essential aspects of the chemistry of Group 14. 14.1 The elements Key points: The lightest elements of the group are nonmetals; tin and lead are metals. All the elements except lead exist as several allotropes. The lightest members of the group, carbon and silicon, are nonmetals, germanium is a metalloid, and tin and lead are metals. This increase in metallic properties on descending a group is a striking feature of the p block and can be understood in terms of the increasing atomic radius and associated decrease in ionization energy down the group (Table 14.1). Because the ionization energies of the heavier elements are low, the metals form cations increasingly readily down the group. As the valence configuration ns2np2 suggests, the 4 oxidation state is dominant in the compounds of the elements. The major exception is lead, for which the most common oxidation state is 2, two less than the group maximum. The relative stability of the low oxidation state is an example of the inert-pair effect (Section 9.5), which is such a striking feature of the heaviest p-block elements. The essentials 351 Table 14.1 Selected properties of the Group 14 elements C Si Ge Sn Pb Melting point/C 3730 (graphite sublimes) 1410 937 232 327 Atomic radius/pm 77 117 122 140 154 73 (2) 93 119 (2) 53 (4) 69 (4) 78 (4) Ionic radius, r(M )/pm n First ionization energy, I/(kJ mol ) 1090 786 762 707 716 Pauling electronegativity 2.5 1.9 2.0 1.9 2.3 154 134 116 1 Electron affinity, Ea /(kJ mol ) 1 107 35 E (M4 + ,M2+ ) / V 0.15 1.69 E (M2+ , M) / V 0.14 0.13 O O C The electronegativities of carbon and silicon are similar to that of hydrogen and they form many covalent hydrogen and alkyl compounds. Carbon and silicon are strong oxophiles and fluorophiles, in the sense that they have high affinities for the hard anions O2 and F, respectively (Section 4.12). Their oxophilic character is evident in the existence of an extensive series of oxoanions, the carbonates and silicates. In contrast, Pb2 forms more stable compounds with soft anions, such as I and S2, than with hard anions, and is therefore classified as chemically soft. Two almost pure forms of carbon, diamond and graphite, are mined. There are many less pure forms, such as coke, which is made by the pyrolysis of coal, and lamp black, which is the product of incomplete combustion of hydrocarbons. Silicon occurs widely distributed in the natural environment and makes up 26 per cent by mass of the Earth’s crust. It occurs as sand, quartz, amethyst, agate, and opal, and is also found in asbestos, feldspar, clays, and micas. Germanium is low in abundance and occurs naturally in the ore germanite, Cu13Fe2Ge2S16, in zinc ores and in coal. Tin occurs as the mineral cassiterite, SnO2, and lead occurs as galena, PbS. Diamond and graphite, the two common crystalline forms of elemental carbon, are strikingly different. Diamond is effectively an electrical insulator; graphite is a good conductor. Diamond is the hardest known natural substance and hence the ultimate abrasive; impure (partially oxidized) graphite is slippery and frequently used as a lubricant. The origin of these widely different physical properties can be traced to the very different structures and bonding in the two allotropes. In diamond, each C atom forms single bonds of length 154 pm with four adjacent C atoms at the corners of a regular tetrahedron (Fig. 14.1); the result is a rigid, covalent, three-dimensional framework. Graphite consists of stacks of planar graphene layers within which each C atom has three nearest neighbours at 142 pm (Fig. 14.2). The bonds between neighbours within the sheets are formed from the overlap of sp2 hybrid orbitals, and the remaining perpendicular p orbitals overlap to form π bonds that are delocalized over the plane. The ready cleavage of graphite parallel to the planes of atoms (which is largely due to the presence of impurities) accounts for its slipperiness. Diamond can be cleaved, but this ancient craft requires considerable expertise as the forces in the crystal are more symmetrical. Diamond and graphite are not the only allotropes of carbon. The fullerenes (known informally as ‘buckyballs’) were discovered in the 1980s and have given rise to a new field within the inorganic chemistry of carbon. All the elements of the group except lead have at least one solid phase with a diamond structure (Fig. 14.1). The cubic phase of tin, which is called grey tin or -tin (-Sn), is not stable at room temperature. It converts to the more stable, common phase, white tin or -tin ( -Sn) in which an Sn atom has six nearest neighbours in a highly distorted octahedral array. When white tin is cooled to 13.2°C, it converts to grey tin. The effects of this transformation were first recognized on organ pipes in medieval European cathedrals, where it was believed to be due to the Devil’s work. Legend has it that Napoleon’s armies Figure 14.1 The cubic diamond structure. Figure 14.2 The structure of graphite. The rings are in register in alternate planes, not adjacent planes. 352 14 The Group 14 elements were defeated in Russia because, as the temperature fell, the white tin buttons on the soldiers’ uniforms were converted to grey tin, which then crumbled away. The gap between the valence and conduction bands (Section 3.19) decreases steadily from diamond, which is classed as a wide-band-gap semiconductor but commonly regarded as an insulator, to tin, which behaves like a metal above its transition temperature. Elemental carbon in the form of coal or coke is used as a fuel and reducing agent in the recovery of metals from their ores. Graphite is used as a lubricant and in pencils, and diamond is used in industrial cutting tools. The band gap and consequent semiconductivity of silicon leads to its many applications in integrated circuits, computer chips, solar cells, and other electronic solid-state devices. Silica (SiO2) is the major raw material used to make glass. Germanium was the first widely used material for the construction of transistors because it was easier to purify than silicon and, having a smaller band gap than silicon (0.72 eV for Ge, 1.11 eV for Si), is a better intrinsic semiconductor. Tin is resistant to corrosion and is used to plate steel for use in tin cans. Bronze is an alloy of tin and copper that typically contains less than 12 per cent by mass of tin; bronze with higher tin content is used to make bells. Solder is an alloy of tin and lead, and has been in use since Roman times. Window glass or float glass is made by floating molten glass on the surface of molten tin. The ‘tin side’ of window glass can be seen as a haze of tin(IV) oxide when viewed with ultraviolet radiation. Trialkyl and triaryltin compounds are in widespread use as fungicides and biocides. The softness and malleability of lead has resulted in its use in plumbing, although this application is now illegal in many countries due to concerns over lead poisoning. Its low melting point contributes to its use in solder and its high density (11.34 g cm3) leads to its use in ammunition and as shielding from ionizing radiation. Lead oxide is added to glass to raise its refractive index and form ‘lead’ or ‘crystal’ glass. 14.2 Simple compounds Key points: All the Group 14 elements form simple binary compounds with hydrogen, oxygen, the halogens, and nitrogen. Carbon and silicon also form carbides and silicides with metals. All the Group 14 elements form tetravalent hydrides, EH4. In addition, carbon and silicon form series of catenated molecular hydrides. Carbon forms an enormous range of hydrocarbon compounds that are best regarded from the viewpoint of organic chemistry. Carbon forms a series of simple hydrocarbons, the alkanes, with the general formula CnH2n2. The stability of the long-chain, catenated hydrocarbons is due to the high CC and CH bond enthalpies (Table 14.2; Section 9.7). Carbon also forms strong multiple bonds in the unsaturated alkenes and alkynes (Table 14.2). The strength of the CC bond and the ability to form multiple bonds are largely responsible for the diversity and stability of carbon compounds. The data in Table 14.2 illustrate how the EE bond enthalpy decreases on descending the group. As a result, the tendency to catenation decreases from C to Pb. Silicon forms a series of compounds analogous to the alkanes, the silanes, but the longest chain contains just seven Si atoms, as heptasilane, Si7H16. The silanes, with their greater number of electrons and stronger intermolecular forces, are less volatile than their hydrocarbon analogues. Thus, whereas propane, C3H8, is a gas under normal conditions, its silicon analogue trisilane, Table 14.2 Selected mean bond enthalpies, B(XY)/(kJ mol1) C—H 412 Si—H 318 Ge—H 288 Ge—O 350 C—O 360 Si—O 466 CO 743 SiO 642 C—C 348 Si—Si 326 Ge—Ge 186 CC 612 C⬅C 837 C—F 486 Si—F 584 Ge—F 466 C—Cl 322 Si—Cl 390 Ge—Cl 344 Sn—H 250 Pb—H 2000ºC ⎯⎯⎯ → CaC2 (s) The formation of graphite intercalation compounds is another example of a direct reaction, but is carried out at much lower temperatures. The intercalation reaction is more facile because no CC covalent bonds are broken when an ion slips between the graphite layers. Reaction of a metal oxide and carbon at a high temperature: CaO(s) + 3C(s) ⎯2000ºC ⎯⎯→ CaC2 (s) + CO(g) Crude calcium carbide is prepared in electric arc furnaces by this method. The carbon serves both as a reducing agent to remove the oxygen and as a source of carbon to form the carbide. Reaction of ethyne (acetylene) with a metalammonia solution: 2 Na(am) + C2 H 2 (g) → Na2C2 (s) + H 2 (g) Figure 14.13 In KC8, a graphite intercalation compound, the potassium atoms lie in a symmetrical array between the sheets. (See Fig. 14.4 for a view parallel to the sheets.) This reaction occurs under mild conditions and leaves the carboncarbon bonds of the starting material intact. As the ethyne molecule is a very weak Brønsted acid (pKa 25), the reaction can be regarded as a redox reaction between a highly active metal and a weak acid to yield H2 (with H the oxidizing agent) and the metal dicarbide. The saline carbides have high electron density on the C atom, so they are readily oxidized and protonated. For example, calcium carbide reacts with the weak acid water to produce ethyne: CaC2 (s) + 2 H 2O(l) → Ca(OH)2 (s) + HC≡ CH (g) Ca2+ C22– This reaction is readily understood as the transfer of a proton from a Brønsted acid (H2O) to the conjugate base (C22) of a weaker acid (HC≡CH). Similarly, the controlled hydrolysis or oxidation of the graphite intercalation compound KC8 restores the graphite and produces a hydroxide or oxide of the metal: 2 KC8 (s) + 2 H 2O(g) → 16C(graphite) + 2 KOH(s) + H 2 (g) (b) Metallic carbides Key point: d-Metal carbides are often hard materials with the carbon atom octahedrally surrounded by metal atoms. Figure 14.14 The calcium carbide structure. Note that this structure bears a similarity to the rock-salt structure. Because C22 is not spherical, the cell is elongated along one axis. This crystal is therefore tetragonal rather than cubic. The d metals provide the largest class of carbides. Examples are Co6Mo6C and Fe3Mo3C. They are sometimes referred to as interstitial carbides because it was long thought that the structures 368 14 The Group 14 elements were the same as those of the metals and that they were formed by the insertion of C atoms in octahedral holes. In fact, the structure of the metal and the metal carbide often differ. For example, tungsten metal has a body-centred structure whereas tungsten carbide (WC) is hexagonal close packed. The name ‘interstitial carbide’ gives the erroneous impression that the metallic carbides are not legitimate compounds. In fact the hardness and other properties of metallic carbides demonstrate that strong metal carbon bonding is present in them. Some of these carbides are economically and technologically useful materials. Tungsten carbide (WC), for example, is used for cutting tools and highpressure apparatus such as that used to produce diamond. Cementite, Fe3C, is a major constituent of steel and cast iron. Metallic carbides of composition MC have an fcc or hcp arrangement of metal atoms with the C atoms in the octahedral holes. The fcc arrangement results in a rock-salt structure. The C atoms in carbides of composition M2C occupy only half the octahedral holes between the close-packed metal atoms. A C atom in an octahedral hole is formally hypercoordinate (that is, has an untypically high coordination number) because it is surrounded by six metal atoms. However, the bonding can be expressed in terms of delocalized molecular orbitals formed from the C2s and C2p orbitals and the d orbitals (and perhaps other valence orbitals) of the surrounding metal atoms. It has been found empirically that the formation of simple compounds in which the C atom resides in an octahedral hole of a close-packed structure occurs when rC /rM < 0.59, where rC is the covalent radius of C and rM is the metallic radius of M. This relationship also applies to metal compounds containing nitrogen or oxygen. Even greater structural diversity than that displayed by the silicates themselves is possible when Al atoms replace some of the Si atoms. The resulting aluminosilicates are largely responsible for the rich variety of the mineral world. We have already seen that in γ-alumina, Al3 ions are present in both octahedral and tetrahedral holes (Section 3.3). This versatility carries over into the aluminosilicates, where Al may substitute for Si in tetrahedral sites, enter an octahedral environment external to the silicate framework, or, more rarely, occur with other coordination numbers. Because aluminium occurs as Al(III), its presence in place of Si(IV) in an aluminosilicate renders the overall charge negative by one unit. An additional cation, such as H, Na, or half as many Ca2, is therefore required for each Al atom that replaces an Si atom. As we shall see, these additional cations have a profound effect on the properties of the materials. Many important minerals are varieties of layered aluminosilicates that also contain metals such as Li, Mg, and Fe: they include clays, talc, and various micas. In one class of layered aluminosilicate the repeating unit consists of a silicate layer with the structure shown in Fig. 14.15. An example of a simple aluminosilicate of this type (simple, that is, in the sense of there being no additional Si O (a) 14.14 Silicides Key points: Siliconmetal compounds (silicides) contain isolated Si, tetrahedral Si4 units, or hexagonal nets of Si atoms. Silicon, like its neighbours boron and carbon, forms a wide variety of binary compounds with metals. Some of these silicides contain isolated Si atoms. The structure of ferrosilicon, Fe3Si, for instance, which plays an important role in steel manufacture, can be viewed as an fcc array of Fe atoms with some atoms replaced by Si. Compounds such as K4Si4 contain isolated tetrahedral cluster anions [Si4]4 that are isoelectronic with P4. Many of the f-block elements form compounds with the formula MSi2 that have the hexagonal layers that adopt the AlB2 structure shown in Fig. 13.8. (b) b) 14.15 Extended siliconoxygen compounds (c) As well as forming simple binary compounds with oxygen, silicon forms a wide range of extended network solids that find a range of applications in industry. Aluminosilicates occur naturally as clays, minerals, and rocks. Zeolite aluminosilicates are widely used as molecular sieves, catalysts, and catalyst support materials. These compounds are discussed further in Chapters 24 and 26. (d) (a) Aluminosilicates Key point: Aluminium may replace silicon in a silicate framework to form an aluminosilicate. The brittle layered aluminosilicates are the primary constituents of clay and some common minerals. Figure 14.15 (a) A net of SiO4 tetrahedra and (b) its tetrahedral representations. (c) Edge view of the above net and (d) its polyhedral representation. The structures (c) and (d) represent a double layer from the mineral chrysotile, for which M is Mg. When M is Al3 and the anions in the bottom layers are replaced by an OH group this structure is close to that of the 1:1 clay mineral kaolinite. The detail SiO4 K+ (a) (b) Figure 14.16 (a) The structure of 2:1 clay minerals such as muscovite mica KAl2(OH)2Si3AlO10, in which K resides between the charged layers (exchangeable cation sites), Si4 resides in sites of coordination number 4, and Al3 in sites of coordination number 6. (b) The polyhedral representation. In talc, Mg2 ions occupy the octahedral sites and O atoms on the top and bottom are replaced by OH groups and the K sites are vacant. elements) is the mineral kaolinite, Al2(OH)4Si2O5, which is used commercially as china clay. The electrically neutral layers are held together by rather weak hydrogen bonds, so the mineral readily cleaves and incorporates water between the layers. A larger class of aluminosilicates has Al3 ions sandwiched between silicate layers (Fig. 14.16). One such mineral is pyrophyllite, Al2(OH)2Si4O10. The mineral talc, Mg3(OH)2Si4O10, is obtained when three Mg2 ions replace two Al3 ions in the octahedral sites. As remarked earlier, in talc (and in pyrophyllite) the repeating layers are neutral, and as a result talc readily cleaves between them. Muscovite mica, KAl2(OH)2Si3AlO10, has charged layers because one Al(III) atom substitutes for one Si(IV) atom in the pyrophyllite structure. The resulting negative charge is compensated by a K ion that lies between the repeating layers and results in greater hardness. There are many minerals based on a three-dimensional aluminosilicate framework. The feldspars, for instance, which are the most important class of rock-forming minerals (and contribute to granite), belong to this class. The aluminosilicate frameworks of feldspars are built up by sharing all vertices of SiO4 or AlO4 tetrahedra. The cavities in this three-dimensional network accommodate ions such as K and Ba2. Two examples are the feldspars orthoclase, KAlSi3O8, and albite, NaAlSi3O8. (b) Microporous solids Key point: Zeolite aluminosilicates have large open cavities or channels giving rise to useful properties such as ion exchange and molecular absorption. The molecular sieves are crystalline aluminosilicates having open structures with apertures of molecular dimensions. These ‘microporous’ substances, which include the zeolites in which cations (typically from Groups 1 or 2) are trapped in an aluminosilicate framework, represent a major triumph of solid-state chemistry, for their synthesis and our understanding of their properties combine challenging determinations of structures, imaginative synthetic chemistry, and important practical applications. The cages are defined by the crystal structure, so they are highly regular and of precise size. Consequently, molecular sieves capture 369 molecules with greater selectivity than high surface area solids such as silica gel or activated carbon, where molecules may be caught in irregular voids between the small particles. Zeolites are used for shape-selective heterogeneous catalysis. For example, the molecular sieve ZSM-5 is used to synthesize 1,2-dimethylbenzene (o-xylene) for use as an octane booster in gasoline. The other xylenes are not produced because the catalytic process is controlled by the size and shape of the zeolite cages and tunnels. This and other applications are summarized in Table 14.6 and discussed in Chapters 24 and 26. Synthetic procedures have added to the many naturally occurring zeolite varieties and have produced zeolites that have specific cage sizes and specific chemical properties within the cages. These synthetic zeolites are sometimes made at atmospheric pressure, but more often they are produced in a highpressure autoclave. Their open structures seems to form around hydrated cations or other large cations such as NR4 ions introduced into the reaction mixture. For example, a synthesis may be performed by heating colloidal silica to 100200°C in an autoclave with an aqueous solution of tetrapropylammonium hydroxide. The microcrystalline product, which has the typical composition [N(C3H7)4]OH(SiO2)48, is converted into the zeolite by burning away the C, H, and N of the quaternary ammonium cation at 500°C in air. Aluminosilicate zeolites are made by including high surface area alumina in the starting materials. A wide range of zeolites has been prepared with varying cage and bottleneck sizes (Table 14.7). Their structures are based on approximately tetrahedral MO4 units, which in the great majority of cases are SiO4 and AlO4. Because the structures involve many such tetrahedral units, it is common practice to abandon the polyhedral representation in favour of one that emphasizes the position of the Si and Al atoms. In this scheme, the Si or Al atom lies at the intersection of four line segments and the O atom bridge lies on the line segment (Fig. 14.17). This framework representation has the advantage of giving a clear impression of the shapes of the cages and channels in the zeolite. Some examples are illustrated in Fig. 14.18. The important zeolites have structures that are based on the ‘sodalite cage’ (Fig 14.3), a truncated octahedron formed by slicing off each vertex of an octahedron (19). The truncation leaves a square face in the place of each vertex and the triangular faces of the octahedron are transformed into regular hexagons. The substance known as ‘zeolite type A’ is based on sodalite cages that are joined by O bridges between the square faces. Eight such sodalite cages are linked in a cubic pattern with a large central cavity called an cage. The cages share octagonal faces, Table 14.6 Some uses of zeolites Function Application Ion exchange Water softeners in detergents Absorption of molecules Selective gas separation Gas chromatography Solid acid Cracking high molar mass hydrocarbons for fuel and petrochemical intermediates Shape-selective alkylation and isomerization of aromatics for petroleum and polymer intermediates 370 14 The Group 14 elements Table 14.7 Composition and properties of some molecular sieves Molecular sieve Composition Bottleneck diameter/pm Chemical properties A Na12[(AlO2)12(SiO2)12].xH2O 400 Absorbs small molecules; ion exchanger, hydrophilic X Na86[(AlO2)86(SiO2)106].xH2O 800 Absorbs mediumsized molecules; ion exchanger, hydrophilic Chabazite Ca2[(AlO2)4(SiO2)8].xH2O 400500 Absorbs small molecules; ion exchanger, hydrophilic; acid catalyst ZSM-5 Na3[(AlO2)3(SiO2)93].xH2O 550 Moderately hydrophilic ALPO-5 AlPO4.xH2O 800 Moderately hydrophobic Silicalite SiO2 600 Hydrophobic (a) (b) Figure 14.18 Two zeolite framework structures: (a) Zeolite-X and (b) ZSM-5. In each case just the SiO4 tetrahedra that form the framework are shown; nonframework atoms such as charge-balancing cations and water molecules are omitted. ■ A brief illustration. To identify the fourfold and sixfold axes in the truncated octahedral polyhedron used to describe the sodalite cage we note that there is one fourfold axis running through each pair of opposite square faces for a total of three fourfold axes. Similarly, a set of four sixfold axes runs through opposite sixfold faces. ■ Figure 14.17 Framework representation of a truncated octahedron (truncation perpendicular to the fourfold axes of the octahedron) and the relationship of Si and O atoms to the framework. Note that a Si atom is at each vertex of the truncated octahedron and an O atom is approximately along each edge. with an open diameter of 420 pm. Thus H2O or other small molecules can fill them and diffuse through octagonal faces. However, these faces are too small to permit the entrance of molecules with van der Waals diameters larger than 420 pm. 19 Truncated octahedron The charge on the aluminosilicate zeolite framework is neutralized by cations lying within the cages. In the type-A zeolite, Na ions are present and the formula is Na12(AlO2)12(SiO2)12.xH2O. Numerous other ions, including d-block cations and NH4, can be introduced by ion exchange with aqueous solutions. Zeolites are therefore used for water softening and as a component of laundry detergent to remove the di- and tri-positive ions that decrease the effectiveness of the surfactant. Zeolites have in part replaced polyphosphates because the latter, which are plant nutrients, find their way into natural waters and stimulate the growth of algae. In addition to the control of properties by selecting a zeolite with the appropriate cage and bottleneck size, the zeolite can be chosen for its affinity for polar or nonpolar molecules according to its polarity (Table 14.7). The aluminosilicate zeolites, which always contain charge-compensating ions, have high affinities for polar molecules such as H2O and NH3. By contrast, the nearly pure silica molecular sieves bear no net electric charge and are nonpolar to the point of being mildly hydrophobic. Another group of hydrophobic zeolites is based on the aluminium phosphate frameworks; AlPO4 is isoelectronic with Si2O4 and the framework is similarly uncharged. One interesting aspect of zeolite chemistry is that large molecules can be synthesized from smaller molecules inside the zeolite cage. The result is like a ship in a bottle because once assembled the molecule is too big to escape. For example, Na ions in a Y-type zeolite may be replaced by Fe2 ions (by ion exchange). The resulting Fe2Y zeolite is heated with phthalonitrile, which The detail diffuses into the zeolite and condenses around the Fe2 ion to form iron phthalocyanine (20), which remains imprisoned in the cage. R O N N N N Fe N 371 Si N (a) (b) N R N R (c) (d) R R R O O O O Si O Si O Si O O O O O Si O Si O Si O O O O R 20 14.16 Organosilicon compounds Key points: Methylchlorosilanes are important starting materials for the manufacture of silicone polymers; the properties of silicone polymers are determined by the degree of cross-linking and may be liquids, gels, or resins. All silicon tetraalkyls and tetraaryls are monomeric with a tetrahedral Si centre. The CSi bond is strong and the compounds are fairly stable. They can be prepared in a variety of ways, examples of which are shown below. SiCl4 + 4 RLi → SiR 4 + 4 LiCl SiCl4 + LiR → RSiCl3 + LiCl The Rochow process provides a cost-effective industrial route to methylchlorosilane, which is an important starting material in the manufacture of silicones: n MeCl + Si/Cu → Men SiCl4− n These methylchlorosilanes, MenSiCl4n, where n 1–3, can be hydrolysed to form silicones or polysiloxanes: Me3SiCl + H 2O → Me3SiOH + HCl 2 Me3SiOH → Me3SiOSiMe3 + H 2O The reaction yields oligomers that contain the tetrahedral silicon group and oxygen atoms that form SiOSi bridges. Hydrolysis of Me2SiCl2 produces chains or rings and the hydrolysis of MeSiCl3 produces a cross-linked polymer (Fig. 14.19). It is interesting to note that most silicon polymers are based on a SiOSi backbone, whereas carbon polymers are generally based on a CC backbone, reflecting the strengths of the SiO and CC bonds (Table 14.2). Silicone polymers have a range of structures and uses. Their properties depend on the degree of polymerization and crosslinking, which are influenced by the choice and mix of reactants, and the use of dehydrating agents such as sulfuric acid and elevated temperatures. The liquid silicones are more stable than hydrocarbon oils. Moreover, unlike hydrocarbons, their viscosity changes only slightly with temperature. Thus silicones are used as lubricants and wherever inert fluids are needed, for example in hydraulic braking systems. Silicones are very hydrophobic R R R R Figure 14.19 The structure of (a) a chain, (b) a ring, and (c) cross-linked silicone; (d) the chemical formula of a fragment. and are used in water-repellent sprays for shoes and other items. The lower molar mass silicones are essential in personal-care products such as shampoos, conditioners, shaving foams, hair gels, and toothpastes, and impart to them a ‘silky’ feel. At the other end of the spectrum of delicacy, silicone greases, oils, and resins are used as sealants, lubricants, varnishes, waterproofing, synthetic rubbers, and hydraulic fluids. 14.17 Organometallic compounds Key points: Tin and lead form tetravalent organo compounds; organotin compounds are used as fungicides and pesticides. Many organometallic compounds of Group 14 are of great commercial importance, although the use of lead is declining owing to its toxicity (see below). Organotin compounds are used to stabilize poly(vinyl chloride) (PVC) as antifouling agents on ships, as wood preservatives, and as pesticides. Generally, organometallic compounds of the group are tetravalent and have low polarity bonds. Their stability decreases from silicon to lead. Organotin compounds differ from organosilicon and organogermanium compounds in several ways. There is a greater occurrence of the 2 oxidation state, a greater range of coordination numbers, and halide bridges are often present. Most organotin compounds are colourless liquids or solids that are stable to air and water. The structures of R4Sn compounds are all similar, with a tetrahedral tin atom (21). The halide derivatives, R3SnX, often contain SnXSn bridges and form chain structures. The presence of bulky R R Sn 21 SnR4 372 14 The Group 14 elements groups may affect the shape. For example, in (SnFMe3)n (22), the SnFSn backbone is in a zig-zag arrangement, in Ph3SnF the chain has straightened, and (Me3SiC)Ph2SnF is a monomer. The haloalkyls are more reactive than the tetraalkyls and are useful in the synthesis of tetraalkyl derivatives. F Sn CH3 22 (SnFMe3)n, Me = CH3 Alkyltin compounds may be prepared in a variety of ways, including by using a Grignard reagent and by metathesis: SnCl4 + 4 RMgBr → SnR 4 + 4 MgBrCl 3SnCl4 + 2 Al2 R6 → 3SnR 4 + 2 Al2Cl6 Organotin compounds have the widest range of uses of all maingroup organometallic compounds and their annual worldwide industrial production exceeds 50 kt. Their major application is in the stabilization of PVC plastics. Without the additive, halogenated polymers are rapidly degraded by heat, light, and atmospheric oxygen to give discoloured, brittle products. The tin stabilizers scavenge labile Cl ions that initiate the loss of HCl, the first step in the degradation process. Organotin compounds also have a wide range of applications relating to their biocidal effects. They are used as fungicides, algaecides, wood preservative, and antifouling agents. However, their widespread use on boats to prevent fouling and attachment of barnacles has caused environmental concerns as high levels of organotin compounds kill some species of marine life and affect the growth and reproduction of others. Many nations now restrict the use of organotin compounds to vessels over 25 m long. Tetraethyl lead used to be made on a huge scale as an antiknock agent in petrol. However, concerns about the levels of lead in the environment has led to it being phased out (Box 14.7). Alkyllead compounds, R4Pb, can be made in the laboratory by using a Grignard reagent or an organolithium compound: 2 PbCl2 + 4 RLi → R 4 Pb + 4 LiCl + Pb 2 PbCl2 + 4 RMgBr → R 4 Pb + Pb + 4 MgBrCl They are all monomeric molecules with tetrahedral geometry around the Pb atom. The halide derivatives may contain bridging halide atoms to form chains. Monomers are favoured by more bulky organic substituents. For example, Pb(CH3)3Cl exists as a chain structure with bridging Cl atoms (23) whereas the mesityl derivative Pb(Me3C6H2)3Cl is a monomer. Cl Pb CH3 23 (PbClMe3)n B OX 14 .7 Lead in the environment Of all the toxic elements in the environment, lead is perhaps the most pervasive. It poisons thousands of people yearly, most of them children in urban areas. The key aspect of lead in the body is that Pb2 can be absorbed through the intestine and is stored in bones because Pb2, having a similar ionic radius to Ca2, can substitute for Ca2 in hydroxyapatite, Ca5(PO4)3(OH). Bone is continuously formed and resorbed, allowing Pb2 to circulate in the blood and reach critical targets in the blood-forming and nerve tissues. Several epidemiological studies have associated blood lead, down to quite low levels, with impairments in growth, hearing, and mental development. Drinking water can be a significant source of lead exposure because of lead in pipes or in the solder used in fitting connections. On contact with oxygenated water, metallic lead can be oxidized and solubilized: 2 Pb + O2 + 4 H+ → 2 Pb2+ + 2 H2O Because the reaction consumes protons, the dissolution rate depends on the pH and is greatest in areas of soft water (low in Ca2 and Mg2), where there has been little neutralization of rainwater’s natural acidity. Districts with soft water sometimes add phosphate to form a protective lead phosphate precipitate that limits further dissolution of lead. 3 Pb2+ + 2 PO34− → Pb3 (PO4 )2 Modern pipes and fittings are lead free, but older lead-based stock is still in use. The major other exposure route is lead-laced dust, which is deposited on food crops or directly ingested, particularly by children, who play in the dust and sometimes eat it. The two major dust sources are leaded paint and leaded fuel. Lead has been added to fuel in the form of tetraethyl or tetramethyl lead since the 1920s to suppress ‘knocking’, the tendency of the fuelair mixture to pre-ignite when compressed in an engine’s cylinder. The lead inhibits pre-ignition by catalysing recombination of radicals. To prevent lead from depositing in the cylinder, dichloroethene or dibromoethene is also added, so that during combustion the lead is volatilized as PbX2 (X Cl or Br). Once in the atmosphere, the PbX2 condenses into fine particles. Although these particles can circulate widely, most of them settle out in the dust near roadways; lead levels in urban dust correlate strongly with traffic congestion. Lead additives started to be phased out in the developed world in the early 1970s, when catalytic converters were introduced for control of emissions because lead particles in the exhaust gas deactivate the catalytic surfaces. However, there are many areas of Africa, Asia, and South America where use of leaded fuels continues, and is even increasing as traffic increases. This is a public health issue because it has become clear that blood lead levels in the population correlate strongly with the use of leaded fuel. The remaining problem involves lead in paint. Lead salts are brightly coloured and have been widely used in pigments and paint bases: PbCrO4 is yellow, Pb3O4 is red, and Pb3(OH)2CO3 is white. As the paint wears away, the lead compound is dispersed in dust. Dust and paint chips are particularly hazardous indoors; lead was banned from indoor paint in 1927 throughout much of Europe but only in 1971 in the USA. Some older housing, particularly in inner cities, still has leaded paint on interior walls. Leaded paints have also been widely used on building exteriors, and weathering can result in high lead levels in the adjacent soils, where children often play. In the USA, about one-fifth of children living in inner-city homes built before 1946 are estimated to have elevated blood lead levels. Problems 373 FURTHER READING M.A. Pitt and D.W. Johnson, Main group supramolecular chemistry, Chem. Soc. Rev., 2007, 36, 1441. A. Schnepf, Metalliod Group 14 cluster compounds: an introduction and perspectives on this novel group of cluster compounds, Chem. Soc. Rev., 2007, 36, 745. H. Berke, The invention of blue and purple pigments in ancient times, Chem. Soc. Rev., 2007, 36, 15. An interesting account of the uses of silicate pigments. R.B. King, Inorganic chemistry of the main group elements. John Wiley & Sons (1994). D.M.P. Mingos, Essential trends in inorganic chemistry. Oxford University Press (1998). A survey of inorganic chemistry from the perspective of structure and bonding. N.C. Norman, Periodicity and the s- and p-block elements. Oxford University Press (1997). Includes coverage of essential trends and features of p-block chemistry. R.B. King (ed.), Encyclopedia of inorganic chemistry. John Wiley & Sons (2005). P.R. Birkett, A round-up of fullerene chemistry. Educ. Chem., 1999, 36, 24. A readable survey of fullerene chemistry. J. Baggot, Perfect symmetry: the accidental discovery of buckminsterfullerene. Oxford University Press (1994). A general account of the story of the discovery of the fullerenes. P.J.F. Harris, Carbon nanotubes and related structures. Cambridge University Press (2002). EXERCISES 14.1 Silicon forms the chlorofluorides SiCl3F, SiCl2F2, and SiClF3. Sketch the structures of these molecules. 14.2 Explain why CH4 burns in air whereas CF4 does not. The enthalpy of combustion of CH4 is −888 kJ mol−1 and the CH and CF bond enthalpies are −412 and −486 kJ mol−1 respectively. 14.3 SiF4 reacts with (CH3)4NF to form [(CH3)4N][SiF5]. (a) Use the VSEPR rules to determine the shape of the cation and anion in the product; (b) Account for the fact that the 19F NMR spectrum shows two fluorine environments. 14.4 Draw the structure and determine the charge on the cyclic anion [Si4O12]n. 14.5 Predict the appearance of the 119Sn-NMR spectrum of Sn(CH3)4. 14.6 Predict the appearance of the 1H-NMR spectrum of Sn(CH3)4. 14.7 Use the data in Table 14.2 and the additional bond enthalpy data given here to calculate the enthalpy of hydrolysis of CCl4 and CBr4. Bond enthalpies/(kJ mol1): OH 463, HCl 431, HBr 366. 14.8 Identify the compounds A to F: H 2O Cl2 A (ii) Sn2(aq) O2(air) → 14.10 Use data from Resource section 3 to determine the standard potential for each of the reactions in Exercise 14.9(b). In each case, comment on the agreement or disagreement with the qualitative assessment you gave for the reactions. 14.11 Give balanced chemical equations and conditions for the recovery of silicon and germanium from their ores. 14.12 (a) Describe the trend in band gap energy, Eg, for the elements carbon (diamond) to tin (grey). (b) Does the electrical conductivity of silicon increase or decrease when its temperature is changed from 20°C to 40°C? RL i B H2O 14.14 Describe the preparation, structure and classification of (a) KC8, (b) CaC2, (c) K3C60. C 14.15 Write balanced chemical equations for the reactions of K2CO3 with HCl(aq) and of Na4SiO4 with aqueous acid. 14.16 Describe in general terms the nature of the [SiO3]n2n ion in jadeite and the silica-alumina framework in kaolinite. RMgBr E (i) Sn2(aq) PbO2(s) (excess) → (air excluded) 14.13 Preferably without consulting reference material, draw a periodic table and indicate the elements that form saline, metallic, and metalloid carbides. F Si 14.9 (a) Summarize the trends in relative stabilities of the oxidation states of the elements of Group 14, and indicate the elements that display the inert-pair effect. (b) With this information in mind, write balanced chemical reactions or NR (for no reaction) for the following combinations, and explain how the answer fits the trends. D 14.17 (a) How many bridging O atoms are in the framework of a single sodalite cage? (b) Describe the (supercage) polyhedron at the centre of the Zeolite A structure in Fig. 14.3. PROBLEMS 14.1 Correct any inaccuracies in the following descriptions of Group 14 chemistry. (a) None of the elements in this group is a metal. (b) At very high pressures, diamond is a thermodynamically stable phase of carbon. (c) Both CO2 and CS2 are weak Lewis acids and the hardness increases from CO2 to CS2. (d) Zeolites are layered materials exclusively composed of aluminosilicates. (e) The reaction of calcium carbide with water yields ethyne and this product reflects the presence of a highly basic C22 ion in calcium carbide. 374 14 The Group 14 elements 14.2 The lightest p-block elements often display different physical and chemical properties from the heavier members. Discuss the similarities and differences by comparison of: (a) The structures and electrical properties of carbon and silicon. (b) The physical properties and structures of the oxides of carbon and silicon. (c) The Lewis acid–base properties of the tetrahalides of carbon and silicon. 14.3 Describe the physical properties of pyrophilite and muscovite mica and explain how these properties arise from the composition and structures of these closely related aluminosilicates. 14.4 There are major commercial applications for semicrystalline and amorphous solids, many of which are formed by Group 14 elements or their compounds. List four different examples of amorphous or partially crystalline solids described in this chapter and briefly state their useful properties. 14.5 The layered silicate compound CaAl2Si2O8 contains a double aluminosilicate layer, with both Si and Al in four-coordinate sites. Sketch an edge-on view of a reasonable structure for the double layer, involving only vertex sharing between the SiO4 and AlO4 units. Discuss the likely sites occupied by Ca2 in relation to the silicaalumina double layer. 14.6 Discuss the solid-state chemistry of silicon with reference to silicon dioxide, mica, asbestos, and silicate glasses. 14.7 One of your friends is studying English and is taking a course in science fiction. A common theme in the sources is silicon-based life forms. Your friend wonders why silicon should be chosen and why all life is carbon based. Prepare a short article that presents arguments for and against silicon-based life. 14.8 Karl Marx remarked in Das Kapital that ‘If we could succeed, at a small expenditure of labour, in converting carbon into diamonds, their value might fall below that of bricks’. Review current methods of synthesizing diamonds and discuss why these developments have not resulted in a large decrease in the value of diamonds. 14.9 The combination of mesoporosity with semiconduction would produce materials with interesting properties. A synthesis of such a material was discussed in ‘Hexagonal mesoporous germanium’ by Gerasimo et al. (Science, 2006, 313, 5788). Summarize the expected advantages of such a material and describe how the mesoporous germanium was synthesized. 14.10 In ‘An atomic seesaw switch formed by tilted asymmetric Sn-Ge dimers on a Ge (001) surface’ Tomatsu et al. describe the synthesis and operation of a molecular switch. Describe how such molecular switches operate and summarize their current and potential applications. (Science, 2007, 315, 1696). The Group 15 elements 18 H He 1 2 Li Be 13 14 15 16 17 B C N O F Ne Si P S Ge As Se The chemical properties of the Group 15 elements are very diverse. Although the simple trends that we observed for Groups 13 and 14 are still apparent, they are complicated by the fact that the Group 15 elements exhibit a wide range of oxidation states and form many complex compounds with oxygen. Nitrogen makes up a large proportion of the atmosphere and is widely distributed in the biosphere. Phosphorus is essential for both plant and animal life. In stark contrast, arsenic is a well-known poison. 15 Part A: The essentials 15.1 The elements 15.2 Simple compounds 15.3 Oxides and oxanions of nitrogen Part B: The detail 15.4 Occurrence and recovery 15.5 Uses 15.6 Nitrogen activation 15.7 Nitrides and azides Sn Sb Te The Group 15 elements—nitrogen, phosphorus, arsenic, antimony, and bismuth—are some of the Pb Bi Po most important elements for life, geology, and industry. They range from gaseous nitrogen to metallic bismuth. The members of this group, the ‘nitrogen group’, are sometimes referred to collectively as the pnictogens (from the Greek for to stifle, a property of nitrogen). This name is neither widely used nor officially sanctioned. As in the rest of the p block, the element at the head of Group 15, nitrogen, differs significantly from its congeners. Its coordination number is generally lower and it is the only member of the group to exist as a gaseous, diatomic molecule under normal conditions. 15.8 Phosphides 15.9 Arsenides, antimonides, and bismuthides 15.10 Hydrides 15.11 Halides 15.12 Oxohalides 15.13 Oxides and oxoanions of nitrogen 15.14 Oxides of phosphorus, arsenic, antimony, and bismuth 15.15 Oxoanions of phosphorus, arsenic, antimony, and bismuth 15.16 Condensed phosphates 15.17 Phosphazenes PART A: THE ESSENTIALS The properties of the Group 15 elements are diverse and more difficult to rationalize in terms of atomic radii and electron configuration than the p-block elements encountered so far. The usual trends of increasing metallic character down a group and stability of low oxidation states at the foot of the group are still evident but they are complicated by the wide range of oxidation states available. 15.1 The elements Key points: Nitrogen is a gas; the heavier elements are all solids that exist in several allotropic forms. All the members of the group other than N are solids under normal conditions. However, the trend to increasing metallic character down the group is not clear-cut because the electrical conductivities of the heavier elements actually decrease from As to Bi (Table 15.1). The normal increase in conductivity down a group reflects the closer spacing of the atomic energy levels in heavier elements and hence a smaller separation of the valence and conduction bands (Section 3.17). The opposite trend in conductivity in this group suggests that there must be a more pronounced molecular character in the solid state. Indeed, the structures of solid As, Sb, and Bi have three nearest-neighbour atoms and three more at significantly larger distances. The ratio of these long and short interactions decreases down the group, indicating the onset of a 15.18 Organometallic compounds of arsenic, antimony, and bismuth FURTHER READING EXERCISES PROBLEMS 376 15 The Group 15 elements Table 15.1 Selected properties of the Group 15 elements Melting point /C N P As Sb Bi 210 44 (white) 613 630 271 590 (red) (sublimes) Atomic radius/pm 74 110 121 141 170 First ionization energy/(kJ mol1) 1402 1011 947 833 704 Electrical conductivity/(106 S m1) 10 3.33 2.50 0.77 3.0 2.2 2.2 2.0 2.0 Electron affinity/(kJ mol ) 8 72 78 103 91 B(EH)/(kJ mol1) 390 322 275 Pauling electronegativity 1 60° P 221 pm 1 P4, Td Figure 15.1 One of the puckered layers of black phosphorus. Note the trigonal pyramidal coordination of the atoms. polymeric, molecular structure. The band structure of Bi suggests a low density of conduction electrons and holes, and it is best classified as a metalloid rather than as a semiconductor or a true metal. The solid elements of Group 15 exist as a number of allotropes. Like the gaseous N2 molecule, P2 has a formal triple bond and a short bond length (189 pm). The strength of π bonds formed by Period 3 elements is weak relative to those of Period 2, so the allotrope P2 is much less favoured than N2. White phosphorus is a waxy solid consisting of tetrahedral P4 molecules (1). Despite the small PPP angle (60º), the molecules persist in the vapour up to about 800ºC, but above that temperature the equilibrium concentration of P2 becomes appreciable. White phosphorus is very reactive and bursts into flame in air to yield P4O10. Red phosphorus can be obtained by heating white phosphorus at 300ºC in an inert atmosphere for several days. It is normally obtained as an amorphous solid, but crystalline materials can be prepared that have very complex three-dimensional network structures. Unlike white phosphorus, red phosphorus does not ignite readily in air. When phosphorus is heated under high pressure, a series of phases of black phosphorus are formed, the thermodynamically most stable form. One of the phases consists of puckered layers composed of pyramidal three-coordinate P atoms (Fig. 15.1). In contrast to the usual practice of choosing the most stable phase of an element as the reference phase for thermodynamic calculations, white phosphorus is adopted because it is more accessible and better characterized than the other forms. Arsenic exists in two solid forms, yellow arsenic and metallic arsenic. Yellow arsenic and gaseous arsenic both consist of tetrahedral As4 molecules. The most stable structure at room temperature of metallic As, and of Sb and Bi, is built from puckered hexagonal layers in which each atom has three nearest neighbours. The layers stack in a way that gives three more distant neighbours in the adjacent net, as described above (Fig. 15.2). Bismuth has recently been found to be radioactive, decaying by emission with a halflife of 1.9 1019 years, which is much longer than the current age of the universe. Nitrogen is readily available as dinitrogen, N2, as it makes up 78 per cent by mass of the atmosphere. The principal raw material for the production of elemental phosphorus is phosphate rock, the insoluble, crushed, and compacted remains of ancient organisms, which consists primarily of the minerals fluorapatite, Ca5(PO4)3F, and hydroxyapatite, Ca5(PO4)3OH. The chemically softer elements As, Sb, and Bi are often found in sulfide ores. Arsenic is found naturally in the ores realgar, As4S4, orpiment, As2S3, arsenolite, As2O3, and arsenopyrite, FeAsS. Antimony occurs naturally as the minerals stibnite, Sb2S3, and ullmanite, NiSbS. 15.2 Simple compounds Figure 15.2 The puckered network structure of bismuth. Each Bi atom has three nearest neighbours; the dark shaded atoms indicate next-nearest neighbours from an adjacent puckered sheet. Lightest atoms are furthest from the viewer. Key point: The Group 15 elements form binary compounds on direct interaction with many elements. Nitrogen achieves oxidation number 5 only with oxygen and fluorine. Oxidation state 5 is common for phosphorus, arsenic, and antimony but rare for bismuth, for which the 3 state is the more stable. The wide variety of possible oxidation states of the Group 15 elements can be understood to a large extent by considering the valence electron configuration of the elements, which is ns2np3. This configuration suggests that the highest oxidation state should be 5, as is 377 The essentials indeed the case. According to the inert-pair effect (Section 9.5), we should also expect the 3 oxidation state to be more stable for Bi, as is in fact observed. Nitrogen has a very high electronegativity (exceeded by only O, F, and Cl) and in many compounds, for example the nitrides, which contain the N3 ion, and ammonia, NH3, nitrogen is in a negative oxidation state. Nitrogen achieves positive oxidation states only in compounds with the more electronegative elements O and F. Nitrogen does achieve the group oxidation state (5), but only under much stronger oxidizing conditions than are necessary to achieve this state for the other elements of the group. The distinctive nature of nitrogen is due in large part to its high electronegativity, its small atomic radius, and the absence of accessible d orbitals. Thus, N seldom has coordination numbers greater than 4 in simple molecular compounds, but the heavier elements frequently reach coordination numbers of 5 and 6, as in PCl5 and AsF6. Nitrogen forms binary compounds, the nitrides, with almost all the elements. The nitrides are classified as saline, covalent, and interstitial. Nitrogen also forms the azides, which contain the N3 ion, in which the average oxidation number of nitrogen is 13 . Like N, P forms compounds with almost every element in the periodic table. There are many varieties of phosphides, with formulas ranging from M4P to MP15. The P atoms may be 3− (4). The arsenides arranged in rings, chains, or cages, for example P73− (2), P82− (3), and P11 and antimonides of the Group 13 elements In and Ga are semiconductors. All the elements form simple hydrides. Ammonia NH3 (5) is a pungent gas that is toxic at high levels of exposure. Ammonia is an excellent solvent for the Group 1 metals, for instance it is possible to dissolve 330 g of Cs in 100 g of liquid ammonia at 50ºC. These highly coloured, electrically conducting solutions contain solvated electrons (Section 11.14). The chemical properties of ammonium salts are similar to those of the Group 1 ions, especially K and Rb. Ammonium salts decompose on heating and ammonium nitrate is a component of some explosives. It is also widely used as a fertilizer. Nitrogen also forms the colourless liquid hydrazine, N2H4. The other hydrides of Group 15 are phosphine (formally phosphane, PH3), arsine (arsane, AsH3), and stibine (stibane, SbH3), which are all poisonous gases. A note on good practice Although phosphane is the correct formal name of phosphine, the latter name is widely used and we adopt it here. However, we shall use the formal names arsane and stibane for these less common compounds and their derivatives. Halogen compounds of P, As, and Sb are numerous and important in synthetic chemistry. Trihalides are known for all the Group 15 elements. However, whereas pentafluorides are known for all members of the group from P to Bi, pentachlorides are known only for P, As, and Sb, and the pentabromide is known only for P. Nitrogen does not reach its group oxidation state (5) in neutral binary halogen compounds, but it does achieve it in NF4. Presumably, an N atom is too small for NF5 to be sterically feasible. The difficulty of oxidizing Bi(III) to Bi(V) by chlorine or bromine is an example of the inert-pair effect (Section 9.5). Bismuth pentafluoride, BiF5, exists, but BiCl5 and BiBr5 do not. Nitrogen forms many oxides and oxoanions, which are treated separately in Section 15.3. Phosphorus, As, Sb, and Bi form oxides and oxoanions in a range of oxidation states from 5 to 1. The most common oxidation state is 5 but the 3 state becomes increasingly more important for bismuth. The complete combustion of phosphorus yields phosphorus(V) oxide, P4O10. Each P4O10 molecule has a cage structure in which a tetrahedron of P atoms is held together by bridging O atoms, and each P atom has a terminal O atom (6). Combustion in a limited supply of oxygen results in the formation of phosphorus(III) oxide, P4O6; this molecule has the same O-bridged framework as P4O10, but lacks the terminal O atoms (7). Arsenic, Sb, and Bi form As2O3, Sb2O3, and Bi2O3. 15.3 Oxides and oxanions of nitrogen – – – P 2 P73– – – P 3 (P82–)n – – – P 3– 4 P11 N .71 0 1p m 7 0 18 °. H 5 Ammonia, NH3, C3v P O 6 P4O10, Td P O Key points: The nitrate ion is a strong but slow oxidizing agent. The intermediate oxidation states of nitrogen are often susceptible to disproportionation. Dinitrogen oxide is unreactive. Nitrogen forms oxo compounds and oxoanions in all oxidation states from 5 to 1. Nitrogen is in the oxidation state 5 in nitric acid, HNO3, which is a major industrial 7 P4O6, Td 378 15 The Group 15 elements chemical used in the production of fertilizers, explosives, and a wide variety of nitrogencontaining chemicals. The nitrate ion, NO3, is a moderately strong oxidizing agent. When concentrated nitric acid is mixed with concentrated hydrochloric acid, the orange, fuming aqua regia is formed, which is one of the few reagents that are able to dissolve platinum and gold. The anhydride of nitric acid is N2O5. It is a crystalline solid of composition [NO2][NO3−]. Nitrogen(IV) oxide, which is commonly called nitrogen dioxide, exists as an equilibrium mixture of the brown NO2 radical and its colourless dimer, N2O4 (dinitrogen tetroxide). The equilibrium constant for the dimerization N2O4(g) 2 NO2(g) is K = 0.115 at 25°C. In nitrous acid, HNO2, nitrogen is present as N(III). Nitrous acid is a strong oxidizing agent. Dinitrogen trioxide, N2O3, the anhydride of nitrous acid, is a blue solid that melts above 100ºC to give a blue liquid that dissociates into NO and NO2. Nitrogen(II) oxide, more commonly nitric oxide, NO, is an odd-electron molecule. However, unlike NO2 it does not form a stable dimer in the gas phase because the odd electron is distributed almost equally over both atoms and not, as in NO2, largely confined to the N atom. Until the late 1980s, no beneficial biological roles were known for NO. However, since then it has been found that NO is generated in vivo, and that it performs functions such as the reduction of blood pressure, neurotransmission, and the destruction of microbes. Thousands of scientific papers have been published on the physiological functions of NO but our fundamental knowledge of its biochemistry is still quite meagre. The average oxidation number of nitrogen in dinitrogen oxide, N2O (specifically, NNO), which is commonly called nitrous oxide, is 1. N2O is a colourless, unreactive gas. One sign of this inertness is that N2O has been used as the propellant gas for instant whipping cream. Similarly, N2O was used for many years as a mild anaesthetic; however, this practice has been discontinued because of undesirable physiological side-effects, particularly mild hysteria, indicated by its common name of laughing gas. It is still used as a 50:50 mixture with oxygen as an analgesic in childbirth and in clinical procedures, such as wound suturing. PART B: THE DETAIL In this section we review the detailed chemistry of the Group 15 elements. We shall see the wide variety of oxidation states achieved by the elements, particularly nitrogen and phosphorus. 15.4 Occurrence and recovery Key points: Nitrogen is recovered by distillation from liquid air; it is used as an inert gas and in the production of ammonia. Elemental phosphorus is recovered from the minerals fluorapatite and hydroxyapatite by carbon arc reduction; the resulting white phosphorus is a molecular solid, P4. Treatment of apatite with sulfuric acid yields phosphoric acid, which is converted to fertilizers and other chemicals. Nitrogen is obtained on a massive scale by the distillation of liquid air. Liquid nitrogen is a very convenient way of storing and handling N2 in the laboratory. Membrane materials that are more permeable to O2 than to N2 are used in laboratory-scale separations from air at room temperature (Fig. 15.3). Phosphorus was first isolated by Hennig Brandt in 1669. Brandt, misinterpreting their colour, was trying to extract gold from urine and sand, and instead extracted a white solid, which glowed in the dark. This element was called phosphorus after the Greek for ‘light bearer’. Today, phosphorus is produced by the action of concentrated sulfuric acid on the mineral fluorapatite to generate phosphoric acid from which elemental phosphorous is subsequently extracted: Ca5 (PO4 )3 F(s) + 5H 2 SO4 (l) → 3H3 PO4 (l) + 5CaSO4 (s) + HF(g) The potential pollutant, hydrogen fluoride, from the fluoride component of the rock is scavenged by reaction with silicates to yield the less reactive SiF62 complex ion. Oxygenpermeable membrane Air Nitrogen Oxygen Oxygen Figure 15.3 Schematic diagram of a membrane separator for nitrogen and oxygen. The detail The product of the treatment of phosphate rock with acid contains d-metal contaminants that are difficult to remove completely, so its use is largely confined to fertilizers and metal treatment. Most pure phosphoric acid and phosphorus compounds are still produced from the element because it can be purified by sublimation. The production of elemental phosphorus starts with crude calcium phosphate (as calcined phosphate rock), which is reduced with carbon in an electric arc furnace. Silica is added (as sand) to produce a slag of calcium silicate: 1500ºC → 2Ca3 (PO4 )2 (s) + 6SiO2 (s) + 10C(s) ⎯⎯⎯⎯ 6CaSiO3 (l) + 10CO(g) + P4 (g) The slag is molten at these high temperatures and so can easily be removed from the furnace. The phosphorus vaporizes and is condensed to the solid, which is stored under water to protect it from reaction with air. Most phosphorus produced in this way is burned to form P4O10, which is then hydrated to yield pure phosphoric acid. Arsenic is usually extracted from the flue dust of copper and lead smelters (Box 15.1). However, it is also obtained by heating the ores in the absence of oxygen: 700ºC → FeS(s) + As(g) FeAsS(s) ⎯⎯⎯ Antimony is extracted by heating the ore stibnite with iron, which produces the metal and iron sulfide: Sb2 S3 (s) + 3Fe(s) → 2Sb(s) + 3FeS(s) Bismuth occurs as bismite, Bi2O3, and bismuthinite, Bi2S3. It is produced as a byproduct of the extraction of copper, tin, lead, and zinc. 15.5 Uses Key points: Nitrogen is essential for the industrial production of ammonia and nitric acid; the major use of phosphorus is in the manufacture of fertilizers. The major nonchemical use of nitrogen gas is as an inert atmosphere in metal processing, petroleum refining, and food processing. Nitrogen gas is used to provide an inert atmosphere in the laboratory, and liquid nitrogen (b.p. 196ºC, 77 K) is a convenient HCN Plastics, pigments CH4 (NH2)2CO Fertilizers, plastics COCl2 O2, H2O NH3 HX 379 X= Fertilizers, explosives HNO3 1 2 SO42–, H2PO4– NH4X Fertilizers Figure 15.4 The industrial uses of ammonia. refrigerant in both industry and the laboratory. The major industrial use of nitrogen is in the production of ammonia by the Haber process (Section 15.6) and conversion to nitric acid by the Ostwald process (Section 15.13). Ammonia provides a route to a wide range of nitrogen compounds, which include fertilizers, plastics, and explosives (Fig. 15.4). Nitrogen plays a crucial role in biology as it is a constituent of amino acids, nucleic acids, and proteins, and the nitrogen cycle is one of the most important processes in the ecosystem (Box 15.2 and Section 27.13). Phosphorus is used in pyrotechnics, smoke bombs, steel making, and alloys. Red phosphorus mixed with sand is used as the striking strip on matchboxes. Sodium phosphate is used as a cleaning agent, a water softener, and to prevent scaling in boilers and pipes. Condensed phosphates are added to detergents as builders that enhance the detergency by softening the water by forming complexes with metal ions. In the natural environment, phosphorus is usually present as phosphate ions. Phosphorus (together with N and K) is an essential plant nutrient. However, as a result of the low solubility of many metal phosphates it is often depleted in soil, and hence hydrogenphosphates are important components of balanced fertilizers. Approximately 85 per cent of the phosphoric acid produced goes into fertilizer manufacture. Phosphorus is also an important constituent of bones and teeth (which are predominantly calcium phosphate), B OX 15 .1 Arsenic in the environment The environmental toxicity of arsenic is a problem of groundwater contamination. The worst occurrence of arsenic pollution is in Bangladesh and the neighbouring Indian province of West Bengal, where hundreds of thousands of people have been diagnosed with arsenicosis. Three major rivers drain into this region, bringing iron-laden sediments from the mountains. The fertile delta is heavily farmed, and organic matter leaches into the shallow aquifer, creating reducing conditions. Arsenic levels are correlated with iron levels in groundwater and the arsenic is thought to be released on the dissolution of iron oxides and hydroxides from the ores. Paradoxically, the problem grew out of a scheme sponsored by the United Nations, starting in the 1960s, to provide clean drinking water (replacing contaminated surface water) by sinking inexpensive tube wells into the aquifer. These wells did in fact improve health greatly by reducing the incidence of water-borne diseases, but the high arsenic content went unrecognized for many years. The tube wells are typically 20100 m deep. Ground water closer to the surface has not had time to develop high concentrations of arsenic and below 100 m the sediment has depleted in arsenic over time. As many as half of the four million tube wells exceed the Bangladesh arsenic standard of 50 ppb (the World Health Organization’s guideline is 10 ppb), whereas levels routinely exceed 500 ppb in the more contaminated areas. There are several schemes to treat the well water to remove arsenic, and new wells could be dug into deeper, uncontaminated aquifers. The World Bank is coordinating a mitigation plan, but the massive effort could take years. Arsenicosis develops over a period of up to 20 years. The first symptoms are keratoses of the skin, which develop into cancers; the liver and kidneys also deteriorate. The early stage is reversible if arsenic ingestion is discontinued, but once cancers develop, effective treatment becomes more difficult. The biochemistry of these effects is uncertain. Arsenate is reduced to As(III) complexes in the body, which probably act by binding sulfhydryl groups. A plausible link to cancer is suggested by the laboratory finding that low levels of arsenic inhibit hormone receptors that turn on cancer suppressor genes. 380 15 The Group 15 elements B OX 15 . 2 The nitrogen cycle Most of the molecules used by biological systems contain nitrogen, including proteins, nucleic acids, chlorophyll, various enzymes and vitamins, and many other cellular constituents. In all these compounds, nitrogen is in its reduced form with an oxidation number of 3. The principal reservoir of nitrogen is N2 in the atmosphere. Therefore, a major challenge to biology (and technology) involves the reduction of N2 for incorporation into essential nitrogen compounds. The nitrogen cycle is shown in Fig. B15.1. The cycle can be viewed as a set of enzymatically catalysed redox reactions that lead to an accessible supply of reduced nitrogen compounds. Microorganisms are almost entirely responsible for the interconversion of inorganic forms of nitrogen. The enzymes that catalyse these conversions have Fe, Mo, and Cu at their active site. Enzymes of the nitrogen cycle are discussed in Section 27.13. Although N2 is the most abundant constituent of the Earth’s atmosphere, its usefulness is limited by its unreactivity. Although small amounts of N2 react with O2 under extreme conditions (for instance, in lightning discharges and high-temperature combustions), such processes provide only a small percentage of the nitrogen requirements of the biosphere. The remainder comes from the process of nitrogen fixation. The biological reduction of nitrogen to ammonia is carried out exclusively by prokaryotes, including both bacteria and archaea. The enzyme system for nitrogen Nitrification NH2OH NO2– O2 NH3 Fixation N2 Denitrification N2O NO2– NO Reduced NO3– Oxidized NO2– Nitrate assimilation Figure B15.1 The nitrogen cycle. cell membranes (phosphate esters of fatty acids), and nucleic acids, including DNA, RNA, and adenosine triphosphate (ATP), the energy-transfer unit of living organisms. The phosphines, PX3, are widely used ligands (Section 7.1). fixation functions anaerobically and O2 rapidly and irreversibly destroys the enzyme. Nevertheless, nitrogen fixation also occurs in aerobic bacteria. In addition, important symbioses and associations exist between higher plants and nitrogen-fixing bacteria (such as rhizobia). In these symbioses, bacteria live within controlled environments in the plant, such as root nodules, that have low O2 levels. The plant provides the bacterium with reduced carbon compounds from photosynthesis, while the bacterium provides fixed nitrogen to the plant. A reduction potential of about 0.28 V is required for biological nitrogen fixation. Reduced ferredoxins or flavoproteins with reduction potentials of 0.4 to 0.5 V are readily available in biological systems (Chapter 27). Whereas these potentials indicate that nitrogen fixation is thermodynamically feasible, this is not the case kinetically. The kinetic barrier to N2 reduction apparently arises from the need to form bound intermediates in the conversion of N2 to ammonia. Organisms invest metabolic energy from adenosine triphosphate (ATP) hydrolysis, for which ∆rG O ⬇ 31 kJ mol1, for conversion to adenosine diphosphate (ADP) and inorganic phosphate (Pi) to produce the key intermediates in the N2 fixation process. The reduction of N2 consumes 16 molecules of ATP for each molecule of N2 reduced. Given the opportunity, most organisms capable of nitrogen fixation use available fixed nitrogen sources (ammonia, nitrate, or nitrite) and repress the synthesis of the elaborate nitrogen fixation system. Once nitrogen is reduced, organisms incorporate the nitrogen into organic molecules, where it enters the biosynthetic pathways of the cell. When organisms die and biomass decays, organonitrogen compounds decompose and release nitrogen to the environment in the form of NH3 or NH4, depending on the conditions. The growing human population and its dependence on synthetic fertilizers have had enormous impact on the nitrogen cycle. Ammonia synthesis is carried out by the HaberBosch process (Section 26.12), which augments the total fixed nitrogen available to life on Earth. Between a third and a half of all nitrogen fixed occurs through technological and agricultural, rather than natural, means. In addition to ammonia itself, nitrate salts are produced industrially from ammonia for use in fertilizers. Both ammonia and nitrates enter the nitrogen cycle as fertilizer, which increases all segments of the natural cycle. The natural reservoirs are inadequate sinks for the excess input. Under such conditions, nitrate or nitrite may accumulate as an undesirable component of ground water or produce eutrophication in lakes, wetlands, river deltas, and coastal areas. Arsenic is used as a dopant in solid-state devices such as integrated circuits and lasers. Although As is a well-known poison it is also an essential trace element in chickens, rats, goats, and pigs, and arsenic deficiency leads to restricted growth (Box 15.3). B OX 15 . 3 Arsenicals ‘Arsenicals’ is the term used to describe chemicals containing arsenic. Arsenic and its compounds are intensely toxic and all the applications of arsenicals are based on this broad spectrum toxicity. Inorganic arsenicals in the form of the mineral realgar and arsenolite were used in ancient times to treat ulcers, skin diseases, and leprosy. In the early 1900s an organoarsenic compound was found to be an effective treatment for syphilis and led to a rapid increase in research in this area. This initial treatment has now been replaced by penicillin, but organoarsenic compounds are still used today to treat trypanosiosis, or sleeping sickness, which is caused by a parasite in the blood. Arsenoamide, C11H12AsNO5S2, is used in veterinary medicines to treat heartworm in dogs. Arsenilic acid, C6H8AsNO3, and sodium arsenilate, NaAsC6H8, are used as antimicrobial agents in animal and poultry feed to prevent the growth of moulds. Another powerful antimicrobial agent is 10,10-oxybisphenoxarsine (OBPA), which is used extensively in the manufacture of plastics. Arsenicals are also used as insecticides and herbicides. Methylarsonic acid, CH3AsO3, is used to control weeds in cotton and turf crops. It is water soluble but is strongly absorbed into the soil and is not easily leached out. The first arsenic-containing insecticide was Paris green, Cu(CH3CO2)2.3Cu(AsO2)2, which was manufactured in 1865 for the treatment of the Colorado potato beetle. Sodium arsenite, NaAsO2, is used in poison baits to control grasshoppers and as a dip to prevent parasites in livestock. The detail Arsenic can be detected by the Marsh test in which the arsenic compound is reacted with zinc powder and sulfuric acid to liberate AsH3: As2O3 6 Zn 6 H2SO4 → 2 AsH3 6 ZnSO4 3 H2O Ignition of AsH3 produces arsenic, which can be observed as a black powder. The development of this test effectively made arsenic poisoning detectable for the first time. Antimony is used in semiconductor technologies to produce infrared detectors and light-emitting diodes. It is used in alloys, where it leads to stronger and harder products. Antimony oxide is used to increase the activity of chlorinated hydrocarbon flame retardants, where it enhances the release of halogenated radicals. In keeping with the general trend down the p block, the 3 oxidation state becomes more favourable relative to 5 on going down the group from P to Bi. Consequently, Bi(V) compounds are useful oxidizing agents. The other major uses of Bi compounds are in medicine. Bismuth subsalicylate, HOC6H4CO2BiO, is used in conjunction with antibiotics and as a treatment for peptic ulcers. Bismuth(III)oxide is used in haemorrhoid creams. E X A MPL E 15 .1 Examining the electronic structure and chemistry of P4 Draw the Lewis structure of P4, and discuss its possible role as a ligand. Answer We use the rules described in Section 2.1 to develop the Lewis structure. There is a total of 4 5 = 20 valence electrons. If each P atom forms a bond to each of the other three P atoms then 12 electrons will be accounted for, leaving eight electrons, or one lone pair on each P atom (8). This structure, together with the fact that the electronegativity of P is moderate (P = 2.06), suggests that P4 might be a moderately good donor ligand. Indeed, though rare, P4 complexes are known. ... P ... P .... P .P .. 8 P4 Self-test 15.1 Consider the Lewis structure of a segment of the structure of bismuth shown in Fig. 15.2. Is this puckered structure consistent with the VSEPR model? and hence the high activation energy required for breaking it. (The strength of this bond also accounts for the lack of nitrogen allotropes.) Another factor is the relatively large size of the HOMOLUMO gap in N2 (Section 2.8b), which makes the molecule resistant to simple electron-transfer redox processes. A third factor is the low polarizability of N2, which does not encourage the formation of the highly polar transition states that are often involved in electrophilic and nucleophilic displacement reactions. Cheap methods of nitrogen activation, its conversion into useful compounds, are highly desirable because they would have a profound effect on the economy, particularly in poorer agricultural economies. In the Haber process for the production of ammonia, H2 and N2 are combined at high temperatures and pressures over an Fe catalyst, as we discuss in detail in Section 15.10. Much of the recent research aimed at achieving more economical ways of activating N2 has been inspired by the way in which bacteria carry out the transformation at room temperature. Catalytic conversion of nitrogen to NH4 involves the metalloenzyme nitrogenase, which occurs in nitrogen-fixing bacteria such as those found in the root nodules of legumes. The mechanism by which nitrogenase carries out this reaction, at an active site containing Fe, Mo, and S, is the topic of considerable research. In this connection, dinitrogen complexes of metals were discovered in 1965, at about the same time that it was realized that nitrogenase contains Mo (Section 27.13). These developments led to optimism that efficient homogeneous catalysts might be developed in which metal ions would coordinate to N2 and promote its reduction. Many N2 complexes have in fact been prepared, and in some cases the preparation is as simple as bubbling N2 through an aqueous solution of a complex: [Ru(NH3 )5 (OH 2 )]2+ (aq) + N 2 (g) → [Ru(NH3 )5 (N 2 )]2+ (aq) + H 2O(l) As with the isoelectronic CO molecule, end-on bonding is typical of N2 when it acts as a ligand (9, Section 22.17). The NN bond length in the Ru(II) complex is only slightly altered from that in the free molecule. However, when N2 is coordinated to a more strongly reducing metal centre, this bond is considerably lengthened by back-donation of electron density into the π orbitals of N2. NH3 15.6 Nitrogen activation Key points: The commercial Haber process requires high temperatures and pressures to yield ammonia, which is a major ingredient in fertilizers and an important chemical intermediate. Nitrogen occurs in many compounds, but N2 itself, with a triple bond between the two atoms, is strikingly unreactive. A few strong reducing agents can transfer electrons to the N2 molecule at room temperature, leading to scission of the NN bond, but usually the reaction needs extreme conditions. The prime example of this reaction is the slow reaction of lithium metal at room temperature, which yields Li3N. Similarly, when Mg (the diagonal neighbor of Li) burns in air it forms the nitride as well as the oxide. The slowness of the reactions of N2 appears to be the result of several factors. One is the strength of the NN triple bond 381 2+ Ru N2 9 [Ru(NH3)5(N2)]2+ A recent advance has been the direct reduction of N2 to ammonia at room temperature and atmospheric pressure with a molybdenum catalyst that contains a tetradentate triamidoamine ligand known as HIPTN3N. Nitrogen coordinates to the Mo centre and is converted to NH3 on addition of a proton source and a reducing agent. 382 15 The Group 15 elements 15.7 Nitrides and azides Nitrogen forms simple binary compounds with other elements; they are classified as nitrides or azides. (a) Nitrides Key point: Nitrides are classified as saline, covalent, or interstitial. The nitrides of metals can be prepared by direct interaction of the element with nitrogen or ammonia or by thermal decomposition of an amide: 6Li(s) + N 2 (g) → 2Li3 N(s) 3Ca(s) + 2 NH3 (l) → Ca3 N 2 (s) + 3H 2 (g) 3Zn(NH 2 )2 (s) → Zn3 N 2 (s) + 4 NH3 (g) Os The average oxidation number of N in the azide ion is − 13 . The ion is isoelectronic with both dinitrogen oxide, N2O, and CO2 and, like these two molecules, is linear. It is a reasonably strong Brønsted base, the pKa of its conjugate acid, hydrazoic acid, HN3, being 4.77. It is also a good ligand towards d-block ions. However, heavy-metal complexes or salts, such as Pb(N3)2 and Hg(N3)2, are shock-sensitive detonators and decompose to produce the metal and nitrogen: Pb(N3)2(s) → Pb(s) 3 N2(g) The compounds of N with H, O, and the halogens are treated separately. The saline nitrides can be regarded as containing the nitride ion, N3. However, the high charge of this ion means that it is highly polarizing (Section 1.9e) and saline nitrides are likely to have considerable covalent character. Saline nitrides occur for lithium, Li3N, and the Group 2 elements, M3N2. The covalent nitrides, in which the EN bond is covalent, possess a wide range of properties depending on the element to which N is bonded. Some examples of covalent nitrides are boron nitride, BN, cyanogen, (CN)2, phosphorus nitride, P3N5, tetrasulfur tetranitride, S4N4, and disulfur dinitride, S2N2. These compounds are discussed in the context of the other elements. The largest category of nitrides consists of the interstitial nitrides of the d-block elements with formula MN, M2N, or M4N. The N atom occupies some or all of the octahedral sites within the cubic or hexagonal close-packed lattice of metal atoms. The compounds are hard and inert, with a metallic lustre and conductivity. They are widely used as refractory materials and find applications as crucibles, high-temperature reaction vessels, and thermocouple sheaths. The nitride ion, N3, is often found as a ligand in d-metal complexes. Its high negative charge, small size, and ability to serve as a good π-donor as well as a σ-donor, means that it can stabilize metals in high oxidation states. The short coordinate bond between the ion and the metal atom is often represented as M⬅N. An example is the complex [Os(N)(NH3)5]2 (10). 2+ N 3 NH 2− + NO3− ⎯175ºC ⎯⎯→ N3− + 3 OH − + NH3 190 ºC → N3− + OH − + NH3 2 NH 2− + N 2O ⎯⎯⎯ NH3 Ionic azides such as NaN3 are thermodynamically unstable but kinetically inert; they can be handled at room temperature. Sodium azide is toxic and is used as a chemical preservative and in pest control. When alkali metal azides are heated or detonated by impact they explode, liberating N2; this reaction is used in the inflation of air bags in cars, in which the heating of the azide is electrical. ■ A brief illustration. A typical airbag contains approximately 50 g of NaN3. To estimate the volume of nitrogen produced when the azide is detonated at room temperature and pressure (20C and 1.0 atm) we need to consider the amount (in moles) of N2 molecules produced in the decomposition reaction 2 NaN3(s) → 2 Na(s) 3 N2(g). Because 50 g of NaN3 contains 0.77 mol NaN3, it liberates 1.2 mol N2. This amount occupies 26 dm3 at 20C and 1.0 atm. As the airbag is restricted in volume, the pressure of nitrogen in the airbag will be high, so providing protection to the driver. ■ Compounds containing the polynitrogen cation, N5 (11), have been synthesized from species containing N3− and N2F ions. For example, N5AsF6 is prepared from N2FAsF6 and HN3 in anhydrous HF solvent: N 2 FAsF6 (sol) + HN3 (sol) → N 5 AsF6 (sol) + HF(l) 166° N 129 pm 111° 110 pm 11 N5+ The compound is a white solid that decomposes explosively above 250ºC. It is a powerful oxidizing agent and ignites organic material even at low temperatures. Salts of dipositive anions can be prepared from metathesis with the salts of monopositive anions in anhydrous HF: 2 N 5SbF6 (sol) + Cs2SnF6 (sol) → (N 5)2 SnF6 (sol) + 2CsSbF6 (s) 10 [Os(NH3)5N]2+ The product is a white solid that is friction-sensitive and decomposes to N5SnF6 above 250ºC. This product, N5SnF6, is stable up to 500ºC. (b) Azides 15.8 Phosphides Key points: Azides are toxic and unstable; they are used as detonators in explosives. Key point: Phosphides may be metal-rich or phosphorus-rich. Azides, in which nitrogen is present as N3, may be synthesized by the oxidation of sodium amide with either NO3− ions or N2O at elevated temperatures: The compounds of phosphorus with hydrogen, oxygen, and the halogens are discussed separately. The phosphides of other elements can be prepared by heating the appropriate element with red phosphorus in an inert atmosphere: The detail nM + mP → M n Pm There are many varieties of phosphides, with formulas ranging from M4P to MP15. They include metal-rich phosphides, in which M:P > 1, monophosphides, in which M:P = 1, and phosphorus-rich phosphides, in which M:P < 1. Metal-rich phosphides are usually very inert, hard, brittle refractory materials and resemble the parent metal in having high electrical and thermal conductivities. The structures have a trigonal prismatic arrangement of six, seven, eight, or nine metal ions around a P atom (12). Monophosphides adopt a variety of structures depending on the relative size of the other atom. For example, AlP adopts the zinc-blende structure, SnP adopts the rock-salt structure, and VP adopts the nickel-arsenide structure (Section 3.9). Phosphorus-rich phosphides have lower melting points and are less stable than metal-rich phosphides and monophosphides. They are semiconductors rather than conductors. M P 12 15.9 Arsenides, antimonides, and bismuthides Key point: Indium and gallium arsenides and antimonides are semiconductors. The compounds formed between metals and arsenic, antimony, and bismuth can be prepared by direct reaction of the elements: Ni(s) + As(s) → NiAs(s) The arsenides and antimonides of the Group 13 elements In and Ga are semiconductors. Gallium arsenide (GaAs) is the more important and is used to make devices such as integrated circuits, light-emitting diodes, and laser diodes. Its band gap is similar to that of silicon and larger than those of other Group 13/15 semiconductors (Table 13.5; Section 24.19). Gallium arsenide is superior to Si for such applications because it has higher electron mobility and the devices produce less electronic noise. Silicon still has major advantages over GaAs in the sense that silicon is cheap and the wafers are stronger than those of GaAs, so processing is easier. Gallium arsenide integrated circuits are commonly used in mobile phones, satellite communications, and some radar systems. 15.10 Hydrides All the Group 15 elements form binary compounds with hydrogen. All the EH3 hydrides are toxic. Nitrogen also forms a catenated hydride, hydrazine, N2H4. (a) Ammonia Key points: Ammonia is produced by the Haber process; it is used to manufacture fertilizers and many other useful nitrogen-containing chemicals. 383 Ammonia is produced in huge quantities worldwide for use as a fertilizer and as a primary source of nitrogen in the production of many chemicals. As already mentioned, the Haber process is used for the entire global production. In this process, N2 and H2 combine directly at high temperature (450ºC) and pressure (100 atm) over a promoted Fe catalyst: N 2 (g) + 3H 2 (g) → 2 NH3 (g) The promoters (compounds that enhance the catalyst’s activity) include SiO2, MgO, and other oxides (Section 26.12). The high temperature and catalyst are required to overcome the kinetic inertness of N2, and the high pressure is needed to overcome the thermodynamic effect of an unfavourable equilibrium constant at the operating temperature. So novel and great were the chemical and engineering problems arising from the then (early twentieth century) uncharted area of large-scale high-pressure technology, that two Nobel Prizes were awarded in connection with the process. One went to Fritz Haber (in 1918), who developed the chemical process. The other went to Carl Bosch (in 1931), the chemical engineer who designed the first plants to realize Haber’s process. The HaberBosch process has had a major impact on civilization because ammonia is the primary source of most nitrogen-containing compounds, including fertilizers and most commercially important compounds of nitrogen. Before the development of the process the main sources of nitrogen for fertilizers were guano (bird droppings) and saltpetre, which had to be mined and transported from South America. In the early twentieth century there were predictions of widespread starvation across Europe, predictions that were never realized because of the widespread availability of nitrogen-based fertilizers. The boiling point of ammonia is 33ºC, which is higher than that of the hydrides of the other elements in the group and indicates the influence of extensive hydrogen bonding. Liquid ammonia is a useful nonaqueous solvent for solutes such as alcohols, amines, ammonium salts, amides, and cyanides. Reactions in liquid ammonia closely resemble those in aqueous solution, as indicated by the following autoprotolysis equilibria: 2 H2O(l) H3O(aq) OH(aq) pKw = 14.00 at 25°C 2 NH3(l) NH4(am) NH2(am) pKam = 34.00 at 33°C Many of the reactions are analogous to those carried out in water. For example, simple acidbase neutralization reactions can be carried out: NH4Cl(am) NaNH2(am) → NaCl(am) 2 NH3(l) Ammonia is a water-soluble weak base: NH3(aq) H2O(l) NH4(aq) OH(aq) pKb = 4.75 The chemical properties of ammonium salts are very similar to those of Group 1 salts, especially of K and Rb. They are soluble in water and solutions of the salts of strong acids, such as NH4Cl, are acidic due to the equilibrium: NH4(aq) H2O(l) H3O(aq) NH3(aq) pKa = 9.25 Ammonium salts decompose readily on heating and for many salts, such as the halides, carbonate, and sulfates, ammonia is evolved: NH 4Cl(s) → NH3 (g) + HCl(g) (NH 4 )2 SO4 (s) → 2 NH3 (g) + H 2 SO4 (l) 384 15 The Group 15 elements When the anion is oxidizing, as in the case of NO3−, ClO4−, and Cr2O72−, the NH4 is oxidized to N2 or N2O: NH 4 NO3 (s) → N 2O(g) + 2 H 2O(g) When ammonium nitrate is heated strongly or detonated, the decomposition of 2 mol NH4NO3(s) in the reaction 2 NH 4 NO3 (s) → 2 N 2 (g) + O2 (g) + 4 H 2O(g) produces 7 mol of gaseous molecules, corresponding to an increase in volume from about approximately 200 cm3 to about 140 dm3, a factor of 700. This features leads to the use of ammonium nitrate as an explosive and nitrate fertilizers are often mixed with materials such as calcium carbonate or ammonium sulfate to make them more stable. Ammonium sulfate and the ammonium hydrogenphosphates, NH4H2PO4 and (NH4)2HPO4, are also used as fertilizers because phosphate is a plant nutrient. Ammonium perchlorate is used as the oxidizing agent in solidfuel rocket propellants. (b) Hydrazine and hydroxylamine Key point: Hydrazine is a weaker base than ammonia and forms two series of salts. Hydrazine, N2H4, is a fuming, colourless liquid with an odour like that of ammonia. It has a liquid range similar to that of water (2114ºC), indicating the presence of hydrogen bonding. In the liquid phase hydrazine adopts a gauche conformation around the NN bond (13). H E X A MPL E 15 . 2 Evaluating rocket fuels Hydrazine, N2H4, and dimethylhydrazine, N2H2(CH3)2, are used as rocket fuels. Given the following data, suggest which would be the more efficient fuel thermochemically. ∆fH O ⲐkJ mol1 50.6 N2H4(l) N2H2(CH3)2(l) 42.0 CO2(g) 394 H2O(g) 242 Answer We must evaluate which combustion reactions release most heat by calculating their (standard) enthalpies of combustion. The combustion reactions are N2H4(l) O2(g) → N2(g) 2 H2O(g) N2H2(CH3)2(l) O2(g) → N2(g) 4 H2O(g) 2 CO2(g) The enthalpy of reaction (in this case, combustion) is calculated from ∆ cH O = ∑ ∆ fH O − ∑ ∆ fH O 95° H It reacts with acids HX to form two series of salts, N2H5X and N2H6X2. The major use of hydrazine and its methyl derivatives, CH3NHNH2 and (CH3)2NNH2, is as a rocket fuel. Hydrazine is also used as a foam-blowing agent and as a treatment in boiler water to scavenge dissolved oxygen and prevent oxidation of pipes. Both N2H4 and N2H5 are reducing agents and are used in the recovery of precious metals. products H H 145 pm 108° N We find 535 kJ mol1 for N2H4 and 1798 kJ mol1 for N2H2(CH3)2. An important factor for selecting rocket fuels is the specific enthalpy (the enthalpy of combustion divided by the mass of fuel), which for these fuels has values 16.7 and 29.9 kJ g1, respectively, indicating that N2H2(CH3)2 is the better fuel even when mass is significant. H 13 Hydrazine, N2H4 Hydrazine is manufactured by the Raschig process, in which ammonia and sodium hypochlorite react in dilute aqueous solution. The reaction proceeds through several steps, which can be simplified to NH3 (aq) + NaOCl(aq) → NH 2Cl(aq) + NaOH(aq) 2 NH3 (aq) + NH 2Cl(aq) → N 2 H 4 (aq) + NH 4Cl(aq) There is a competing side reaction that is catalysed by d-metal ions: N 2 H 4 (aq) + 2 NH 2Cl(aq) → N 2 (g) + 2 NH 4Cl(aq) Gelatine is added to the reaction mixture to trap the d-metal ions by forming a complex with them. The dilute aqueous solution of hydrazine so produced is converted to a concentrated solution of hydrazine hydrate, N2H4.H2O, by distillation. This product is often preferred commercially as it is cheaper than hydrazine and has a wider liquid range. Hydrazine is produced by distillation of the hydrate in the presence of a drying agent such as solid NaOH or KOH. Hydrazine is a weaker base than ammonia: N2H4(aq) H2O(l) N2H5(aq) OH(aq) reactants pKb1 = 7.93 N2H5(aq) H2O(l) N2H62(aq) OH(aq) pKb2 = 15.05 Self-test 15.2 Refined hydrocarbons and liquid hydrogen are also used as rocket fuel. What are the advantages of dimethylhydrazine over these fuels? Hydroxylamine, NH2OH (14), is a colourless, hygroscopic solid with a low melting point (32ºC). It is usually available as one of its salts or in aqueous solution. It is a weaker base than either ammonia or hydrazine: NH2OH(aq) H2O(l) NH3OH(aq) OH(aq) pKb = 8.18 Anhydrous hydroxylamine can be prepared by adding sodium butoxide, NaC4H9O (NaOBu), to a solution of hydroxylamine hydrochloride in 1-butanol. The NaCl produced is filtered off and the hydroxylamine precipitated by the addition of ether. [NH3OH]Cl(sol) + NaOBu → NH 2OH(sol) + NaCl(s) + BuOH(l) The major commercial use of hydroxylamine is in the synthesis of caprolactam, which is an intermediate in the manufacture of nylon. 147 pm N H O 14 Hydroxylamine, NH2OH The detail 385 (c) Phosphine, arsane, and stibane (a) Nitrogen halides Key points: Unlike liquid ammonia, liquid phosphine, arsane, and stibane do not associate through hydrogen bonding; their much more stable alkyl and aryl analogues are useful soft ligands. Key point: Except for NF3, nitrogen trihalides have limited stability and nitrogen triiodide is dangerously explosive. In contrast to the commanding role that ammonia plays in nitrogen chemistry, the highly poisonous hydrides of the heavier nonmetallic elements of Group 15 (particularly phosphine, PH3, and arsane, AsH3) are of minor importance in the chemistry of their respective elements. Both phosphine and arsane are used in the semiconductor industry to dope Si or to prepare other semiconductor compounds, such as GaAs, by chemical vapour deposition. These thermal decomposition reactions reflect the positive Gibbs energy of formation of these hydrides. The commercial synthesis of PH3 uses the disproportionation of white phosphorus in basic solution: P4 (s) + 3OH − (aq) + 3H 2O(l) → PH3 (g) + 3H 2 PO2− (aq) Arsane and stibane may be prepared by the protolysis of compounds that contain an electropositive metal in combination with arsenic or antimony: Zn3 E 2 (s) + 6 H3O+ (aq) → 2 EH3 (g) + 3Zn2+ (aq) + 6 H 2O(l) (E = As,Sb) Phosphine and arsane are poisonous gases that readily ignite in air, but the much more stable organic derivatives PR3 and AsR3 (R = alkyl or aryl groups) are widely used as ligands in metal coordination chemistry. In contrast with the hard donor properties of ammonia and alkylamine ligands, the organophosphines and organoarsanes, such as P(C2H5)3 and As(C6H5)3, are soft ligands and are therefore often incorporated into metal complexes having central metal atoms in low oxidation states. The stability of these complexes correlates with the soft acceptor nature of the metals in low oxidation states, and the stability of soft-donor soft-acceptor combinations (Section 4.12). All the Group 15 hydrides are pyramidal, but the bond angle decreases down the group: NH3 ,107.8º PH3 , 93.6º AsH3 , 91.8º SbH3 , 91.3º The large change in bond angle has been attributed to a decrease in the extent of sp3 hybridization from NH3 to SbH3 but is more likely to be due to steric effects. The EH bonding pairs of electrons will repel each other. This repulsion is greatest when the central element, E, is small, as in NH3, and the H atoms will be as far away from each other as possible in a near-tetrahedral arrangement. As the size of the central atom is increased down the series, the repulsion between bonding pairs decreases and the bond angle is close to 90º. It is evident from the boiling points plotted in Fig. 10.6 that PH3, AsH3, and SbH3 are subject to little, if any, hydrogen bonding with themselves; however, PH3 and AsH3 can be protonated by strong acids, such as HI, to form phosphonium and arsonium ions, PH4 and AsH4, respectively. 15.11 Halides All the elements form a trihalide with at least one halogen. Phosphorus, arsenic, and antimony form stable pentahalides. Nitrogen trifluoride, NF3, is the only exergonic binary halogen compound of nitrogen. This pyramidal molecule is not very reactive. Thus, unlike NH3, it is not a Lewis base because the strongly electronegative F atoms make the lone pair of electrons unavailable: whereas the polarity of the NH bond in NH3 is NH, that of the NF bond in NF3 is NF. Nitrogen trifluoride can be converted into the N(V) species NF4 by the reaction NF3 (l) + 2 F2 (g) + SbF3 (l) → [NF4+ ]SbF6− Nitrogen trichloride, NCl3, is a highly endergonic, explosive, yellow oil. It is prepared commercially by the electrolysis of an aqueous solution of ammonium chloride and was once used as an oxidizing bleach for flour. The electronegativities of nitrogen and chlorine are the same and the NCl bond is almost nonpolar. Nitrogen tribromide, NBr3, is an explosive, deep red oil. Nitrogen triiodide, NI3, is highly explosive. Nitrogen is more electronegative than both bromine and iodine, and so the NX bonds are polar in the sense NX and, formally, the oxidation numbers are 3 for the N and 1 for each halogen. (b) Halides of the heavy elements Key points: Whereas the halides of nitrogen have limited stability, their heavier congeners form an extensive series of compounds; the trihalides and pentahalides are useful starting materials for the synthesis of derivatives by metathetical replacement of the halide. The trihalides and pentahalides of Group 15 elements other than nitrogen are used extensively in synthetic chemistry and their simple empirical formulas conceal an interesting and varied structural chemistry. The trihalides range from gases and volatile liquids, such as PF3 (b.p. 102ºC) and AsF3 (b.p. 63ºC), to solids, such as BiF3 (m.p. 649ºC). A common method of preparation is direct reaction of the element and halogen. For phosphorus, the trifluoride is prepared by metathesis of the trichloride and a fluoride: 2 PCl3 (l) + 3ZnF2 (s) → 2 PF3 (g) + 3ZnCl2 (s) The trichlorides PCl3, AsCl3, and SbCl3 are useful starting materials for the preparation of a variety of alkyl, aryl, alkoxy, and amino derivatives because they are susceptible to protolysis and metathesis: ECl3 (sol) + 3EtOH(l) → E(OEt)3 (sol) + 3HCl(sol) (E = P,As,Sb) ECl3 (sol) + 6 Me2 NH(sol) → E(NMe2 )3 (sol) + 3 [M Me2 NH 2 ]Cl(sol) (E = P,As,Sb) Phosphorus trifluoride, PF3, is an interesting ligand because in some respects it resembles CO. Like CO, it is a weak donor but a strong π acceptor, and complexes of PF3 exist that are the analogues of carbonyls, such as Ni(PF3)4, the analogue of Ni(CO)4 (Section 22.18). The π-acceptor character is attributed to a PF antibonding LUMO, which has mainly P p-orbital character. The trihalides also act as mild Lewis acids towards Lewis bases such as trialkylamines and halides. Many halide complexes have been isolated, such as the simple mononuclear species AsCl4− (15) and SbF52− (16). More complex dinuclear and 386 15 The Group 15 elements polynuclear anions linked by halide bridges, such as the polymeric chain ([BiBr3]2)n in which Bi(I) is surrounded by a distorted octahedron of Br atoms, are also known. l][B C [P l] C 4 l C B O=PCl3 4 F P K 3 HO2 F K l C P 2– – H N s A l C P(NH2)4Cl 6 5 HC N l 3 H N h P 4 –(N l) – C P = 2 2 n Sb l C = N h P F 15 AsCl4– , C4v 3 Figure 15.5 The uses of phosphorus pentachloride. 16 SbF52–, C4v The pentahalides vary from gases, such as PF5 (b.p. 85ºC) and AsF5 (b.p. 53ºC), to solids, such as PCl5 (sublimes at 162ºC) and BiF5 (m.p. 154ºC). The five-coordinate gas-phase molecules are trigonal bipyramidal. In contrast to PF5 and AsF5, SbF5 is a highly viscous liquid in which the molecules are associated through F-atom bridges. In solid SbF5 these bridges result in a cyclic tetramer (17), which reflects the tendency of Sb(V) to achieve a coordination number of 6. A related phenomenon occurs with PCl5, which in the solid state exists as [PCl4][PCl6−]. In this case, the ionic contribution to the lattice enthalpy provides the driving force for the transfer of a Cl ion from one PCl5 molecule to another. Another contributing factor may be the more efficient packing of the PCl4 and PCl6 units compared to the less efficient stacking of PCl5 units. The pentafluorides of P, As, Sb, and Bi are strong Lewis acids (Section 4.10). SbF5 is a very strong Lewis acid; it is much stronger, for example, than the aluminium halides. When SbF5 or AsF5 is added to anhydrous HF, a superacid is formed. (see Section 4.15): SbF5 (l) + 2 HF(l) → H 2F + (sol) + SbF6− (sol) Of the pentachlorides, PCl5 and SbCl5 are stable whereas AsCl5 is very unstable. This difference is a manifestation of the alternation effect (Section 9.2c). The instability of AsCl5 is attributed to the increased effective nuclear charge arising from the poor shielding of the 3d electrons, which leads to a ‘d-block contraction’ and a lowering of the energy of the 4s orbitals in As. Consequently, it is more difficult to promote a 4s electron to form AsCl5. Sb F 17 (SbF5)4 The pentahalides of P and Sb are very useful in syntheses. Phosphorus pentachloride, PCl5, is widely used in the laboratory and in industry as a starting material and some of its characteristic reactions are shown in Fig. 15.5. Note, for example, that reaction of PCl5 with Lewis acids yields PCl4 salts, and simple Lewis bases like F give six-coordinate complexes such as PF6−. Compounds containing the NH2 group lead to the formation of PN bonds, and the interaction of PCl5 with either H2O or P4O10 yields OPCl3. 15.12 Oxohalides Key points: Nitrosyl and nitryl halides are useful halogenating agents; phosphoryl halides are important industrially in the synthesis of organophosphorus derivatives. Nitrogen forms all the nitrosyl halides, NOX, and the nitryl halides, NO2X. The nitrosyl halides and NO2F are prepared by direct interaction of the halogen with NO or NO2, respectively: 2 NO(g) + Cl2 (g) → 2 NOCl(g) 2 NO2 (g) + F2 (g) → 2 NO2 F(g) They are all reactive gases and the oxofluorides and oxochlorides are useful fluorinating and chlorinating agents. Phosphorus readily forms the phosphoryl halides POCl3 and POBr3 by the reaction of the trihalides PX3 with O2 at room temperature. The fluorine and iodine analogues are prepared by the reaction of POCl3 with a metal fluoride or iodide: POCl3 (l) + 3NaF(s) → POF3 (g) + 3NaCl(s) All the molecules are tetrahedral and contain a PO bond. POF3 is gaseous, POCl3 is a colourless liquid, POBr3 is a brown solid, and POI3 is a violet solid. They are all readily hydrolysed, fume in air, and form adducts with Lewis acids. They provide a route to the synthesis of organophosphorus compounds, which are manufactured on a large scale for use as plasticizers, oil additives, pesticides, and surfactants. For example, reaction with alcohols and phenols gives (RO)3PO and Grignard reagents (Section 12.13) yield RnPOCl3-n: 3ROH(l) + POCl3 (l) → (RO)3 PO(sol) + 3HCl(sol) n RMgBr(sol) + POCl3 (sol) → R n POCl3- n (sol) + n MgBrCl(s) 387 The detail 15.13 Oxides and oxoanions of nitrogen Key point: Reactions of nitrogenoxygen compounds that liberate or consume N2 are generally very slow at normal temperatures and pH = 7. We can infer the redox properties of the compounds of the elements in Group 15 in acidic aqueous solution from the Frost diagram in Fig. 15.6. The steepness of the slopes of the lines on the far right of the diagram show the thermodynamic tendency for reduction of the 5 oxidation states of the elements. They show, for instance, that Bi2O5 is potentially a very strong oxidizing agent, which is consistent with the inert-pair effect and the tendency of Bi(V) to form Bi(III). The next strongest oxidizing agent is NO3−. Both As(V) and Sb(V) are milder oxidizing agents, and P(V), in the form of phosphoric acid, is a very weak oxidant. The redox properties of nitrogen are important because of its widespread occurrence in the atmosphere, the biosphere, industry, and the laboratory. Nitrogen chemistry is quite complex, partly because of the large number of accessible oxidation states but also because reactions that are thermodynamically favourable are often slow or have rates that depend crucially on the identity of the reactants. As the N2 molecule is kinetically inert, redox reactions that consume N2 are slow. Moreover, the formation of N2 is often slow and may be sidestepped in aqueous solution (Fig. 15.7). As with several other p-block elements, the barriers to reaction of high oxidation state oxoanions, such as NO3−, are greater than for low oxidation state oxoanions, such as NO2−. We should also remember that low pH enhances the oxidizing power of oxoanions (Section 5.6). Low pH also often accelerates their oxidizing reactions by protonation, and this step is thought to facilitate subsequent NO bond breaking. Table 15.2 summarizes some of the properties of the nitrogen oxides, and Table 15.3 does the same for the nitrogen oxoanions. Both tables will help us to navigate through the details of their properties. (a) Nitrogen(V) oxides and oxoanions Key points: The nitrate ion is a strong but slow oxidizing agent at room temperature; strong acid and heating accelerate the reaction. +6 N Bi NE/V +4 +2 Sb As P 0 N –2 P –3 –2 3 –1 0 1 2 Oxidation number, N 4 5 Figure 15.6 Frost diagram for the elements of the nitrogen group in acidic solution. The species with oxidation number 3 are NH3, PH3, and AsH3, and those with oxidation numbers 2 and 1 are N2H4 and NH2OH, respectively. The positive oxidation states refer to the most stable oxo or hydroxo species in acidic solution, and may be oxides, oxoacids, or oxoanions. Oxidation number +5 +4 Ion Molecule NO3– H2O,∆ OH– +3 NO2– +2 HSO3–,H+ H NO N2O –3 O2, cat. N2 0 –2 O2 NO +1 –1 NO2, N2O4 NH3OH N2H5+ NH4+ H2, cat. H+ –H+ H+ N2H4 OCl– NH3 –H+ Figure 15.7 The interconversion of important nitrogen species. The most common source of N(V) is nitric acid, HNO3, which is a major industrial chemical used in the production of fertilizers, explosives, and a wide variety of nitrogen-containing chemicals. It is produced by modern versions of the Ostwald process, which make use of an indirect route from N2 to the highly oxidized compound HNO3 via the fully reduced compound NH3. Thus, after nitrogen has been reduced to the 3 state as NH3 by the Haber process, it is oxidized to the 4 state: 4 NH3 (g) + 7 O2 (g) → 6 H 2O(g) + 4 NO2 (g) ∆ rG O = − 308.0 kJ (mol NO2 )−1 The NO2 then undergoes disproportionation into N(II) and N(V) in water at elevated temperatures: 3NO2 (aq) + H 2O(l) → 2 HNO3 (aq) + NO(g) ∆ rG O = − 5.0 kJ (mol HNO3)−1 All the steps are thermodynamically favourable. The byproduct NO is oxidized with O2 to NO2 and recirculated. Such an indirect route is used because the direct oxidation of N2 to NO2 is thermoO dynamically unfavourable, with ∆rG (NO2, g) = 51 kJ mol1. In part, this endergonic character is due to the great strength of the N⬅N bond. Standard potential data imply that the NO3 ion is a moderately strong oxidizing agent. However, its reactions are generally slow in dilute acid solution. Because protonation of an O atom promotes NO bond breaking, concentrated HNO3 (in which NO3 is protonated) undergoes more rapid reactions than the dilute acid (in which HNO3 is fully deprotonated). It is also a thermodynamically more potent oxidizing agent at low pH. A sign of this oxidizing character is the yellow colour of the concentrated acid, which indicates its instability with respect to decomposition into NO2: 4 HNO3 (aq) → 4 NO2 (aq) + O2 (g) + 2 H 2O(l) 388 15 The Group 15 elements Table 15.2 Oxides of nitrogen Oxidation number Formula Name 1 N2O Nitrous oxide (dinitrogen oxide) 2 NO Nitric oxide (nitrogen monoxide) 3 N2O3 Dinitrogen trioxide Structure (gas phase) Comments 119 pm Colourless gas, not very reactive 115 pm Colourless, reactive, paramagnetic gas Blue solid (m.p. 101C); dissociates into NO and NO2 in the gas phase Planar 119 pm 4 NO2 Nitrogen dioxide Brown, reactive, paramagnetic gas 134° 118 pm 4 N2O4 Colourless liquid (m.p. 11C); in equilibrium with NO2 in the gas phase Dinitrogen tetroxide Planar 5 N2O5 Dinitrogen pentoxide (dinitrogen pentoxide) This decomposition is accelerated by light and heat. The reduction of NO3 ions rarely yields a single product, as so many lower oxidation states of nitrogen are available. For example, a strong reducing agent such as zinc can reduce a substantial proportion of dilute HNO3 as far as oxidation state 3: HNO3 (aq) + 4 Zn(s) + 9 H + (aq) → NH 4+ (aq) + 3H 2O(l) + 4 Zn2+ (aq) A weaker reducing agent, such as copper, proceeds only as far as oxidation state 4 in the concentrated acid: 2 HNO3 (aq) + Cu(s) + 2 H + (aq) → 2 NO2 (g) + Cu2+ (aq) + 2 H 2O(l) Planar Colourless, ionic solid [NO2][NO3] (m.p. 32C); unstable With the dilute acid, the 2 oxidation state is favoured, and NO is formed: 2 NO3− (aq) + 3Cu(s) + 8H + (aq) → 2 NO(g) + 3Cu2+ (aq) + 4 H 2O(l) Aqua regia is a mixture of concentrated nitric acid and concentrated hydrochloric acid, which is yellow due to the presence of the decomposition products NOCl and Cl2. It loses its potency as these volatile products are formed: HNO3 (aq) + 3HCl(aq) → NOCl(g) + Cl2 (g) + 2 H 2O(l) Aqua regia is Latin for ‘royal water’ and was so called by alchemists because of its ability to dissolve the noble metals gold and platinum. Gold will dissolve to a very small extent in concentrated nitric acid. In aqua regia the Cl ions present react immediately The detail 389 Table 15.3 Nitrogen-oxygen ions Oxidation number Formula Common name Structure Comments 2– 1 N2O 3 − 2 2− 2 Hyponitrite Usually acts as a reducing agent – NO Nitrite Weak base; acts as an oxidizing agent and a reducing agent 124 pm 115⬚ 3 NO Nitrosonium (nitrosyl cation) Oxidizing agent and Lewis acid; π-acceptor ligand – 122 pm 5 NO3− Nitrate 5 NO2+ Nitronium (nitryl cation) with the Au3 ions formed to produce [AuCl4] and thereby remove Au3 from the product side of the oxidation reaction: [AuCl4 ]−(aq) + Au(s) + NO3− (aq) + 4 Cl −(aq) + 4 H + (aq) → NO(g) + 2 H 2O(l) The anhydride of nitric acid is N2O5. It is a crystalline solid with the more accurate formula [NO2][NO2−] and can be prepared by dehydration of nitric acid with P4O10: 4 HNO3 (l) + P4O10 (s) → 2 N 2O5 (s) + 4 HPO3 (l) The solid sublimes at 320ºC and the gaseous molecules dissociate to give NO2 and O2. The compound is a strong oxidizing agent and can be used for the synthesis of anhydrous nitrates: N 2O5 (s) + Na(s) → NaNO3 (s) + NO2 (g) Very weak base; an oxidizing agent 115 pm Oxidizing agent, nitrating agent, Lewis acid (b) Nitrogen(IV) and nitrogen(III) oxides and oxoanions Key point: The intermediate oxidation states of nitrogen are often susceptible to disproportionation. Nitrogen(IV) oxide exists as an equilibrium mixture of the brown NO2 radical and its colourless dimer, N2O4 (dinitrogen tetroxide): N2O4(g) 2 NO2(g) K = 0.115 at 25°C This readiness to dissociate is consistent with the NN bond in N2O4 (18) being long and weak and arises because the molecular orbital occupied by the unpaired electron is spread almost equally over all three atoms in NO2 rather than being concentrated on the N atom. This structure is in contrast to the isoelectronic oxalate ion, C2O42, where the CC bond is stronger because in CO2 the electron is more concentrated on the C atom. EX AMPLE 15.3 Correlating trends in the stabilities of N(V) and Bi(V) O Compounds of N(V) and Bi(V) are stronger oxidizing agents than the 5 oxidation states of the three intervening elements. Correlate this observation with trends in the periodic table. Answer We need to consider some of the periodic trends discussed earlier in Chapter 9. The light p-block elements are more electronegative than the elements immediately below them in the periodic table; accordingly, these light elements are generally less easily oxidized themselves and are thus good oxidizing agents. Nitrogen is generally a good oxidizing agent in its positive oxidation states. Bismuth is much less electronegative, but favours the 3 oxidation state in preference to the 5 state on account of the inert-pair effect. Self-test 15.3 From trends in the periodic table, decide whether phosphorus or sulfur is likely to be the stronger oxidizing agent. N 175 pm 135° 121 pm 18 N2O4, D2h Nitrogen(IV) oxide is a poisonous oxidizing agent that is present in low concentrations in the atmosphere, especially in photochemical smog. In basic aqueous solution it disproportionates into N(III) and N(V), forming NO2− and NO3− ions: 2 NO2 (aq) + 2OH −(aq) → NO2− (aq) + NO3− (aq) + H 2O(l) 390 15 The Group 15 elements In acidic solution (as in the Ostwald process) the reaction product is N(II) in place of N(III) because nitrous acid itself readily disproportionates: 3 HNO2 (aq) → NO3− (aq) + 2 NO(g) + H3O+ (aq) E O = +0.05 V, K = 50 Nitrous acid, HNO2, is a strong oxidizing agent: HNO2 (aq) + H + (aq) + e− → NO(g) + H 2O(l) E O = +1.00 V and its reactions as an oxidizing agent are often more rapid than its disproportionation. The rate at which nitrous acid oxidizes is increased by acid as a result of its conversion to the nitrosonium ion, NO: HNO2(aq) H (aq) → H2NO (aq) → NO (aq) H2O(l) 2 The nitrosonium ion is a strong Lewis acid and forms complexes rapidly with anions and other Lewis bases. The resulting species may not themselves be susceptible to oxidation (as in the case of SO2− and F ions, which form [O3SONO] (19) and ONF (20), 4 respectively). Thus there is good experimental evidence that the reaction of HNO2 with I ions leads to the rapid formation of INO: INO(aq) I −(aq) + NO+ (aq) → followed by the rate-determining second-order reaction between two INO molecules: Nitrosonium salts containing poorly coordinating anions, such as [NO][BF4], are useful reagents in the laboratory as facile oxidizing agents and as a source of NO. Dinitrogen trioxide, N2O3, the anhydride of nitrous acid, is a blue solid that melts above 100ºC to give a blue liquid that dissociates to give NO and NO2: N 2O3 (l) → NO(g) + NO2 (g) The brown colour of NO2 means that the liquid becomes progressively more green as the dissociation proceeds. – O S N 113 pm O 19 O3SONO – (d) Low oxidation state nitrogenoxygen compounds Key points: Dinitrogen oxide is unreactive, for kinetic reasons; the isoelectronic azide ion, N3, forms many metal complexes. Dinitrogen oxide, N2O is a colourless, unreactive gas and is produced by the comproportionation of molten ammonium nitrate. Care must be taken to avoid an explosion in this reaction, in which the cation is oxidized by the anion: 250ºC NH 4 NO3 (l) ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → N 2O(g) + 2 H 2O(g) Standard potential data suggest that N2O should be a strong oxidizing agent in acidic and basic solutions: N 2O(g) + 2 H + (aq) + 2e− → N 2 (g) + H 2O(l) E O = +1.77 V at pH = 0 N 2O(g) + H 2O(l) + 2e− → N 2 (g) + 2OH − (aq) E O = +0.94 V at pH = 14 However, kinetic considerations are paramount, and the gas is unreactive towards many reagents at room temperature. I2 (aq) + 2 NO(g) 2 INO(aq) → N Because NO is endergonic, it should be possible to find a catalyst to convert the pollutant NO to the natural atmospheric gases N2 and O2 at its source in exhausts. It is known that Cu in a zeolite catalyses the decomposition of NO, and a reasonable understanding of the mechanism has been developed; however, this system is not yet sufficiently robust for a practical automobile exhaust catalytic converter. 152 pm 110° F 20 ONF (c) Nitrogen(II) oxide Key points: Nitric oxide is a strong π-acceptor ligand, and a troublesome pollutant in urban atmospheres; the molecule acts as a neurotransmitter. Nitrogen(II) oxide reacts with O2 to generate NO2, but in the gas phase the rate law is second-order in NO because a transient dimer, (NO)2, is produced that subsequently collides with an O2 molecule. Because the reaction is second-order, atmospheric NO (which is produced in low concentrations by coal-fired power plants and by internal combustion engines) is slow to convert to NO2. E X A MPL E 15 . 4 Comparing the redox properties of nitrogen oxoanions and oxo compounds Compare (a) NO3− and NO2− as oxidizing agents, (b) NO2, NO, and N2O with respect to their ease of oxidation in air, (c) N2H4 and H2NOH as reducing agents. Answer We need to refer to the Frost diagram for nitrogen, which is included in Fig. 15.6, and use the interpretation described in Section 5.13. (a) Both NO3− and NO2.− ions are strong oxidizing agents. The reactions of the former are often sluggish but are generally faster in acidic solution. The reactions of NO2− ions are generally faster and become even faster in acidic solution, where the NO is a common identifiable intermediate. (b) NO2 is stable with respect to oxidation in air. N2O and NO are thermodynamically susceptible to oxidation. However, the reaction of N2O with oxygen is slow, and at low NO concentrations the reaction between NO and O2 is slow because the rate law is second-order in NO. (c) Hydrazine and hydroxylamine are both good reducing agents. In basic solution hydrazine becomes a stronger reducing agent. Self-test 15.4 Summarize the reactions that are used for the synthesis of hydrazine and hydroxylamine. Are these reactions best described as electron-transfer processes or nucleophilic displacements? 15.14 Oxides of phosphorus, arsenic, antimony, and bismuth Key points: The oxides of phosphorus include P4O6 and P4O10, both of which are cage compounds with Td symmetry; on progressing from arsenic to bismuth, the 5 oxidation state is more readily reduced to 3. Phosphorus forms phosphorus(V) oxide, P4O10, and phosphorus(III) oxide, P4O6. It is also possible to isolate the The detail 391 Table 15.4 Latimer diagrams for phosphorus intermediate compositions having one, two, or three O atoms terminally attached to the P atoms. Both principal oxides can be hydrated to yield the corresponding acids, the P(V) oxide giving phosphoric acid, H3PO4, and the P(III) oxide giving phosphonic acid, H3PO3. As remarked in Section 4.5, phosphonic acid has one H atom attached directly to the P atom; it is therefore a diprotic acid and better represented as PHO(OH)2. In contrast to the high stability of phosphorus(V) oxide, arsenic, antimony, and bismuth more readily form oxides with oxidation number 3, specifically As2O3, Sb2O3, and Bi2O3. In the gas phase, the arsenic(III) and antimony(III) oxides have the molecular formula E4O6, with the same tetrahedral structure as P4O6. Arsenic, Sb, and Bi do form oxides with oxidation state 5, but Bi(V) oxide is unstable and has not been structurally characterized. This is another example of the consequences of the inert-pair effect. Note the coordination number of 6 for Sb and 4 for the lighter elements P and As, with their smaller atoms. cells. There, unlike P, it is reduced to an As(III) species, which is thought to be the actual toxic agent. This toxicity may stem from the affinity of As(III) for sulfur-containing amino acids. The enzyme arsenite oxidase, which contains a Mo cofactor, is produced by certain bacteria and is used to reduce the toxicity of As(III) by converting it to As(V). 15.15 Oxoanions of phosphorus, arsenic, 15.16 Condensed phosphates antimony, and bismuth Key points: Important oxoanions are the P(I) species hypophosphite, H2PO2, the P(III) species phosphite, HPO32, and the P(V) species phosphate, PO43. The existence of PH bonds and the highly reducing character of the two lower oxidation states is notable. Phosphorus(V) also forms an extensive series of O-bridged polyphosphates. In contrast to N(V), P(V) species are not strongly oxidizing. As(V) is more easily reduced than P(V). It can be seen from the Latimer diagram in Table 15.4 that elemental P and most of its compounds other than P(V) are strong reducing agents. White phosphorus disproportionates into phosphine, PH3 (oxidation number 3), and hypophosphite ions (oxidation number 1) in basic solution: P4 (s) + 3OH − (aq) + 3H 2O(l) → PH3 (g) + 3H 2 PO2− (aq) Table 15.5 lists some common P oxoanions (Box 15.4). The approximately tetrahedral environment of the P atom in their structures should be noted, as should the existence of PH bonds in the hypophosphite and phosphite anions. The synthesis of various P(III) oxoacids and oxoanions, including HPO32and alkoxophosphanes, is conveniently performed by solvolysis of phosphorus(III) chloride under mild conditions, such as in cold tetrachloromethane solution: PCl3 (l) + 3 H 2O(l) → H3 PO3 (sol) + 3 HCl(sol) PCl3 (l) + 3 ROH(sol) + 3 N(CH3 )3 (sol) → P(OR)3 (sol) + 3 [HN(CH3 )3 ]Cl(sol) Reductions with H2PO2− and HPO32− are usually fast. One of the commercial applications of this lability is the use of H2PO2− to reduce Ni2(aq) ions and so coat surfaces with metallic Ni in the process called ‘electrodeless plating’. Ni2+ (aq) + 2 H 2 PO2− (aq) + 2 H 2O ( l ) → Ni(s) + 2 H 2 PO3− (aq) + H 2 (g) + 2 H + (aq) The Frost diagram for the elements shown in Fig. 15.6 reveals similar trends in aqueous solution, with −oxidizing character following the order PO34− ≈ AsO34− < Sb (OH )6 ≈ Bi ( V ) . The thermodynamic tendency and kinetic ease of reducing AsO43− is thought to be key to its toxicity towards animals. Thus, As(V) as AsO43− readily mimics PO43−, and so may be incorporated into –0.93 Acidic solution +0.38 –0.50 –0.28 –1.12 Basic solution –0.51 –0.06 H3PO4 → H4P2O6 → H3PO3 → H3PO2 → P → PH3 –0.50 –1.57 –2.05 –0.89 PO43− → HPO23− → H2PO2− → P → PH3 –1.73 Key point: Dehydration of phosphoric acid leads to the formation of chain or ring structures that may be based on many PO4 units. When phosphoric acid, H3PO4, is heated above 200ºC, condensation occurs resulting in the formation of POP bridges between two neighbouring PO43− units. The extent of this condensation depends upon the temperature and duration of heating. H 4 P2O7 (l) + H 2O(g) 2 H3 PO4 (l) → H 5 P3O10 (l) + H 2O(g) H3 PO4 (l) + H 4 P2O7 (l) → The simplest condensed phosphate is thus H4P2O7. The most commercially important condensed phosphate is the sodium salt of the triacid, Na5P3O10 (21). It is widely used in detergents for laundry and dishwashers, and in other cleaning products and in water treatment (Box 15.5). Polyphosphates are also used in various ceramics and as food additives. Triphosphates such as adenosine triphosphate (ATP) are of vital importance in living organisms (Section 26.8). – O – – – P – 5– 21 P3O10 A range of condensed phosphates occurs with chain lengths ranging from those based on two PO4 units to polyphosphates having chain lengths of several thousand units. Di-, tri-, tetra-, and pentapolyphosphates have been isolated but higher members of the series always contain mixtures. However, the average chain length can be determined by the usual methods used in polymer analysis or by titration. Just as the three successive acidity constants for phosphoric acid differ, so do the acidity constants for the two types of OH group of the polyphosphoric acids. The terminal OH groups, of which there are two per molecule, are weakly acidic. The remaining OH groups, of which there is one per P atom, are strongly acidic because they are situated 392 15 The Group 15 elements Table 15.5 Some phosphorus oxoanions Oxidation number Formula Name Structure Comments – 1 H2PO − 2 Hypophosphite Facile reducing agent (dihydrodioxophosphate) 2– 3 HPO2− 3 Facile reducing agent Phosphite 4– 4 P2O64− Hypophosphate Basic 3– 5 PO 3− 4 Phosphate Strongly basic 4– 5 P2O74− Diphosphate Basic; longer chain B OX 15 . 4 Phosphates and the food industry Phosphorus in the form of phosphates is essential to life and phosphate fertilizers in forms such as bone, fish, and guano have been used since ancient times. The phosphate industry started in the mid-nineteenth century when sulfuric acid was used to decompose bones and phosphate minerals to make the phosphate more readily available. The development of more economical routes led to the diversification of the industrial applications of phosphoric acid and phosphate salts. More than 90 per cent of world production of phosphoric acid is used to make fertilizers but there are several other applications. One of the most important is the food industry. A dilute solution of phosphoric acid is nontoxic and has an acidic taste. It is used extensively in beverages to give a tart taste, as a buffering agent in jams and jellies, and as a purifying agent in sugar refining. The phosphates and hydrogenphosphates have many applications in the food industries. Sodium dihydrogenphosphate, NaH2PO4, is added to animal feeds as a dietary supplement. The disodium salt, Na2HPO4, is used as an emulsifier for processing cheese. It interacts with the protein casein and prevents separation of the fat and water. The potassium salts are more soluble and more expensive than the sodium salts. The dipotassium salt, K2HPO4, is used as an anti-coagulant in coffee creamer. It interacts with the protein and prevents coagulation by the coffee acids. Calcium dihydrogenphosphate monohydrate, Ca(H2PO4)2.H2O, is used as a raising agent in bread, cake mixes, and self-raising flour. Together with NaHCO3 it produces CO2 during the baking process but it also reacts with the protein in the flour to control the elasticity and viscosity of the dough or mixture. The largest use of calcium monohydrogenphosphate, CaHPO4.2 H2O, is as a dental polish in nonfluoride toothpaste. Calcium diphosphate, Ca2P2O7, is used in fluoride toothpaste. Calcium phosphate, Ca3PO4, is added to sugar and salt to improve their flow. The detail 393 B OX 15 . 5 Polyphosphates The most widely used polyphosphate is sodium tripolyphosphate, Na5P3O10. Its major use is as a ‘builder’ for synthetic detergents used in domestic laundry products, car shampoos, and industrial cleaners. Its role in these applications is to form stable complexes with the calcium and magnesium ions in hard water, effectively making them unavailable for precipitation, the process called ‘sequestration’. It also acts as a buffer and prevents the flocculation of dirt and the redeposition of soil particles. Food-grade sodium tripolyphosphate is used in the curing of hams and bacon. It interacts with the proteins and leads to good moisture retention during curing. It is also used to improve the quality of processed chicken and seafood products. The technical-grade product is used as a water softener, by sequestration as above, and in the paper pulping and textile industries, where it is used to help break down cellulose. opposite the strongly electron-withdrawing O groups. The ratio of weakly to strongly acidic protons gives an indication of average chain length. The long-chain polyphosphates are viscous liquids or glasses. E X A MPL E 15 . 5 Determining the chain length of a polyphosphoric acid by titration A sample of a polyphosphoric acid was dissolved in water and titrated with dilute NaOH(aq). Two stoichiometric points were observed at 16.8 and 28.0 cm3. Determine the chain length of the polyphosphate. Answer We need to determine the ratio of the two different types of OH group. The strongly acidic OH groups are titrated by the first 16.8 cm3. The two terminal OH groups are titrated by the remaining 28.0 16.8 cm3 = 11.2 cm3. Because the concentrations of analyte and titrant are such that each OH group requires 5.6 cm3 of the titrant (because 11.2 cm3 is used to titrate two such groups), we conclude that there are (16.8 cm3)/(5.6 cm3) = 3 strongly acidic OH groups per molecule. A molecule with two terminal OH groups and three further OH groups is a tripolyphosphate. Potassium tripolyphosphate is more soluble and more expensive than the sodium analogue and is used in liquid detergents. For some applications an effective trade-off between solubility and cost can be achieved by using sodium potassium tripolyphosphate, Na3K2P3O10. The use of polyphosphates in detergents has been implicated in excessive algae growth and eutrophication of some natural waters. This has led to restrictions in their use in many countries and the reduction in their use in home laundry products. However, phosphate is still widely used as an essential fertilizer and more of it enters rivers and lakes by run-off from farm land than from detergents. Many analogues of phosphorusoxygen compounds exist in which the O atom is replaced by the isolobal NR or NH group, such as P4(NR)6 (24), the analogue of P4O6 (Section 22.20c). Other compounds exist in which OH or OR groups are replaced by the isolobal NH2 or NR2 groups. An example is P(NMe2)3, the analogue of P(OMe)3. Another indication of the scope of PN chemistry, and a useful point to remember, is that PN is structurally equivalent to SiO. For example, various phosphazenes, which are chains and rings containing R2PN units (25), are analogous to the siloxanes (Chapter 14) and their R2SiO units (26). R P N CH3 P N Self-test 15.5 When titrated against base a sample of polyphosphate gave end points at 30.4 and 45.6 cm3. What is the chain length? If NaH2PO4 is heated and the water vapour allowed to escape, then the tricyclo anion P3O93 (22) is formed. If this reaction is carried out in a closed system, the product is ‘Maddrell’s salt’, a crystalline material that contains long chains of PO4 units. The tetracyclo anion (23) is formed when P4O10 is treated with cold aqueous solutions of NaOH or NaHCO3. O P O – 24 P4(NR)6 25 (Me2PN)3, Me = CH3 CH3 O Si – – – P – 26 (Me2SiO)3, Me = CH3 – – 22 P3O93– 4– 23 P4O12 15.17 Phosphazenes Key points: The range of PN compounds is extensive, and includes cyclic and polymeric phosphazenes, (PX2N)n; phosphazenes form highly flexible elastomers. The cyclic phosphazene dichlorides are good starting materials for the preparation of the more elaborate phosphazenes. They are easily synthesized: n PCl5 + n NH 4Cl → (Cl2 PN)n + 4n HCl n = 3 or 4 A chlorocarbon solvent and temperatures near 130ºC produce the cyclic trimer (27) and tetramer (28), and when the trimer is heated to about 290ºC it changes to polyphosphazene 394 15 The Group 15 elements B OX 15 .6 Biomedical applications of polyphosphazenes Biodegradable polymers are attractive biomedical materials because they survive for only a limited time in vivo. Polyphosphazenes are proving to be very useful in this respect as they degrade to harmless byproducts and their physical properties can be tuned by altering the substituents on the P atoms. They are used as bio-inert housing materials for implantation of devices, as structural materials for construction of heart valves and blood vessels, and as biodegradable supports for in vivo bone regeneration. The best polyphosphazenes for this last application form fibres in which the PN backbone consists of alkyoxy groups that form bonds to Ca2 ions. The polymer fibre becomes populated with the patient’s osteoblasts (Box 15.6). The Cl atoms in the trimer, tetramer, and polymer are readily displaced by other Lewis bases (bone-making cells). The polymer degrades as the osteoblasts multiply and fill the space between the fibres. Polyphosphazenes have been designed that hydrolyse at a specific rate and maintain their strength as the erosion process proceeds. Polyphosphazenes are also used as drug delivery systems. The bioactive molecule is trapped within the structure of the polymer or incorporated into the PN backbone and the drug is released when the polymer degrades. The rate of degradation can be controlled by altering the structure of the polymer backbone, thus giving control over the rate of drug delivery. Drugs that can be delivered in this way include cisplatin, dopamine, and steroids. cides. However, because of their high toxicity they no longer have major commercial applications. [(CF3CF2O)2 PN]n + 2nCl − (Cl2 PN)n + 2nCF3CF2O− → Like silicone rubber, the polyphosphazenes remain rubbery at low temperatures because, like the isoelectronic SiOSi group, the molecules are helical and the PNP groups are highly flexible. C6H5 As Cl P Cl CH3 N As P N 29 As(CH3)3 27 (Cl2PN)3 28 (Cl2PN)4 A phosphorusnitrogen compound that was particularly useful in the laboratory until its carcinogenic properties were recognized is the aprotic solvent hexamethylphosphoramide, ((CH3)2N)3PO, which is sometimes designated HMPA. The large bis(triphenylphosphine)iminium cation, [Ph3PNPPh3], which is commonly abbreviated as PPN, is very useful in forming salts of large anions. The salts of this cation are usually soluble in polar aprotic solvents such as HMPA, dimethylformamide, and even dichloromethane. ■ A brief illustration. To prepare [NP(OCH3)2]4 from PCl5, NH4Cl, and NaOCH the cyclic chlorophosphazene is synthesized first: 130 C 4PCl5 + 4NH4 Cl ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → (Cl2PN)4 + 16HCl Then, because Cl atoms are readily replaced by strong Lewis bases, such as alkoxides, the chlorophosphazene is used as follows: (Cl2PN)4 + 8NaOCH3 → [(CH3O)2PN]4 + 8NaCl ■ 15.18 Organometallic compounds of arsenic, antimony, and bismuth Oxidation states 3 and 5 are encountered in many of the organometallic compounds of arsenic, antimony, and bismuth. An example of a compound with an element in the 3 oxidation state is As(CH3)3 (29) and an example of the 5 state is As(C6H5)5 (30). Organoarsenic compounds were once widely used to treat bacterial infections and as herbicides and fungi- 30 As(C6H5)5 (a) Oxidation state 3 Key points: The stability of the organometallic compounds decreases in the order As > Sb > Bi; the aryl compounds are more stable than the alkyl compounds. Organometallic compounds of arsenic(III), antimony(III), and bismuth(III) can be prepared in an ether solvent by using a Grignard reagent, an organolithium compound, or an organohalide: AsCl3 (et) + 3RMgCl(et) → AsR3 (et) + 3MgCl2 (et) ∆ ⎯⎯ → AsRBr2 (et) + AsR 2 Br(et) 2 As(et) + 3RBr(et) ⎯Cu/ AsR 2 Br (et) + R'Li(et) → AsR 2 R'(et) + LiBr(et) The compounds are all readily oxidized but are stable to water. The MC bond strength decreases for a given R group in the order As > Sb > Bi. Consequently, the stability of the compounds decreases in the same order. In addition, the aryl compounds, such as (C6H5)3As, are generally more stable than the alkyl compounds. The halogen-substituted compounds RnMX3n have been prepared and characterized. All the compounds act as Lewis bases and form complexes with d-metal ions. The basicity decreases in the order As > Sb > Bi. Many complexes of alkyl- and arylarsanes have been prepared but fewer stibane complexes are known. A useful ligand, for example, is the bidentate compound known as diars (31). Because of their soft-donor character, many aryl- and alkylarsane complexes of the soft species Rh(I), Ir(I), Pd(II), and Pt(II) have been prepared. However, hardness criteria are only approximate, so we should not be surprised to see phosphine and arsane complexes of some metals in higher oxidation states. 395 The detail For example, the unusual 4 oxidation state of palladium is stabilized by the diars ligand (32). Me 2+ As As(CH3)2 C6H4 As H C Me 3 Cl Me Pd Me CH3 As(CH3)2 31 C6H4(As(CH3)2)2, diars 32 [PdCl2(diars)2]2+ The synthesis of diars provides a good illustration of some common reactions in the synthesis of organoarsenic compounds. The starting material is (CH3)2AsI. This compound is not conveniently prepared by metathesis reaction between AsI3 and a Grignard or similar carbanion reagent because that reaction is not selective to partial substitution on the As atom when the organic group is compact. Instead, the compound can be prepared by the direct action of a haloalkane, CH3I, on metallic arsenic: As As As As As As As As Me Me Me Me 34 (AsMe)n 33 As5(CH3)5 As well as forming single MC bonds, As, Sb, and Bi also form MC bonds. A well-studied group of compounds are the arylometals in which a metal atom forms part of a heterocyclic six-membered benzene-like ring (35). Arsabenzene, C5H5As, is stable up to 200ºC, stibabenzene, C5H5Sb, can be isolated but readily polymerizes, and bismabenzene, C5H5Bi, is very unstable. These compounds exhibit typical aromatic character although arsabenzene is 1000 times more reactive than benzene. A related group of compounds is arsole, stibole, and bismuthole, C4H5M, in which the metal atom forms part of a five-membered ring (36). 4 As(s) + 6CH3 I(l) → 3 (CH3 )2 AsI(sol) + AsI3 (sol) In the next step, the action of sodium on (CH3)2AsI is used to produce [(CH3)2As]: (CH3 )2 AsI(sol) + 2 Na(sol) → Na(CH3 )2 As + NaI(s) The resulting powerful nucleophile [(CH3)2As] is then used to displace chlorine from 1,2-dichlorobenzene: Cl As(CH 3 )2 Cl As(CH 3 )2 2 NaCl 2 Na[(CH3)2As] Polyarsane compounds, (RAs)n, can be prepared in ether by reduction of a pentavalent organometallic compound, R5As, or by treating an organohaloarsenic compound with Li: (RAs)n (et) + 2n LiX(et) n RAsX 2 (et) + 2n Li(et) → The compound R2AsAsR2 is very reactive because the AsAs bond is readily cleaved. It reacts with oxygen, sulfur, and species containing CC bonds, and forms complexes with d-metal species in which the AsAs bond may be cleaved or left intact: Me Me2 AsAsMe2 As Mn(CO)4 As Me M 36 C4H6M (b) Oxidation state 5 Key point: The tetraphenylarsonium ion is a starting material for the preparation of other As(V) organometallic compounds. The trialkylarsanes act as nucleophiles towards haloalkanes to produce tetraalkylarsonium salts, which contain As(V): As(CH3 )3 (sol) + CH3Br(sol) → [As(CH3 )4 ]Br(sol) This type of reaction cannot be used for the preparation of the tetraphenylarsonium ion, [AsPh4] because triphenylarsane is a much weaker nucleophile than trimethylarsane. Instead, a suitable synthetic reaction is: + Ph Ph3As=O Br− + MgO As + PhMgBr Ph Ph Ph Me (OC )4 Mn Mn2 (C O)10 M 35 C5H5M Me Polyarsanes of up to six units have been characterized. Polymethylarsane exists as a yellow, puckered cyclic pentamer (33) and as a purpleblack ladder-like structure (34). The strength of the MM bond decreases in the order As > Sb > Bi, so although arsenic forms catenated organometallic compounds, only R2BiBiR2 has been isolated. This reaction may look unfamiliar, but it is simply a metathesis in which the Ph anion replaces the formal O2 ion attached to the As atom, resulting in a compound in which the arsenic retains its 5 oxidation state. The formation of the highly exergonic compound MgO also contributes to the Gibbs energy of this reaction, and its formation drives the reaction forward. The tetraphenylarsonium, tetraalkylammonium, and tetraphenylphosphonium cations are used in synthetic inorganic chemistry as bulky cations to stabilize bulky anions. The tetraphenylarsonium ion is also a starting material for the preparation of other As(V) organometallic compounds. For instance, the 396 15 The Group 15 elements action of phenyllithium on a tetraphenylarsonium salt produces pentaphenylarsenic (30), a compound of As(V): AsPh5 (sol) + LiBr(s) [AsPh4 ]Br(sol) + LiPh(sol) → Pentaphenylarsenic, AsPh5, is trigonal bipyramidal, as expected from VSEPR considerations. We have seen (Section 2.3) that a square-pyramidal structure is often close in energy to the trigonalbipyramidal structure, and the antimony analogue, SbPh5, is in fact square pyramidal (37). A similar reaction under carefully controlled conditions yields the unstable compound As(CH3)5. Ph Sb 37 SbPh5, Ph = C6H5 FURTHER READING R.B. King, Inorganic chemistry of the main group elements. John Wiley & Sons (1994). J. Emsley, The shocking history of phosphorus: a biography of the devil’s element. Pan (2001). D.M.P. Mingos, Essential trends in inorganic chemistry. Oxford University Press (1998). A survey of inorganic chemistry from the perspective of structure and bonding. W.T. Frankenberger, The environmental chemistry of arsenic. Marcel Dekker (2001). R.B. King (ed.), Encyclopedia of inorganic chemistry. John Wiley & Sons (2005). H.R. Allcock, Chemistry and applications of polyphosphazenes. John Wiley & Sons (2002). G.J. Leigh, The world’s greatest fix: a history of nitrogen and agriculture. Oxford University Press (2004). N.N. Greenwood and A. Earnshaw, Chemistry of the elements. Butterworth-Heinemann (1997). EXERCISES 15.1 List the elements in Groups 15 and indicate the ones that are (a) diatomic gases, (b) nonmetals, (c) metalloids, (d) true metals. Indicate those elements that display the inert-pair effect. 15.9 Starting with NH3(g) and other reagents of your choice, give the chemical equations and conditions for the synthesis of (a) HNO3, (b) NO2 , (c) NH2OH, (d) N3. 15.2 (a) Give complete and balanced chemical equations for each step in the synthesis of H3PO4 from hydroxyapatite to yield (a) high-purity phosphoric acid and (b) fertilizer-grade phosphoric acid. (c) Account for the large difference in costs between these two methods. 15.10 Write the balanced chemical equation corresponding to the standard enthalpy of formation of P4O10(s). Specify the structure, physical state (s, l, or g), and allotrope of the reactants. Do either of the reactants differ from the usual practice of taking as reference state the most stable form of an element? 15.3 Ammonia can be prepared by (a) the hydrolysis of Li3N or (b) the high-temperature, high-pressure reduction of N2 by H2. Give balanced chemical equations for each method starting with N2, Li, and H2, as appropriate. (c) Account for the lower cost of the second method. 15.11 Without reference to the text, sketch the general form of the Frost diagrams for phosphorus (oxidation states 0 to 5) and bismuth (0 to 5) in acidic solution and discuss the relative stabilities of the 3 and 5 oxidation states of both elements. 15.4 Show with an equation why aqueous solutions of NH4NO3 are acidic. 15.5 Carbon monoxide is a good ligand and is toxic. Why is the isoelectronic N2 molecule not toxic? 15.6 Compare and contrast the formulas and stabilities of the oxidation states of the common nitrogen chlorides with the phosphorus chlorides. 15.7 Use the VSEPR model to predict the probable shapes of (a) PCl4, (b) PCl4 , (c) AsCl5. 15.8 Give balanced chemical equations for each of the following reactions: (a) oxidation of P4 with excess oxygen, (b) reaction of the product from part (a) with excess water, (c) reaction of the product from part (b) with a solution of CaCl2 and name the product. 15.12 Are reactions of NO2 as an oxidizing agent generally faster or slower when pH is lowered? Give a mechanistic explanation for the pH dependence of NO2 oxidations. 15.13 When equal volumes of nitric oxide (NO) and air are mixed at atmospheric pressure a rapid reaction occurs, to form NO2 and N2O4. However, nitric oxide from an automobile exhaust, which is present in the parts per million concentration range, reacts slowly with air. Give an explanation for this observation in terms of the rate law and the probable mechanism. 15.14 Give balanced chemical equations for the reactions of the following reagents with PCl5 and indicate the structures of the products: (a) water (1:1), (b) water in excess, (c) AlCl3, (d) NH4Cl. 15.15 Use data in Resource section 3 to calculate the standard potential of the reaction of H3PO2 with Cu2. Are HPO22 and H2PO22 useful as oxidizing or reducing agents? Problems 15.16 Identify the compounds A, B, C, and D. 15.18 Identify the nitrogen compounds A, B, C, D, and E. D E LiAlH4 As Cl2 A 397 Cl2/h␯ Cu/H+ B NO O2 A H2O B+C Zn/H+ 3 RMgBr C 15.17 Sketch the two possible geometric isomers of the octahedral [AsF4Cl2]⫺ and explain how they could be distinguished by 19 F-NMR. D F 15.19 Use the Latimer diagrams in Resource section 3 to determine which species of N and P disproportionate in acid conditions. PROBLEMS 15.1 Explain how you could use 31P-NMR to distinguish between PF3 and POF3. 15.2 The tetrahedral P4 molecule may be described in terms of localized 2c,2e bonds. Determine the number of skeletal valence electrons and from this decide whether P4 is closo, nido, or arachno (these terms are specified in Section 13.3). If it is not closo, determine the parent closo polyhedron from which the structure of P4 could be formally derived by the removal of one or more vertices. 15.3 Describe sewage treatment methods that result in a decrease in phosphate levels in wastewater. Outline a laboratory method that could be used to monitor phosphate levels in water. 15.4 On account of their slow reactions at electrodes, the potentials of most redox reactions of nitrogen compounds cannot be measured in an electrochemical cell. Instead, the values must be determined from other thermodynamic data. Illustrate such a calculation by using ∆ fG O (NH3 , aq) = −26.5 kJ mol −1 to calculate the standard potential of the N2/NH3 couple in basic aqueous solution. 15.5 A compound containing pentacoordinate nitrogen has been characterized (A. Frohmann, J. Riede, and H. Schmidbaur, Nature, 1990, 345, 140). Describe (a) the synthesis, (b) the structure of the compound, (c) the bonding. 15.6 Two articles (A. Lykknes and L. Kvittingen, ‘Arsenic: not so evil after all?’, J. Chem. Educ., 2003, 80, 497 and J. Wang and C.M. Chien, ‘Arsenic in drinking water—a global environmental problem’, J. Chem. Educ., 2004, 81, 207) present opposing perspectives on the toxic nature of arsenic. Use these references to produce a critical assessment of the beneficial and detrimental effects of arsenic. 15.7 A paper published by N. Tokitoh et al. (Science, 1997, 277, 78) gives an account of the synthesis and characterization of a stable bismuthene, containing Bi=Bi double bonds. Give the equations for the synthesis of the compound. Name and sketch the structure of the steric protecting group that was used. Why was the isolation of the product simple? What methods were used to determine the structure of the compound? 15.8 A paper published by Y. Zhang et al. (Inorg. Chem., 2006, 45, 10446) describes the synthesis of phosphazene cations as precursors for polyphosphazenes. Polyphosphazenes are prepared by ring opening polymerization of the cyclic (NPCl2)3 and the reaction is initiated by phosphazene cations. Discuss which Lewis acids were used to produce the cations and give the reaction scheme for the ring opening polymerization of (NPCl2)3. 15.9 In their paper ‘Catalytic reduction of dinitrogen to ammonia at single molybdenum center’ (Science, 2003, 301, 5629) D. Yandulov and R. Schrock describe the catalytic conversion of nitrogen to ammonia at room temperature and atmospheric pressure. Discuss why this development could be important commercially. Review nonbiological methods of nitrogen activation. 16 The Group 16 elements Part A: The essentials 18 H 16.1 The elements 16.2 Simple compunds 16.3 Ring and cluster compounds Part B: The detail 16.4 Oxygen 16.5 Reactivity of oxygen 16.6 Sulfur 16.7 Selenium, tellurium, and polonium 16.8 Hydrides 16.9 Halides 16.10 Metal oxides 16.11 Metal sulfides, selenides, tellurides, and polonides 16.12 Oxides 16.13 Oxoacids of sulfur 16.14 Polyanions of sulfur, selenium, and tellurium He 1 2 Li Be 13 14 15 16 17 B C N O F Ne P S Cl As Se Br Group 16 contains two of the most important elements for life. Oxygen is most commonly found as water and in the atmosphere, both of which are essential for almost all life forms. Dioxygen is produced from water by photosynthesis and recycled in respiration by higher organisms. Sulfur is also essential to all life forms and even selenium is required in trace amounts. Sulfur and selenium exhibit a tendency to catenation and form rings and chains. The Group 16 elements oxygen, sulfur, selenium, tellurium, and polonium are often called the chalcogens. The name derives from the Greek Bi Po At word for bronze, and refers to the association of sulfur and its congeners with copper. As in the rest of the p block, the element at the head of the group, oxygen, differs significantly from the other members of the group. The coordination numbers of its compounds are generally lower, and it is the only member of the group to exist as diatomic molecules under normal conditions. Sb Te I 16.15 Polycations of sulfur, selenium, and tellurium 16.16 Sulfur⫺nitrogen compounds FURTHER READING EXERCISES PROBLEMS PART A: THE ESSENTIALS All the members of Group 16 other than oxygen are solids under normal conditions and, as we have seen previously, metallic character generally increases down the group. In this section we discuss the essential features of the chemistry of the Group 16 elements. 16.1 The elements Key points: Oxygen is the most electronegative element in Group 16 and is the only gas; all the elements occur in several allotropic forms. Oxygen, sulfur, and selenium are nonmetals, tellurium is a metalloid, and polonium is a metal. Allotropy and polymorphism are important features of Group 16 and sulfur occurs in more natural allotropes and polymorphs than any other element. The group electron configuration of ns2np4 suggests a group maximum oxidation number of ⫹6 (Table 16.1). Oxygen never achieves this maximum oxidation state, although the other elements do in some circumstances. The electron configuration also suggests that stability may be achieved with an oxidation number of ⫺2, which is overwhelmingly common for O. The most remarkable feature of S is that it forms stable compounds with oxidation numbers between ⫺2 and ⫹6. In addition to its distinctive physical properties, O is significantly different chemically from the other members of the group (Section 9.8). It is the second most electronegative element in the periodic table and significantly more electronegative than its congeners. The essentials 399 Table 16.1 Selected properties of the elements Covalent radius/pm O S Se Te Po 74 104 117 137 140 140 184 198 221 First ionization energy/(kJ mol ) 1310 1000 941 870 812 Melting point/⬚C ⫺218 113 (␣) 217 450 254 Boiling point/⬚C ⫺183 445 685 990 960 Ionic radius/pm ⫺1 Pauling electronegativity ⫺1 Electron affinity/(kJ mol ) 3.4 2.6 2.6 2.1 2.0 141 200 195 190 183 ⫺844 ⫺532 The first value is for X(g) ⫹ e⫺(g) → X⫺(g), the second value is for X⫺(g) ⫹ e⫺(g) → X2⫺(g). This high electronegativity has an enormous influence on the chemical properties of the element. The small atomic radius of O and absence of accessible d orbitals also contribute to its distinctive chemical character. Thus, O seldom has a coordination number greater than 3 in simple molecular compounds, but its heavier congeners frequently reach coordination numbers of 5 and 6, as in SF6. Dioxygen (O2) oxidizes many elements and reacts with many organic and inorganic compounds under suitable conditions. Only the noble gases He, Ne, and Ar do not form oxides directly. The oxides of the elements are discussed in each relevant chapter and will not be revisited here. Even though the O⫽O bond energy of ⫹494 kJ mol⫺1 is high, many exothermic combustion reactions occur because the resulting E⫺O covalent bond enthalpies or MOn lattice enthalpies are also high. One of the most important reactions of dioxygen is the coordination to the oxygen-transport protein haemoglobin (Section 27.7b). Oxygen is the most abundant element in the Earth’s crust at 46 per cent by mass and is present in all silicate minerals. It comprises 86 per cent by mass of the oceans and 89 per cent of water. The average human is two-thirds oxygen by mass. Dioxygen, which is derived completely from the water-splitting action of photosynthetic organisms, makes up 21 per cent by mass of the atmosphere (Box 16.1). Oxygen is also the third most abundant element in the Sun, and the most abundant element on the surface of the Moon (46 per cent by mass). Oxygen also occurs as ozone, O3, a highly reactive pungent B OX 16 .1 The oxygen atmosphere In the course of the evolution of the Earth’s atmosphere, the proliferation of oxygen-evolving photosynthesis eventually resulted in the presence of O2 in the atmosphere at the present level of 21 per cent by volume. Photosynthesis also produced organic carbon compounds, which became the materials of cellular biomass (Box 14.3). Oxygen was a toxic constituent of the atmosphere of the early Earth and led to the extinction of many species. Some species retreated to habitats deeper in the soil or waters, where anaerobic conditions remained and where their descendents remain today. Other organisms adapted differently and evolved to exploit this now abundant and powerful oxidant. These organisms are the aerobes, among which were our ancestors. The shift from an anaerobic atmosphere to an oxygenic atmosphere had a profound effect on the composition of the waters. Sulfur, which was present largely in the form of sulfide in anaerobic waters, was oxidized to sulfate. Metal ion concentrations also changed dramatically. Two of the metals most profoundly affected were molybdenum and iron. In Earth’s modern oceans, molybdenum is the most abundant d metal (at 0.01 ppm). However, before the oxygenation of the oceans and atmosphere, molybdenum was present as insoluble solids, mainly MoO2 and MoS2. Oxidation of these solids produced the molybdate ion: 2MoS2 (s) + 7O2 (g) + 2 H2O(l) → 2 MoO24− (aq) + 4SO2 (g) + 4H+ (aq) The highly soluble molybdate ion became available to aquatic organisms and is now transported into cells by methods that differ dramatically from those used to acquire the more widespread cationic d-metal species in the marine environment. In addition, competition between molybdate and sulfate, both tetrahedral dinegative ions, remains a challenge for modern organisms. Iron suffered the opposite fate to molybdenum. In the ancient oceans, the element was present as Fe(II). Iron(II) hydroxide and sulfide are essentially soluble and so iron would have been readily available to aquatic organisms. However, on oxygenation of the atmosphere, the oxidation of Fe2⫹ to Fe3⫹ led to the precipitation of iron(III) hydroxides and oxides. Massive banded formations containing magnetite (Fe3O4) and hematite (Fe2O3) in Canada and Australia are testimony to the precipitation of the iron from the oceans between 2 and 3 Ga ago. 400 16 The Group 16 elements gas that is crucial to the protection of life on Earth, for it shields the surface from solar ultraviolet radiation. Sulfur occurs as deposits of the native element, in meteorites, volcanoes, and hot springs, as the ores galena, PbS, and barite, BaSO4, and as Epsom salts, MgSO4.7 H2O. It also occurs as H2S in natural gas and as organosulfur compounds in crude oil. That sulfur can exist in a large number of allotropic forms can be explained by the ability of S atoms to catenate because of the high S⫺S bond energy of 265 kJ mol⫺1, which is exceeded only by C⫺C (330 kJ mol⫺1) and H⫺H (436 kJ mol⫺1). All the crystalline forms of sulfur that can be isolated at room temperature consist of Sn rings. The striking difference between O⫺O and S⫺S single bond energies has important consequences. The O⫺O bond enthalpy is 146 kJ mol⫺1 and peroxides are powerful oxidizing agents: in contrast, the S⫺S bond enthalpy of 265 kJ mol⫺1 is so high that it is used in biology to stabilize protein structure by forming permanent linkages (RS⫺SR) between cysteine residues on different protein strands and different regions of one strand. The chemically soft elements Se and Te occur in metal sulfide ores and their principal source is the electrolytic refining of copper. Polonium occurs in 27 known isotopes and all are radioactive. 16.2 Simple compounds Key point: The elements of Group 16 form simple binary compounds with hydrogen, halogens, oxygen, and metals. The most important hydride of any element is that of oxygen, namely water. The properties and reactions of water and of reactions in water are of paramount importance to inorganic chemists and are discussed throughout this text. Water is the only Group 16 hydride that is not a poisonous, malodorous gas. Its melting and boiling points (0⬚C and 100⬚C, respectively) are both very high compared to compounds of similar molecular mass and the analogous molecules in Group 16 (Table 16.2). This high boiling point is due to extensive hydrogen bonding between H and the highly electronegative oxygen, O⫺H ... O (Section 10.6). Oxygen also forms hydrogen peroxide, H2O2, which is also a liquid (from 0⬚C to 150⬚C) on account of extensive hydrogen bonding. Oxygen forms oxides with most metals, and peroxides and superoxides with Group 1 and 2 metals. When the oxidation number of the metal is lower than ⫹4 the oxide is commonly ionic. When the oxidation number of the metal is ⫹4 or greater the oxide is molecular. Sulfur forms sulfides, S2⫺, and disulfides, S22⫺, with metals. Selenium and tellurium form selenides and tellurides, Se2⫺ and Te2⫺. Sulfur, Se, Te, and Po have a rich halogen chemistry, and some of the most common halides are summarized in Table 16.3. Sulfur forms very unstable iodides, but the iodides of Te and Po are more robust, which is an example of a large anion stabilizing a large cation (Section 3.15a). Of the halogens, only F brings out the maximum group oxidation state of the chalcogen elements, but the low oxidation state fluorides of Se, Te, and Po are unstable with respect to disproportionation into the element and a higher oxidation Table 16.2 Selected properties of the Group 16 hydrides Property H2O H2S H2Se H2Te H2Po Melting point/⬚C 0.0 ⫺85.6 ⫺65.7 ⫺51 ⫺36 37 Boiling point/⬚C 100.0 ⫺60.3 ⫺41.3 ⫺4 ∆ fH O / (kJ mol−1 ) ⫺285.6 (l) ⫺20.1 ⫹73.0 ⫹99.6 Bond length/pm 96 134 146 169 Bond angle/⬚ 104.5 92.1 91 90 14.00 6.89 3.89 2.64 14.15 11 10.80 Acidity constants pKa1 pKa2 The essentials Table 16.3 Some halides of sulfur, selenium, and tellurium Oxidation number Formula 21 eTX2X (I,rB = ⫹1 S2F2 ⫹2 ⫹4 ) Structure Remarks Halide bridges Silver-grey Two isomers: S2Cl2 Reactive SCl2 Reactive SF4 Gas SeX4 (X ⫽ F, Cl, Br) SeF4 liquid TeX4 (X ⫽ F, Cl, Br l) TeF4 solid eT I ⫹5 S2F4 Se2F10 Reactive 1 Te2I S 143 pm O ⫹6 SF6, SeF6 TeF6 120° Colourless gases 2 SO2, C2v Liquid (b.p. 36⬚C) S 2 4 1p m O state fluoride. A series of catenated subhalides exist for the heavy members of the group. For example, Te2I and Te2Br consist of ribbons of edge-shared Te hexagons with halogen bridges (1). The molecules of the two common oxides of S, SO2 (b.p. ⫺10⬚C) and SO3 (b.p. 44.8⬚C), are angular (2) and trigonal planar (3), respectively, in the gas phase. In the solid, sulfur trioxide exists as cyclic trimers (4). Sulfur dioxide is a poisonous gas with a sharp, choking odour. The major use of SO2 is in the manufacture of sulfuric acid, where it is first oxidized to SO3. It is also used as a bleach, disinfectant, and food preservative. Sulfur trioxide, SO3, is made on a huge scale by catalytic oxidation of SO2. It is seldom isolated but immediately converted to sulfuric acid, H2SO4. Because sulfur trioxide is extremely corrosive, anhydrous SO3 is seldom handled in the laboratory. It is available as oleum, or fuming sulfuric acid, H2S2O7, which is a solution of 25⫺65 per cent SO3 by mass in concentrated sulfuric acid. Sulfur trioxide reacts with water to give H2SO4 in a vigorous °1 0 2 3 SO3, D3h O S 4 (SO3)n, C3v 401 402 16 The Group 16 elements and very exothermic reaction. The reaction with metal oxides to produce sulfates is used to scrub undesirable SO3 from effluent gases from industrial processes. Selenium, Te, and Po all form a dioxide and trioxide. Sulfuric acid, H2SO4, is a dense viscous liquid. It is a strong acid (for the first deprotonation step), is a useful nonaqueous solvent, and exhibits extensive autoprotolysis (Section 4.1). Concentrated sulfuric acid extracts water from organic matter to leave a charred, carbonaceous residue. Sulfuric acid forms two series of salts, the sulfates, SO42⫺, and the hydrogensulfates, HSO4⫺. Sulfurous acid, H2SO3, has never been isolated. Aqueous solutions of SO2, which are referred to as ‘sulfurous acid’, are better considered as hydrates, SO2.nH2O. Two series of salts, the sulfites, SO32⫺, and the hydrogensulfites, HSO3⫺, are known and are moderately strong reducing agents, becoming oxidized to sulfates, SO42⫺, or dithionates, S2O62⫺. O S 5 Tetrathionate ion, S4O62– 2– 16.3 Ring and cluster compounds Key point: Ring and chain compounds of Group 16 elements are anionic or cationic. Neutral heteroatomic ring and chain compounds also are formed with other p-block elements. O S 6 Pentathionate ion, S5O62– 2– 7 S32–, C2v Sulfur forms many polythionic acids, H2SnO6, with up to six S atoms, such as the tetrathionate, S4O62⫺ (5), and pentathionate, S5O62⫺ (6), ions. Many polysulfides of electropositive elements have been characterized. They all contain the Sn2⫺ ions, where n ⫽ 2⫺6, as in (7). The smaller polyselenides and polytellurides resemble the polysulfides. The structures of the larger ones are more complex and depend to some extent on the nature of the cation. The polyselenides up to Se92⫺ are chains but larger molecules 2⫺ , which has a Se atom at the centre of two six-membered form rings such as Se11 rings in a square-planar arrangement (8). The polytellurides may be bicyclic, as in Te72⫺ (9). Many cationic chain, ring, and cluster compounds of the p-block elements have been prepared. The majority of them contain S, Se, or Te. The square-planar ions E42⫹ (E ⫽ S, Se, Te; 10) are stabilized by the occupation of molecular orbitals. Each E atom has six valence electrons, giving 24 ⫺ 2 ⫽ 22 electrons in all. There are two lone pairs on each E atom, which leaves six electrons to occupy the available molecular orbitals. Of those orbitals, one is bonding, two are nonbonding, and one is antibonding. The electrons occupy the first three, leaving the antibonding orbital unoccupied. Neutral heteroatomic ring and cluster compounds of the p-block elements include the cyclic tetrasulfurtetranitride, S4N4 (11) which decomposes explosively. Disulfurdinitride, S2N2 (12) is even less stable but polymerizes to form a superconducting polymer, (SN)n, that is stable up to 240°C. 2– 2– e S Te 8 Se112– 9 Te72– 2+ N Se S S m p 6 1 ° .6 9 8 .4° 0 9 N 10 Se42+ 11 S4N4 12 S2N2 The detail 403 PART B: THE DETAIL 2σu In this section we discuss the detailed chemistry of the Group 16 elements and observe the rich variation in the structures of the compounds they form. 1πg 2p 16.4 Oxygen Dioxygen is a biogenic gas (that is, one that has been produced by the action of organisms) and most of it is the result of photosynthesis, although some is produced by the action of ultraviolet radiation on water. It is colourless, odourless, and soluble in water to the extent of 3.08 cm3 per 100 cm3 water at 25⬚C and atmospheric pressure. This solubility falls to below 2.0 cm3 in seawater but is still sufficient to support aerobic marine life. The solubility of O2 in organic solvents is approximately ten times greater than in water. The high solubility of O2 makes it necessary to purge all solvents used in the synthesis of oxygensensitive compounds. Oxygen is readily available as O2 from the atmosphere and is obtained on a massive scale by the liquefaction and distillation of liquid air. The main commercial motivation is to recover O2 for steel making, in which it reacts exothermically with coke (carbon) to produce carbon monoxide. The high temperature is necessary to achieve a fast reduction of iron oxides by CO and carbon (Section 5.16). Pure oxygen, rather than air, is advantageous in this process because energy is not wasted in heating the nitrogen. About 1 tonne (1 t ⫽ 103 kg) of oxygen is needed to make 1 tonne of steel. Oxygen is also required by industry in the production of the white pigment TiO2 by the chloride process: Energy Key points: Oxygen has two allotropes, dioxygen and ozone. Dioxygen has a triplet ground state and oxidizes hydrocarbons by a radical chain mechanism. Reaction with an excited state molecule can produce a fairly long-lived singlet state that can react as an electrophile. Ozone is an unstable and highly aggressive oxidizing agent. 1πu 2σg 1σu 2s 2s 1σg Figure 16.1 The molecular orbital diagram for O2. when it is appropriate to specify the spin state.1 The singlet state 1∑⫺g , with paired electrons in the same two π orbitals as in the ground state, is higher in energy by 1.61 eV (155 kJ mol⫺1), and another singlet state 1∆g (‘singlet delta’), with electrons paired in one π orbital, lies between these two terms at 0.95 eV (91.7 kJ mol⫺1) above the ground state. Of the two singlet states, the latter has much the longer excited state lifetime (its return to the ground state is spin-forbidden) and O2(1∆g) survives long enough to participate in chemical reactions. When it is needed for reactions, O2(1∆g) can be generated in solution by energy transfer from a photoexcited molecule. Thus [Ru(bpy)3]2⫹ can be excited by absorption of blue light (452 nm) to give an electronically excited state, denoted [Ru(bpy)3]2⫹ (Section 20.7), and this state transfers energy to O2(3∑⫺g ): TiCl4 (l) + O2 (g) → TiO2 (s) + 2Cl 2 (g) Oxygen is used in many oxidation processes, for example the production of oxirane (ethylene oxide) from ethene. Oxygen is also supplied on a large scale for sewage treatment, renewal of polluted waterways, paper-pulp bleaching, and as an artificial atmosphere in medical and submarine applications. Liquid oxygen is very pale blue and boils at ⫺183⬚C. Its colour arises from electronic transitions involving pairs of neighbouring molecules: one photon from the red⫺yellow⫺green region of the visible spectrum can raise two O2 molecules to an excited state to form a molecular pair. Under high pressure the colour of solid oxygen changes from light blue to orange and then to red at approximately 10 GPa. The molecular orbital description of O2 implies the existence of a double bond; however, as we saw in Section 2.8, the outermost two electrons occupy different antibonding π orbitals with parallel spins; as a result, the molecule is paramagnetic (Fig. 16.1). The term symbol for the ground state is 3∑⫺g , and henceforth the molecule will be denoted O2(3∑⫺g ) 2p 冤Ru (bpy )3 冥2+ + O2 ( ∑ ) → 冤Ru (bpy ) 冥 3 − g 3 2+ O2 (∆) 1 g Another efficient way to generate O2(1∆g) is through the thermal decomposition of an ozonide: RO RO P RO + O3 –78°C O O RO P O RO OR RO RO P O RO + O 2( 1∆ g) In contrast to the radical character of many O2(3∑⫺g ) reactions, O2(1∆g) reacts as an electrophile. This mode of reaction is feasible because O2(1∆g) has an empty π orbital, rather than two that are each occupied by a single electron. For example, O2(1∆g) The symbols ∑ , ⌸ , and ⌬ are used for linear molecules such as dioxygen in place of the symbols S, P, and D used for atoms. The Greek letters represent the magnitude of the total orbital angular momentum around the internuclear axis. 1 404 16 The Group 16 elements adds across a diene, thus mimicking the Diels⫺Alder reaction of butadiene with an electrophilic alkene: O O2 (1⌬ g) O Singlet oxygen is implicated as one of the biologically hazardous products of photochemical smog. The other allotrope of oxygen, ozone, O3, boils at ⫺112⬚C and is an explosive and highly reactive endoergic blue gas (⌬fG O ⫽ ⫹163 kJ mol⫺1). It decomposes into dioxygen 2O3 (g) → 3O2 (g) but this reaction is slow in the absence of a catalyst or ultraviolet radiation. Ozone has a pungent odour; this property is reflected in its name, which is derived from the Greek ozein, to smell. The O3 molecule is angular, in accord with the VSEPR model (13) and has bond angle 117⬚; it is diamagnetic. Gaseous ozone is blue, liquid ozone is blue⫺black, and solid ozone is violet⫺black. Ozone is produced from electrical discharges or ultraviolet radiation acting on O2. This second method is used to produce low concentrations of ozone for the preservation of foodstuffs. The ability of O3 to absorb strongly in the 220⫺290 nm region of the spectrum is vital in preventing the harmful ultraviolet rays of the Sun from reaching the Earth’s surface (Box 17.3). Ozone reacts with unsaturated polymers, causing undesirable cross-linking and degradation. .. O .. .. .. O .. .. .. O .. .. O .. O .. O .. 13 O3, C2v Reactions of ozone typically involve oxidation and transfer of an O atom. Ozone is very unstable in acidic solution and much more stable in basic conditions: O3 (g) + 2 H + (aq) + 2e− → O2 (g) + H 2O(l) E O = +2.08 V − − O3 (g) + H 2O(l) + 2e → O2 (g) + 2OH (aq) E O = +1.25 V Ozone is exceeded in oxidizing power only by F2, atomic O, the OH radical, and perxenate ions (Section 18.5). Ozone forms ozonides with Group 1 and 2 elements (Sections 11.8 and 12.8). They are prepared by passing gaseous ozone over the powdered hydroxide, MOH or M(OH)2, at temperatures below ⫺10⬚C. The ozonides are red⫺brown solids that decompose on warming: MO3 (s) → MO2 (s) + 12 O2 (g) The ozonide ion, O3⫺, is angular, like O3, but with the slightly larger bond angle of 119.5⬚. 16.5 Reactivity of oxygen Key point: The reactions of dioxygen are often thermodynamically favourable but sluggish. Oxygen is by no means an inert molecule, yet many of its reactions are sluggish (a point first made in connection with overpotentials in Section 5.18). For example, a solution of Fe2⫹ is only slowly oxidized by air even though the reaction is thermodynamically favourable. Several factors contribute to the appreciable activation energy of many reactions of O2. One factor is that, with weak reducing agents, single-electron transfer to O2 is mildly unfavourable thermodynamically: O2 (g) + H + (aq) + e− → HO2 (g) O2 (g) + e− → O2− (aq) E O = −0.13 V at pH = 0 E O = −0.33 V at pH = 14 A single-electron reducing reagent must exceed these potentials for the reaction to be thermodynamically viable and must exceed them to achieve a significant rate. Second, the ground state of O2, with both π orbitals singly occupied, is neither an effective Lewis acid nor an effective Lewis base, and therefore has little tendency to undergo displacement reactions with p-block Lewis bases or acids. Finally, the high bond energy of O2 (497 kJ mol⫺1) results in a high activation energy for reactions that depend on its dissociation. Radical chain mechanisms can provide reaction paths that circumvent some of these activation barriers in combustion processes at elevated temperatures, and radical oxidations also occur in solution. In metalloenzymes (Section 27.10) O2 coordinates to metals such as Fe and Cu, and the enzyme catalyses the four-electron reduction of O2 to water. 16.6 Sulfur Key points: Sulfur is extracted as the element from underground deposits. It has many allotropic and polymorphic forms, including a metastable polymer, but its most stable form is the cyclic S8 molecule. Sulfur can be extracted from deposits of the element by the Frasch process, in which underground deposits are forced to the surface using superheated water and steam, and compressed air. The extracted S is molten and is allowed to cool in large basins. The process is energy intensive and commercial success depends on access to cheap water and energy. Extraction from natural gas and crude oil by the Claus process has become increasingly important. In this process, H2S is first oxidized in air at 1000⫺1400⬚C. This step produces some SO2 that then reacts with the remaining H2S at 200⫺350⬚C over a catalyst: 2 H 2 S (g) + SO2 (g) → 3S(l) + 2 H 2O(l) Unlike O, S (and all the heavier members of the group) tends to form single bonds with itself rather than double bonds because of the poor π overlap of its orbitals, which are held apart by the bulky atomic cores of neighbouring atoms. As a result, it aggregates into larger molecules or extended structures and hence is a solid at room temperature. Sulfur vapour, which is formed at high temperatures, consists partially of paramagnetic disulfur molecules, S2, that resemble O2 in having a triplet ground state and a formal double bond. The common yellow orthorhombic polymorph, ␣-S8, consists of crown-like eight-membered rings (14) and all other forms of S eventually revert to this form. Orthorhombic ␣-sulfur is an electrical and thermal insulator. When it is heated to 93⬚C, the packing of the S8 rings is modified and monoclinic ␤-S8 forms. When molten sulfur that has been heated above 150⬚C is cooled slowly, monoclinic ␥-sulfur is formed. This polymorph consists of S8 rings like the ␣ and ␤ forms but the packing of the rings is more efficient, resulting in a higher density. The detail A note on good practice The various molecular entities that S forms are allotropes of the element. The various crystalline forms that these entities exist as are polymorphs. S 14 S8 Table 16.4 Properties of selected sulfur allotropes and polymorphs Allotrope Melting point/C Appearance S3 Gas Cherry red S6 50d Orange red S7 39d Yellow -S8 113 Yellow -S8 119 Yellow -S8 107 Pale yellow S10 0d Yellow green S12 148 Pale yellow S18 128 Lemon yellow S20 124 Pale yellow S∞ 104 Yellow S 15 Sn Most of the sulfur produced is used to manufacture sulfuric acid, H2SO4, which is one of the most important manufactured chemicals. Sulfuric acid has many uses, including the synthesis of fertilizers and in dilute aqueous solution as the electrolyte in leadacid batteries (Box 14.6). Sulfur is a component of gunpowder (a mixture of potassium nitrate, KNO3, carbon, and sulfur). It is also used in the vulcanization of natural rubber. 16.7 Selenium, tellurium, and polonium It is possible to synthesize and crystallize sulfur rings with from six to 20 S atoms (Table 16.4). An additional complexity is that some of these allotropes exist in several crystalline forms. For example, S7 is known in four crystalline forms and S18 is known in two. Orthorhombic sulfur melts at 113C; the yellow liquid darkens above 160C and becomes more viscous as the sulfur rings break open and polymerize. The resulting helical Sn polymers (15) can be drawn from the melt and quenched to form metastable rubber-like materials that slowly revert to -S8 at room temperature. In the gas phase, S2 and S3 are observed. S3 is a cherry red, angular molecule like ozone. The more stable species is the violet S2 molecule that, like O2, is doubly bonded with a bond dissociation energy of 421 kJ mol1. Sulfur reacts directly with many elements at room or elevated temperatures. It ignites in F2 to form SF6, reacts rapidly with Cl2 to form S2Cl2, and dissolves in Br2 to give S2Br2, which readily dissociates. It does not react with liquid I2, which can therefore be used as a low-temperature solvent for sulfur. Atomic sulfur, S, is extremely reactive, and triplet and singlet states are possible with different reactivities, as with O. d, decomposes. 405 Key points: Selenium and tellurium crystallize in helical chains; polonium crystallizes in a primitive cubic form. Selenium can be extracted from the waste sludge from sulfuric acid plants. Selenium and tellurium can be extracted from copper sulfide ores, where they occur as the copper selenide or telluride. The extraction method depends on the other compounds or elements present. The first step usually involves oxidation in the presence of sodium carbonate: Cu2 Se(aq) + Na2CO3 (aq) + 2O2 (g) → 2CuO(s) + Na2 SeO3 (aq) + CO2 (gg) The solution containing Na2SeO3 and Na2TeO3 is acidified with sulfuric acid. The Te precipitates out as the dioxide, leaving selenous acid, H2SeO3, in solution. Selenium is recovered by treatment with SO2: H 2 SeO3 (aq) + 2 SO2 (g) + H 2O(l) → Se(s) + 2 H 2 SO4 (aq) Tellurium is liberated by dissolving the TeO2 in aqueous sodium hydroxide followed by electrolytic reduction: TeO2 (s) + 2 NaOH(aq) → Na2 TeO3 (aq) + H 2O(l) → Te(ss) + 2 NaOH(aq) + O2 (g) As with S, three polymorphs of Se exist that contain Se8 rings and differ only in the packing of the rings to give , , and forms of red selenium. The most stable form at room temperature is metallic grey selenium, a crystalline material composed of helical chains. The common commercial form of the element is amorphous black selenium; it has a very complex structure comprising rings containing up to 1000 Se atoms. Another amorphous form of Se, obtained by deposition of the vapour, is used as the photoreceptor in the xerographic photocopying process. Selenium is an essential element for humans, but, as with many essential elements, there is only a narrow range of concentration between the minimum daily requirement and toxicity. An early indication of Se poisoning is a garlicky smell on the breath, which is due to methylated selenium. Selenium exhibits both photovoltaic character, where light is converted directly into electricity, and photoconductive character. The photoconductivity of grey selenium arises from the ability of incident light to excite electrons across its reasonably small band gap (2.6 eV in the crystalline material, 1.8 eV in the amorphous material). These properties make Se useful in the production of photocells and exposure meters for photographic use, as well as solar cells. Selenium is also a p-type semiconductor (Section 3.20) and is used in electronic and solid-state applications. It is also used in photocopier toner and in the glass industry to make red glasses and enamels. 406 16 The Group 16 elements Tellurium crystallizes in a chain structure like that of grey selenium. Polonium crystallizes in a primitive cubic structure and a closely related higher temperature form above 36C. We remarked in Section 3.5 that the primitive cubic structure represents inefficient packing of atoms, and Po is the only element that adopts this structure under normal conditions. Tellurium and Po are both highly toxic; the toxicity of Po is enhanced by its intense radioactivity. Mass for mass, it is about 2.5 1011 times as toxic as hydrocyanic acid. All 29 isotopes of polonium are radioactive. It has been found in tobacco as a contaminant and in uranium ores. It can be produced in small amounts (gram quantities) through irradiation of 209Bi (atomic number 83) with neutrons, which gives 210Po (atomic number 84): 209 83 Bi + 11 n → 210 84 Po + e− Metallic Po can then be separated from the remaining Bi by fractional distillation or electrodeposited on to a metal surface. Selenium, Te, and Po combine directly with most elements, although less readily than O or S. The occurrence of multiple bonds is lower than with O or S, as is the tendency towards catenation (compared with S) and the number of allotropes. The unexpected difficulty of oxidizing Se to Se(VI) (Fig. 16.2) is due to the lanthanide-like contraction (Section 9.2a) in radius following the 3d elements. 16.8 Hydrides The impact of hydrogen bonding is seen clearly in the hydrides of the Group 16 elements. The hydrides of oxygen are water and hydrogen peroxide, which are both liquids. The hydrides of the heavier elements are all toxic, foul-smelling gases. (a) Water Key point: Hydrogen bonding in water results in a high boiling liquid and a highly structured arrangement in the solid, ice. At least nine distinct forms of ice have been identified. At 0C and atmospheric pressure, hexagonal ice Ih forms (Fig. 10.7) but between 120 and 140C the cubic form, Ic, is produced. At very high pressures several higher density polymorphs are formed, 6 Se Te NoxE °/V 4 f H (H 2O,l) = −286 kJ mol −1 O This reaction is very exothermic and provides the basis for the development of the hydrogen economy and hydrogen fuel cells (Fig. 5.1, Box 10.2, and Section 24.1). Water is the most widely used solvent not only because it is so widely available but also because of its high relative permittivity (dielectric constant), wide liquid range, and—through a combination of its polar character and ability to form hydrogen bonds—solvating ability. Many anhydrous and hydrated compounds dissolve in water to give hydrated cations and anions. Some predominantly covalent compounds, such as ethanol and ethanoic (acetic) acid, are soluble in water or miscible with it because of hydrogen-bonded interactions with the solvent. Many other covalent compounds react with water in hydrolysis reactions; examples are discussed in the appropriate chapters. In addition to simple dissolution and hydrolysis reactions, the importance of aqueous solution chemistry can be seen in redox reactions (Chapter 5) and acidbase reactions (Chapter 4). Water also acts as a Lewis base ligand in metal complexes (Section 7.1). The deprotonated forms, OH and particularly the oxide ion O2 are important ligands for stabilizing higher oxidation states, examples of which are found in the simple oxo-cations of early d-block elements, such as the vanadyl ion, VO2. (b) Hydrogen peroxide Key point: Hydrogen peroxide is susceptible to decomposition by disproportionation at elevated temperatures or in the presence of catalysts. Hydrogen peroxide is a very pale blue, viscous liquid. It has a higher boiling point than water (150C) and a greater density (1.445 g cm3 at 25C). It is miscible in water and is usually handled in aqueous solution. The Frost diagram for oxygen (Fig. 16.2) shows that H2O2 is a good oxidizing agent, but it is unstable with respect to disproportionation: H 2O2 (l) → H 2O(l) + 12 O2 (g) rG = −119 kJ mol −1 O This reaction is slow but is explosive when catalysed by a metal surface or alkali dissolved from glass. For this reason, hydrogen peroxide and its solutions are stored in plastic bottles and a stabilizer is added. This reaction can be considered in terms of the reduction half-reactions H 2O2 (aq) + H + (aq) + e− → H 2O(l) H + (aq) + 12 O2 (g) + e− → 12 H 2O2 (aq) E = +1.68 V E = +0.70 V O O Any substance with a standard potential in the range 0.701.68 V that has suitable binding sites will catalyse this reaction. As can be inferred from these standard potentials, hydrogen peroxide is a very powerful oxidizing agent in acid solution: 0 Se S –2 O –4 –2 H 2 (g) + 12 O2 (g) → H 2O(l) 1 2 S 2 Te some of which are based on silica-like structures (Section 14.10). Water is formed by the direct interaction of the elements: –1 0 1 2 3 4 Oxidation number, Nox 5 6 Figure. 16.2 Frost diagram for the elements of Group 16 in acidic solution. The species with oxidation number 2 are H2E. For oxidation number 1 the compound is H2O2. The positive oxidation numbers refer to the oxo acids or oxoanions. 2Ce3+ (aq) + H 2O2 (aq) + 2 H + (aq) → 2Ce4+ (aq) + 2 H 2O(l) However, in basic solution hydrogen peroxide can act as a reducing agent: 2Ce4+ (aq) + H 2O2 (aq) + 2OH − (aq) → 2Ce3+ (aq) + 2 H 2O(l) + O2 (g) The detail The underlying reason for the oxidizing nature of hydrogen peroxide lies in the weakness of the OO single bond (146 kJ mol1). Hydrogen peroxide reacts with d-metal ions such as Fe2 to form the hydroxyl radical in the Fenton reaction: Fe2+ (aq) + H 2O2 (aq) → Fe3+ (aq) + OH − (aq) + OH(aq) The Fe3 product can react with a second H2O2 to regenerate Fe2, so that the production of hydroxyl radical is catalytic. The hydroxyl radical is one of the strongest oxidizing agents known O (E 2.85 V) and the reaction is used to oxidize organic matter. In living cells, its reaction with DNA has potentially lethal consequences. 407 (c) Hydrides of sulfur, selenium, and tellurium Key points: The extent of hydrogen bonding is much less for these hydrides than for water; all the hydrides are gases. Hydrogen sulfide, H2S, is toxic, its toxicity made more hazardous by the fact that it tends to anaesthetize the olfactory nerves, making intensity of smell a dangerously inaccurate guide to concentration. Hydrogen sulfide is produced by volcanoes and by some micro-organisms (Box 16.3). It is an impurity in natural gas and must be removed before the gas is used. Pure H2S can be prepared by direct combination of the elements above 600C: H 2 (g) + S(l) → H 2 S(s) E X A MPL E 16 .1 Deciding whether an ion can catalyse H2O2 disproportionation Is Fe3 thermodynamically capable of catalysing the decomposition of H2O2? Answer For Fe3 to catalyse the decomposition of H2O it needs to act in the reaction 2 Fe3+ ( aq) + H2O2 ( aq) → 2 Fe2+ ( aq) + O2 ( g) + 2 H+ ( aq) Hydrogen sulfide is easily generated in the laboratory by trickling dilute hydrochloric or phosphoric acid on to FeS: FeS(s) + 2 HCl(aq) → H 2 S(g) + FeCl2 (aq) It can also be prepared by hydrolysis of aluminium sulfide, which is easily generated by ignition of a mixture of the elements: 2 Al(s) 3 S(s) → Al2S3(s) Al2S3(s) 3 H2O(l) → Al2O3(s) 3 H2S(g) and then be regenerated in the reaction 2 Fe2+ ( aq) + H2O2 ( aq) + 2 H+ ( aq) → 2 Fe3+ ( aq) + 2 H2O (l) It is readily soluble in water, and is a weak acid: So the net reaction is simply the decomposition of H2O2: H2S(aq) H2O(l) H3O(aq) HS(aq) pKa1 6.89 2 H2O2(aq) → O2(g) 2 H2O(l) HS(aq) H2O(l) H3O(aq) S2(aq) pKa2 14.15 The first reaction is the difference between the half-reactions Fe3(aq) e → Fe2(aq) E 0.77 V O2(g) 2 H(aq) 2 e → H2O2(aq) E O O 0.70 V O and therefore Ecell 0.07 V. This reaction is therefore spontaneous (K 1). The second reaction is the difference of the half-reactions H2O2(aq) 2 H(aq) 2 e → 2 H2O(l) E Fe3(aq) e → Fe2(aq) E O O Acidic solutions of H2S are mild reducing agents and deposit elemental S on standing. In a similar way, H2Se can be made by direct combination of the elements, by the reaction of FeSe with hydrochloric acid, or by hydrolysis of Al2Se3: 1.76 V H 2 (g) + Se(s) → H 2 Se(g) FeSe(s) + 2 HCl(aq) → H 2Se(g) + FeCl2 (aq) 0.77 V Al2Se3(s) 3 H2O(l) → Al2O3(s) 3 H2Se(g) and therefore Ecell 0.99 V and this reaction is also spontaneous (K 1). Because both reactions are spontaneous (in the sense K 1), catalytic decomposition is thermodynamically favoured. In fact, the rates also are high, so Fe3 is a highly effective catalyst for the decomposition of H2O2, and in its manufacture great pains are taken to minimize contamination by iron. O Self-test 16.1 Determine whether the decomposition of H2O2 is spontaneous in the presence of either Br or Cl. Hydrogen peroxide is a slightly stronger acid than water: H2O2(aq) H2O(l) H3O(aq) HO2(aq) pKa 11.65 Deprotonation occurs in other basic solvents such as liquid ammonia, and NH4OOH has been isolated and found to consist of NH4 and HO2 ions. When solid NH4OOH melts (at 25C), the melt contains hydrogen-bonded NH3 and H2O2 molecules. The oxidizing ability of hydrogen peroxide and the harmless nature of its byproducts lead to its many applications. It is used in water treatment to oxidize pollutants, as a mild antiseptic, and as a bleach in the textile, paper, and hair-care industries (Box 16.2). On the other hand, H2Te is made by hydrolysis of Al2Te3 or by the action of hydrochloric acid on Mg, Zn, or Al tellurides. Al2 Te3 (s) + 6 H 2O(l) → 3H 2 Te(g) + 2 Al(OH)3 (aq) MgTe(s) + 2 HCl(aq) → H 2 Te(g) + MgCl2 (aq) The solubilities of H2Se and H2Te in water are similar to that of H2S. The acidity constants of the hydrides (which are protic) increase from H2S to H2Te (Table 16.2). Like their sulfur analogue, aqueous solutions of H2Se and H2Te are readily oxidized and deposit elemental selenium and tellurium on standing. 16.9 Halides Key points: The halides of oxygen have limited stability but its heavier congeners form an extensive series of halogen compounds; typical formulas are EX2, EX4, and EX6. The oxidation number of O is 2 in all its compounds with the halogens other than F. Oxygen difluoride, OF2, is the highest 408 16 The Group 16 elements B OX 16 . 2 Environmentally friendly bleach Hydrogen peroxide is rapidly replacing chlorine and hypochlorite bleaches in industrial applications as it is environmentally benign, producing only water and oxygen. The major consumers of hydrogen peroxide bleach are the paper, textile, and wood pulp industries. A growing market is in de-inking of recycled paper and in the manufacture of kraft paper (strong brown paper). Approximately 85 per cent of all cotton and wool is bleached with hydrogen peroxide. One of its advantages over chlorine-based bleaches is that it does not affect many modern dyes. It is also used to decolour oils and waxes. Hydrogen peroxide is used to treat domestic and industrial effluent and sewage. It minimizes odours by preventing the production of H2S by anaerobic reactions in sewers and pipes. It also acts as a source of oxygen in sewage sludge treatment plants. Other industrial uses of hydrogen peroxide are the epoxidation of soybean and linseed oils to produce plasticizers and stabilizers for the plastics industry and as a propellant for torpedoes and missiles. It has been speculated that the sinking of the Russian submarine Kursk in 2000 was due to an explosion involving the hydrogen peroxide used to fuel its torpedoes. Hydrogen peroxide is increasingly being used as a green oxidant. It can be used in aqueous solution if generated in situ and the only byproduct is water. B OX 16 . 3 The sulfur cycle Sulfur is essential to all life forms through its presence in the amino acids cysteine and methionine, and in many key active site structures, including the inorganic sulfide in FeS proteins, and all molybdenum and tungsten enzymes. Moreover, many organisms obtain energy by the oxidation or reduction of inorganic sulfur compounds. The resultant transformations constitute the sulfur cycle. The redox extremes of sulfur chemistry are demonstrated by sulfate, the most oxidized form, and by H2S and its ionized forms, HS or S2, the most reduced forms. Many classes of organisms occupy ecological niches defined by the sulfur. The sulfate-reducing bacteria (SRBs), such as Desulfovibrio, reduce sulfate to sulfide under anaerobic conditions, oxidizing organic compounds in the process. Sulfide oxidizing organisms, such as Thiobacilli, are generally, but not always, aerobic and use O2 to oxidize sulfide, polysulfide ions, elemental sulfur, or thiosulfate to sulfate. Figure B16.1 is an incomplete version of the sulfur cycle, highlighting some of the known participating molecules. Sulfur oxidizing bacteria (e.g. Thiobacillus) SO2– 4 SO32– Inorganic sulfur, S(0) Bacterial oxidation Bacterial reduction Sulfite oxidase Biosynthesis (activated) Bio-oxidation Organic sulfur (e.g. cysteine) S2– (H2S and metal sulfides) Sulfur-reducing bacteria (e.g. Desulfurvibrio) Heterotrophic bacteria (e.g. Thiobacillus) Sulfur-oxidizing bacteria (e.g. Thiobacillus) Figure B16.1 The sulfur cycle. SRBs use sulfate as their electron acceptor and generate sulfide under anaerobic conditions. These anaerobic bacteria are found in environments where both SO2− and reduced organic matter are found, for example in 4 anoxic marine sediments and in the rumen of sheep and cattle. SRBs are important in sulfide ore formation, bio-corrosion, the souring of petroleum fluoride of oxygen and hence contains O in its highest oxidation state (2). The structures of the sulfur halides S2F2, SF4, SF6, and S2F10 (Table 16.3) are all in line with the VSEPR model. Thus, SF4 has under anaerobic conditions, the CuMo antagonism in ruminants, and many other physiological, ecological, and biogeochemical contexts. The reduction of sulfate is carried out in two steps: SO24− + 8 e − + 10 H+ → H2S + 4 H2O First, the relatively unreactive sulfate must be activated. This step is achieved through reaction with ATP to form adenosine phosphosulfate (APS) and pyrophosphate. The further hydrolysis of pyrophosphate rH O = −30.5 kJ mol−1 ensures that the APS formation reaction goes to the right: ( ) ATP + SO24− → APS + P2O74− The enzyme APS reductase carries out the catalytic reduction of the sulfate intermediate to sulfite: APS + 2 e − + H+ → AMP + HSO3− Conversion of sulfite to sulfide is then catalysed by the enzyme sulfite reductase: HSO3− + 6 e − + 7 H+ → H2S + 3 H2O The oxidative part of the sulfur cycle is the province of bacteria that gain energy from various interconversions. Some Thiobacilli species can oxidize sulfide in ores, for example iron sulfides. The oxidation of sulfide to sulfate produces an acidic environment in which some species of Thiobacilli thrive and can alter the pH to produce acidic conditions favourable to their own metabolic processes. Acid mine drainage water can have a microbially produced pH as low as 1.5 and Thiobacilli are used commercially to mobilize metals from sulfide ores. For example, Thiobacillus ferrooxidans not only oxidizes the sulfur in iron sulfide deposits but also oxidizes the iron(II) to the soluble iron(III): 4 FeS2 + 15 O2 + 2 H2O → 4 Fe3+ + 8 SO24− + 4 H+ Thiobacilli live solely on inorganic materials: they use energy obtained from sulfide oxidation to drive all their cellular reactions, including the fixation of carbon from CO2. ten valence electrons around the S atom, two of which form a lone pair in an equatorial position of a trigonal bipyramid. We have already mentioned the theoretical evidence that the molecular orbitals bonding the F atoms to the central atom in SF6 The detail primarily use the sulfur 4s and 4p orbitals, with the 3d orbitals playing a relatively unimportant role (Section 2.11a). The same seems to be true of SF4 and S2F10. Sulfur hexafluoride is a gas at room temperature. It is very unreactive and its inertness stems from the suppression, presumably by steric protection of the central S atom, of thermodynamically favourable reactions, such as the hydrolysis SF6 (g) + 4 H 2O(l) → 6 HF(aq) + H 2 SO4 (aq) The less sterically crowded SeF6 molecule is easily hydrolysed and is generally more reactive than SF6. Similarly, the sterically less hindered molecule SF4 is reactive and undergoes rapid partial hydrolysis: SF4 (g) + H 2O(l) → OSF2 (aq) + 2HF(aq) Both SF4 and SeF4 are selective fluorinating agents for the conversion of COOH into CF3 and C O and P O groups into CF2 and PF2 groups: 409 M(I): M2O oxides often have a rutile or antifluorite structure (6,3)- and (8,4)-coordination, respectively. M(II): MO oxides usually have the rock-salt structure (6,6)-coordination, M(III): M2O3 oxides often have (6,4)-coordination. At the other extreme, MO4 compounds are molecular: the tetrahedral compound osmium tetroxide, OsO4, is an example. The structures of the oxides with metals in high oxidation states and the oxides of nonmetallic elements often have multiple bond character. Deviations from these simple structures are common with p-block metals, where the less symmetric packing of O2 ions around the metal can often be rationalized in terms of the existence of a stereochemically active lone pair, as in PbO (Section 14.11). Another common structural motif for nonmetals and some metals in high oxidation states is a bridging oxygen atom, as in EOE, in angular and linear structures. 2 R 2CO(l) + SF4 (g) → 2 R 2CF2 (sol) + SO2 (g) Sulfur chlorides are commercially important. The reaction of molten S with Cl2 yields the foul-smelling and toxic substance disulfur dichloride, S2Cl2, which is a yellow liquid at room temperature (b.p. 138C). Disulfur dichloride and its further chlorination product sulfur dichloride, SCl2, an unstable red liquid, are produced on a large scale for use in the vulcanization of rubber. In this process, S atom bridges are introduced between polymer chains so the rubber object can retain its shape. 16.10 Metal oxides Key points: The oxides formed by metals include the basic oxides with high oxygen coordination number that are formed with most M and M2 ions. Oxides of metals in intermediate oxidation states often have more complex structures and are amphoteric. Metal peroxides and superoxides are formed between O2 and alkali metals and alkaline earth metals. Terminal E O linkages and EOE bridges are common with nonmetals and with metals in high oxidation states. The O2 molecule readily removes electrons from metals to form a variety of metal oxides containing the anions O2 (oxide), O2 (superoxide), and O22 (peroxide). Even though the existence of O2 can be rationalized in terms of a closed-shell noble gas electron configuration, the formation of O2(g) from O2(g) is highly endothermic, and the ion is stabilized in the solid state by its strong Coulombic interaction with the surrounding cations. Alkali metals and alkaline earth metals often form peroxides or superoxides (Sections 11.8 and 12.8), but peroxides and superoxides of other metals are rare. Among the metals, only some of the noble metals do not form thermodynamically stable oxides. However, even where no bulk oxide phase is formed, an atomically clean metal surface (which can be prepared only in an ultrahigh vacuum) is quickly covered with a surface layer of oxide when it is exposed to traces of oxygen. Structural trends in the metal oxides are not readily summarized but for oxides in which the metal has oxidation number 1, 2, or 3, the O2 ion is generally in a site of high coordination number: 16.11 Metal sulfides, selenides, tellurides, and polonides Key point: Monatomic and polyatomic sulfide ions are known as discrete anions and as ligands. Many metals occur naturally as their sulfide ores. The ores are roasted in air to form the oxide or the water-soluble sulfate, from which the metals are extracted. The sulfides can be prepared in the laboratory or industry by a number of routes; direct combination of the elements, reduction of a sulfate, or precipitation of an insoluble sulfide from solution by addition of H2S: Fe(s) + S(s) → FeS(s) MgSO4 (s) + 4 C(s) → MgS(s) + 4 CO (g) M 2+ (aq) + H 2 S(g) → MS(s) + 2 H + (aq) The solubilities of the metal sulfides vary enormously. The Group 1 and 2 sulfides are soluble, whereas the sulfides of the heavy elements of Group 11 and 12 are among the least soluble compounds known. The wide variation enables selective separation of metals to take place on the basis of the solubilities of the sulfides. The Group 1 sulfides, M2S, adopt the antifluorite structure (Section 3.9). The Group 2 elements and some of the f-block elements form monosulfides, MS, with a rock-salt structure. The first-row d-block metals form monosulfides with the NiAs structure, whereas the heavier elements have a greater tendency to covalence and adopt a zinc-blende structure. The d-metals form disulfides that have a layered structure or contain discrete S22 ions. These compounds are discussed in Chapter 19. The selenides and tellurides are the most common naturally occurring sources of the elements. Group 1 and 2 selenides, tellurides, and polonides are prepared by direct interaction of the elements in liquid ammonia. They are water-soluble solids that are rapidly oxidized in air to give the elements, with the exception of the polonides, for which they are among the most stable compounds of the element. The selenides and tellurides of Li, Na, and K adopt the antifluorite structure; those of the heavier elements of Group 1 adopt the rock-salt structure. Selenides, tellurides, and polonides of the d metals are also prepared by direct interaction of the elements and are nonstoichiometric. 410 16 The Group 16 elements Two examples are compounds of approximate stoichiometry Ti2Se and Ti3Se. 16.12 Oxides Oxides of Group 16 elements are described in the relevant group chapters. In this section, we concentrate on compounds formed between oxygen and its congeners in Group 16. (a) Sulfur oxides and oxohalides a Lewis acid and formed a complex with the base, F or (CH3)3N. Both the complexes still have a lone pair on S, and the resulting four electron pairs form a tetrahedron around the S atom, yielding the trigonal pyramidal complexes (17) and (18). As the OH ion is a stronger Lewis base than either F or N(CH3)3, it will form a complex with the SO2 in preference to either of them. Therefore, exposure of either complex to OH will yield the hydrogensulfite ion, HSO3 which has been found to exist in two isomers, (19) and (20). Self-test 16.2 Draw the Lewis structures and identify the point groups of (a) SO3(g) and (b) SO3F. Key points: Sulfur dioxide is a mild Lewis acid towards p-block bases, and OSCl2 is a useful drying agent. Sulfur dioxide and sulfur trioxide are both Lewis acids, with the S atom the acceptor site, but SO3 is the much the stronger and harder acid. The high Lewis acidity of SO3 accounts for its occurrence as a cyclic trimeric O-bridged solid at room temperature and pressure (4). Sulfur dioxide is manufactured on a large scale by combustion of sulfur or H2S or by roasting sulfide ores in air: – F S S S O O O O O 16 SO2, C2v 17 SO2F–, Cs 4 FeS(s) + 7 O2 (g) → 4 SO2 (g) + 2 Fe2O3 (s) It is soluble in water and gives a solution commonly referred to as sulfurous acid, H2SO3, but which is in fact a complex mixture of numerous species (Box 16.4). Sulfur dioxide forms weak complexes with simple p-block Lewis bases. For example, although it does not form a stable complex with H2O, it does form stable complexes with stronger Lewis bases, such as trimethylamine and F ions. Sulfur dioxide is a useful solvent for acidic substances (see Box 16.5). O S N – O S R H 18 SO2NR3 19 HSO – 3 E X A M PL E 16 . 2 Deducing the structures and properties of SO2 complexes H – Suggest the probable structures of SO2F and (CH3)3NSO2, and predict their reactions with OH. Answer A good starting point for the discussion of shape is to draw a Lewis structure. The Lewis structure of SO2 is shown in (16). We know that SO2 can act as either a Lewis acid or a Lewis base, but in both cases SO2 has acted as O S 20 HSO3– B OX 16 . 4 Acid rain The main components of acid rain are nitric and sulfuric acids produced by interaction of the oxides with hydroxyl radicals: HO. NO2 → HNO3 HO. SO2 → HSO3. HSO3. O2 H2O → H2SO4 HO2. The resulting hydroperoxyl radicals produce additional hydroxyl radicals: HO2. X → XO HO. X NO or SO2 Sulfuric and nitric acid molecules form hydrogen bonds and interact strongly with one another, with metal oxides and gases in the atmosphere, and with water, to form particles. These small particles are a major health threat in polluted air. Recent studies have convincingly associated increased concentrations of particulate matter in the size range of 2.5 μm or less with increased mortality from pulmonary, and especially heart, disease. These particles are small enough to lodge deep in the lungs, and they can carry noxious chemicals on their surface. In addition to their physiological effects, these particles affect the ecosystem because of the acids they contain. As the acidity of rainfall increases, the protons increasingly wash alkali metal (Na, K) and alkaline earth (Ca2, Mg2) ions from the soil, where they are held in ion exchange sites in clay and humus or in limestone. Depletion of these nutrients limits plant growth. The same chemistry also erodes marble statues and buildings. Granite-lined lakes (which have low buffer capacity) can be acidified, leading to the disappearance of fish and other aquatic life. Because emissions from combustion sources can travel long distances, acid rain is a regional problem, with large areas at risk, particularly downwind of coal-fired power plants, which have tall smokestacks to disperse the NO and SO2 exhaust gases. These environmental and health effects combine to make NO and SO2 a main focus of air pollution regulatory activity. 411 The detail B OX 16 . 5 Synthesis in nonaqueous solvents The nonaqueous solvents liquid ammonia, liquid sulfur dioxide, and sulfuric acid show an interesting set of contrasts as they range from a good Lewis base with a resistance to reduction (ammonia) to a strong Brønsted acid with resistance to oxidation (sulfuric acid). Liquid ammonia (b.p. 33C) and the solutions it forms may be handled in an open Dewar flask in an efficient fume hood (Fig. B16.2), in a vacuum line when the liquid is kept below its boiling point, or (with caution) in sealed heavywalled glass tubes at room temperature (the vapour pressure of ammonia is about 10 atm at room temperature). Liquid ammonia solutions of the s-block metals are excellent reducing agents. One example of their application is in the preparation of a complex of Ni in its unusual 1 oxidation state: 2 K(am) + 2 [Ni(CN)4 ]2− (am) → [Ni2 (CN)6 ]4 − (am) + 2 KCN(am) Liquid sulfur dioxide (b.p. 10°C) can be handled like liquid ammonia. An example of its use is the reaction This reaction is performed in a sealed thick-walled borosilicate glass tube, which is sometimes called a Carius tube (Fig. B16.3). The hydroxylamine is introduced into the tube and the sulfur dioxide is condensed into the tube after it has been cooled to about 45C. After the tube has been sealed, it is allowed to warm to room temperature behind a blast shield in a hood. The reaction is complete after several days at room temperature; the tube is cooled (to reduce the pressure) and broken open. The product is known as sulfamic acid and is used in the synthesis of compounds that taste sweet. Sulfuric acid and solutions of SO3 in sulfuric acid (fuming sulfuric acid, H2S2O7) are used as oxidizing acidic media for the preparation of polychalcogen cations: 8 Se(s) + 5 H2SO4 (l) → Se28+ (sol) + 2 H3O+ (sol) + 4 HSO−4 ( sol) + SO2(g) NH2OH(s) + SO2 (l) → H2NSO2OH(s) Metal tank Needle valve Sulfur dioxide gas Liquid ammonia Cold bath Liquid sulfur dioxide Unsilvered Dewar vessel Figure B16.2 The transfer of liquid ammonia from a pressurized tank. (Based on W.L. Jolly, The synthesis and characterization of inorganic compounds. Waveland Press, Prospect Heights (1991).) (b) Oxides of selenium and tellurium Key points: Selenium and tellurium dioxides are polymorphic; selenium dioxide is thermodynamically less stable than SO2 or TeO2, and selenium trioxide, SeO3, is thermodynamically less stable than SeO2. The dioxides of Se, Te, and Po can be prepared by direct reaction of the elements. Selenium dioxide is a white solid that sublimes at 315C. It has a polymeric structure in the solid state (21). It is thermodynamically less stable than SO2 or TeO2 and is reduced to selenium on reaction with NH3, N2H4, or aqueous SO2. Seal here Dewar vessel Figure B16.3 The condensation of sulfur dioxide into a Carius tube. Tellurium dioxide occurs naturally as the mineral tellurite, -TeO2, which has a layer structure in which TeO4 units form dimers (22). Synthetic -TeO2 consists of similar TeO4 units that share all vertices to form a three-dimensional rutile-like structure (23). Polonium dioxide exists as the yellow form with the fluorite structure and the red tetragonal form. O Te Te 3 SeO2(s) 4NH3(l) → 3 Se(s) 2 N2(g) 6 H2O(l) O It is used an oxidizing agent in organic chemistry. 22 (TeO4)2 in β -TeO2 Se 23 TeO4 in α -TeO2 O 21 SeO2 Selenium trioxide, unlike SO3 or TeO3, is thermodynamically less stable than the dioxide (Table 16.5). It is a white hygroscopic solid that sublimes at 100C and decomposes at 165C. In the solid state the structure is based on Se4O12 tetramers (24) 412 16 The Group 16 elements Table 16.5 Standard enthalpies of formation, O ∆ f H / (kj mol−1) , of sulfur, selenium, and tellurium oxides SO2 297 SO3 432 SeO2 230 SeO3 184 TeO2 325 TeO3 348 Sulfur (like N and P) forms many oxoacids. These exist in aqueous solution or as the solid salts of the oxoanions (Table 16.6). Many of them are important in the laboratory and in industry. (a) Redox properties of the oxoanions Key points: The oxoanions of sulfur include the sulfite ion, SO32, which is a good reducing agent, the rather unreactive sulfate ion, SO42, and the strongly oxidizing peroxodisulfate ion, O3SOOSO32. As with sulfur, the redox reactions of selenium and tellurium oxoanions are often slow. O Se O 24 (SeO3)4, C4v but it is monomeric in the vapour phase. Tellurium trioxide exists as the yellow -TeO3, which is prepared by dehydration of Te(OH)6, and the more stable -TeO3, which is made by heating -TeO3 or Te(OH)6 in oxygen. (c) Chalcogen oxohalides Key points: The most important oxohalides are those of sulfur; selenium and tellurium oxofluorides are known and the ‘teflate’ ion is a useful ligand. Many chalcogen oxohalides are known. The most important are the thionyl dihalides, OSX2, and the sulfuryl dihalides, O2SX2. One laboratory application of thionyl dichloride is the dehydration of metal chlorides: MgCl2 .6H 2O(s) + 6OSCl2 (l) → MgCl2 (s) + 6SO2 (g) + 12 HCl(g) The compound F5TeOTeF5 and its selenium analogue are known, and the OTeF5 ion, which is known informally as ‘teflate’, is a bulky, electronegative anion. It is a well-established ligand for high oxidation state d-metal and main-group element complexes such as [Ti(OTeF5)6]2 (25), [Xe(OTeF5)6], and [M(C5H5)2(OTeF5)2], where M Ti, Zr, Hf, W, and Mo. 2– Te 16.13 Oxoacids of sulfur Sulfur’s common oxidation numbers are 2, 0, 2, 4, and 6, but there are also many SS bonded species that are assigned odd and fractional average oxidation numbers. A simple example is the thiosulfate ion, S2O32, in which the average oxidation number of S is 2, but in which the environments of the two S atoms are quite different. The thermodynamic relations between the oxidation states are summarized by the Frost diagram (Fig. 16.2). As with many other p-block oxoanions, many of the thermodynamically favourable reactions are slow when the element is in its maximum oxidation state (6), as in SO42. Another kinetic factor is suggested by the fact that oxidation numbers of compounds containing a single S atom generally change in steps of 2, which requires an O atom transfer path for the mechanism. In some cases a radical mechanism operates, as in the oxidation of thiols and alcohols by peroxodisulfate, in which OO bond cleavage produces the transient radical anion SO4. We saw in Section 5.6 that the pH of a solution has a marked effect on the redox properties of oxoanions. This strong dependence is true for SO2 and SO32 because the former is easily reduced in acidic solution and is therefore an oxidizing agent, whereas the latter in basic solution is primarily a reducing agent: SO2 (aq) + 4 H + (aq) + 4 e− → S(s) + 2 H 2O(l) E = +0.50 V O SO24− (aq) + H 2O(l) + 2e− → SO32− (aq) + 2 OH − (aq) E = −0.94 V O The principal species present in acidic solution is SO2, not H2SO3, but in more basic solution HSO3 exists in equilibrium with HSO3 and HOSO2. The oxidizing character of SO2 accounts for its use as a mild disinfectant and preservative for foodstuffs, such as dried fruit and wine. The peroxodisulfate ion, O3SOOSO32, is a powerful and useful oxidizing agent: 2– O F Ti O O O S O O S O O O + 2e– 2– 2 O S O O O E° = +2.01 V 25 [Ti(OTeF5)6]2– although this reactivity reflects the properties of O rather than S because it arises from the weakness of the OO bond, as we have discussed above for hydrogen peroxide. The detail 413 Table 16.6 Some sulfur oxoanions Oxidation number Formula Name Structure Remarks One S atom 4 2– SO2− 3 Sulfite Basic, reducing agent 2– 6 SO2− 4 Sulfate Weakly basic Two S atoms 2– 2 S2O 2− 3 Thiosulfate Moderately strong reducing agent 2– 3 S2O2− 4 Dithionite Strong reducing agent 2– 4 S2O2− 5 Disulfite 2– 5 S2O2− 6 Dithionate Resists oxidation and reduction Polysulfur oxanions Variable 2– Sn O22n−+2 n 3, trithionate 3 n 20 The oxoanions of selenium and tellurium are a much less diverse and extensive group. Selenic acid is thermodynamically a strong oxidizing acid: SeO24− (aq) + 4 H + (aq) + 2e− → H 2 SeO3 (aq) + H 2O(l) E = +1.15 V O However, like SO42 and in common with the behaviour of oxoanions of other elements in high oxidation states, the reduction of SeO42 is generally slow. Telluric acid exists as Te(OH)6 and also as (HO)2TeO2 in solution. Again, its reduction is thermodynamically favourable but kinetically sluggish. (b) Sulfuric acid Key points: Sulfuric acid is a strong acid; it is a useful nonaqueous solvent due to its extensive autoprotolysis. Sulfuric acid is a dense viscous liquid. It dissolves in water in a highly exothermic reaction: H 2SO4 (l) → H 2SO4 (aq) O r H = −880 kJ mol −1 It is a strong Brønsted acid in water (pKa1 2) but not for its second deprotonation (pKa2 1.92). Anhydrous H2SO4 has a very 414 16 The Group 16 elements high relative permittivity and high electrical conductivity consistent with extensive autoprotolysis: 2 H2SO4(l) H3SO4(sol) HSO4(sol) K 2.7 × 104 The equilibrium constant for this autoprotolysis is greater than that of water by a factor of more than 1010. This property leads to the use of sulfuric acid as a nonaqueous, protic solvent (Box 16.5). Bases (proton acceptors) increase the concentration of HSO4 ions in anhydrous sulfuric acid: they include water and salts of weaker acids, such as nitrates: H 2O(l) + H 2 SO4 (sol) → H3O+ (sol) + HSO4− (sol) NO3− (s) + H 2 SO4 (l) → HNO3 (sol) + HSO4− (sol) Another example of this kind is the reaction of concentrated sulfuric acid with concentrated nitric acid to produce the nitronium ion, NO2 , which is responsible for the nitration of aromatic species: HNO3 (aq) + 2 H 2 SO4 (aq) → NO2+ (aq) + H3O+ (aq) + 2 HSO4− (aq) The number of species that are acidic in sulfuric acid is much smaller than in water because the acid is a poor proton acceptor. For example, HSO3F is a weak acid in sulfuric acid: HSO3F(sol) H2SO4(l) H3SO4(sol) SO3F(sol) As well as undergoing autoprotolysis, H2SO4 dissociates into H2O and SO3, which react further with H2SO4 to give a number of products: H2O H2SO4 H3O HSO4 SO3 H2SO4 H2S2O7 H2S2O7 H2SO4 H3SO4 HS2O7 Consequently, rather than being a single substance, anhydrous sulfuric acid is made up of a complex mixture of at least seven characterized species. Sulfuric acid is one of the most important chemicals produced on an industrial scale. Over 80 per cent of it is used to manufacture fertilizers. It is also used to remove impurities from petroleum, to ‘pickle’ (clean) iron and steel before electroplating, as the electrolyte in leadacid batteries (Box 14.6), and in the manufacture of many other bulk chemicals, such as hydrochloric and nitric acids. Concentrated sulfuric acid is manufactured by the contact process. The first stage is the oxidation of a sulfur compound to SO2. Most plants use elemental sulfur but metal sulfides and H2S are also used: S(s) + O2 (g) → SO2 (g) 4 FeS(s) + 7 O2 (g) → 2 Fe2O3 + 4 SO2 (g) 2 H 2 S(g) + 3O2 (g) → 2SO2 (g) + 2 H 2O(g) The second stage is the oxidation of SO2 to SO3. This reaction is carried out at high temperatures and pressures over a V2O5 catalyst supported on silica beads: 2 SO2 (g) + O2 (g) → 2 SO3 (g) The SO3 is then passed into the bottom of a packed column and washed by running oleum, H2S2O7, from the top of the column. The gas is then washed in a second column with 98 per cent by mass H2SO4. The SO3 reacts with the 2 per cent of water to produce sulfuric acid, H2SO4: SO3 (g) + H 2O(sol) → H 2 SO4 (l) (c) Sulfurous acid and disulfurous acid Key points: Sulfurous and disulfurous acids have never been isolated. However, salts of both acids exist; sulfites are moderately strong reducing agents and are used as bleaches; disulfites rapidly decompose in acidic conditions. Although an aqueous solution of SO2 is referred to as ‘sulfurous acid’, H2SO3 has never been isolated and the predominant species present are the hydrates SO2.nH2O. The first and second proton donation are therefore best represented as follows: SO2.nH2O(aq) 2H2O(l) H3O(aq) HSO3(aq) n H2O(l) pKa 1.79 HSO3(aq) H2O(l) H3O(aq) SO32 (aq) pKa 7.00 Anhydrous sodium sulfite, Na2SO3, is produced on an industrial scale and used as a bleach in the pulp and paper industry, as a reducing agent in photography, and as an oxygen scavenger in boiler treatments. Disulfurous acid, H2S2O5 (26), does not exist in the free state but salts are readily obtained from a concentrated solution of hydrogensulfites: 2 HSO3− (aq) S2O52− (aq) + H 2O(l) Acidic solutions of disulfites rapidly decompose to give HSO3 and SO32. O S H 26 Disulfurous acid, H2S2O5 (d) Thiosulfuric acid Key points: Thiosulfuric acid decomposes but the salts are stable; thiosulfate ion is a moderately strong reducing agent. Aqueous thiosulfuric acid, H2S2O3 (27), decomposes rapidly in a complex process that produces a number of products, such as S, SO2, H2S, and H2SO4. The anhydrous acid is more stable and decomposes slowly to H2S and SO3. In contrast to the acid, the thiosulfate salts are stable and can be prepared by boiling the sulfites or hydrogensulfites with elemental sulfur or by oxidation of polysulfides: 8 K2 SO3 (aq) + S8 (s) → 8 K2 S2O3 (aq) 2CaS2 (s) + 3O2 (g) → 2CaS2O3 (s) The detail 415 sodium amalgam. Sodium dithionite is an important reducing agent in biochemistry. Neutral and acidic solutions of dithionite disproportionate into HSO3 and S2O32: H O S 2S2O42− (aq) + H 2O(l) → 2 HSO3− (aq) + S2O32− (aq) Dithionates, S2O62, are prepared by oxidation of the corresponding sulfite. Strong oxidizing agents such as MnO4 oxidize dithionate to sulfate: 27 Thiosulfuric acid, H2S2O3 The thiosulfate ion, S2O2 , is a moderately strong reducing agent: 3 1 2 S4O62− (aq) + e− → S2O32− (aq) E = +0.09 V O The reaction with iodine is the basis of iodometric titrations in analytical chemistry: I (aq) + e → I (aq) E = +0.54 V 2S2O32− (aq) + I2 (aq) → S4O62− (aq) + 2 I− (aq) − 1 2 2 − O Stronger oxidizing agents, such as chlorine, oxidize thiosulfate to sulfate, which has led to the use of thiosulfate to remove excess chlorine in the bleaching industries. SO24− (aq) + 2 H + (aq) + e− → 12 S2O62− (aq) + H 2O(l) E = −0.25 V O Strong reducing agents such as sodium amalgam reduce it to SO2 : 3 1 2 S2O62− (aq) + 2 H + (aq) + e− → H 2 SO3 (aq) E = +0.57 V O Neutral and acidic solutions of dithionate slowly decompose to SO2 and SO42: S2O62− (aq) → SO2 (g) + SO24− (aq) (e) Peroxosulfuric acids Key point: Peroxodisulfate salts are strong oxidizing agents. Peroxomonosulfuric acid, H2SO5 (28), is a crystalline solid that can be prepared by the reaction of H2SO4 with peroxodisulfates or as a byproduct of the synthesis of H2S2O8 by electrolysis of H2SO4. The salts are unstable and decompose, producing H2O2. Peroxodisulfuric acid, H2S2O8 (29), is also a crystalline solid. Its ammonium and potassium salts are prepared on an industrial scale by the oxidation of ammonia and potassium sulfates. They are strong oxidizing and bleaching agents. 1 2 S2O82− (aq) + H + (aq) + e− → HSO4− (aq) E = +2.12 V S S O 28 Peroxomonosulfuric acid, H2SO5 O H 30 Dithionous acid, H2S2O4 H 31 Dithionic acid, H2S2O6 (g) Polythionic acids O When K2S2O8 is heated, ozone and oxygen are evolved. H S O S O H Key point: Polythionic acids can be prepared with up to six S atoms. Many polythionic acids, H2SnO6, were first identified by the study of ‘Wackenroder’s solution’, which consists of H2S in aqueous SO2. Among those first characterized are the tetrathionate, S4O62 (5), and pentathionate, S5O62 (6), ions. More recently, a wide variety of preparative routes have been developed, many of which are complicated by numerous redox and catenation reactions. Typical examples are oxidation of thiosulfates with I2 or H2O2, and reaction of polysulfanes, H2Sn, with SO3 to yield H2Sn2O6, where n 26: H 2 Sn (aq) + 2SO3 (aq) → H 2 Sn +2O6 (aq) 29 Peroxodisulfuric acid, H2S2O8 16.14 Polyanions of sulfur, selenium, and tellurium (f) Dithionous and dithionic acids Key point: Dithionite salts may disproportionate whereas dithionate salts may be oxidized or reduced. Neither anhydrous dithionous acid, H2S2O4 (30), nor dithionic acid, H2S2O6 (31), can be isolated. However, the dithionous and dithionate salts are stable crystalline solids. Dithionites, S2O42, can be prepared by the reduction of sulfites with zinc dust or Key points: Sulfur forms polyanions with up to six sulfur atoms; polyselenides form chains and rings, and polytellurides form chains and bicyclic structures. Many polysulfides of electropositive elements have been characterized. They all contain the Sn2 ions where n 26, as in (7), (32), and (33). Typical examples are Na2S2, BaS2, Na4S4, K2S4, and Cs2S6. They can be prepared by heating stoichiometric amounts of S and the element in a sealed tube. 416 16 The Group 16 elements Many cationic chain, ring, and cluster compounds of the p-block elements have been prepared. The majority of them contain S, Se, or Te. Because these cations are oxidizing agents and Lewis acids, the preparative conditions are quite different from those used to synthesize the highly reducing polyanions. For example, S8 is oxidized by AsF5 in liquid sulfur dioxide to yield the S84 ion: 2– 2– S S SO 2 S8 + 3 AsF5 ⎯⎯⎯ → [S8 ][ AsF6 ]2 + AsF3 33 S62– 32 S42– The polysulfides can also act as ligands. An example is [Mo2(S2)6]2 (34), which is formed from ammonium polysulfide and MoO42; it contains side-bonded S22 ligands. The larger polysulfides bond to metal atoms, forming chelate rings, as in [WS(S4)2]2 (35), which contains chelating S4 ligands. The mineral iron pyrites, also known as ‘Fools’ gold’, has the formula FeS2 and consists of Fe2 and discrete S22 anions in a rock-salt structure. 2– S S o M 2– W 34 [Mo2(S2)6]2– 35 [WS(S4)2]2– More polyselenides and polytellurides have been characterized than polysulfides. Structurally, the smaller polyanion solids resemble the polysulfides. The structures of the larger polyanion solids are more complex and depend to some extent on the nature of the cation. The polyselenides up to Se92 are chains but 2 , which has a Se atom larger molecules form rings such as Se11 at the centre of two six-membered rings in a square-planar arrangement (8). The polytellurides are structurally more complex and there is a greater occurrence of bicyclic arrangements, such as Te72 (9) and Te82 (36). d-Metal complexes of larger polyselenides and polytellurides are known, such as [Ti(Cp)2Se5] (37). It appears that in polysulfides, polyselenides, and polytellurides electron density is concentrated at the ends of an En2 chain, which accounts for coordination through the terminal atoms, as shown in (35) and (37). Cp Ti HSO F 3 4 Se + S2O6 F2 ⎯⎯⎯ ⎯ → [Se4 ][SO3 F]2 The E42 ions have square-planar (D4h) structures (10). In the molecular orbital model of the bonding, the cations have a closed-shell configuration in which six electrons fill the a2u and eg orbitals, leaving the higher energy antibonding b2u orbital vacant. In contrast, most of the larger ring systems can be understood in terms of localized 2c,2e bonds. For these larger rings, the removal of two electrons brings about the formation of an additional 2c,2e bond, thereby preserving the local electron count on each element. This change is readily seen for the oxidation of S8 to S82 (38). An X-ray single-crystal structure determination shows that the transannular bonds in S82 are long compared with the other bonds. Long transannular bonds are common in these types of compounds. S 204 pm 2+ 283 pm 38 S82+ 16.16 Sulfurnitrogen compounds Key points: Neutral heteroatomic ring and cluster compounds of the p-block elements include P4S10 and cyclic S4N4. Disulfurdinitride transforms into a polymer that is superconducting at very low temperatures. Sulfurnitrogen compounds have structures that can be related to the polycations discussed above. The oldest known, and easiest to prepare, is the pale yelloworange tetrasulfurtetranitride, S4N4 (11), which is made by passing ammonia through a solution of SCl2: 2– Te A solvent is used that is more acidic than the polycations, such as fluorosulfuric acid. Sulfur, Se, and Te each form ions of the type E42. For example, Se42 is formed by oxidation of elemental Se by the strongly oxidizing peroxide compound FO2SOOSO2F: Se 6 SCl2 (l) + 16 NH3 (g) → S4 N 4 (s) + 41 S8 (s) + 12 NH 4Cl(sol) Tetrasulfurtetranitride is endergonic (fG 536 kJ mol1) and may decompose explosively. The ‘cradle-like’ molecule is an eight-membered ring with the four N atoms in a plane and bridged by S atoms that project above and below the plane. The short SS distance (258 pm) suggests that there is a weak interaction between pairs of S atoms. Lewis acids such as BF3, SbF5, and SO3 form 1:1 complexes with one of the N atoms and in the process the S4N4 ring rearranges (39). O 36 Te82– 37 [Ti(Cp)2Se5], Cp = C5H5 16.15 Polycations of sulfur, selenium, and tellurium Key point: Polyatomic cations of S, Se, and Te can be produced by the action of mild oxidizing agents on the elements in strong acid media. The detail O S 417 N S N 40 (SN)n 39 S4N4SO3 Disulfurdinitride, S2N2 (12), is formed (together with Ag2S and N2) when S4N4 vapour is passed over hot silver wool. It is even more sensitive than its precursor and explodes above room temperature. When allowed to stand at 0C for several days, disulfurdinitride transforms into a bronze-coloured zig-zag polymer of composition (SN)n (40), which is much more stable than its precursor, not exploding until 240C. The compound exhibits metallic conductivity along the chain axis and becomes superconducting below 0.3 K. The discovery of this superconductivity was important because it was the first example of a superconductor that had no metal constituents. Halogenated derivatives have been synthesized that have even higher conductivity. For example, partial bromination of (SN)n produces blueblack single crystals of (SNBr0.4)n, which has room temperature conductivity an order of magnitude greater than (SN)n. Treatment of S4N4 with ICl, IBr, and I2 produces highly conducting, nonstoichiometric polymers with conductivities greater by 16 orders of magnitude than (SN)n. The compound S4N2 can be prepared by heating S4N4 with sulfur in CS2 at 120C and increased pressure: CS /120ºC 2 S4 N 4 + 4 S ⎯⎯⎯⎯ → 2S4 N 2 It forms dark red, needle-like crystals that melt to a dark red liquid at 25C. It decomposes explosively at 100C. FURTHER READING J.S. Thayer, Relativistic effects and the chemistry of the heaviest maingroup elements, J. Chem. Ed., 2005, 82, 1721. This paper explains the reasons for the different properties of the heaviest elements in each group compared to the lighter elements in terms of relativistic effects. R.B. King, Inorganic chemistry of the main group elements. John Wiley & Sons (1994). D.M.P. Mingos, Essential trends in inorganic chemistry. Oxford University Press (1998). A survey of inorganic chemistry from the perspective of structure and bonding. R.B. King (ed.), Encyclopedia of inorganic chemistry. John Wiley & Sons (2005). N. Saunders, Oxygen and the elements of group 16. Heinemann (2003). P. Ball, H2O: a biography of water. Phoenix (2004). An entertaining look at the chemistry and physics of water. R. Steudel, Elemental sulfur and sulfur-rich compounds. SpringerVerlag (2003). N.N. Greenwood and A. Earnshaw, Chemistry of the elements. Butterworth-Heinemann (1997). EXERCISES 16.1 State whether the following oxides are acidic, basic, neutral, or amphoteric: CO2, P2O5, SO3, MgO, K2O, Al2O3, CO. 16.5 Rank the following species from the strongest reducing agent to the strongest oxidizing agent: SO24− , SO32− ,O3SO2 SO32−. 16.2 (a) Use standard potentials (Resource section 3) to calculate the standard potential of the disproportionation of H2O2 in acid solution. (b) Is Cr2 a likely catalyst for the disproportionation of H2O2? (c) Given the Latimer diagram 0.13 +1.51 O2 ⎯⎯⎯⎯ → HO2 ⎯⎯⎯⎯ → H 2O2 16.6 Predict which oxidation states of Mn will be reduced by sulfite ions in basic conditions. in acidic solution, calculate rG for the disproportionation of hydrogen superoxide (HO2) into O2 and H2O2, and compare the result with its value for the disproportionation of H2O2. 16.3 Which hydrogen bond would be stronger: S—H ... O or O—H ... S? 16.8 Use the standard potential data in Resource section 3 to predict which oxoanions of sulfur will disproportionate in acidic conditions. O 16.4 Which of the solvents ethylenediamine (which is basic and reducing) or SO2 (which is acidic and oxidizing) might not react with (a) Na2S4, (b) K2Te3? 16.7 (a) Give the formula for Te(VI) in acidic aqueous solution and contrast it with the formula for S(VI). (b) Offer a plausible explanation for this difference. 16.9 Use the standard potential data in Resource section 3 to predict whether SeO32 is more stable in acidic or basic solution. 16.10 Predict whether any of the following will be reduced by thiosulfate ions, S2O32, in acidic conditions: VO2, Fe3, Cu, Co3. 418 16 The Group 16 elements 16.11 SF4 reacts with BF3 to form [SF3][BF4]. Use VSEPR theory to predict the shapes of the cation and anion. 16.12 Tetramethylammonium fluoride (0.70 g) reacts with SF4 (0.81 g) to form an ionic product. (a) Write a balanced equation for the reaction and (b) sketch the structure of the anion. (c) How many lines would be observed in the 19F-NMR spectrum of the anion? 16.13 Identify the sulfur-containing compounds A, B, C, D, E, and F. F O2 S H2S Cl2 A excess NH3 B Ag/∆ C K2SO3 D E I2 PROBLEMS 16.1 The bond lengths in O2, O2+ , and O2− are 121, 112, and 149 2 pm, respectively. Describe the bonding in these molecules in terms of molecular orbital theory and use this description to rationalize the differences in bond lengths. 16.2 Correct any inaccuracies in the following statements and, after correction, provide examples to illustrate each statement. (a) Elements in the middle of Group 16 are easier to oxidize to the group oxidation number than are the lightest and heaviest members. (b) In its ground state, O2 is a triplet and it undergoes DielsAlder electrophilic attack on dienes. (c) The diffusion of ozone from the stratosphere into the troposphere poses a major environmental problem. 16.3 Write a comparative account of the properties of sulfuric, selenic, and telluric acids. 16.4 In their paper ‘Formation of tellurium nanotubes through concentration depletion at the surfaces of seeds’ (Advanced Materials, 2002, 14, 279) Mayers and Xia describe the synthesis of tellurium nanotubes. Describe how their method differs from those used to produce carbon nanotubes. What were the dimensions of tellurium nanotubes produced? What applications do the authors envisage for tellurium nanotubes? 16.5 In November 2006 the former KGB agent Alexander Litvinenko was found to have been poisoned by radioactive polonium-210. Write a review of the chemical and radiological properties of Po and discuss its toxicity. 16.6 A mechanistic study of reaction between chloramine and sulfite has been reported (B.S. Yiin, D.M. Walker, and D.W. Margerum, Inorg. Chem., 1987, 26, 3435). Summarize the observed rate law and the proposed mechanism. Accepting the proposed mechanism, why should SO2(OH) and HSO3− display different rates of reaction? Explain why it was not possible to distinguish the reactivity of SO2(OH) from that of HSO3− . 16.7 Tetramethyltellurium, Te(CH3)4, was prepared in 1989 (R.W. Gedrige, D.C. Harris, K.R. Higa, and R.A. Nissan, Organometallics, 1989, 8, 2817), and its synthesis was soon followed by the preparation of the hexamethyl compound (L. Ahmed and J.A. Morrison, J. Am. Chem. Soc., 1990, 112, 7411). Explain why these compounds are so unusual, give equations for their syntheses, and speculate on why these synthetic procedures are successful. In relation to the last point, speculate on why reaction of TeF4 with methyllithium does not yield tetramethyltellurium. 16.8 The bonding in the square-planar Se2+ ion is described in 4 Section 16.15. Explore this proposition in more detail by carrying out computations, using software of your choice, on S2+ with SS 4 bond distances of 200 pm (sulfur is recommended because its semiempirical parameters are more reliable than those of Se). From the output (a) draw the molecular orbital energy level diagram, (b) assign the symmetry of each level, and (c) sketch the highest energy molecular orbital. Is a closed-shell molecule predicted? 16.9 The nature of the sulfur cycle in ancient times has been investigated (J. Farquhar, H. Bao, and M. Thiemen, Science, 2000, 289, 756). What three factors influence the modern-day cycle? When did the authors establish that a significant change in the cycle had occurred and how were the differences between the ancient and modern cycles explained? 16.10 H. Keppler has investigated the concentration of sulfur in volcano magma (Science, 1999, 284, 1652). In what forms is sulfur erupted from volcanoes? What concentration of sulfur was found in the magma erupted from Mount Pinatubo in 1991? Discuss whether this concentration was expected and how any deviation from the expected value was explained. The Group 17 elements 18 17 All the Group 17 elements are nonmetals. As with the elements in Groups 15 and 16, we shall see that the oxoanions of the halogens are oxidizing agents that often react by atom transfer. There is a useful correlation between the oxidation number of the central atom and the rates of redox reactions. A wide range of oxidation states is observed for most of the halogens. The reactions of the dihalogens are often fast. Part A: The essentials The Group 17 elements, fluorine, chlorine, bromine, iodine, and astatine, are known as the halogens Te I Xe from the Greek for ‘salt giver’. Fluorine and chlorine are poisonous gases, bromine is a toxic, volatile Po At Rn liquid, and iodine is a sublimable solid. They are among the most reactive nonmetallic elements. The chemical properties of the halogens are extensive and their compounds have been mentioned many times already and were reviewed in Section 9.9c. Therefore, in this chapter we highlight their systematic features, their compounds with oxygen, and discuss the interhalogens. 17.6 Reactivity trends H He 1 2 Li Be 13 14 15 16 17 B C N O F Ne S Cl Ar Se Br Kr 17.1 The elements 17.2 Simple compounds 17.3 The interhalogens Part B: The detail 17.4 Occurrence, recovery, and uses 17.5 Molecular structure and properties 17.7 Pseudohalogens 17.8 Special properties of fluorine compounds 17.9 Structural features 17.10 The interhalogens 17.11 Halogen oxides 17.12 Oxoacids and oxoanions 17.13 Thermodynamic aspects of oxoanion redox reactions 17.14 Trends in rates of oxoanion redox reactions PART A: THE ESSENTIALS As we discuss the elements of the penultimate group in the p block, we shall see that many of the systematic themes that were helpful for discussing preceding groups again prove useful. For instance, the VSEPR model can be used to predict the shapes of the wide range of molecules that the halogens form among themselves, with oxygen, and with xenon. 17.1 The elements Key points: Except for fluorine and the highly radioactive astatine, the halogens exist with oxidation numbers ranging from 1 to 7; the small and highly electronegative fluorine atom is effective in oxidizing many elements to high oxidation states. The atomic properties of the halogens are listed in Table 17.1; they all have the valence electron configuration ns2np5. The features to note include their high ionization energies and their high electronegativities and electron affinities. Their electron affinities are high because the incoming electron can occupy an orbital of an incomplete valence shell and experience a strong nuclear attraction: recall that Zeff increases progressively across the period (Section 1.6). We have seen when discussing the earlier groups in the p block that the element at the head of each group has properties that are distinct from those of its heavier congeners. The anomalies are much less striking for the halogens, and the most notable difference is that F has a lower electron affinity than Cl. Intuitively, this feature seems to be at odds with the high electronegativity of F, but it stems from the larger electron–electron repulsion in the 17.15 Redox properties of individual oxidation states 17.16 Fluorocarbons FURTHER READING EXERCISES PROBLEMS 420 17 The Group 17 elements F(g) + Na+(g) + e–(g) Table 17.1 Selected properties of the elements +78.99 1 2 F2(g) + Na+(g) + e–(g) Covalent radius/pm –334.38 1 2 F2(g) + Na(s) (a) At 71 99 114 133 140 181 196 220 1251 1139 1008 926 Melting point/C 220 101 7.2 114 302 Boiling point/C 188 34.7 58.8 184 4.0 3.2 3.0 2.6 2.2 Electron affinity/(kJ mol ) 328 349 325 295 270 E O (X2,X)/V 3.05 1.36 1.09 0.54 1 –573.65 I 131 Pauling electronegativity –1505.59 Br 1681 1 F–(g) + Na+(g) Cl First ionization energy/(kJ mol ) Ionic radius/pm +609.36 F NaF(s) Cl(g) + Na+(g) + e–(g) +121.88 1 2 Cl2(g) + Na+(g) + e–(g) –354.81 – (g) + Na+(g) Cl +609.36 1 2 Cl2(g) + Na(s) –787.38 –411.15 (b) NaCl(s) Figure 17.1 Thermochemical cycles for (a) sodium fluoride and (b) sodium chloride. All values are in kilojoules per mole (kJ mol1). compact F atom as compared with the larger Cl atom. This electron–electron repulsion is also responsible for the weakness of the FF bond in F2. Despite this difference in electron affinity, the enthalpies of formation of metal fluorides are generally much greater than those of metal chlorides because the low electron affinity of F is more than offset by the high lattice enthalpies of ionic compounds containing the small F ion (Fig. 17.1) and the strengths of bonds in covalent species (for example, the fluorides of metals in high oxidation states). Fluorine is a pale yellow gas that reacts with most inorganic and organic molecules and the noble gases Kr, Xe, and Rn (Section 18.6). Consequently, it is very difficult to handle but it can be stored in steel or monel metal (a nickel/copper alloy) as these alloys form a passivating metal fluoride surface film. Chlorine is a green–yellow toxic gas. Bromine is the only liquid nonmetallic element at room temperature and pressure and is dark red, toxic, and volatile. Iodine is a purple–grey solid that sublimes to a violet vapour. The violet colour persists when it is dissolved in nonpolar solvents such as CCl4. However, in polar solvents it dissolves to give red–brown solutions, indicating the presence of polyiodide ions such as I3 (Section 17.10c). The halogens are so reactive that they are found naturally only as compounds. They occur mainly as halides, but the most easily oxidized element, I, is also found as sodium or potassium iodate, KIO3, in alkali metal nitrate deposits. Because many chlorides, bromides, and iodides are soluble, these anions occur in the oceans and in brines. The primary source of F is calcium fluoride, which has low solubility in water and is often found in sedimentary deposits (as fluorite, CaF2). Chlorine occurs as sodium chloride in rock salt. Bromine is obtained by the displacement of the element from brine by chlorine. Iodine accumulates in seaweed, from which it can be extracted. B OX 17.1 Preparation of anhydrous metal halides Because anhydrous halides are common starting materials for inorganic syntheses, their preparation and the removal of water from impure commercial halides is of considerable practical importance. The principal synthetic methods involve the direct reaction of a metal and a halogen, and the reaction of halogen compounds with metals or metal oxides. An example of direct reaction is shown in Fig. B17.1, in which the gaseous halogen reacts with the metal in a heated tube. When the halide is volatile at the temperature of the reaction, it sublimes to the exit end of the tube. Tube furnace N2 Ga Br2/N2 GaBr3 Fig. B17.1 A tube furnace used for the synthesis of anhydrous gallium bromide. It is sometimes advantageous to use a halogen compound for the synthesis of a metal halide. For example, the reaction of ZrO2 with CCl4 vapour in a heated tube is a good method for the preparation of ZrCl4. A contribution to the Gibbs energy of reaction comes from the production of the C O bond in phosgene, COCl2: ZrO2 (s) + 2CCl4 (g) → ZrCl4 (s) + 2COCl2 (g) Anhydrous halides can often be prepared by the removal of water from hydrated metal chlorides. Simply heating the sample in a dry gas stream is usually not satisfactory for metals with high charge-to-radius ratio because significant hydrolysis may lead to the oxide or oxohalide: 2CrCl3 .6H2O(s) → Cr2O3 (s) + 6HCl(g) + 9H2O(g) This hydrolysis can be suppressed by carrying out the dehydration in a heated tube in a stream of hydrogen halide, or by dehydration with thionyl chloride (b.p. 79C), which produces the desired anhydrous chloride and volatile HCl and SO2 byproducts: FeCl3 .6H2O(s) + 6SOCl2 (l) → FeCl3 (s) + 6SO2 (g) + 12HCl(g) The essentials 17.2 Simple compounds Key points: All the halogens form hydrogen halides; HF is a liquid and HCl, HBr, and HI are gases. All the Group 17 elements form oxo compounds and oxoanions. Because fluorine is the most electronegative of all elements, it is never found in a positive oxidation state (except in the transient gas-phase species F2). With the possible exception of At, the other halogens occur with oxidation numbers ranging from 1 to 7. Compounds of Br(VII) are very unstable compared to those of Cl and I, this being yet another example of the alternation effect (Section 9.2c). The dearth of chemical information on At stems from its lack of any stable isotopes and the relatively short half-life (8.3 hours) of the most long-lived of its 33 known isotopes. Astatine solutions are intensely radioactive and can be studied only in high dilution. Astatine appears to exist as the anion At and as At(I) and At(III) oxoanions; no evidence for At(VII) has yet been obtained. The high electronegativity of F leads to enhanced Brønsted acidity in compounds containing the element compared to nonfluorine analogues. A combination of high electronegativity and small atomic radius, which means that many atoms can pack round a central atom, results in F being able to stabilize high oxidation states of most elements, for example UF6 and IF7. (Oxygen, however, can often achieve more because it counts for 2 in the calculation of the oxidation number whereas F counts for only 1.) The metal fluorides in low oxidation states tend to be predominantly ionic whereas extensive covalent character is found in metal chlorides, bromides, and iodides. The synthesis of fluorocarbon compounds, such as polytetrafluoroethene (PTFE), is of great technological importance because these compounds are useful in applications ranging from coatings for nonstick cookware and halogen-resistant laboratory vessels to the volatile fluorocarbons used as refrigerants in air conditioners and refrigerators. Fluorocarbon derivatives have also been the topic of considerable exploratory synthetic research because their derivatives often have unusual properties. Hydrofluorocarbons are used as replacements for chlorofluorocarbons as refrigerants and propellants. They are also used as anaesthetics, where they are less flammable than their nonfluorinated counterparts. Tetrafluorethane, CHF2CHF2, is an important solvent for the extraction of natural products such as vanilla and taxol, which is used in chemotherapy. All the Group 17 elements form protic molecular hydrides, the hydrogen halides. Largely on account of its ability to participate in extensive hydrogen bonding, the properties of HF contrast starkly with those of the other hydrogen halides (Table 17.2). Because hydrogen bonding (Section 10.2) is extensive, HF is a volatile liquid whereas HCl, HBr, and HI are gases at room temperature. Hydrogen fluoride has a wide liquid range, a high relative permittivity, and high conductivity. All the hydrogen halides are Brønsted acids: aqueous HF is a weak acid (‘hydrofluoric acid’) whereas HCl, HBr, and HI are all essentially fully deprotonated in water (Table 17.2). Although hydrofluoric acid is a weak acid it is one of the most toxic and corrosive substances known, being able to attack glass, metals, concrete, and organic matter. It is much more hazardous to handle than other acids because it is very readily absorbed through the skin, and even brief contact can lead to severe burning and necrosis of the skin and deep tissue, and damage to bone by decalcification by formation of CaF2 from calcium phosphate. Table 17.2 Selected properties of the hydrogen halides HF HCl HBr HI Melting point/C 84 114 89 51 Boiling point/C 20 85 67 35 Relative permittivity 83.6 (at 0C) 9.3 (at 95C) 7.0 (at 85C) 3.4 (at 50C) Electrical conductivity/ (S cm1) c. 106 (at 0C) c. 109 (at 85C) c. 109 (at 85C) c. 1010 (at 50C) fG O /(kJ mol1) 273.2 95.3 54.4 1.72 Bond dissociation energy/(kJ mol1) 567 431 366 298 pKa 3.45 c. 7 c. 9 c. 11 421 422 17 The Group 17 elements Table 17.3 Selected oxides of chlorine Oxidation number 1 3 4 6 7 Formula Cl2O Cl2O3 ClO2 Cl2O4 Cl2O6 Cl2O7 Colour brown–yellow dark brown yellow pale yellow dark red colourless State gas solid gas liquid liquid liquid Many binary compounds of the halogens and oxygen are known, but most are unstable and not commonly encountered in the laboratory. We shall mention only a few of the most important. Oxygen difluoride, OF2, is the most stable oxide of F but decomposes above 100C. Chlorine occurs with many different oxidation numbers in its oxides (Table 17.3). Some of these oxides are odd-electron species, including ClO2, in which Cl has the unusual oxidation number 4, and Cl2O6, which exists as a mixed oxidation state ionic solid, [ClO2] [ClO4]. All the chlorine oxides are endergonic (fG O > 0) and unstable. They all explode when heated. There are fewer oxides of Br than Cl. The best characterized compounds are Br2O, Br2O3, and BrO2. The oxides of I are the most stable halogen oxides. The most important of these is I2O5. Both BrO and IO (which are odd-electron species) have been implicated in ozone depletion and both are produced naturally by volcanic activity. All the Group 17 elements form oxoanions and oxoacids. The wide range of oxoanions and oxoacids of the halogens presents a challenge to those who devise systems of nomenclature. We shall use the common names, such as chlorate for ClO3, rather than the systematic names, such as trioxidochlorate(V). Table 17.4 lists the oxoanions of Cl with common and systematic nomenclature. The strength of the acids can be predicted by using Pauling’s rules (Section 4.5b). All the oxoanions are strong oxidizing agents. ■ A brief illustration. To use Pauling’s rules we write perchloric acid, HClO4, as O3Cl(OH). The rules then predict that pKa 8 5p, with p 3. Thus the pKa of perchloric acid is predicted to be 7 (corresponding to a strong acid). ■ 17.3 The interhalogens Key point: All the halogens form compounds with other members of the group. The interhalogens are formed between Group 17 elements. The binary interhalogens are molecular compounds with formulas XY, XY3, XY5, and XY7, where the heavier, less electronegative halogen X is the central atom. They also form ternary interhalogens of the type XY2Z and XYZ2, where Z is also a halogen atom. The interhalogens are of special importance as highly reactive intermediates and for providing useful insights into bonding. Table 17.4 Halogen oxoanions Oxidation number Formula Name Point group Shape Remarks 1 ClO Hypochlorite [monoxidochlorate(I)] C∞v Linear Good oxidizing agent 2 ClO2− Chlorite [dioxidochlorate(III)] C2v Angular Strong oxidizing agent, disproportionates 5 ClO3− Chlorate [trioxidochlorate(V)] C3v Pyramidal Oxidizing agent 7 ClO 4− Perchlorate [tetraoxidochlorate(VII)] Td Tetrahedral Oxidizing agent, very weak ligand IUPAC names in square brackets. The essentials 2π 2π 1π Signal intensity The diatomic interhalogens, XY, have been made for all combinations of the elements, but many of them do not survive for long. All the F interhalogen compounds are exergonic (fG O < 0). The least labile interhalogen is ClF, but ICl and IBr can also be obtained in pure crystalline form. Their physical properties are intermediate between those of their component elements. For example, the deep red -ICl (m.p. 27C, b.p. 97C) is intermediate between yellowish–green Cl2 (m.p. 101C, b.p. 35C) and dark purple I2 (m.p. 114C, b.p. 184C). Photoelectron spectra indicate that the molecular orbital energy levels in the mixed dihalogen molecules lie in the order 3 2 < 1 4 < 2 4, which is the same as in the homonuclear dihalogen molecules (Fig. 17.2). An interesting historical note is that ICl was discovered before Br2 in the early nineteenth century, and, when later the first samples of the dark red–brown Br2 (m.p. 7C, b.p. 59C) were prepared, they were mistaken for ICl. Most of the higher interhalogens are fluorides (Table 17.5). The only neutral interhalogen with the central atom in a 7 oxidation state is IF7, but the cation ClF6, a compound of Cl(VII), is known. The absence of a neutral ClF7 reflects the destabilizing effect of nonbonding electron repulsions between F atoms (indeed, coordination numbers greater than six are not observed for other p-block central atoms in Period 3). The lack of BrF7 might be rationalized in a similar way, but in addition we shall see later that bromine is reluctant to achieve its maximum oxidation state. This is another manifestation of the alternation effect (Section 9.2c). In this respect, it resembles some other Period 4 p-block elements, notably arsenic and selenium. The shapes of interhalogen molecules (1), (2), and (3) are largely in accord with the VSEPR model (Section 2.3). For example, the XY3 compounds (such as ClF3) have five valence electron pairs around the X atom in a trigonal–bipyramidal arrangement. The Y atoms attach to the two axial pairs and one of the three equatorial pairs, and then the two axial bonding pairs move away from the two equatorial lone pairs. As a result, XY3 molecules have a C2v bent T shape. There are some discrepancies: for example, ICl3 is a Cl-bridged dimer. The Lewis structure of XF5 has five bonding pairs and one lone pair on the central X atom and, as expected from the VSEPR model, XF5 molecules are square pyramidal. As already mentioned, the only known XY7 compound is IF7, which is predicted to be pentagonal bipyramidal. The experimental evidence for its actual structure is inconclusive. As with other hypervalent molecules, the bonding in IF7 can be explained without invoking d-orbital participation by adopting a molecular orbital model in which bonding and nonbonding orbitals are occupied but antibonding orbitals are not. Polymeric interhalogens may also be formed and may be cationic or anionic. Examples of cationic polyhalides are I3 (4) and I5 (5). Anionic polyhalides are most numerous for iodine. The I3 ion is the most stable but others with the general formula [(I2)nI] are formed. Other anionic polyhalides include Cl3 and BrF4. 3σ 16 14 12 10 8 Ionization energy, I/eV Figure 17.2 Photoelectron spectrum of ICl. The 2π levels give rise to two peaks because of spin-orbit interaction in the positive ion. Table 17.5 Representative interhalogens XY XY3 XY5 ClF ClF3 ClF5 BrF BrF3 BrF5 IF (IF3)n IF5 BrCl I2Cl6 ICl IBr very unstable F F F I l C Br 1 ClF3, C2v 3 IF7, D5h 2 BrF5, C4v I 90° 290 pm 268 pm 0 12° 4 I+3, C2v 268 pm 5 I5+ 423 XY7 IF7 424 17 The Group 17 elements PART B: THE DETAIL Key points: Fluorine, chlorine, and bromine are prepared by electrochemical oxidation of halide salts; chlorine is used to oxidize Br and I to the corresponding dihalogen. All the dihalogens (except the radioactive At2) are produced commercially on a large scale, with chlorine production by far the greatest, followed by fluorine. The principal method of production of the elements is by electrolysis of the halides (Section 5.18). The strongly positive standard potentials E O (F2,F) 2.87 V and E O (Cl2,Cl) 1.36 V indicate that the oxidation of F and Cl ions requires a strong oxidizing agent. Only electrolytic oxidation is commercially feasible. An aqueous electrolyte cannot be used for fluorine production because water is oxidized at a much lower potential (1.23 V) and any fluorine produced would react rapidly with water. The isolation of elemental fluorine is achieved by electrolysis of a 1:2 mixture of molten KF and HF in a cell like that shown in Fig. 17.3. It is important to keep fluorine and the byproduct, hydrogen, separate because they react violently. Most commercial chlorine is produced by the electrolysis of aqueous sodium chloride solution in a chloralkali cell (Fig. 17.4). The half-reactions are Anode half-reaction: 2Cl – (aq) → Cl 2 (g) + 2 e – Cathode half-reaction: 2 H 2O(l) + 2 e – → 2OH – (aq) + H 2 (g) The oxidation of water at the anode is suppressed by using an electrode material that has a higher overpotential for O2 evolution than for Cl2 evolution (Section 5.18). The best anode material seems to be RuO2 (Section 19.8). This process is the basis of HF inlet F2 outlet H2 outlet 35 per cent NaOH(aq) Anode (+) 17.4 Occurrence, recovery, and uses H2 outlet H2 Cl2 Brine feed Membrane M Me emb mbrra an ne e In this section we look in some detail at the chemistry of the halogens. Most elements form halides with the halogens and these have been dealt with in individual group chapters so our focus here is particularly on the interhalogens and the halogen oxides. nt Spent e brine Na Carbon anode (+) HF/KF electrolyte D Dilute N NaOH(aq) Figure 17.4 Schematic diagram of a chloralkali cell using a cation transport membrane, which has high permeability to Na ions and low permeability to OH and Cl ions. the chloralkali industry, which produces sodium hydroxide on a massive scale (see Box 11.2): 2 NaCl(s) + 2 H 2O(l) → 2 NaOH(aq) + H 2 (g) + Cl2 (g) Bromine is obtained by the chemical oxidation of Br ions in seawater. A similar process is used to recover iodine from certain natural brines that are rich in I. The more strongly oxidizing halogen, chlorine, is used as the oxidizing agent in both processes, and the resulting Br2 and I2 are driven from the solution in a stream of air: ⎯→ 2Cl − (aq) + X2 (g) Cl2 (g) + 2 X − (aq) ⎯air (X = Br or I) E X A M PL E 17.1 Analysing the recovery of Br2 from brine Show that from a thermodynamic standpoint bromide ions can be oxidized to Br2 by Cl2 and by O2, and suggest a reason why O2 is not used for this purpose. Answer We need to consider the relevant standard potentials and recall that a redox couple can be driven in the direction of oxidation by a couple with a more positive standard potential. The two half-reactions we need to consider for oxidation by chlorine are Cl2(g) 2 e → 2 Cl(aq) Br2(g) 2 e → 2 Br(aq) E O 1.358 V E O 1.087 V Because E O (Cl2,Cl) E O (Br2,Br), chlorine can be used to oxidize Br in the reaction Cl2 (g) + 2Br − (aq) → 2Cl− (aq) + Br2 (g) Steel cathode (–) Cathode (–) Ecell O 0.271 V To encourage the formation of bromine, the Br2 is removed in a steamair mixture. Oxygen would be thermodynamically capable of carrying out this reaction in acidic solution: O2(g) 4 H(aq) 4 e → 2 H2O(l) Br2(l) 2 e → 2 Br E O 1.229 V E O 1.087 V Resulting in Figure 17.3 Schematic diagram of an electrolysis cell for the production of fluorine from KF dissolved in liquid HF. O2 (g) + 4Br − (aq) + 4H+ (aq) → 2H2O(l) + 2Br2 (l) Ecell O 0.142 V The detail Self-test 17.1 One source of iodine is sodium iodate, NaIO3. Which of the reducing agents SO2(aq) or Sn2(aq) would seem practical from the standpoints of thermodynamic feasibility and plausible judgements about cost? Standard potentials are given in Resource section 3. All the Group 17 elements undergo thermal or photochemical dissociation in the gas phase to form radicals. These radicals take part in chain reactions, such as: /hv → X⋅+ X⋅ X2 ⎯⎯⎯ H 2 + X⋅→ HX + H ⋅ H ⋅+ X2 → HX + X⋅ A reaction of this type between chlorine and methane is used in the industrial synthesis of chloroform, CH3Cl, and dichloromethane, CH2Cl2. Compounds of F are used throughout industry. Fluorine as F ions is added to some domestic water supplies and toothpaste to prevent tooth decay (Box 17.2). It is used as UF6 in the nuclear power industry for the separation of the isotopes of uranium. Hydrogen fluoride is used to etch glass and as a nonaqueous solvent. Chlorine is widely used in industry to make chlorinated hydrocarbons and in applications in which a strong oxidizing agent is needed, including disinfectants and bleaches. These applications are in decline, however, because some organic Cl compounds are carcinogenic and chlorofluorocarbons (CFCs) are implicated in the destruction of ozone in the stratosphere (Box 17.3). Hydrofluorocarbons (HFCs) are now replacing CFCs in applications such as refrigeration and air conditioning. Organobromine compounds are used in synthetic organic chemistry: the CBr bond is not as strong as the CCl bond and Br can be more readily displaced (and recycled). Iodine is an essential element and iodine deficiency is a cause of goitre, the enlargement of the thyroid gland. For this reason, small amounts of potassium iodide are added to table salt. 17.5 Molecular structure and properties Key points: The FF bond is weak relative to the ClCl bond; bond strengths decrease down the group from chlorine. Among the most striking physical properties of the halogens are their colours. In the vapour they range from the almost colourless F2, through yellow–green Cl2 and red–brown Br2, to purple I2. The progression of the maximum absorption to longer wavelengths reflects the decrease in the HOMO–LUMO gap on descending the group. In each case, the optical absorption spectrum arises primarily from transitions in which an electron is promoted from the highest filled 2 g and 1 g orbitals into the vacant antibonding 2 u orbital (Fig. 17.5). Except for F2, the analysis of the UV absorption spectra gives precise values for the dihalogen bond dissociation energies (Fig. 17.6). It is found that bond strengths decrease down the group from Cl2. The UV spectrum of F2, however, is a broad 2σu 1πg 3p Energy but the reaction is not favourable at pH 7, when Ecell O 0.15 V. Even though the reaction is thermodynamically favourable in acidic solution, it is doubtful that the rate would be adequate because an overpotential of about 0.6 V is associated with the reactions of O2 (Section 5.18). Even if the oxidation by O2 in acidic solution were kinetically favourable, the process would be unattractive because of the cost of acidifying large quantities of brine and then neutralizing the effluent. 425 3p 2σg 1πu 1σu 3s 3s 1σg Figure 17.5 Schematic molecular orbital energy level diagram for Cl2 (similarly Br2 and I2). For F2 the order of the u and upper g orbitals is reversed. B OX 17. 2 Fluoridation of water and dental health In the first half of the twentieth century extensive tooth decay was prevalent among most of the population in the developed world. In 1901, F.S. McKay, a Colorado dentist, noticed that many of his patients had a mottled brown stain on their tooth enamel. He also noticed that these patients seemed to be less affected by tooth decay. He suspected that something in the local water supply was responsible. It was not until the 1930s, when analytical science had advanced, that McKay’s suspicions were confirmed and high levels (12 ppm) of fluoride ions were identified in the local water supply. Studies confirmed inverse relationship between the appearance of the mottled brown stain (which is termed fluorosis) and the occurrence of dental cavities. It was found that fluoride ion levels of up to 1 ppm decreased the occurrence of tooth cavities without causing fluorosis. This observation led to widespread fluoridation of drinking water supplies in the western world, which has drastically reduced the level of tooth decay in all sectors of the population. Fluoride has also been added to toothpastes, mouthwashes, and even to table salt. Research suggests that fluoride prevents dental cavities by inhibition of demineralization and inhibition of bacterial activity in dental plaque. Enamel and dentine are composed of hydroxyapatite, Ca5(PO4)3OH. The hydroxyapatite is dissolved by acids present in food, or produced by bacterial action on food. The fluoride ions form fluoroapatite, Ca5(PO4)3F, with the enamel, which is less soluble in acid than the hydroxyapatite. The fluoride ions are also taken up by the bacteria, where they disrupt enzyme activity and reduce acid production. The fluoridation of public water supplies is controversial. Opponents have claimed that it is linked to increased risk of cancer, Down’s syndrome, and heart disease, although, thus far, there has been no convincing evidence to support these claims. Opponents also claim that mass fluoridation infringes civil liberties and that the long-term effects of increased fluoride levels in river habitats is not yet known. 426 17 The Group 17 elements B OX 17. 3 Chlorofluorocarbons and the ozone hole The ozone layer extends from 10 km to 50 km above the Earth’s surface and plays a crucial role in protecting us from the harmful effects of the Sun’s ultraviolet rays by absorbing radiation at wavelengths below 300 nm, thereby attenuating the spectrum of sunlight at ground level. Ozone is produced naturally by the action of UV radiation on O2 in the upper atmosphere: hν O2 ⎯⎯→ O O O O2 → O3 H2O + XONO2 → HOX + HNO3 When ozone absorbs an ultraviolet photon, it dissociates: where X Cl or Br. They also react with co-adsorbed HCl or HBr (formed by attack of Cl and Br on the methane escaping from the troposphere): HX + XONO2 → X 2 + HNO3 hν O3 ⎯⎯→ O2 O The resulting O atom can then remove ozone in the reaction O3 O → O2 O2 These reactions constitute the principal steps of the oxygenozone cycle that maintains an equilibrium (but seasonally varying) concentration of ozone. If all the atmospheric O3 were condensed into a single layer at 1 atm and 25C, it would cover the Earth to a depth of about 3 mm. The stratosphere also contains naturally occurring species, such as the hydroxyl radical and nitric oxide, that catalyse the destruction of ozone by reactions such as X + O3 → XO + O2 XO + O → X + O2 However, the main concern about the loss of ozone centres on Cl and Br atoms introduced artificially by industrial activities, which catalyse O3 destruction very efficiently. Chlorine and bromine are carried into the stratosphere as part of organohalogen molecules, RHal, which release the halogen atoms when the CHal bond is fragmented by far-UV photons. The ozone-destroying potential of these molecules was pointed out in 1974 by Mario Molina and Sherwood Rowland, who won the 1995 Nobel Prize in chemistry (together with Paul Crutzen) for their work. International action followed 13 years later (in the form of the 1987 Montreal Protocol), and was given added impetus by the discovery of the ‘ozone hole’ over Antarctica, which provided dramatic evidence of the Bond dissociation enthalpy/(kJ mol–1) vulnerability of atmospheric ozone. This hole surprised even the scientists working on the problem; its explanation required additional chemistry, involving the polar stratospheric clouds that form in winter. The ice crystals in these clouds adsorb molecules of chlorine or bromine nitrate, ClONO2 or BrONO2, which form when stratospheric ClO or BrO combine with NO2. Once on the ice surface, these molecules react with water: 243 193 159 F2 151 Cl2 Br2 I2 Figure 17.6 Bond dissociation enthalpies of the halogens in kilojoules per mole (kJ mol1). The nitric acid, being very hygroscopic, enters the ice crystals and the HOX or X2 molecules are released during the dark polar winter. When the sunlight strengthens in the spring, these molecules photolyse, releasing high concentrations of ozone-destroying radicals. ν HOX ⎯h⎯ → HO ⋅ + X⋅ hν X 2 ⎯⎯→ 2 X ⋅ To threaten the ozone layer, organohalogen molecules must survive their migration from the Earth’s surface. Those containing H atoms are mostly broken down in the troposphere (the lowest region of the atmosphere) by reaction with HO radicals. Even so, they may be a problem if released in sufficient amounts. There is currently a major controversy over the use of bromomethane, CH3Br, as an agricultural fumigant. However, the greatest potential for ozone destruction rests with molecules that lack H atoms, the chlorofluorocarbons (CFCs), which have been used in many industrial applications, and their brominated analogues (the halons), which are used to extinguish fires. These compounds have no tropospheric sink and eventually reach the stratosphere unaltered. They are the main focus of the international regulatory regime worked out in 1987 (and amended in 1990 and 1992). Most CFCs and halons have been phased out of production, and their atmospheric concentrations are beginning to decline. The CFCs present an additional problem as they are also potent greenhouse gases. continuum that lacks structure because absorption is accompanied by dissociation of the F2 molecule. The lack of discrete absorption bands makes it difficult to estimate the dissociation energy spectroscopically, and thermochemical methods are complicated by the highly corrosive nature of this reactive halogen. When these problems were solved, the FF bond enthalpy was found to be less than that of Br2 and thus out of line with the trend in the group. However, the low FF bond enthalpy is consistent with the low single-bond enthalpies of NN, OO, and various combinations of N, F, and O (Fig. 17.7). The simplest explanation (like the explanation of the low electron affinity of fluorine) is that the bond is weakened by the strong repulsions between nonbonding electrons in the small F2 molecule. In molecular orbital terms, the molecule has numerous electrons in strongly antibonding orbitals. Chlorine, bromine, and iodine all crystallize in lattices of the same symmetry (Fig. 17.8), so it is possible to make a detailed comparison of distances between bonded and nonbonded adjacent atoms (Table 17.6). The important conclusion is that nonbonded distances do not increase as rapidly as the bond lengths. 427 The detail C F Cl Br I –1 D/(kJ mol ) 600 400 A A High vacuum (a) A A A A A A (b) 200 0 0.4 E 0.6 100/(R/pm) F B (c) 0.8 B G To pump D 1 To fluorine tank Figure 17.9 A typical metal vacuum system for handling fluorine and reactive fluorides. Nickel tubing is used throughout. (A) Monel valves, (B) nickel U-traps, (C) monel pressure gauge, (D) nickel container, (E) PTFE reaction tube, (F) nickel reaction vessel, and (G) nickel canister filled with soda lime to neutralize HF and react with F2 and fluorine compounds. Figure 17.7 Dissociation enthalpies of (a) carbonhalogen, (b) hydrogenhalogen, and (c) halogenhalogen bonds plotted against the reciprocal of the bond length. materials for the construction of apparatus to contain fluorine and oxidizing fluorine compounds (Fig. 17.9). Few laboratories have the equipment and expertise for research involving elemental F2. The standard potentials for the halogens (Table 17.1) indicate that F2 is a much stronger oxidizing agent than Cl2. The decrease in oxidizing strength continues in more modest steps from Cl2 through Br2 to I2. Although the half-reaction Figure 17.8 Solid chlorine, bromine, and iodine have similar structures. The closest nonbonded interactions are relatively less compressed in Cl2 and Br2 than in I2. Table 17.6 Bonding and shortest nonbonding distances for solid dihalogens Element Temperature/C Bond length/pm Nonbonding distance/pm Ratio Cl2 160 198 332 1.68 Br2 106 227 332 1.46 I2 163 272 350 129 1 2 is favoured by a high electron affinity (which suggests that F should have a lower standard potential than Cl), the process is favoured by the low bond enthalpy of F2 and by the highly exothermic hydration of the small F ion (Fig. 17.10). The net outcome of these three competing effects of size is that F is the most strongly oxidizing element of the group. 17.7 Pseudohalogens Key points: Pseudohalogens and pseudohalides mimic halogens and halides, respectively; the pseudohalogens exist as dimers and form molecular compounds with nonmetals and ionic compounds with alkali metals. This observation suggests the presence of weak intermolecular bonding interactions that strengthen on going from Cl2 to I2. Solid iodine is a semiconductor and under high pressure exhibits metallic conductivity. F(g) + Na+(g) + e–(g) Cl(g) + Na+(g) + e–(g) +78.99 1 2 F2(g) + Na+(g) + e–(g) +121.88 –334.38 17.6 Reactivity trends +609.36 Key points: Fluorine is the most oxidizing halogen; the oxidizing power of the halogens decreases down the group. Fluorine, F2, is the most reactive nonmetal and is the strongest oxidizing agent among the halogens. The rapidity of many of its reactions with other elements may in part be due to a low kinetic barrier associated with the weak FF bond. Despite the thermodynamic stability of most metal fluorides, fluorine can be handled in containers made from some metals, such as Ni, because a substantial number of them form a passive metal fluoride surface film on contact with fluorine gas. Fluorocarbon polymers, such as polytetrafluoroethene (PTFE), are also useful X2 (g) + e – → X – (aq) 1 2 – 1 2 Cl2(g) + Na+(g) + e–(g) –354.81 – + +609.36 Cl (g) + Na (g) F (g) + Na (g) F2(g) + Na(s) –572.75 1 2 Cl2(g) + Na(s) –926.72 –407.27 (a) NaF(aq) (b) –783.50 NaCl(aq) Figure 17.10 Thermochemical cycles for the enthalpy of formation of (a) aqueous sodium fluoride and (b) aqueous sodium chloride. The hydration is much more exothermic for F than for Cl. All values are in kilojoules per mole (kJ mol1). 428 17 The Group 17 elements Table 17.7 Pseudohalides, pseudohalogens, and corresponding acids Pseudohalide Pseudohalogen E O /V Acid pKa CN NCCN 0.27 HCN 9.2 Cyanide Cyanogen Si Hydrogen cyanide 0.77 NCS NCSSCN Thiocyanate Dithiocyanogen HNCS HNCO Cyanate Isocyanic acid CNO HCNO Fulminate Fulminic acid NNN HNNN Azide Hydrazoic acid 1.9 3.5 3.66 7 (CH3)3SiCl 4.92 A number of compounds have properties so similar to those of the halogens that they are called pseudohalogens (Table 17.7). For example, like the dihalogens, cyanogen, (CN)2, undergoes thermal and photochemical dissociation in the gas phase; the resulting CN radicals are isolobal with halogen atoms and undergo similar reactions, such as a chain reaction with hydrogen: heat or light NC—CN ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 2CN· HCN + H· H 2 + CN· → HCN + CN· H· + NC—CN → Overall: H2C2N2 → 2 HCN Another similarity is the reduction of a pseudohalogen: (CN)2 (aq)(aq) + e — → CN — (aq) The anion formally derived from a pseudohalogen is called a pseudohalide ion. An example is the cyanide anion, CN. Covalent pseudohalides similar to the covalent halides of the p-block elements are also common. They are often structurally similar to the corresponding covalent halides (compare (6) and (7)), and undergo similar metathesis reactions. As with all analogies, the concepts of pseudohalogen and pseudohalide have many limitations. For example, pseudohalogen ions are not spherical, so the structures of their ionic compounds often differ: NaCl is fcc but NaCN is similar to CaC2 (Section 11.12). The pseudohalogens are generally less electronegative than the lighter halogens and some pseudohalides have more versatile donor properties. The thiocyanate ion, SCN, for instance, acts as an ambidentate ligand with a soft base site, S, and a hard base site, N (Section 4.15). 17.8 Special properties of fluorine compounds Key points: Fluorine substituents promote volatility, increase the strengths of Lewis and Brønsted acids, and stabilize high oxidation states. The boiling points in Table 17.8 demonstrate that molecular compounds of F tend to be highly volatile, in some cases even more volatile than the corresponding hydrogen compounds (compare, for example, PF3, b.p. 101.5C, and PH3, b.p. 87.7C) and in all cases much more volatile than the Cl analogues. The volatilities of the compounds are a result of variations in the strength of the dispersion interaction (the interaction between instantaneous transient electric dipole moments), which is strongest for highly polarizable molecules. The electrons in the small F atoms are gripped tightly by the nuclei, and consequently F compounds have low polarizabilities and hence weak dispersion interactions. There are some opposite effects on volatility that can be traced to hydrogen bonding. The structure of solid HF is a planar zigzag chain polymer of FH…F units. Although liquid HF has a lower density and viscosity than water, which suggests the absence of an extensive three-dimensional network of H bonds, in the gas phase HF forms H-bonded oligomers, (HF)n with n up to 5 or 6. As with H2O and NH3, the properties of HF, such as a wide liquid range, make it an excellent nonaqueous solvent. Hydrogen fluoride undergoes autoprotolysis: 2 HF(l) H2F(sol) F(sol) CN Si CH3 Hydrogen thiocyanate NCO 1 2 Cl CH3 pKauto 12.3 It is a much weaker acid (pKa 3.45 in water) than the other hydrogen halides. Although this difference is sometimes attributed to the formation of an ion pair (H3OF), theoretical considerations show that its poor proton donor properties are a Table 17.8 Normal boiling points (in C) of compounds of fluorine and their analogues 6 (CH3)3SiCN F2 188.2 H2 252.8 Cl2 CF4 127.9 CH4 161.5 CCl4 34.0 76.7 PF3 101.5 PH3 87.7 PCl3 75.5 The detail direct result of the very strong HF bond. Carboxylic acids act as bases in anhydrous HF and are protonated: HCOOH(l) + 2 HF(l) → HC(OH)2+ (sol) + HF2– (sol) An important characteristic is the ability of an F atom in a compound to withdraw electrons from the other atoms present and, if the compound is a Brønsted acid, to enhance its acidity. An example of this effect is the increase by three orders of magnitude in the acidity of trifluoromethanesulfonic acid, HOSO2CF3 (pKa 3.0 in nitromethane), over that of methanesulfonic acid, HOSO2CH3 (pKa 6.0 in nitromethane). The presence of F atoms in a molecule also results—for the same reason—in an enhanced Lewis acidity. For example, we saw in Sections 4.15 and 15.11b that SbF5 is one of the strongest Lewis acids of its type and much stronger than SbCl5. Some examples of high oxidation state compounds of F are IF7, PtF6, BiF5, KAgF4, UF6, and ReF7. Rhenium(VII) heptafluoride is the only example of a thermally stable metal heptafluoride and uranium(VI) hexafluoride is important in the separation of U isotopes in the preprocessing of nuclear fuels. All these compounds are examples of the highest oxidation state attainable for these elements, the rare oxidation state Ag(III) being perhaps the most notable. Another example is the stability of PbF4, compared to all other Pb(IV) halides. A related phenomenon is the tendency of fluorine to disfavour low oxidation states. Thus, solid copper(I) fluoride, CuF, is unstable but CuCl, CuBr, and CuI are stable with respect to disproportionation. Similar trends were discussed in Section 3.11 in terms of a simple ionic model in which the small size of the F ion in combination with a small, highly charged cation results in a high lattice enthalpy. As a result, there is a thermodynamic tendency for CuF to disproportionate and form copper metal and CuF2 (because Cu2 is doubly charged and its ionic radius is smaller than that of Cu and it has a greater lattice enthalpy). Compounds that accept F ions are Lewis acids and compounds that donate F are Lewis bases: SbF5 (s) + HF(l) → SbF6 (sol) + H (sol) XeF6 (s) + HF(l) → XeF5+ (sol) + HF2 − (sol) − Ionic fluorides dissolve in HF to give highly conducting solutions. The fact that chlorides, bromides, and iodides react with HF to give the corresponding fluoride and HX provides a preparative route to anhydrous fluorides: TiCl4 (l) + 4 HF(l) → TiF4 (s) + 4 HCl(g) 17.9 Structural features Metal difluorides, MF2, where M is a Group 2 or d-metal, generally adopt the CaF2 or rutile structures and are described well by the ionic model. In contrast, whereas the Group 2 dichlorides, dibromides, and diiodides may be described by the ionic model, the d-metal analogues adopt the CdI2 or CdCl2 layer structures and their bonding is not described well by either the ionic or covalent models. Many metal trifluorides have three-dimensional ionic structures but the trichlorides, tribromides, and triodides have layered structures. The compounds NbF3 and FeF3 (at high temperature) adopt the ReO3 structure type (Section 3.6) and many other metal trifluorides (including AlF3, ScF3, and CoF3) have a slightly distorted variant of this structure type. 429 As the oxidation number of the metal atom increases, the halides become more covalent. Thus all metal hexahalides such as MoF6 and WCl6, are molecular covalent compounds. For intermediate oxidation states (such as MF4 and MF5) the structures normally consist of linked MF6 polyhedra. Titanium tetrafluoride has a structure based on columns of triangular Ti3F15 units formed from three TiF6 octahedra (8) whereas NbF5 is built from four NbF6 octahedra forming a square unit of composition Nb4F20 (9). Ti F 8 Ti3F15 F Nb 9 Nb4F20 Although not as important in applications as complex oxides, complex solid fluorides and chlorides such as the ternary phases MMFn and MMCln, and the quaternary compounds MMMFn, have structures similar to their oxide counterparts. As F has an oxidation number of 1 compared to 2 for O2, the compositionally equivalent fluorides or chlorides generally contain d metals in lower oxidation states than the equivalent oxide. Thus, ternary fluorides of stoichiometry ABF3 with, for example, A K, Rb, and Cs and M a dipositive d-metal ion, adopt the perovskite structure (Section 3.9). One example is KMnF3, which precipitates when potassium fluoride is added to Mn(II) solutions. Molten cryolite, Na3AlF6, is used to dissolve aluminium oxide in electrochemical extraction of aluminium. Its structure is related to that of perovskite (ABO3) with Na in the A sites and a mixture of Na and Al in the B sites: the formula Na(Al½Na½)F3, which is equivalent to Na3AlF6. Mixed-anion compounds containing halides are also well characterized and include the superconducting cuprate Sr2xNaxCuO2F2. 17.10 The interhalogens The halogens form many compounds between themselves with formulas ranging from XY to XY7 (Table 17.9). Their structures can usually be predicted accurately by the VSEPR rules and verified by techniques such as 19F-NMR. 430 17 The Group 17 elements Table 17.9 Properties of interhalogens XY XY3 XY5 XY7 ClF ClF3 ClF5 Colourless Colourless Colourless m.p.156C m.p. 76C m.p.103C b.p.100C b.p. 12C b.p.13C BrF BrF3 BrF5 Light brown Yellow Colourless m.p. ≈240C m.p. 9C m.p.61C b.p.20C b.p. 126C b.p. 41C IF (IF3)n IF5 IF7 Yellow Colourless Colourless Dec. 28C m.p. 9C b.p. 105C m.p. 6.5 (triple point) F Br 10 BrF5, C4v F Br Subl. 5C BrCl Red-brown m.p. ≈66C b.p. 5C -ICl, -ICl I2Cl6 Ruby red solid, black liquid Bright yellow m.p. 27C, m.p. 14C m.p. 101C (16 atm) b.p. 97100C IBr Black solid m.p. 41C b.p.≈116C Very unstable. dec., decomposes; subl., sublimes. E X A MPLE 17. 2 Predicting the shape of an interhalogen molecule. BrF5 is a fluxional molecule that rapidly interconverts between a square pyramidal (10) and a trigonal-bipyramidal structure (11). If these two structures could be isolated explain how 19F-NMR could be used to differentiate between them. Answer We need to identify the number of different 19F environments in each case and consider how each is coupled to other F environments. In the case of the square pyramid there are two different F environments of four and one F atoms, respectively. The 19F-NMR would show a signal equivalent to four F atoms split into a doublet by coupling to the other one F atom. A second signal equivalent to one F atom would be split into a quintuplet by coupling. The trigonal-bipyramidal structure has two F environments equivalent to three and two F atoms, respectively. Their resonances would be split into a triplet and quartet by spinspin coupling. 19 Self-test 17.2 Predict the F-NMR pattern for IF7. 11 BrF5, D3h (a) Chemical properties Key point: Fluorine–containing interhalogens are typically Lewis acids and strong oxidizing agents. All the interhalogens are oxidizing agents. In general, the rates of oxidation of interhalogens do not bear a simple relation to their thermodynamic stabilities. As with all the known interhalogen fluorides, ClF3 is an exergonic compound, so thermodynamically it is a weaker fluorinating agent than F2 itself. However, the rate at which it fluorinates substances generally exceeds that of fluorine, so it is in fact an aggressive fluorinating agent towards many elements and compounds. The fluorides ClF3 and BrF3 are much more aggressive fluorinating agents than BrF5, IF5, and IF7; iodine pentafluoride, for instance, is a convenient mild fluorinating agent that can be handled in glass apparatus. One use of ClF3 as a fluorinating agent is in the formation of a passivating metal fluoride film on the inside of the nickel apparatus used in fluorine chemistry. Both ClF3 and BrF3 react vigorously (often explosively) with organic matter, burn asbestos, and expel oxygen from many metal oxides: 2Co3O4 (s) + 6ClF3 (g) → 6CoF3 (s) + 3Cl2 (g) + 4O2 (g) Bromine trifluoride autoionizes in the liquid state: 2 BrF3(l) BrF2(sol) BrF4(sol) This Lewis acid-base behaviour is shown by its ability to dissolve a number of halide salts: CsF(s) + BrF3 (l) → Cs + (sol) + BrF4 − (sol) Bromine trifluoride is a useful solvent for ionic reactions that must be carried out under highly oxidizing conditions. The Lewis acid character of BrF3 is shared by other interhalogens, which react with alkali metal fluorides to produce anionic fluoride complexes. 431 The detail (b) Cationic interhalogens Energy Key point: Cationic interhalogen compounds have structures in accord with the VSEPR model. ClF3 + SbF5 → [ClF ][SbF ] + 2 – 6 This formulation is idealized because X-ray diffraction of solid compounds that contain these cations indicates that the F abstraction from the cations is incomplete and that the anions remain weakly associated with the cations by fluorine bridges (12). Table 17.10 lists a variety of interhalogen cations that are prepared in a similar manner. SbF5 F 158 pm F SbF5 F Cl 243 pm – : Nonbonding I I I : : Under special strongly oxidizing conditions, such as in fuming sulfuric acid, I2 is oxidized to the blue paramagnetic diiodinium cation, I2 . The dibrominium cation, Br2 , is also known. The bonds of these cations are shorter than those of the corresponding neutral dihalogens, which is the expected result for loss of an electron from a π orbital and the accompanying increase in bond order from 1 to 1.5 (see Fig. 17.5). Three higher polyhalogen cations, Br5 , I3 , and I5 , are known, and X-ray diffraction studies of the iodine species have established the structures shown in (4) and (5). The angular shape of I3 is in line with the VSEPR model because the central I atom has two lone pairs of electrons. Another class of polyhalogen cations of formula XFn is obtained when a strong Lewis acid, such as SbF5, abstracts F from interhalogen fluorides: Antibonding Bonding (a) (b) Figure 17.11 Some representations of the I3 polyiodide ion. (a) The interaction. (b) Lewis and VSEPR rationalization of the linear structure, where the five electron pairs are arranged around the central atom in a trigonal-pyramidal array. is the most stable member of this series. In combination with a large cation, such as [N(CH3)4], it is symmetrical and linear with a longer II bond than in I2. However, the structure of the triiodide ion, like that of the polyiodides in general, is highly sensitive to the identity of the counterion. For example, Cs, which is smaller than the tetramethylammonium ion, distorts the I3 ion and produces one long and one short II bond (13). The ease with which the ion responds to its environment is a reflection of the weakness of bonds that just manage to hold the atoms together. An example of sensitivity to the cation is provided by NaI3, which can be formed in aqueous solution but decomposes when the water is evaporated: remove water Na+ (aq) + I3— (aq) ⎯⎯⎯⎯⎯ → NaI(s) + I2 (ss) F I 310pm 12 (ClF2)(SbF6)2 282 pm – (c) Polyhalides Key points: Polyiodides, such as I3 , are formed by adding I2 to I; they are stabilized by large cations. Some of the most stable polyhalogen anions contain fluorine as the substituent; their structures usually conform to the VSEPR model. A deep brown colour develops when I2 is added to a solution of I ions. This colour is characteristic of the polyiodides, which include triiodide ions, I3, and pentaiodide ions, I5. These polyiodides are Lewis acid-base complexes in which I and I3 act as the bases and I2 acts as the acid (Fig. 17.11). The Lewis structure of I3 has three equatorial lone pairs on the central I atom and two axial bonding pairs in a trigonal-bipyramidal arrangement. This hypervalent Lewis structure is consistent with the observed linear structure of I3, which is described in more detail below. An I3 ion can interact with other I2 molecules to yield larger mononegative polyiodides of composition [(I2)nI]. The I3 ion Table 17.10 Representative interhalogen cations Compound Shape ClF2 , BrF2 , ICl2 Bent ClF4 , BrF4 , IF4 See-saw 6 6 6 ClF , BrF , IF Octahedral 13 I3– A more extreme example is NI3.NH3, which is a black powder formed when iodine crystals are added to concentrated ammonia solution. The free NI3 can be prepared from reacting iodine monofluoride with boron nitride: 3IF(g) + BN(s) → NI3 (s) + BF3 (g) Nitrogen triiodide and the ammoniate are extremely unstable and detonate at the slightest touch or vibration: 2 NI .NH (s) → N (g) + 3I (s) + 2 NH (g) 3 3 2 2 3 Although the formula of nitrogen triiodide is usually written NI3, it would be more accurate to write I3N, as the compound is thought to consist of I and N3 ions and its sensitivity to shock is due to the redox instability of these ions. This behaviour is also another example of the instability of large anions in combination with small cations, which, as we saw in Section 3.15, can be rationalized by the ionic model. The existence and structures of the higher polyiodides are sensitive to the counterion for similar reasons, and large cations are necessary to stabilize them in the solid state. In fact, entirely different shapes are observed for polyiodide ions in combination with various large cations, as the structure of the anion is determined in large measure by the manner in which the ions 432 17 The Group 17 elements 317 94° 334 343 324 334 2– 4 – – I (2I + I2) I5–(I– + 2I2) 267 280 280 318 314 Br 318 267 F I2– 4 318 290 290 87° 290 I7– (I3– + 2I2) 344 324 I– (I– + 4I ) 9 2 291 273 14 BrF4– 290 344 Figure 17.12 Some representative polyiodide structures and their approximate description in terms of I, I3 , and I2 building blocks. Bond lengths and angles vary with the identity of the cation. Table 17.11 Representative interhalogen anions Compound Shape ClF2 , IF2 , ICl2 , IBr2 Linear ClF4 , BrF4 , IF4 , ICl4 Square planar 6 pack together in the crystal. The bond lengths in a polyiodide ion often suggest that it can be regarded as a chain of associated I, I2, I3, and sometimes I42 units (Fig. 17.12). Solids containing polyiodides exhibit electrical conductivity, which may arise either from the hopping of electrons (or holes) or by an ion relay along the polyiodide chain (Fig. 17.13). Some dinegative polyiodides are known. They contain an even number of I atoms and their general formula is [I(I2)nI]. They have the same sensitivity to the cation as their mononegative counterparts. Although polyhalide formation is most pronounced for iodine, other polyhalides are also known. They include Cl3− , Br3− , and BrI2− , which are known in solution and (in partnership with large cations) as solids too. Even F3− has been detected spectroscopically at low temperatures in an inert matrix. This technique, known as matrix isolation, makes use of the co-deposition of the reactants with a large excess of noble gas at very low temperatures (in the region of 4–14 K). The solid noble gas forms an inert matrix within which the F3 ion can sit in chemical isolation. In addition to complex formation between dihalogens and halide ions, some interhalogens can act as Lewis acids towards halide ions. The reaction results in the formation of polyhalides that, in contrast to the chain-like polyiodides, are assembled around a central halogen acceptor atom in a high oxidation state. As mentioned earlier, for instance, BrF3 reacts with CsF to form CsBrF4, which contains the square-planar BrF4− anion (14). Many of these interhalogen anions have been synthesized (Table 17.11). 6 ClF , BrF Octahedral IF6 Trigonally distorted octahedron 8 IF Square antiprism Their shapes generally agree with the VSEPR model, but there are some interesting exceptions. Two such exceptions are ClF6 and BrF6 , in which the central halogen has a lone pair of electrons, but the apparent structure is octahedral. The ion IF6 participates in an extended array through IF…I interactions. E X A M PL E 17. 3 Proposing a bonding model for I complexes In some cases the interaction of I2 with strong donor ligands leads to the formation of cationic complexes such as bis(pyridine)iodine(1), [pyIpy]. Propose a bonding model for this linear complex from (a) the standpoint of the VSEPR model and (b) simple molecular orbital considerations. Answer (a) The Lewis electron structure places 10 electrons around the central I in [pyIpy], six from the iodine cation and four from the lone pairs on the two pyridine ligands. According to the VSEPR model, these pairs should form a trigonal bipyramid. The lone pairs will occupy the equatorial positions and consequently the complex should be linear. (b) From a molecular orbital perspective, the orbitals of the NIN array can be pictured as being formed from an iodine 5p orbital and an orbital of symmetry from each of the two ligand atoms. Three orbitals can be constructed: 1 (bonding), 2 (nearly nonbonding), and 3 (antibonding). There are four electrons to accommodate (two from each ligand atom; the iodine 5p orbital is empty). The resulting configuration is 1 22 2, which is net bonding. Self-test 17.3 From the perspective of structure and bonding, indicate several polyhalides that are analogous to [pyIpy], and describe their bonding. I I I– I I I I I I I I I I I I I– I I I I I I I I I I I I I– I I I I 17.11 Halogen oxides Figure 17.13 One possible mode of charge transport along a polyiodide chain is the shift of long and short bonds, resulting in the effective migration of an I ion along a chain. Three successive stages in the migration are shown. Note that the iodide ion from the I3 on the left is not the same one emerging on the right. Key points: The only fluorine oxides are OF2 and O2F2; chlorine oxides are known for Cl oxidation numbers of 1, 4, 6, and 7; the strong and facile oxidizing agent ClO2 is the most commonly used halogen oxide. Oxygen difluoride (FOF; m.p. 224C, b.p. 145C), the most stable binary compound of O and F, is prepared by passing fluorine through dilute aqueous hydroxide solution: 2 F2 (g) + 2OH – (aq) → OF2 (g) + 2 F − (aq) + H 2O(l) The detail The pure difluoride survives in the gas phase above room temperature and does not react with glass. It is a strong fluorinating agent, but less so than fluorine itself. As suggested by the VSEPR model, the OF2 molecule is angular. Dioxygen difluoride (FOOF; m.p. 154C, b.p. 57C) can be synthesized by photolysis of a liquid mixture of the two elements. It is unstable in the liquid state and decomposes rapidly above 100C, but can be transferred (with some decomposition) as a low-pressure gas in a metal vacuum line. Dioxygen difluoride is an even more aggressive fluorinating agent than ClF3. For example, it oxidizes plutonium metal and its compounds to PuF6, which is an intermediate in reprocessing of nuclear fuels, in a reaction that ClF3 cannot accomplish: Pu (s) + 3O2F2 (g) → PuF6 (g) + 3O2 (g) Chlorine dioxide is the only halogen oxide produced on a large scale. The reaction used is the reduction of ClO3− with HCl or SO2 in strongly acidic solution: acid 2ClO3– (aq) + SO2 (g) ⎯⎯⎯ → 2ClO2(g) + SO2– (aq) 4 Because chlorine dioxide is a strongly endergonic compound (fG O 121 kJ mol1), it must be kept dilute to avoid explosive decomposition and is therefore used at the site of production. Its major uses are to bleach paper pulp and to disinfect sewage and drinking water. Some controversy surrounds these applications because the action of chlorine (or its product of hydrolysis, HClO) and chlorine dioxide on organic matter produces low concentrations of chlorocarbon compounds, some of which are potential carcinogens. However, the disinfection of water undoubtedly saves many more lives than the carcinogenic byproducts may take. Chlorine bleaches are being replaced by oxygen-based bleaches such as hydrogen peroxide (Box 16.1). The most well known oxides of bromine are given below: Oxidation number 1 3 4 Formula Br2O Br2O3 BrO2 Colour Dark brown Orange Pale yellow State Solid Solid Solid The structure of BrO2 has been found to be a mixed Br(I)/Br(VII) oxide, BrOBrO3. All the bromine oxides are thermally unstable above 40C and explode on heating. The most stable halogen oxides are those formed by iodine. The most important of these is I2O5 (15), which is used to oxidize carbon monoxide quantitatively to carbon dioxide in the analysis of CO in blood and in air. The compound is a white, hygroscopic solid. It dissolves in water to give iodic acid, HIO3. The less stable iodine oxides I2O4 and I4O9 are both yellow solids that decompose on heating to give I2O5: 5I2O4 (s) → 4 I2O5 (s) + I2 (g) 4 I4O9 (s) → 6 I2O5 (s) + 2 I2 (g) + 3O2 (g) 17.12 Oxoacids and oxoanions Key points: The halogen oxoanions are thermodynamically strong oxidizing agents; perchlorates of oxidizable cations are unstable. The strengths of the oxoacids vary systematically with the number of O atoms on the central atom (Table 17.12; see Pauling’s rules in Section 4.5b). Periodic acid, H5IO6, is the I(VII) analogue of perchloric acid. It is a weak acid (pKa1 3.29), which can be explained as soon as we note that its formula is (HO)5IO and that there is only one I O group. The O atoms in the conjugate base H4IO6 are very labile on account of the rapid equilibration: H4IO6(aq) IO4(aq) 2 H2O(l) K 40 IO4 is the dominant ion. The tendency to have an expanded coordination shell is shared by the oxoacids of the neighbouring Group 16 element tellurium, which in its maximum oxidation state forms the weak acid Te(OH)6. The halogen oxoanions, like many oxoanions, form metal complexes, including the metal perchlorates and periodates discussed here. In this connection we note that, because HClO4 is a very strong acid and H5IO6 is a weak acid, it follows that ClO4− is a very weak base and H4IO6 is a relatively strong base. In view of the low Brønsted basicity and single negative charge of the perchlorate ion, ClO4, it is not surprising that it is a weak Lewis base with little tendency to form complexes with cations in aqueous solution. Therefore, metal perchlorates are often used to study the properties of hexaaqua ions in solution. The ClO4 ion is used as a weakly coordinating ion that can readily be displaced from a complex by other ligands, or as a medium-sized anion that might stabilize solid salts containing large cationic complexes with easily displaced ligands. However, the ClO4 ion is a treacherous ally. Because it is a powerful oxidizing agent, solid compounds of perchlorate should be avoided whenever there are oxidizable ligands or ions present (which is commonly the case). In some cases the danger lies in wait, as the reactions of ClO4 are generally slow and it is possible to prepare many metastable perchlorate complexes or salts that may be handled with deceptive ease. However, once reaction has been initiated by mechanical action, heat, or static electricity, these compounds can detonate with disastrous consequences. Such explosions have injured chemists who may Table 17.12 Acidities of chlorine oxoacids Acid p/q pKa HOCl 0 7.53 (weak) HOClO 1 2.00 HOClO2 2 1.2 HOClO3 3 10 (strong) I O 15 I2O5 433 434 17 The Group 17 elements have handled a compound many times before it unexpectedly exploded. Some readily available and more docile weakly basic anions may be used in place of ClO4; they include trifluoridomethanesulfonate [SO3CF3], tetrafluoridoborate BF4, and hexafluoridophosphate [PF6]. For many years it was believed that perbromate did not exist. However, it was prepared in 1968 by a radiochemical route based on the -decay of 83Se and chemical syntheses have now been devised. The perbromate ion is more oxidizing than any other oxohalide. The instability of perbromate compared to perchlorate and periodate is an example of the reluctance of post-3d elements to achieve their highest possible oxidation state and is a manifestation of the alternation effect (Section 9.2c). In contrast to perchlorate, periodate is a rapid oxidizing agent and a stronger Lewis base. These properties lead to the use of periodate as an oxidizing agent and stabilizing ligand for metal ions in high oxidation states. Some of the high oxidation states it can be used to form are very unusual: they include Cu(III) in a salt containing the [Cu(HIO6)2]5 complex and Ni(IV) in an extended complex containing the [Ni(IO6)] unit. The periodate ligand is bidentate in these complexes, and in the last example it forms a bridge between Ni(IV) ions. 17.13 Thermodynamic aspects of oxoanion redox reactions Key point: The oxoanions of halogens are strong oxidizing agents, especially in acidic solution. The thermodynamic tendencies of the halogen oxoanions and oxoacids to participate in redox reactions have been extensively studied. As we shall see, we can summarize their behaviour with a Frost diagram that is quite easy to rationalize. It is a very different story with the rates of the reactions, which vary widely. Their mechanisms are only partly understood despite many years of investigation. Recent progress in the understanding of some of these mechanisms stems from advances in techniques for fast reactions and interest in oscillating reactions (Box 17.4). We saw in Section 5.13 that, if in a Frost diagram a species lies above the line joining its two neighbours of higher and lower oxidation numbers, then it is unstable with respect to disproportionation into them. From the Frost diagram for the halogen oxoanions and oxoacids in Fig. 17.14 we can see that many of the oxoanions in intermediate oxidation states are susceptible to disproportionation. Chlorous acid, HClO2, for instance, lies above the line joining its two neighbours, and is liable to disproportionation: 2 HClO2 (aq) → ClO−3 (aq) + HClO(aq) + H+ (aq) E o cell = +0.52 V Although BrO2− is well characterized, the corresponding I(III) species is so unstable that it does not exist in solution, except perhaps as a transient intermediate. We also saw in Section 5.13 that the more positive the slope for the line from a lower to higher oxidation state species in a Frost diagram, the stronger the oxidizing power of the couple. A glance at Fig. 17.14 shows that all three Frost diagrams have steep positively sloping lines, which immediately shows that all the oxidation states except the lowest (Cl, Br, and I) are strongly oxidizing. Finally, basic conditions decrease reduction potentials for oxoanions as compared with their conjugate acids (Section 5.5). This decrease is evident in the less steep slopes of the lines in the Frost diagrams for the oxoanions in basic solution. The numerical comparison for ClO4− ions in 1 m acid compared with 1 m base makes this clear: At pH = 0: ClO4− (aq) + 2 H + (aq) + 2e− → ClO3− (aq) + H 2O(l) o E = +1.20 V At pH = 14: ClO4− (aq) + H 2O(l) + 2e− → ClO3− (aq) + 2OH − (aq) o EB = + 0.37 V The reduction potentials show that perchlorate is thermodynamically a weaker oxidizing agent in basic solution than in acidic solution. B OX 17. 4 Oscillating reactions Clock reactions and oscillating reactions are an active topic of research and provide fascinating lecture demonstrations. Most oscillating reactions are based on the reactions of halogen oxoanions, apparently because of the variety of oxidation states and their sensitivity to changes in pH. In 1895, H. Landot discovered that a mixture of sulfite, iodate, and starch in acidic aqueous solution remains nearly colourless for an initial period and then suddenly switches to the dark purple of the I2starch complex. When the concentrations are properly adjusted, the reaction oscillates between nearly colourless and opaque blue. The reactions leading to this oscillation are the reduction of iodate to iodide by sulfite, where all reactants are colourless: IO3− (aq) + 3 SO23− (aq) → I− (aq) + 3 SO24− (aq) A comproportionation reaction between I and IO3 then produces I2, which forms an intensely coloured complex with starch: IO−3 (aq) + 6 H+ (aq) + 5 I− (aq) → 3 H2O(l) + 3 I2 (starch) Under some conditions the I2starch complex is the final state, but adjustment of concentrations may lead to bleaching of the complex by sulfite reduction of iodine to the colourless I(aq) ion: 3I2 (starch) + 3 SO32− (aq) + 3 H2O(l) → 6I− (aq) + 6 H+ (aq) + 3 SO24− (aq) The reaction may then oscillate between colourless and blue as the I2 /I ratio changes. The detailed analysis of the kinetic conditions for oscillating reactions is pursued by chemists and chemical engineers. In the former case the challenge is to use kinetic data determined separately for the individual steps to model the observed oscillations with a view to testing the validity of the overall scheme. As oscillating reactions have been observed in commercial catalytic processes, the concern of the chemical engineer is to avoid large fluctuations or even chaotic reactions that might degrade the process. Oscillating reactions are of more than industrial interest, for they also maintain the rhythm of the heartbeat and their interruption can result in fibrillation and death. The detail 12 12 12 BrO4– ClO4– 10 435 10 10 H5IO6 8 6 HClO2 4 2 ClO4– HClO ClO2– ClO 0 BrO4– 4 – – 1 2 3 4 5 6 Oxidation number, N 7 HIO 0 Br2 –2 Br –1 0 4 2 BrO3– BrO– 0 IO3– 6 IO4– HBrO – Cl2 Cl– –2 –1 0 6 2 ClO3– 8 BrO3– NE°/V ClO3– NE°/V NE°/V 8 1 2 3 4 5 6 Oxidation number, N 7 I –2 –1 0 I2 IO3– IO– 1 2 3 4 5 6 Oxidation number, N 7 Figure 17.14 Frost diagrams for chlorine, bromine, and iodine in acidic solution (red line) and in basic solution (blue line). 17.14 Trends in rates of oxoanion redox reactions Key points: Oxidation by halogen oxoanions is faster for the lower oxidation states; rates and thermodynamics of oxidation are both enhanced by an acidic medium. Mechanistic studies show that the redox reactions of halogen oxoanions are complex. Nevertheless, despite this complexity, a few discernible patterns help to correlate the trends in rates of reaction. These correlations have practical value and give some clues about the mechanisms that may be involved. The oxidation of many molecules and ions by halogen oxoanions becomes progressively faster as the oxidation number of the halogen decreases. Thus the rates observed are often in the order ClO < ClO < ClO ≈ ClO ≈ Cl2 BrO < BrO ≈ BrO ≈ Br2 IO4– < IO3– < I2 – 4 – 4 – 3 – 3 – 2 – – O S – For example, aqueous solutions containing Fe2 and ClO4− are stable for many months in the absence of dissolved oxygen, but an equilibrium mixture of aqueous HClO and Cl2 rapidly oxidizes Fe2. Oxoanions of the heavier halogens tend to react most rapidly, particularly for the elements in their highest oxidation states: ClO4− < BrO4− < IO4− As we have remarked, perchlorates in dilute aqueous solution are usually unreactive, but periodate oxidations are fast enough to be used for titrations. The mechanistic details are often complex, but the existence of both four- and six-coordinate periodate ions shows that the I atom in periodate is accessible to nucleophiles. We have already seen that the thermodynamic tendency of oxoanions to act as oxidizing agents increases as the pH is lowered. It is found that their rates are increased too. Thus, kinetics and equilibria unite to bring about otherwise difficult oxidations. The oxidation of halides by BrO3 ions, for instance, is second-order in H: Rate = kr [BrO3− ][X − ][H + ]2 and so the rate increases as the pH is decreased. The acid is thought to protonate the oxo group in the oxoanion, so aiding oxygen–halogen bond scission. Another role of protonation is to increase the electrophilicity of the halogen. An example is HClO, where, as described below, the Cl atom may be viewed as an electrophile towards an incoming reducing agent (16). An illustration of the effect of acidity on rate is the use of a mixture of H2SO4 and HClO4 in the final stages of the oxidation of organic matter in certain analytical procedures. HOCl 16 17.15 Redox properties of individual oxidation states Key point: Dihalogen molecules disproportionate in aqueous solution. Hypochlorite is a facile oxidizing agent; hypohalite and halite ions undergo disproportionation. Chlorate ions undergo disproportionation in solution but bromates and iodates do not. With the general redox properties of the halogens now outlined, we can consider the characteristic properties and reactions of specific oxidation states. Although we are dealing here with halogen oxides, it is convenient to mention, for the sake of completeness, the redox properties of halogen(0) species. Figure 17.15 summarizes some of the reactions that interconvert the oxoanions and oxoacids of chlorine in its various oxidation states. One point to note is the major role of disproportionation and electrochemical reactions in the scheme. For example, the figure includes the production of Cl2 by the electrochemical oxidation of Cl, which was discussed in Section 17.4. 436 17 The Group 17 elements Oxidation number Disproportionation is much less favourable in acidic solution, as would be expected from the fact that H is a product of the reaction: H2O Cl2O7 ClO4– +7 Cl2(aq) H2O(l) HClO(aq) H(aq) Cl(aq) K 3.9 × 104 +H+, –H2O +6 +5 –e– ClO3– SO2 or Cl– ClO2 +4 +3 HClO2, ClO2– H+ +2 +1 HClO, ClO– H2O Cl2O Na2CO3 – H2O(OH ) 0 –1 –e– Cl– H2SO4 –H Cl2 H2 HCl Figure 17.15 The interconversion of oxidation states of some important chlorine species. Disproportionation is thermodynamically favourable for basic solutions of Cl2, Br2, and I2. The equilibria in basic aqueous solution are: X2(aq) 2 OH(aq) XO(aq) X(aq) H2O(l) K= [XO– ][X – ] [X2 ][OH – ]2 with K 7.5 × 1015 for X Cl, 2 × 108 for Br, and 30 for I. A note on good practice When writing the associated equilibrium expressions in aqueous solution, we conform to the usual convention that the activity of water is 1, so H2O does not appear in the equilibrium constant. Because the redox reactions of Cl2 are often fast, Cl2 in water is widely used as an inexpensive and powerful oxidizing agent. The equilibrium constants for the hydrolysis of Br2 and I2 in acid solution are smaller than for Cl2, and both elements are unchanged when dissolved in slightly acidified water. Because F2 is a much stronger oxidizing agent than the other halogens, it produces mainly O2 and H2O2 when in contact with water. As a result, hypofluorous acid, HFO, was discovered long after the other hypohalous acids. The aqueous Cl(I) species hypochlorous acid, HClO (the angular molecular species HOCl), and hypochlorite ions, ClO, are facile oxidizing agents that are used as household bleach and disinfectant, and as laboratory oxidizing agents (Box 17.5). The ready access to the unobstructed, electrophilic Cl atom in HClO appears to be one feature that leads to the very fast redox reactions of this compound. These rates contrast with the much slower redox reactions of perchlorate ions, in which access to the Cl atom is blocked by the surrounding O atoms. A note on good practice Hypohalous acids are widely denoted HOX to emphasize their structure. We have adopted the formula HXO to emphasize their relationship to the other oxoacids, HXOn. Hypohalite ions undergo disproportionation. For instance, ClO disproportionates into Cl and ClO3: 3 ClO(aq) 2 Cl(aq) ClO3(aq) K 1.5 × 1027 This reaction (which is used for the commercial production of chlorates) is slow at or below room temperature for ClO but is much faster for BrO. It is so fast for IO that this ion has been detected only as a reaction intermediate. Chlorite ions, ClO2, and bromite ions, BrO2, are both susceptible to disproportionation. However, the rate is strongly dependent on pH and ClO2 (and to a lesser extent BrO2) can be handled in basic solution with only slow decomposition. B OX 17. 5 Chlorine-based bleaches Substances that are used as bleaches are powerful oxidizing agents. As mentioned in Section 17.2, the oxidizing power of the halogen oxoanions increases as the oxidation number of the halogen decreases. It is not surprising then that the chlorine-based bleaches contain Cl in a low oxidation state. Chlorine disproportionates in water to produce the oxidizing hypochlorite ion, ClO, and Cl. Solutions of up to 15 per cent by mass of sodium hypochlorite are used as industrial bleaches in the paper, textiles, and laundry industries and for disinfecting swimming pools. Household bleach is a more dilute (5 per cent) solution of NaClO. A 0.5 per cent aqueous solution of NaClO is used by dentists during root-canal work, where it is used kill pathogens and dissolve necrotic tissue. Other hypochlorite salts are also used as oxidants. Calcium hypochlorite, Ca(ClO)2, is used as a disinfectant in dairies, breweries, food processing, and bottling plants. It is also used in domestic mildew removers. Bleaching powder is a mixture of Ca(ClO)2 and CaCl2 and is used for large-scale applications such as disinfecting seawater, reservoirs, and sewers. It is also used as a decontaminant in areas where chemical weapons, such as mustard gas, have been deployed. Chlorine dioxide gas is widely used as a bleach in the wood pulp industry where it produces whiter and stronger paper than other bleaches because, unlike oxidizing bleaches such as chlorine, ozone, and hydrogen peroxide, it does not attack the cellulose and therefore preserves the mechanical strength of the pulp. Chlorine-based bleaches lead to the production of toxic chlorinated organic compounds. The most toxic polychlorinated phenols, such as dioxins, are mostly produced by ClO2, but the levels can be drastically reduced by substituting some of the ClO2 with Cl2. The detail By contrast, chlorous acid, HClO2, and bromous acid, HBrO2, both disproportionate rapidly. Iodine(III) is even more elusive, and HIO2 has been identified only as a transient species in aqueous solution. The Frost diagram for Cl shown in Fig. 17.14 indicates that chlorate ions, ClO3, are unstable with respect to disproportionation in both acidic and basic solution: 4ClO−3 (aq) 3 ClO−4 (aq) + Cl− (aq) ∆rG = −24 kJ mol−1 O K 1.4 × 10 25 Because HClO3 is a strong acid, and this reaction is slow at both low and high pH, ClO3 ions can be handled readily in aqueous solution. Bromates and iodates are thermodynamically stable with respect to disproportionation. Of the three XO4 ions, BrO4 is the strongest oxidizing agent. That perbromate is out of line with its adjacent halogen congeners fits a general pattern for anomalies in the chemistry of p-block elements of Period 4. However, the reduction of periodate in dilute acid is faster than that of perchlorate or perbromate and periodates are therefore used in analytical chemistry as oxidizing titrants and also in syntheses, such as the oxidative cleavage of diols: O HOC(CH 3) 2C (C H 3)2 OH (H3 C) 2C O OH I – IO4 (aq) (H3 C) 2C O OH O – 2 (CH3)2CO + IO3(aq) + H2O ■ A brief illustration. To confirm that perbromate is the most oxidizing perhalate ion we need to consider the slope of the lines joining the perhalate ions to their neighbours in the Frost diagram. The more positive the slope of the line, the stronger the oxidizing power of the couple. Inspection of the diagram reveals that the line joining the 437 BrO4 /BrO3 couple is the most positive. In fact the E O values for the ClO4 /ClO3 , BrO4 /BrO3 , and IO4 /IO3 couples in acidic conditions are 1.201, 1.853, and 1.600 V, respectively, confirming that perbromate is the strongest oxidizing agent. ■ 17.16 Fluorocarbons Key point: Fluorocarbon molecules and polymers are resistant to oxidation. Fluorocarbons find many useful applications (Box 17.6). The direct reaction of an aliphatic hydrocarbon with an oxidizing metal fluoride leads to the formation of strong CF bonds (456 kJ mol1) and produces HF as a byproduct: RH(l) + 2CoF3 (s) → RF(sol) + 2CoF2 (s) + HF(sol) R = alkyl or aryl When R is aryl, CoF3 yields the cyclic saturated fluoride: C6H6 (l) + 18 CoF3 (s) → C6F12 (l) + 18 CoF2 (s) + 6 HF(l) The strongly oxidizing fluorinating agent used in these reactions, CoF3, is regenerated by the reaction of CoF2 with fluorine: 2CoF2 (s) + F2 (g) → 2CoF3 (s) Another important method of CF bond formation is halogen exchange by the reaction of a nonoxidizing fluoride, such as HF, with a chlorocarbon in the presence of a catalyst, such as SbF3: CCl4 ( l )+ HF ( l ) → CCl3F ( l )+ HCl (g) CHCl3 ( l ) + 2 HF ( l ) → CHClF2 ( l ) + 2 HCl (g) These processes used to be performed on a large scale to produce the chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs) that were used as refrigerant fluids, the propellant in spray cans, and in the blowing agent in plastic foam products. These applications have been banned in some countries and are being phased out worldwide because of the role of CFCs B OX 17.6 PTFE: a high-performance polymer Polytetrafluoroethene, PTFE, is a unique product in the plastics industry. It is chemically inert, thermally stable over a wide temperature range (196 to 260C), is an excellent electrical insulator, and has a low coefficient of friction. It is a white solid that is manufactured by the polymerization of tetrafluoroethene: n CF2 CF2 → (CF2CF2)n PTFE is an expensive polymer because of the cost of synthesizing and purifying the monomer by a multistage process: CH4 (g) + 3 Cl2 (g) → CHCl3 (g) + 3 HCl (g) CHCl3 (g) + 2 HF (g) → CHClF2 (g) + 2 HCl (g) → CF2= CF2 (g) + 2 HCl (g) 2 CHClF2 (g) ⎯∆⎯ The hydrogen fluoride is generated by the action of sulfuric acid on fluorite: CaF2(s) H2SO4(l) → CaSO4(s) 2 HF(l) As the process employs HF and HCl, the reactors have to be lined with platinum. Many byproducts are produced, which leads to complex purification of the final product. The tetrafluoroethene is polymerized in two ways: solution polymerization with vigorous agitation produces a resin known as granular PTFE; emulsion polymerization with a dispersing agent and gentle agitation produces small particles known as dispersed PTFE. The molten polymer does not flow, so the usual methods of processing cannot be used. Instead processes similar to those used for metals are applied. For example, the dispersed form can be cold extruded (which is a method used for processing lead). The remarkable properties of PTFE arise from the protective sheath that the F atoms form around the carbon polymer backbone. The F atoms are just the right size to form a smooth sheath. This smooth sheath reduces the disruption of intermolecular forces at the surface, leading to a low coefficient of friction and the familiar nonstick properties. The polymer is used in a wide range of applications. Its low electrical conductivity leads to its use in electrical tapes, wires, and coaxial cable. Its mechanical properties make it an ideal material for seals, piston rings, and bearings. It is used as a packaging material, in hose lines, and as thread-sealant tape. Familiar applications are as the nonstick coating on cookware, and as the porous fabric Gore-Tex®. 438 17 The Group 17 elements and HCFCs in ozone depletion. They are being replaced by hydrofluorocarbons (HFCs) after investment by the chemical industry because, in contrast to the simple one-step synthesis of CFCs and HCFCs, HFC production is a complex multistage process. For example, the preferred route to CF3CH2F, which is one of the preferred CFC replacements is HF +Cl 2 CCl2 =CCl 2 ⎯⎯⎯⎯ → CClF2CCl2F ⎯isomerize ⎯⎯ ⎯ → CF3CCl3 The polymerization of tetrafluoroethene is carried out with a radical initiator: ( ⋅ n C2F4 ⎯ROO ⎯⎯ → –CF2 –CF2 – ( –CF –CF – ) H −800ºC 2CHClF2 ⎯600 ⎯⎯⎯ → C2F4 + 2 HCl n Polytetrafluoroethene (PTFE) is sold under many trade names, one of which is Teflon (DuPont). Its depolymerization at high temperatures is the most convenient method of preparing tetrafluoroethene in the laboratory: HF ⎯⎯ → CF3CCl2F ⎯⎯2⎯ → CF3CH 2F When heated, chlorodifluoromethane is converted to the useful monomer, C2F4: ) 2 2 n 600ºC ⎯⎯⎯ → n C2F4 Although tetrafluoroethene is not highly toxic, a byproduct, 1,1,3,3,3-pentafluoro-2-trifluoromethyl-1-propene, is toxic and its presence dictates care in handling crude tetrafluoroethene. FURTHER READING M. Schnürch, M. Spina, A.F. Khan, M.D. Mihovilovic, and P. Stanetty, Halogen dance reactions—A review, Chem. Soc. Rev., 2007, 36, 1046. S. Purser, P.R. Moore, S. Swallow, and V. Gouverneur, Fluorine in medicinal chemistry, Chemical Society Reviews, 2008, 37, 2, 320. A.G. Massey, Main group chemistry. Wiley (2000). D.M.P. Mingos, Essential trends in inorganic chemistry. Oxford University Press (1998). R.B. King (ed.), Encyclopedia of inorganic chemistry. Wiley (2005). N.N. Greenwood and A. Earnshaw, Chemistry of the elements. Butterworth-Heinemann (1997). P. Schmittinger, Chlorine: principles and industrial practice. Wiley– VCH (2000). M. Howe-Grant, Fluorine chemistry. Wiley (1995). EXERCISES 17.1 Preferably without consulting reference material, write out the halogens as they appear in the periodic table, and indicate the trends in (a) physical state (s, l, or g) at room temperature and pressure, (b) electronegativity, (c) hardness of the halide ion, (d) colour. 17.2 Describe how the halogens are recovered from their naturally occurring halides and rationalize the approach in terms of standard potentials. Give balanced chemical equations and conditions where appropriate. 17.3 Sketch a choralkali cell. Show the half-cell reactions and indicate the direction of diffusion of the ions. Give the chemical equation for the unwanted reaction that would occur if OH migrated through the membrane and into the anode compartment. 17.4 Sketch the form of the vacant orbital of a dihalogen molecule and describe its role in the Lewis acidity of the dihalogens. 17.5 Which dihalogens are thermodynamically capable of oxidizing H2O to O2? 17.6 Nitrogen trifluoride, NF3, boils at 129C and is a very weak Lewis base. By contrast, the lower molar mass compound NH3 boils at 33C and is well known as a Lewis base. (a) Describe the origins of this very large difference in volatility. (b) Describe the probable origins of the difference in basicity. shapes of IF3 and the cation and anion in X, (c) predict how many F-NMR signals would be observed in IF3 and X. 19 17.9 Use the VSEPR model to predict the shapes of SbCl5, FClO3, and [ClF6]. 17.10 Indicate the product of the reaction between ClF5 and SbF5. 17.11 Sketch all the isomers of the complexes MCl4F2 and MCl3F3. Indicate how many fluorine environments would be indicated in the 19 F-NMR spectrum of each isomer. 17.12 (a) Use the VSEPR model to predict the probable shapes of [IF6] and IF7. (b) Give a plausible chemical equation for the preparation of [IF6][SbF6]. 17.13 Predict the shape of the doubly chlorine-bridged I2Cl6 molecule by using the VSEPR model and assign the point group. 17.14 Predict the structure and identify the point group of ClO2F. 17.15 Predict whether each of the following solutes is likely to make liquid BrF3 act as a Lewis acid or a Lewis base: (a) SbF5, (b) SF6, (c) CsF. 17.16 Predict the appearance of the 19F-NMR spectrum of IF5. 17.17 Predict whether each of the following compounds is likely to be dangerously explosive in contact with BrF3 and explain your answer: (a) SbF5, (b) CH3OH, (c) F2, (d) S2Cl2. 17.7 Based on the analogy between halogens and pseudohalogens write: (a) the balanced equation for the probable reaction of cyanogen, (CN)2, with aqueous sodium hydroxide, (b) the equation for the probable reaction of excess thiocyanate with the oxidizing agent MnO2(s) in acidic aqueous solution, (c) a plausible structure for trimethylsilyl cyanide. 17.18 The formation of Br3 from a tetraalkylammonium bromide and Br2 is only slightly exergonic. Write an equation (or NR for no reaction) for the interaction of [NR4][Br3] with I2 in CH2Cl2 solution and give your reasoning. 17.8 Given that 1.84 g of IF3 reacts with 0.93 g of [(CH3)4N]F to form a product X, (a) identify X, (b) use the VSEPR model to predict the 17.20 Write plausible Lewis structures for (a) ClO2 and (b) I2O6 and predict their shapes and the associated point group. 17.19 Explain why CsI3(s) is stable with respect to decomposition but NaI3(s) is not. Problems 439 17.22 (a) Describe the expected trend in the standard potential of an oxoanion in a solution with decreasing pH. (b) Demonstrate this phenomenon by calculating the reduction potential of ClO4 at pH 7 and comparing it with the tabulated value at pH 0. 17.25 (a) For which of the following anions is disproportionation thermodynamically favourable in acidic solution: ClO, ClO2 , ClO3 , and ClO4 ? (If you do not know the properties of these ions, determine them from a table of standard potentials as in Resources section 3.) (b) For which of the favourable cases is the reaction very slow at room temperature? 17.23 With regard to the general influence of pH on the standard potentials of oxoanions, explain why the disproportionation of an oxoanion is often promoted by low pH. 17.26 Which of the following compounds present an explosion hazard? (a) NH4ClO4, (b) Mg(ClO4)2, (c) NaClO4, (d) [Fe(OH2)6] [ClO4]2. Explain your reasoning. 17.24 Which oxidizing agent reacts more readily in dilute aqueous solution, perchloric acid or periodic acid? Give a mechanistic explanation for the difference. 17.27 Use standard potentials to predict which of the following will be oxidized by ClO ions in acidic conditions: (a) Cr3, (b) V3, (c) Fe2, (d) Co2. 17.21 (a) Give the formulas and the probable relative acidities of perbromic acid and periodic acid. (b) Which is the more stable? PROBLEMS 17.1 Many of the acids and salts corresponding to the positive oxidation numbers of the halogens are not listed in the catalogue of a major international chemical supplier: (a) KClO4 and KIO4 are available but KBrO4 is not, (b) KClO3, KBrO3, and KIO3 are all available, (c) NaClO2 and NaBrO2.3H2O are available but salts of IO2 are not, (d) only ClO salts are available but the bromine and iodine analogues are not. Describe the probable reason for the missing salts of the oxoanions. 17.4 Given the bond lengths and angles in I5 (5), describe the bonding in terms of two-centre and three-centre bonds and account for the structure in terms of the VSEPR model. 17.2 Identify the incorrect statements among the following descriptions and provide correct statements. (a) Oxidation of the halides is the only commercial method of preparing the halogens F2 through I2. (b) ClF4 and I5 are isolobal and isostructural. (c) Atomtransfer processes are common in the mechanisms of oxidations by the halogen oxoanions and an example is the O atom transfer in the oxidation of SO32 by ClO. (d) Periodate appears to be a more facile oxidizing agent than perchlorate because the former can coordinate to the reducing agent at the I(VII) centre, whereas the Cl(VII) centre in perchlorate is inaccessible to reducing agents. 17.6 The use of templates to synthesize long-chain polyiodide ions has been described (A.J. Blake et al., Chem. Soc. Rev., 1998, 27, 195). (a) According to the authors what is the longest polyiodide that has been characterized? (b) How does the nature of the cation influence the structure of the polyanion? (c) What was the templating agent used for the synthesis of I7 and I12 ? (c) Which spectroscopic method was used for the characterization of the polyanions in this study? 17.3 The reaction of I ions is often used to titrate ClO, giving deeply coloured I3 ions, along with Cl and H2O. Although never proved, it was thought that the initial reaction proceeds by O atom transfer from Cl to I. However, it is now believed that the reaction proceeds by Cl atom transfer to give ICl as the intermediate (K. Kumar, R.A. Day, and D.W. Margerum, Inorg. Chem., 1986, 25, 4344). Summarize the evidence for Cl atom transfer. 17.5 Until the work of K.O. Christe (Inorg. Chem., 1986, 25, 3721), F2 could be prepared only electrochemically. Give chemical equations for Christe’s preparation and summarize the reasoning behind it. 17.7 Review published studies on the fluoridation of drinking water in your country. Summarize both the reasons for continuing fluoridation and the main concerns expressed by those opposed to it. 17.8 Write a review of the environmental problems associated with the use of chlorine-based bleaches in industry and suggest possible solutions. 18 The Group 18 elements 18 Part A: The essentials H 18.1 The elements He 1 18.2 Simple compounds 2 Li Be Part B: The detail 13 14 15 16 17 B C N O F Ne 18.3 Occurrence and recovery Cl Ar 18.4 Uses 18.5 Synthesis and structure of xenon fluorides 18.6 Reactions of xenon fluorides 18.7 Xenonoxygen compounds 18.8 Xenon insertion compounds 18.9 Organoxenon compounds 18.10 Coordination compounds 18.11 Other compounds of noble gases FURTHER READING EXERCISES PROBLEMS 3.97 4 The Group 18 elements, helium, neon, argon, krypton, xenon, and radon, are all monatomic gases. They are the least reactive elements and have been I Xe given and then lost various collective names over the years as different aspects of their properties At Rn have been identified and then disproved. Thus they have been called the rare gases and the inert gases, and are currently called the noble gases. The first name is inappropriate because argon is by no means rare (it is substantially more abundant than CO2 in the atmosphere). The second has been inappropriate since the discovery of compounds of xenon. The name ‘noble gases’ is now accepted because it gives the sense of low but significant reactivity. Br Kr PART A: THE ESSENTIALS In this section we survey the limited chemistry of the noble gases and concentrate in particular on the well-characterized compounds of xenon. 2 log(abundance/ppmV) The final group in the p block contains six elements that are so unreactive that they form only a very limited range of compounds. The existence of the Group 18 elements was not suspected until late in the nineteenth century and their discovery led to the redrawing of the periodic table and played a key role in the development of bonding theories. 1.26 0.72 0 18.1 The elements 0.06 He Ne Ar Kr Xe Rn –1.07 –2 –4 –6 –6 Figure 18.1 Abundances of the noble gases in the Earth’s crust. The values are log (atmospheric parts per million by volume). Key point: Of the noble gases, only xenon forms a significant range of compounds with fluorine and oxygen. All the Group 18 elements are very unreactive. Their unreactivity can be understood in terms of their atomic properties (Table 18.1) and in particular their ground-state valence electron configurations, ns2np6. The features to note include their high ionization energies and negative electron affinities. The first ionization energy is high because the effective nuclear charge is high at the far right of the period. The electron affinities are negative because an incoming electron needs to occupy an orbital belonging to a new shell. Helium makes up 23 per cent by mass of the Universe and the Sun, and is the second most abundant element after hydrogen; it is rare in the atmosphere because its atoms travel fast enough to escape from the Earth. All the other noble gases occur in the atmosphere. The crustal abundances of argon (0.94 per cent by volume) and neon (1.5 × 103 per cent) make these two elements more plentiful than many familiar elements, such as arsenic and bismuth, in the Earth’s crust (Fig. 18.1). Xenon and radon are the rarest elements of The essentials Table 18.1 Selected properties of the elements He Ne Ar Kr Xe Covalent radius/pm 99 160 192 197 217 Melting point/°C 272 249 189 157 112 71 62 Boiling point/°C 269 246 186 152 108 Electron affinity/(kJ mol1) 48.2 115.8 96.5 96.5 77.2 First ionization energy/(kJ mol1) 2373 2080 1520 1350 1170 Rn 1036 the group. Radon is a product of radioactive decay and, as its atomic number exceeds that of lead, is itself unstable; it accounts for around 50 per cent of background radiation. 18.2 Simple compounds Key point: Xenon forms fluorides, oxides, and oxofluorides. The most important oxidation numbers of Xe in its compounds are 2, 4, and 6. Compounds with XeF, XeO, XeN, XeH, XeC, and Xemetal bonds are known and Xe can behave as a ligand. The chemical properties of xenon’s lighter congener krypton are much more limited. The study of radon chemistry, like that of astatine, is inhibited by the high radioactivity of the element. Xenon reacts directly with fluorine to produce XeF2 (1), XeF4 (2), and XeF6 (3). Solid XeF6 is more complex than its gas-phase structure and contains fluoride ion-bridged XeF5 cations. The xenon fluorides are strong oxidizing agents and form complexes with F ions, such as XeF5. Xenon forms xenon trioxide, XeO3 (4), and xenon tetraoxide, XeO4 (5), which both decompose explosively. The white crystalline perxenates of several alkali metals have been prepared and contain the XeO64 ion. Xenon also forms several oxofluorides. The oxofluoride XeOF2 has a T-shape (6) and XeO3F2 is a trigonal bipyramid (7). The square-pyramidal molecule XeOF4 (8) is remarkably similar to IF5 in its physical and chemical properties. F Xe 1 XeF2 Xe Xe F Xe O F 2 XeF4 4 XeO3 3 XeF6 F F Xe O Xe O O Xe F O 5 XeO4 6 XeOF2 7 XeO3F2 Xe 8 XeOF4 441 442 18 The Group 18 elements Xenon forms a number of hydrides, such as HXeH, HXeOH, and HXeOXeH. Xenon, argon, and krypton form clathrates when frozen with water at high pressure (see Box 14.2). The noble gas atoms are guests within the three-dimensional ice structure and have the composition E.6 H2O. Clathrates provide a convenient way of handling the radioactive isotopes of krypton and xenon (Section 10.6a). PART B: THE DETAIL In this section we describe the detailed chemistry of the Group 18 elements. Although they are the least reactive of all elements, the noble gases, especially Xe, form a surprisingly wide range of compounds with hydrogen, oxygen, and the halogens. 18.3 Occurrence and recovery Key points: The noble gases are monatomic; radon is radioactive. Because they are so unreactive and are rarely concentrated in nature, the noble gases eluded recognition until the end of the nineteenth century. Indeed, Mendeleev did not make a place for them in his periodic table because the chemical regularities of the other elements, on which his table was based, did not suggest their existence. However, in 1868 a new spectral line observed in the spectrum of the Sun did not correspond to a known element. This line was eventually attributed to helium, and in due course the element itself and its congeners were found on Earth. The noble gases were given names that reflect their curious nature; helium from the Greek helios for sun, neon from the Greek neos for new, argon from argos meaning inactive, krypton from kryptos meaning hidden, and xenon from xenos meaning strange. Radon was named after radium as it is a product of the radioactive decay of that element. Helium atoms are too light to be retained by the Earth’s gravitational field, so most of the He on Earth (5 parts per million by volume) is the product of -emission in the decay of radioactive elements. High He concentrations of up to 7 per cent by mass are found in certain natural gas deposits (mainly in the USA and eastern Europe) from which it can be recovered by low-temperature distillation. Some He arrives from the Sun as a solar wind of particles. Neon, Ar, Kr, and Xe are extracted from liquid air by low-temperature distillation. The elements are all monatomic gases at room temperature. In the liquid phase they form low concentrations of dimers that are held together by dispersion forces. The low boiling points of the lighter noble gases (Table 18.1) follow from the weakness of these forces between the atoms and the absence of other forces. When helium (specifically 4He, not the rarer isotope 3 He) is cooled below 2.178 K it undergoes a transformation into a second liquid phase known as helium-II. This phase is classed as a superfluid, as it flows without viscosity. Solid He is formed only under pressure. 18.4 Uses Key points: Helium is used as an inert gas and as a light source in lasers and electric discharge lamps; liquid helium is a very low temperature refrigerant. On account of its low density and nonflammability, He is used in balloons and lighter-than-air craft. Its very low boiling point leads to its wide use in cryogenics and as a very low temperature refrigerant; it is the coolant for superconducting magnets used for NMR spectroscopy and magnetic resonance imaging. It is also used as an inert atmosphere for growing crystals of semiconducting materials, such as Si. It is mixed with O2 in a 4:1 ratio to provide an artificial atmosphere for divers, where its lower solubility than nitrogen minimizes the danger of causing the ‘bends’, or decompression sickness. The most widespread use of Ar is to provide an inert atmosphere for the production of air-sensitive compounds and as an inert gas blanket to suppress the oxidation of metals when they are welded. Argon is also used as a cryogenic refrigerant and to fill the gap between the panes in sealed double-glazed windows as its low thermal conductivity reduces heat loss. Xenon has anaesthetic properties but is not very widely used because it is approximately 2000 times more expensive than N2O. Isotopes of Xe are also used in medical imaging (Box 18.1). Radon, which is a product of nuclear power plants and of the radioactive decay of naturally occurring Th and U, is a health hazard because of the ionizing nuclear radiation it produces. It is usually a minor contributor to the background radiation arising from cosmic rays and terrestrial sources. However, in regions where the soil, underlying rocks, or building materials contain significant concentrations of U, excessive amounts of the gas have been found in buildings. Because of their lack of chemical reactivity, the noble gases are extensively used in various light sources, including conventional sources (neon signs, fluorescent lamps, and xenon flash lamps) and lasers (heliumneon, argon-ion, and krypton-ion lasers). Argon is used as the inert atmosphere in incandescent light bulbs, where it reduces burning of the filament. In each case, an electric discharge through the gas ionizes some of the atoms and promotes both ions and neutral atoms into excited states that then emit electromagnetic radiation on return to a lower state. 18.5 Synthesis and structure of xenon fluorides Key point: Xenon reacts with fluorine to form XeF2, XeF4, and XeF6. The reactivity of the noble gases has been investigated sporadically ever since their discovery, but all early attempts to coerce them into compound formation were unsuccessful. Until the 1960s the only known compounds were the unstable diatomic species such as He2 and Ar2, which were detected only spectroscopically. However, in March 1962, Neil Bartlett, then at the University of British Columbia, observed the reaction The detail 443 B OX 18 .1 Using 129Xe-NMR in medicine and materials Magnetic resonance imaging (MRI) is widely used in medicine to produce high-quality images of soft matter within the body. The technique relies on the protons of water molecules in tissue to provide the NMR signal, but protons are difficult to image in some parts of the body, particularly the lungs and the brain. However, other NMR-active nuclei can also be used for MRI, most notably 129Xe, which is 26.4 per cent abundant with I 21 . Because the extent of polarization of nuclear spin in 129Xe is too low to give a good signal, it is enhanced by using low temperatures or increasing the applied magnetic field. Alternatively, the polarization can be enhanced by a factor of about 105 by spin-exchange with a polarized alkali metal. MRI images of the lungs can be obtained by inhaling this of a noble gas. Bartlett’s report, and another from Rudolf Hoppe’s group in the University of Munster a few weeks later, set off a flurry of activity throughout the world. Within a year, a series of xenon fluorides and oxo compounds had been synthesized and characterized. The field is somewhat limited, but compounds with bonds to nitrogen, carbon, and metals have been prepared. Bartlett’s motivation for studying xenon was based on the observations that PtF6 can oxidize O2, to give the solid O2PtF6, and that the ionization energy of xenon is similar to that of molecular oxygen. Indeed, reaction of xenon with PtF6 did give a solid, but the reaction is complex and the complete formulation of the product (or products) remains unclear. The direct reaction of xenon and fluorine leads to a series of compounds with oxidation numbers 2 (XeF2), 4 (XeF4), and 6 (XeF6). The structures of XeF2 and XeF4 are well-established from diffraction and spectroscopic methods. Similar measurements on XeF6 in the gas phase, however, led to the conclusion that this molecule is fluxional. Infrared spectra and electron diffraction on XeF6 show that a distortion occurs about a threefold axis, suggesting that a triangular face of F atoms opens up to accommodate a lone pair of electrons, as in (3). One interpretation is that the fluxional process arises from the migration of the lone pair from one triangular face to another. Solid XeF6 consists of F-bridged XeF5 units and in solution it forms Xe4F24 tetramers. The gaseous and solid structures bear a molecular and electronic structural resemblance to the isoelectronic polyhalide anions I3 and ClF4 (Section 17.10c). The xenon fluorides are synthesized by direct reaction of the elements, usually in a nickel reaction vessel that has been passivated by exposure to F2 to form a thin protective NiF2 coating. This treatment also removes surface oxide, which would react with the xenon fluorides. The synthetic conditions indicated in the following equations show that formation of the higher halides is favoured by a higher proportion of fluorine and higher total pressure: 400 ºC ,1atm → XeF2 (g) (Xein excess) Xe(g) + F2 (g) ⎯⎯⎯⎯⎯ 600 ºC , 6 atm Xe(g) + 2 F2 (g) ⎯⎯⎯⎯⎯ → XeF4 (g) (Xe : F2 = 1:5) 300 ºC , 60 atm Xe(g) + 3F2 (g) ⎯⎯⎯⎯⎯ → XeF6 (g) (Xe : F2 = 1:20) A simple ‘window-sill’ synthesis is also possible. Xenon and fluorine are sealed in a glass bulb (rigorously dried to prevent the formation of HF and the attendant etching of the glass) and hyperpolarized 129Xe into the lungs. The xenon is then transferred from the lungs to the blood and then on to other tissues. Consequently, images of the circulatory system, the brain, and other vital organs can be obtained. 129 Xe-NMR is also used in materials chemistry, where it is applied to examine the structures of mesoporous materials such as zeolites and ceramics, and to soft matter, such as polymer melts and elastomers. The technique is also useful for spectroscopic characterization of compounds of xenon. The fact that 129Xe is only 26.4 per cent abundant does not affect the Xe spectra but does produce satellite peaks with relative intensities 13:74:13 in the spectra of other NMR nuclei, especially 19F. the bulb is exposed to sunlight, whereupon beautiful crystals of XeF2 slowly form in the bulb. It will be recalled that F2 undergoes photodissociation (Chapter 17), and in this synthesis the photochemically generated F atoms react with Xe atoms. 18.6 Reactions of xenon fluorides Key points: Xenon fluorides are strong oxidizing agents and form com; they are used in the prepaplexes with, F such as XeF5 , XeF7 , and XeF2 8 ration of compounds containing XeO and XeN bonds. The reactions of the xenon fluorides are similar to those of the high oxidation state interhalogens (Section 17.10), and redox and metathesis reactions dominate. One important reaction of XeF6 is metathesis with oxides: XeF6 (s) + 3H 2O(l) → XeO3 (aq) + 6 HF(g) 2 XeF6 (s) + 3SiO2 (s) → 2 XeO3 (s) + 3SiF4 (g) Another striking chemical property of the xenon fluorides is their strong oxidizing power: 2 XeF2 (s) + 2 H 2O(1) → 2 Xe(g) + 4 HF(g) + O2 (g) XeF4 (s) + Pt(s) → Xe(g) + PtF4 (s) As with the interhalogens, the xenon fluorides react with strong Lewis acids to form xenon fluoride cations: XeF2 (s) + SbF5 (l) → [XeF]+ SbF6 These cations are associated with the counterion by F bridges. Another similarity with the interhalogens is the reaction of XeF4 with the Lewis base F in acetonitrile (cyanomethane, CH3CN) solution to produce the XeF5 ion: XeF4 [N(CH3)4]F → [N(CH3)4][XeF5] The XeF5 ion is pentagonal planar (9), and in the VSEPR model the two electron pairs on Xe occupy axial positions on opposite sides of the plane. Similarly, it has been known for many years that reaction of XeF6 with an F source produces the XeF7 or XeF82 ions depending on the proportion of fluoride. Only the shape of XeF82 is known: it is a square antiprism (10), which is difficult to reconcile with the simple VSEPR model because this shape does not provide a site for the lone pair on Xe. 444 18 The Group 18 elements 18.7 Xenonoxygen compounds – Xenon oxides are endergonic ( fG o > 0 ) and cannot be prepared by direct interaction of the elements. The oxides and oxofluorides are prepared by the hydrolysis of xenon fluorides: Xe F Key point: The xenon oxides are unstable and highly explosive. XeF6 (s) + 3H 2O(l) → XeO3 (s) + 6 HF(aq) 3XeF4 (s) + 6 H 2O(l) → XeO3 (s) + 2 Xe(g) + 32 O2 (g) + 12 HF(aq) XeF6 (s) + H 2O(l) → XeOF4 (s) + 2 HF(aq) 9 XeF5– 2– F Xe The pyramidal xenon trioxide, XeO3 (4) presents a serious hazard because this endergonic compound is highly explosive. It is a very O strong oxidizing agent in acidic solution, with E (XeO3,Xe) 2.10 V. In basic aqueous solution the Xe(VI) oxoanion HXeO4 slowly decomposes in a coupled disproportionation and water , and xenon: oxidation to yield a Xe(VIII) perxenate ion, XeO4 6 2 HXeO4 (aq) + 2OH(aq) → XeO64(aq) + Xe(g) + O2 (g) + 2 H 2O(l) The perxenates of several alkali metal ions have been prepared by treatment of XeO3 with ozone in basic conditions. These compounds are white crystalline solids with octahedral XeO4 6 units (12). They are powerful oxidizing agents in acidic aqueous solution: 10 XeF82– The xenon fluorides are the gateway to the preparation of compounds of the noble gases with elements other than F and O. The reaction of nucleophiles with a xenon fluoride is one useful strategy for the synthesis of such bonds. For instance, the reaction XeF2 + HN(SO2F)2 → FXeN(SO2F)2 + HF is driven forward by the stability of the product HF and the energy of formation of the XeN bond (11). A strong Lewis acid such as AsF5 can extract F from the product of this reaction to yield the cation [XeN(SO2F)2]. Another route to XeN bonds is the reaction of one of the fluorides with a strong Lewis acid: XeO64(aq) + 3 H + (aq) → HXeO4 (aq) + 1 2 O2 (g) + H 2O(l) Treating Ba2XeO6 with concentrated sulfuric acid produces the only other known oxide of xenon, XeO4 (5), which is an explosively unstable gas. 4– Xe O XeF2 + AsF5 → [XeF]+ [AsF6 ]− followed by the introduction of a Lewis base, such as CH3CN, to yield [CH3CNXe] [AsF6]. ■ A brief illustration. The structure of many xenon compounds can be successfully predicted by using the VSEPR model. The Lewis structure of the perxenate ion is shown in (13). With six electron pairs around the Xe atom, the VSEPR model predicts an octahedral arrangement of bonding electron pairs and an octahedral overall structure. ■ S O Xe N 12 XeO64– F 4– O O O Xe O O O 11 FXeN(SO2F)2 13 XeO64– The detail Xenon forms the oxofluorides XeOF2 (6), XeO3F2 (7), and XeOF4 (8). When alkali metal fluorides are dissolved in XeOF4, solvated fluoride ions are formed of composition F.3XeOF4. Attempted removal of XeOF4 from the solvate yields XeOF5 (14), which is a pentagonal pyramid. O H Xe 445 O 15 HXeOH – O Xe F H 16 O(XeH)2 Xe H 14 XeOF5– Xe C 17 HXeCCH ■ A brief illustration. The structures of compounds of xenon can be probed using 129Xe-NMR spectroscopy. For example, the 129Xe-NMR spectrum of XeOF4 (8) consists of a quintuplet of peaks. These peaks correspond to the single Xe environment, which is coupled to four equivalent 19F atoms. ■ E X A MPL E 18 .1 Describing the synthesis of a noble gas compound Describe a procedure for the synthesis of potassium perxenate, starting with xenon and other reagents of your choice. Answer We know that XeO compounds are endergonic, so they cannot be prepared by direct reaction of xenon and oxygen, so we need to look for an indirect method. As was described in the text, the hydrolysis of XeF6 yields XeO3, which undergoes disproportionation in basic solution to yield perxenate, XeO4 . Thus XeF6 could be synthesised by the reaction of 6 xenon and excess F2 at 300°C and 6 MPa in a stout nickel container. The resulting XeF6 might then be converted to the perxenate in one step by exposing it to aqueous KOH solution. The resulting potassium perxenate (which turns out to be a hydrate) could then be crystallized. Self-test 18.1 Write a balanced equation for the decomposition of xenate ions in basic solution for the production of perxenate ions, xenon, and oxygen. 18.8 Xenon insertion compounds Key point: Xenon can insert into HY bonds. A number of noble gas hydrides have been isolated at low temperatures with the general formula HEY, where E is the Group 18 element and Y is an electronegative element or fragment. The first species identified were HXeCl, HXeBr, HXeI, and HKrCl. More recently characterized species include HKrCN and HXeC3N. They are all prepared by UV photolysis of the HY precursors in the solid noble gas at low temperatures. When xenon reacts in this way with water, both HXeOH (15) and the HXeO radical are produced. The latter can react further with another Xe atom and hydrogen to form HXeOXeH (16), which is the smallest molecule to contain two Xe atoms. Xenon atoms have been successfully inserted into HC bonds of hydrocarbons by photolysis and annealing of a solid mixture of C2H2 and Xe. The noble gas hydrides HXeCCH (17) and HXeCCXeH (18) have been prepared in this way. H Xe C 18 HXeCCXeH These insertion reactions may hold the key to explaining the so-called ‘missing xenon’ phenomenon. Relative to the other noble gases the amount of xenon in the atmosphere is depleted by a factor of 20. One theory is that xenon in the Earth’s interior can form stable compounds. The formation of these insertion compounds indicates that xenon compound formation under extreme conditions may indeed be possible. In fact, recent studies have indicated that Xe will exchange with Si in silicate minerals under high temperature and pressure. 18.9 Organoxenon compounds Key point: Organoxenon compounds can be prepared by xenodeborylation of an organoboron compound. The first compound containing XeC bonds was reported in 1989. Since then a wide variety of organoxenon compounds have been prepared. The most useful routes to organoxenon compounds are through the fluorides XeF2 and XeF4. Organoxenon(II) salts can be prepared from organoboranes by xenodeborylation, the substitution of B by Xe. For example, tris(pentafluorophenyl)borane reacts with XeF2 in dichloromethane to produce arylxenon(II) fluoroborates (19): CH Cl 2 2 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → (C6F5)3 B + XeF2 ⎯ [C6F5Xe]+ + [(C6F5)n BF4n ] n = 1, 2 When this reaction is carried out in anhydrous HF, all the C6F5 groups are transferred to the Xe: → 3[C F Xe] (C F ) B + 3XeF ⎯⎯⎯ HF 6 5 3 2 6 5 [BF4 ] + 2[F ( HF )n ] A general route that can be used to introduce other organic groups uses organodifluoroboranes, RBF2: CH Cl 2 2 RBF2 + XeF2 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → [RXe] + [BF4 ] 446 18 The Group 18 elements In addition to xenodeborylation, organoxenon(II) compounds (20) can be prepared from C6F5SiMe3: CH Cl 2 2 3C6F5SiMe3 + 2 XeF2 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → Xe(C6F5)2 + C6F5XeF + 3Me3SiF Green crystals of [Xe2][Sb4F21] are also produced at 60°C. The XeXe bond length is 309 pm and is the longest homonuclear bond known for a main-group element. Au 2+ Organoxenon(II) compounds are thermally unstable and decompose above 40°C. Xe 21 [AuXe4]2+ F F F F F F Xe F F F F Xe + 19 C6F5Xe F F F F F 20 Xe(C6H5)2 The first organoxenon(IV) compound was prepared by the reaction between XeF4 and C6F5BF2 in CH2Cl2: Many complexes of the noble gases are transient species that have been characterized by matrix isolation. The complex Fe(CO)4Xe is formed when Fe(CO)5 is photolysed in solid xenon at 12 K. Similarly, M(CO)5E is formed when M(CO)6 (M Cr, Mo, or W) is photolysed in solid argon, krypton, or xenon at 20 K, where E Ar, Kr, or Xe, respectively. An alternative method of synthesizing these complexes is to generate them in an argon, krypton, or xenon atmosphere. The Xe complex has also been isolated in liquid xenon. The stabilities of the complexes decrease in the order W Mo ≈ Cr and Xe Kr Ar. The complexes are octahedral (22) and the bonding is thought to involve interactions between the p orbitals on the noble gas and orbitals on the equatorial CO groups. Thus, noble gases can be thought of as potential ligands, and indeed have been fully characterized as such (including by NMR; Section 22.16). CH Cl 2 2 C6F5BF2 + XeF4 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → [C6F5XeF2 ][BF4 ] Organoxenon(IV) compounds are less thermally stable than the analogous Xe(II) compounds. In all Xe(II) and Xe(IV) salts the Xe atom is bonded to a C atom that is part of a π system. Extended π systems, such as in aryl groups, increase the stability of the XeC bond. This stability is further favoured by the presence of electron-withdrawing substituents (such as fluorine) on the aryl group. CO M E 18.10 Coordination compounds Key points: Argon, xenon, and krypton form coordination compounds that are usually studied by matrix isolation; the stability of the complexes decreases in the order Xe Kr Ar. Coordination compounds of noble gases have been known since the mid-1970s. The first stable noble-gas coordination compound to be synthesized was [AuXe4]2[SbF11]2, which contains a square-planar [AuXe4]2 cation (21). The compound is made by reduction of AuF3 by HF/SbF5 in elemental xenon to yield dark red crystals that are stable up to 78°C. Alternatively, addition of Xe to a solution of Au2 in HF/SbF5 gives a dark red solution that is stable up to 40°C. This solution is stable at room temperature under a xenon pressure of 1 MPa (about 10 atm). During the reduction of Au3 to Au2 the extreme Brønsted acidity of HF/SbF5 (Section 15.11) is essential and the overall reaction indicates the role of protons: HF/SbF 5 AuF3 + 6 Xe + 3H + ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → [AuXe4 ]2+ + Xe2+ + 3HF 22 [M(CO)5E] E = Ar, Kr, Xe When [Rh(5-Cp)(CO)2] or [Rh(5-Cp)(CO)2] is photolysed in supercritical xenon or krypton at room temperature, the complexes [Rh(5-Cp)(CO)E] and [Rh(5-Cp)(CO)E] are formed, where E Xe or Kr. The 5-Cp complexes are less stable than the 5-Cp analogues and the Kr complexes are less stable than those of Xe. 18.11 Other compounds of noble gases Key point: Krypton and radon fluorides are known but their chemical properties are much less extensive than those of xenon. Radon has a lower ionization energy than Xe, so it can be expected to form compounds even more readily. Evidence exists for the formation of RnF2, and cationic compounds, such as The detail [RnF][SbF6], but detailed characterization is frustrated by their radioactivity. Krypton has a much higher ionization energy than Xe (Table 18.1) and its ability to form compounds is more limited. Krypton difluoride, KrF2, is prepared by passing an electric discharge or ionizing radiation through a fluorinekrypton mixture at low temperatures (196°C). As with XeF2, the krypton compound is a colourless volatile solid and the molecule is linear. It is an endergonic and highly reactive compound that must be stored at low temperatures. When monomeric HF is photolysed in solid argon and annealed to 18 K, HArF is formed. This compound is stable up to 27 K and contains the HAr and F ions. The related molecular ions, HHe, HNe, and HXe have been observed by spectroscopy. The heavier noble gases form clathrates. Argon, Kr, and Xe form clathrates with quinol (1,4-C6H4(OH)2) with one gas atom to three quinol molecules. They also form clathrate hydrates 447 with water in a ratio of one gas atom to 46 H2O molecules. Helium and Ne are too small to form stable clathrates. Titan, Saturn’s moon, has a dense atmosphere in which levels of Kr and Xe are depleted in comparison to Ar. It is believed that the Kr and Xe are trapped in clathrates whereas the smaller Ar atoms are trapped less effectively. Endohedral fullerene complexes, in which the ‘guest’ atom or ion sits inside a fullerene cage (Section 14.6b), have been n n and C70 (n 1, 2, or 3) with He and C60 with observed for C60 Ne. Molecular orbital calculations indicate that Ar should be cage, although this complex has not able to penetrate the C60 yet been observed. Other than the fullerene complexes, those identified transiently in high-energy molecular beams, or van der Waals complexes in the gas phase, there are no known compounds of He. However, theoretical calculations predict HeBeO to be exergonic. FURTHER READING W. Grochala, Atypical compounds of gases which have been called ‘noble’, Chem. Soc. Rev., 2007, 36, 1632. M. Ozima and F.A. Podosec, Noble gas geochemistry. Cambridge University Press (2002). A.G. Massey, Main group chemistry. Wiley (2000). P. Lazlo and G.J. Schrobilgen, Angew. Chem., Int. Ed. Engl., 1988, 27, 479. An enjoyable account of the early failure and final success in the quest for compounds of the noble gases. D.M.P. Mingos, Essential trends in inorganic chemistry. Oxford University Press (1998). M.S. Albert, G.D. Cates, B. Driehuys, W. Happer, B. Saam, C.S. Springer, and A. Wishnia, Biological magnetic resonance imaging using laserpolarized 129Xe, Nature, 370, 199–201 (1994). R.B. King (ed.), Encyclopedia of inorganic chemistry. Wiley (2005). J. Holloway, Twenty-five years of noble gas chemistry. Chemistry in Britain, 1987, 658. A good summary of the development of the field. H. Frohn and V.V. Bardin, Organometallics, 2001, 20, 4750. A readable review of the organo compounds of the noble gases. EXERCISES 18.1 Explain why helium is present in low concentration in the atmosphere even though it is the second most abundant element in the universe. 18.6 (a) Give a Lewis structure for XeF7. (b) Speculate on its possible structures by using the VSEPR model and analogy with other xenon fluoride anions. 18.2 Which of the noble gases would you choose as (a) the lowest temperature liquid refrigerant, (b) an electric discharge light source requiring a safe gas with the lowest ionization energy, (c) the least expensive inert atmosphere? 18.7 Use molecular orbital theory to calculate the bond order of the diatomic species E2 with E He and Ne. 18.3 By means of balanced chemical equations and a statement of conditions, describe a suitable synthesis of (a) xenon difluoride, (b) xenon hexafluoride, (c) xenon trioxide. 18.4 Draw the Lewis structures of (a) XeOF4, (b) XeO2F2, and (c) XeO62. 18.5 Give the formula and describe the structure of a noble gas species that is isostructural with (a) ICl , (b) IBr2 , (c) BrO , (d) ClF. 4 3 18.8 Identify the xenon compounds A, B, C, D, and E. D H2O C xs F2 Xe F2 A MeBF2 B 2F2 E 18.9 Predict the appearance of the 129Xe-NMR spectrum of XeOF3. 18.10 Predict the appearance of the 19F-NMR spectrum of XeOF4. PROBLEMS 18.1 The first compound containing an XeN bond was reported by R.D. LeBlond and K.K. DesMarteau (J. Chem. Soc., Chem. Commun., 1974, 554). Summarize the method of synthesis and characterization. (The proposed structure was later confirmed in an X-ray crystal structure determination.) organometallic noble gas complexes. (b) How did the methods used by these authors differ from the usual techniques of matrix isolation? (c) Place the complexes [MnCp(CO)2Xe], [RhCp(CO)Xe], [MnCp(CO)2Kr], [Mo(CO)5Kr], and [W(CO)5Kr] in order of increasing stability towards CO substitution. 18.2 (a) Use the references in the paper by O.S. Jina, X.Z. Sun, and M.W. George (J. Chem. Soc., Dalton Trans., 2003, 1773) to produce a review of the use of matrix isolation in the characterization of 18.3 In their paper ‘Xenon as a complex ligand: the tetra xenon gold(II) cation in AuXe24 (Sb2 F11 )2 ’, S. Seidel and K. Seppelt, Science, 2000, 290, 117 describe the first synthesis of a stable noble 448 18 The Group 18 elements gas coordination compound. Give details of the synthesis and characterization of the compound. 18.4 The synthesis and characterization of the XeOF5 anion has been described by A. Ellern and K. Seppelt (Angew. Chem., Int. Ed. Engl, 1995, 34, 1586). (a) Summarize the similarities between XeOF4 and IF5. (b) Give possible reasons for the differences in structure between XeOF5 and IF6 . (c) Summarize how XeOF5 was prepared. 18.5 In their paper ‘Helium chemistry: theoretical predictions and experimental challenge’, (J. Am. Chem. Soc., 1987, 109, 5917) W. Koch and co-workers used quantum mechanical calculations to demonstrate that helium can form strong bonds with carbon in cations. (a) Give the range of bond lengths calculated for these HeC cations. (b) What are the requirements for an element to form a strong bond with He? (c) To which branch of science do the authors suggest that this work would be particularly relevant? 18.6 In their paper ‘Observation of superflow in solid helium’ (Science, 2004, 305, 5692, 1941) Kim and Chan described the observed superfluidity of solid helium. Define superfluidity, describe the experiment carried out by the authors to demonstrate the property, and summarize their explanation of superfluidity. The d-block elements The metallic elements are the most numerous of the elements and of 3 4 5 6 7 8 9 10 11 12 the metallic elements the d-block Ca cS T i V C r Mn e F C o N i u Z C n Ga elements are the most important: their chemical properties are central Sr Y Z r b Mo T N h P c u R R d A g C d In to both industry and contemporary research. We shall see systematic trends across the block and trends a Ba L Tl f H aT W e R O s Ir t A P u H g in their chemical properties within each group. One of the most imporRa A s R g c f D R b S g B h H s Mt D tant trends is the variation in stability of the oxidation states because these stabilities are closely related to the ease of recovery of metals from their ores and to the ways in which the various metals and their compounds are handled in the laboratory. It will be seen that simple binary compounds, such as metal halides and oxides, often follow systematic trends; however, when the metal is in a low oxidation state, interesting variations such as metalmetal bonding may be encountered. Mg Al The two terms d-block metal and transition metal are often used interchangeably; however, they do not mean the same thing. The name transition metal originally derived from the fact that their chemical properties were transitional between those of the s and p blocks. Now, however, the IUPAC definition of a transition element is that it is an element that has an incomplete d subshell in either the neutral atom or its ions. Thus the Group 12 elements (Zn, Cd, Hg) are members of the d block but are not transition elements. In the following discussion, it will be convenient to refer to each row of the d block as a series, with the 3d series the first row of the block (Period 4), the 4d series the second row (Period 5), and so on. It will prove important to note the intrusion of the f block, the inner transition elements, the lanthanoids, into the 5d series. Elements towards the left of the d block are often referred to as early and those towards the right are referred to as late. The elements The chemical properties of the d-block elements (or d metals as we shall commonly call them) will occupy this and the next three chapters. It is therefore appropriate to begin with a survey of their occurrence and properties. The trends in their properties and the correlation with their electronic structures and therefore location in the periodic table should be kept in mind throughout these chapters. 19.1 Occurrence and recovery Key point: Chemically soft members of the block occur as sulfide minerals and are partially oxidized to obtain the metal; the more electropositive ‘hard’ metals occur as oxides and are extracted by reduction. 19 The elements 19.1 Occurrence and recovery 19.2 Physical properties Trends in chemical properties 19.3 Oxidation states across a series 19.4 Oxidation states down a group 19.5 Structural trends 19.6 Noble character Representative compounds 19.7 Metal halides 19.8 Metal oxides and oxido complexes 19.9 Metal sulfides and sulfide complexes 19.10 Nitrido and alkylidyne complexes 19.11 Metalmetal bonded compounds and clusters FURTHER READING EXERCISES PROBLEMS 450 19 The d-block elements The elements on the left of the 3d series occur in nature primarily as metal oxides or as metal cations in combination with oxoanions (Table 19.1). Of these elements, titanium ores are the most difficult to reduce, and the element is widely produced by heating TiO2 with chlorine and carbon to produce TiCl4, which is then reduced by molten magnesium at about 1000ºC in an inert-gas atmosphere. The oxides of Cr, Mn, and Fe are reduced with carbon (Section 5.16), a much cheaper reagent. To the right of Fe in the 3d series, Co, Ni, Cu, and Zn occur mainly as sulfides and arsenides, which is consistent with the increasingly soft Lewis acid character of their dipositive ions. Sulfide ores are usually roasted in air either to the metal directly (for example, Ni) or to an oxide that is subsequently reduced (for example, Zn). Copper is used in large quantities for electrical conductors; electrolysis is used to refine crude copper to achieve the high purity needed for high electrical conductivity. The difficulty of reducing the early 4d and 5d metals Mo and W is apparent from Table 19.1. It reflects the tendency of these elements to have stable high oxidation states, as discussed later. The platinum metals (Ru and Os, Rh and Ir, and Pd and Pt), which are found at the lower right of the d block, occur as sulfide and arsenide ores, usually in association with larger quantities of Cu, Ni, and Co. They are collected from the sludge that forms during the electrolytic refinement of copper and nickel. Gold (and to some extent silver) is found in its elemental form. The role that certain d metals play in the environment is summarized in Box 19.1. 19.2 Physical properties Key points: The properties of the d metals are largely derived from their electronic structure, with the strength of metallic bonding peaking at Group 6; the lanthanide contraction is responsible for some of the anomalous behaviour of the metals in the 5d series. The d block of the periodic table contains the metals most important to modern society. It contains the immensely strong and light titanium, the major components of most steels (Fe, Cr, Mn, Mo), the highly electrically conducting copper, the malleable gold and platinum, and the very dense osmium and iridium. To a large extent these properties derive from the nature of the metallic bonding that binds the atoms together. Table 19.1 Mineral sources and methods of recovery of some commercially important d metals Metal Principal minerals Method of recovery Titanium Ilmenite, FeTiO3 Rutile, TiO2 TiO2 2 C 2 Cl2 → TiCl4 2 CO followed by reduction of TiCl4 with Na or Mg Chromium Chromite, FeCr2O4 FeCr2O4 4 C → Fe 2 Cr 4 CO Molybdenum Molybdenite, MoS2 2 MoS2 7 O2 → 2 MoO3 4 SO2 followed by either MoO3 2 Fe → Mo Fe2O3 or MoO3 3 H2 → Mo 3 H2O Tungsten Scheelite, CaWO4 Wolframite, FeMn(WO4)2 CaWO4 2 HCl → WO3CaCl2H2O followed by WO3 3 H2 → W 3 H2O Manganese Pyrolusite, MnO2 MnO2 2 C → Mn 2 CO Iron Haematite, Fe2O3 Magnetite, Fe3O4 Limonite, FeO(OH) Fe2O3 3 CO → 2 Fe 3 CO2 Cobalt CoAsS Smaltite, CoAs2 Linnaeite, Co3S4 Byproduct of copper and nickel production Nickel Pentlandite, (Fe,Ni)6S8 NiS O2→ Ni SO2 Copper Chalcopyrite, CuFeS2 Chalocite, Cu2S 2 CuFeS2 2 SiO2 5 O2 → 2 Cu 2 FeSiO3 4 SO2 Note (a) (b) (c) (a) The ironchromium alloy is used directly for stainless steel. (b) The reaction is carried out in a blast furnace with Fe2O3 to produce alloys. (c) NiS is formed by smelting the ore and is then separated by physical processes. NiO is used in a blast furnace with iron oxides to produce steel. Nickel is purified by electrolysis or the Mond process via Ni(CO)4, Box 22.1. The elements 451 BOX 19.1 Toxic metals in the environment The biosphere evolved in close association with all the elements in the periodic table, and has had a long time to adapt to them. Metals are released from rocks by weathering, and are processed by a variety of mechanisms, including biological ones. Indeed, many metals have been harnessed for essential biochemical functions (Chapter 27), and other biochemical systems have evolved to sequester metals and keep them out of harm’s way. However, the natural biogeochemical cycles have been greatly perturbed by mining: circulating levels of most metals have increased dramatically since pre-industrial times. Many metals and metalloids, including Be, Mn, Cr, Ni, Cd, Hg, Pb, Se, and As, are hazardous in occupational or environmental settings. Exposure to these elements varies widely, and depends on the pattern of their industrial use, and on their environmental chemistry. The metals of greatest environmental concern are heavy metals, such as mercury, which, being chemically very soft, bind tightly to thiol groups in proteins. Growth or maintenance of organism Non-essential element Essential element (copper) Essential element (iron) Concentration of element Figure B19.1 The idealized doseresponse curves for a non-essential element and two essential elements. The effect of a substance on a living organism is often plotted as a doseresponse curve, such as those in Fig. B19.1. The shape of the doseresponse curves for various metals varies widely, depending on physiological variables, and on the chemistry of the metals. For example, iron and copper are both essential elements but have very different doseresponse curves: toxicity occurs at a much lower concentration for copper than for iron. It is reasonable to associate this toxicity with the higher affinity of copper ions for sulfur and nitrogen ligands, which makes copper more likely to interfere with crucial sites in proteins. Nevertheless, iron is harmful at higher doses because it can catalyse the production of oxygen radicals, and also because excess iron can stimulate the growth of bacteria. The ability of elements to exist in alternative redox states can be of crucial importance to the hazard they present. For example, the solubilities of different oxidation states of a particular metal are often very different. Many metals are found as insoluble sulfides in the Earth’s crust and are thus immobile. However, when they are disturbed and brought into contact with air, which oxidizes them, they can become very mobile. An example is mercury(II) sulfide, which is oxidized to the mobile mercury(II) sulfate on exposure to air. Another effect of oxidation is exemplified by iron, which occurs widely as the mineral pyrite, FeS2. Mining operations often lead to acid drainage due to the reaction FeS2 + 154 O2 + 72 H2O → Fe ( OH)3 + 2SO24− + 4H+ In this process (which is generally catalysed by certain bacteria), S2 is 2 oxidized to sulfuric acid and Fe2 is oxidized to Fe3, which hydrolyses to Fe(OH)3, yielding additional H ions. Streams emanating from iron mines, many of them abandoned, are thus often highly acidic. Redox status can have opposite effects on solubility for different metals. Thus, whereas manganese is mobile under reducing conditions, other metals are mobilized under oxidizing conditions. For instance, Cr(III) is immobile, forming an insoluble oxide or kinetically inert complexes with soil constituents; in addition, there is no mechanism for transporting free Cr(III) ions across biological membranes. However, Cr(III) is solubilized upon . Chromate(VI) is highly toxic, and oxidation to the chromate(VI) ion, CrO2 4 is implicated in cancer. The CrO2 ion has the same shape and charge as 4 SO2 , and can be ferried across biological membranes by sulfate transport 4 proteins. Once inside a cell, chromate is converted to Cr(III) complexes by endogenous reductants, such as ascorbate. In the process, reactive oxygen species are generated, which can damage DNA; in addition, the Cr(III) reduction product can complex and cross-link DNA. Cr(III) accumulates inside the chromate-exposed cells, as it does not have a mechanism for transport through biological membranes. For other metals, toxicity is controlled by the interplay of membrane transport and intracellular binding with the redox state. For example, mercury salts are relatively harmless because membranes, which present a barrier to ionic species, are impermeable to Hg2. Likewise, metallic mercury is not absorbed through the gut, so it is not toxic when swallowed. However, mercury vapour is highly toxic (that is why all mercury spills must be rigorously cleaned up) because the neutral atoms readily pass through the lungs’ membranes and also across the bloodbrain barrier. Once in the brain, the Hg(0) is oxidized to Hg(II) by the vigorous oxidizing activity of brain cell mitochondria, and the Hg(II) binds tightly to crucial thiolate groups of neuronal proteins. Mercury is a powerful neurotoxin, but only if it gets inside nerve cells. Even more hazardous than mercury vapour are organomercury compounds, particularly methylmercury. Thus, CH3Hg is taken up through the gut because it is complexed by the stomach’s chloride, forming CH3HgCl, which, being electrically neutral, can pass through a membrane. Once inside cells, CH3Hg binds to thiolate groups and accumulates. The environmental toxicity of mercury is associated almost entirely with eating fish. Methylmercury is produced by the action of sulfate-reducing bacteria on Hg2 in sediments, and accumulates as little fish are eaten by bigger fish further up the aquatic food chain. Fish everywhere have some level of mercury present. Mercury levels can increase markedly if sediments are contaminated by additional mercury. The worst known case of environmental mercury poisoning occurred in the 1950s in the Japanese fishing village of Minamata. A polyvinyl chloride plant, using Hg2 as a catalyst, discharged mercury-laden residues into the bay, where fish accumulated methylmercury to levels approaching 100 ppm. Thousands of people were poisoned by eating the fish, and a number of infants suffered mental disabilities and motor disturbance from exposure in utero. This disaster led to strict standards for fish consumption. Limited consumption is advised for fish at the top of the food chain, such as pike and bass in fresh waters, and swordfish and tuna in the oceans. Regulatory action has been aimed at reducing mercury discharges and emissions, and industrial point sources have been largely controlled. Chloralkali plants, producing the large-volume industrial chemicals Cl2 and NaOH by electrolysis of NaCl, were a major source as a mercury pool electrode was used to transfer metallic sodium to a separate hydroxidegenerating compartment. However, this is now accomplished by separating the two electrode compartments with a cation-exchange membrane, which prevents migration of the anions. Combustion can vent mercury to the atmosphere if the fuel contains mercury compounds. Whereas municipal waste and hospital incinerators have been equipped with filters to reduce (Continued) 452 19 The d-block elements BOX 19.1 (Continued) mercury emissions to the air, coal contains small amounts of mercury minerals. Because of the huge quantities burned, coal is a major contributor to environmental mercury. Mercury is a global problem because mercury vapour and volatile organomercurial compounds can travel long distances in the atmosphere. Eventually elemental mercury is oxidized and organomercurials are decomposed, both to Hg2, by reaction with ozone or by atmospheric hydroxyl or halogen radicals. The Hg2 ions are solvated by water molecules and deposited in rainfall. Thus mercury deposition can occur far from the emission source, and is distributed fairly uniformly around the globe. For example, it is estimated that only a third of North American mercury emissions are in fact deposited in the USA, and this accounts for only half of the mercury deposition in the USA. Even the gold fields of Brazil contribute because miners use mercury to extract the gold, which is recovered by heating the resultant amalgam to drive off the mercury. This practice is estimated to account for 2 per cent of global mercury emissions (half of South American emissions). The picture is further complicated by the fact that much of the deposited mercury is recirculated through processes that produce volatile compounds or mercury vapour. For example, much of the biomethylation activity of the sulfate-reducing bacteria produces dimethylmercury, (CH3)2Hg, which, being volatile, is vented to the atmosphere. Other bacteria have an enzyme (methylmercury lyase) that breaks the methylmercury bond of CH3Hg, and another enzyme (methylmercury reductase) that reduces the resulting Hg(II) to Hg(0); this is a protective mechanism for the microorganisms, ridding them of mercury as volatile Hg(0). Metallic bonding was covered in Chapter 3, which introduced the concept of band structure. Generally speaking, the same band structure is present for all the d-block metals and arises from the overlap of the (n1)s orbitals to give an s band and of the nd orbitals to give a d band. The principal differences between the metals is the number of electrons available to occupy these bands: Ti (3d24s2) has four bonding electrons, V (3d34s2) five, Cr (3d54s1) six, and so on. The lower, net bonding region of the valence band is therefore progressively filled with electrons on going to the right across the block, which results in stronger bonding, until around Group 7 (at Mn, Tc, Re) when the electrons begin to populate the upper, net antibonding part of the band. This trend in bonding strength is reflected in the increase in melting point from the low-melting alkali metals (effectively only one bonding electron for each atom, resulting in melting points typically less than 100ºC) up to Cr, and its decline thereafter to the low-melting Group 12 metals (mercury being a liquid at room temperature, Fig. 19.1 and Section 9.2). The strength of metallic bonding in tungsten is such that its melting point (3410ºC) is exceeded by only one other element, carbon. The radii of d-metal ions depend on the effective charge of the nucleus, and ionic radii generally decrease on moving to the right as the atomic number increases. The radius of the metal atoms in the solid element is determined by a combination of the strength of the metallic bonding and the size of the ions. Thus, the separations of the centres of the atoms in the solid generally follow a similar pattern to the melting points: they decrease to the middle of the d block, followed by an increase back up to Group 12, with the smallest separations occurring in and near Groups 7 and 8. The atomic radii of the elements in the 5d series (Hf, Ta, W,...) are not much bigger than those of their 4d-series congeners (Zr, Nb, Mo,...). In fact, the atomic radius of Hf Melting temperature/°C W 3000 Mo Hf V 2000 Be 1000 Mg Ba Sn 0 Li Na K Zn 10 Rb Cs 30 50 Atomic number, Z Figure 19.1 The melting points of the metals in Groups 115. Bi 70 Hg 90 Trends in chemical properties 25 Density, ρ/(g cm–3) Ir 20 15 Rh Hf 10 Cu Bi Sn 5 0 Ga Mg Rb Li Na K 10 30 Ba Cs 50 70 90 Atomic number, Z Figure 19.2 The densities of the elements in Groups 115. is smaller than that of Zr even though it appears in a later period. To understand this anomaly, we need to consider the effect of the lanthanoids (the first row of the f block). The intervention of the lanthanoid elements in Period 6 corresponds to the occupation of the poorly shielding 4f orbitals. Because the atomic number has increased by 32 between Zr in Period 5 and its congener Hf in Period 6 without a corresponding increase in shielding, the overall effect is that the atomic radii of the 5d-series elements are much smaller than expected. This reduction in radius is the lanthanide contraction introduced in Section 1.9a. The lanthanide contraction also affects the ionization energies of the 5d-series elements, making them higher than expected on the basis of a straightforward extrapolation. Some of the metals—specifically Au, Pt, Ir, and Os—have such high ionization energies that they are unreactive under normal conditions. Atomic mass increases with atomic number, and the combination of this increase with the changes in the radii of the metal atoms in the metal lattice means that the mass densities of the elements reach a peak with Ir (density 22.65 g cm3). Figure 19.2 illustrates this trend. Trends in chemical properties Many of the d metals display a wide range of oxidation states, which leads to a rich and fascinating chemistry. They also form an extensive range of coordination compounds (Chapters 7, 20, and 21) and organometallic compounds (Chapter 22). Many of the trends discussed in Chapter 9 are applicable to the d metals; in particular we should recall that ionization energies generally decrease down a group and increase across a period. 19.3 Oxidation states across a series The range of oxidation states of the d metals accounts for the interesting electronic properties of many solid compounds (Chapters 24 and 25), their ability to participate in catalysis (Chapter 26), and their subtle and interesting role in biochemical processes (Chapter 27). In this section we focus on trends in the stabilities of oxidation states within the d block. (a) High oxidation states Key point: The group oxidation state can be achieved by elements that lie towards the left of the d block but not by elements on the right. Oxygen is usually more effective than fluorine at bringing out the highest oxidation states because less crowding is involved. The group oxidation state (in which the oxidation number is equal to the Group number) can be achieved by elements that lie towards the left of the d block but not by elements on the right (Section 9.5). For example, Sc, Y, and La in Group 3, with configurations nd1(n1)s2, are found in aqueous solution only in oxidation state 3, which corresponds to the loss of all their outermost electrons, and the majority of their complexes contain the 453 454 19 The d-block elements elements in this state. The group oxidation state is never achieved after Group 8 (Fe, Ru, and Os). This limit on the maximum oxidation state correlates with the increase in ionization energy and hence noble character from left to right across each series in the d block. The trend in thermodynamic stability of the group oxidation states of the 3d-series elements is illustrated in Fig. 19.3, which shows the Frost diagram for species in aqueous acidic solution. We see that the group oxidation states of Sc, Ti, and V fall in the lower part of the diagram. This location indicates that the element and any species in intermediate oxidation states are readily oxidized to the group oxidation state. By contrast, species in the group oxidation state for Cr and Mn (6 and 7, respectively) lie in the upper part of the diagram. This location indicates that they are very susceptible to reduction. The Frost diagram shows that the group oxidation state is not achieved in Groups 812 of the 3d series (Fe, Co, Ni, Cu, and Zn), and also shows the oxidation states that are most stable under acid conditions; namely Ti3, V3, Cr3, Mn2, Fe2, Co2, and Ni2. The binary compounds of the 3d-series elements with the halogens and with oxygen also illustrate the trend in stabilities of the group oxidation states. The earliest metals can achieve their group oxidation states in compounds with chlorine (for example, ScCl3 and TiCl4), but the more strongly oxidizing halogen fluorine is necessary to achieve the group oxidation state of V (Group 5) and Cr (Group 6), which form VF5 and CrF6, respectively. Beyond Group 6 in the 3d series, even fluorine cannot produce the group oxidation state, and MnF7 and FeF8 have not been prepared. Oxygen brings out the group oxidation state for many elements more readily than does fluorine because fewer O atoms than F atoms are needed to achieve the same oxidation number, thus decreasing steric crowding. For example, the group oxidation state of 7 for Mn is achieved in manganate(VII) salts, such as potassium permanganate, KMnO4. As can be inferred from the Frost diagram in Fig. 19.3, chromate(VI) CrO42, manganate(VII) MnO4, and ferrate(VI) FeO42 are strong oxidizing agents and become stronger from CrO42 to FeO42. This trend is another illustration of the decreasing stability of the maximum attainable oxidation state for Groups 6, 7, and 8. Yet another example of the greater difficulty of oxidizing an element to the right of Cr to its group oxidation state is that the air oxidation of MnO2 in molten potassium hydroxide does not take Mn to its group oxidation state but instead yields the deep green compound potassium manganate(VI), K2MnO4. The disproportionation of MnO42 in acidic aqueous solution yields manganese(IV) oxide (MnO2) and the deep purple manganate(VII) ion MnO4: 3 MnO42(aq) 4 H(aq) → 2 MnO4(aq) MnO2(s) 2 H2O(l) (b) Intermediate oxidation states in the 3d series Key point: Oxidation state 3 is common to the left of the 3d series and 2 is common for metals from the middle to the right of the block. +6 H2FeO4 +4 HMnO4 H2MnO4 CoO2 H3MnO4 +2 Cu2+ NE°/V Cu+ Ni2+ 0 H2Cr2O7 FeO2+ Fe3+ MnO2 Co 3+ Zn2+ Mn 2+ VO2+ Cr3+ VO 2+ Fe 2+ Cr+ Mn2+ –2 –4 Figure 19.3 A Frost diagram for the first series of the d-block elements in acidic solution (pH 0). The broken line connects species in their group oxidation states. Co3+ V2+ V3+ Ti2+ Ti3+ TiO2+ Ca2+ –6 0 +1 Sc3+ +3 +5 +2 +4 Oxidation number, N +6 +7 Trends in chemical properties 4 Co3(aq) 2 H2O(l) → 4 Co2(aq) 4 H(aq) O2 cell Despite this instability, the slowness of the evolution of H2 makes it possible to work with aqueous solutions of these dipositive ions in the absence of air, and as a result they are useful reducing agents. Beyond Cr (for Mn2, Fe2, Co2, Ni2, and Cu2), M(II) is stable with respect to reaction with water, and only Fe2 is oxidized by air. Water is incompatible with many metal ions because it can serve as an oxidizing agent. A wider range of M2 ions is therefore known in solids than in aqueous solution. For example, TiCl2 can be prepared by the reduction of TiCl4 with hexamethyldisilane, (CH3)3SiSi(CH3)3(l) TiCl4(l) → TiCl2(s) 2 (CH3)3SiCl(l) but it is oxidized by both water and air to give TiO2. With the exception of ScCl2, these dihalides are well described by structures containing isolated M2 ions (Table 19.2). The fluorides have a rutile structure (Section 3.9) and most of the heavier halides are layered; Table 19.2 Structures of the d-metal dihalides F Cl Ti V Cr Mn Fe Co Ni R R† R† R R R R† L † L L L L L L L L R Br L L L L L L L I L L L L L L L R rutile, L layered (CdI2, CdCl2 or related structures). † The structure is distorted from ideal. Adapted from A.F. Wells, Structural inorganic chemistry. Oxford University Press (1984). 36 Cu Third 34 32 30 28 26 22 20 18 Second 16 14 12 Ecell O 0.58 V Species such as Co(III) are stabilized as oxido compounds (for instance, CoO(OH)), which are formed under basic conditions (Section 19.8), or when complexed by good donor ligands such as amines. Aqua ions of Ni3 and Cu3 have not been prepared. In contrast, M(II) becomes increasingly common from left to right across the series. For example, among the early members of the 3d series, Sc2(aq) (Group 3) is unknown and Ti2(aq) (Group 4) is formed only upon bombarding solutions of Ti3 with electrons in the technique known as pulse radiolysis. For Groups 5 and 6, V2(aq) and Cr2(aq) are thermodynamically unstable with respect to oxidation by H ions: E O 0.26 V 2 V2(aq) 2 H(aq) → 2 V3(aq) H (g) 2 38 Ionization energy, I/eV Most unipositive d-metal ions (M) disproportionate (to M and M2) because the bonds in the solid metal are so strong and, for the 3d metals, the 2 oxidation state is generally the lowest one it is necessary to consider in aqueous solution. Exceptions are mainly confined to metalmetal bonded and organometallic compounds, such as the metal carbonyls Ni(CO)4 and Mo(CO)6 discussed in Chapter 22, in which the metal is in oxidation state 0. Figure 19.4 shows the second and third ionization energies of the 3d metals, and we can see the expected increase across the period, in line with increasing nuclear charge. The anomalous values for manganese and iron are a result of the very stable d5 configurations of the Mn2 and Fe3 ions. The dipositive aqua ions, M2(aq) (specifically, the octahedral complexes [M(OH2)6]2), play an important role in the chemistry of the 3d-series metals. Many of these ions are coloured as a result of dd transitions in the visible region of the spectrum (Section 20.4). For example, Cr2(aq) is blue, Fe2(aq) is green, Co2(aq) is pink, Ni2(aq) is green, and Cu2(aq) is blue. The 3 oxidation state is common at the left of the period, and is the only oxidation state normally encountered for scandium. Careful inspection of the Frost diagrams in Fig. 19.3 gives the balance of stability between the 3 and 2 oxidation states. Titanium, vanadium, and chromium all form a wide range of compounds in oxidation state 3, and under normal conditions the 3 oxidation state is more stable than the 2 state. Manganese(II) is especially stable due to its half-filled d shell, and relatively few Mn(III) compounds are known. Beyond manganese, many Fe(III) complexes are known, but are often oxidizing. In acid solution, Co3(aq) is powerfully oxidizing and O2 is evolved: 455 Ti V Cr Mn Fe Co Ni Cu Figure 19.4 The second (red) and third (blue) ionization energies of the 3d metals. 456 19 The d-block elements in both cases the metal is in an octahedral site. This shift in structure is understandable because the rutile structure type is associated with ionic bonding and the layered structures are associated with more covalent bonding. E X A M PL E 19.1 Judging trends in redox stability in the d block On the basis of trends in the properties of the 3d-series elements, suggest possible M2 aqua ions for use as reducing agents and write a balanced chemical equation for the reaction of one of these ions with O2 in acidic solution. Answer We need to identify an element that has an accessible, but oxidizable, M(II) oxidation state. The M(II) state is most stable for the late 3d-series elements with ions of the metals on the left of the series, such as V2(aq) and Cr2(aq) being too strongly reducing to be used easily in water. By contrast, the Fe2(aq) ion is only weakly reducing and the Co2(aq), Ni2(aq), and Cu2(aq) ions not reducing in water. The Latimer diagram for iron indicates that Fe3 is the only accessible higher oxidation state of iron in acidic solution: +0.77 −0.44 Fe3+ ⎯⎯⎯ → Fe2+ ⎯⎯⎯ → Fe The chemical equation for the oxidation is then 4 Fe2(aq) O2(g) 4 H(aq) → 4 Fe3(aq) 2 H2O(l) Self-test 19.1 Refer to the appropriate Latimer diagram in Resource section 3 and identify the oxidation state and formula of the species that is thermodynamically favoured when an acidic aqueous solution of V2 is exposed to oxygen. (c) Intermediate oxidation states in the 4d and 5d series Key points: Complexes of M(II) with -donor ligands are common for the 3d-series metals but complexes of M(II) of the 4d- and 5d-series metals are less common; they generally contain -acceptor ligands. 2+ H N 3 Ru H O In contrast to the 3d-series, the 4d- and 5d-series metals only rarely form simple M2(aq) ions. A few examples have been characterized, including [Ru(OH2)6]2, [Pd(OH2)4]2, and [Pt(OH2)4]2. However, the 4d- and 5d-series metals do form many M(II) complexes with ligands other than H2O; they include the very stable d6 octahedral complexes, such as (1), and the much rarer square-pyramidal d6 complexes, such as (2), which form with bulky ligands. Palladium(II) and platinum(II) form many square-planar d8 complexes, such as [PtCl4]2. Much of the rationalization of this behaviour relates to the bonding of the ligands, and cannot be fully understood without reference to the concepts introduced in Chapter 20; however, examples of complexes are included here for completeness. The Ru(II) complex in (1) is obtained by reducing RuCl3.3H2O with zinc in the presence of ammonia; it is a useful starting material for the synthesis of a range of ruthenium(II) pentaammine complexes that have -acceptor ligands, such as CO, as the sixth ligand: 2 [Ru(NH3)5(OH2)]2(aq) L(aq) → [Ru(NH3)5L]2(aq) H2O(l) (L CO, N2, N2O) 1 [Ru(OH2)(NH3)5]2+ These Ru and the related Os pentaammine species are strong donors, and consequently the complexes they form with the acceptors CO and N2 are stable. 19.4 Oxidation states down a group PPh3 Ru Cl 2 [Ru(Cl)2(PPh3)3] Key points: In Groups 410, the highest oxidation state of an element becomes more stable on descending a group, with the greatest change in stability occurring between the first two rows of the d block; ease of oxidation of the metal does not correlate with the highest available oxidation state. The stability of an oxidation state of an element changes on descending a group. Figure 19.5 is the Frost diagram for the chromium group: note that Mo(VI) and W(VI) lie below Cr(VI) in H2Cr2O7, indicating that the maximum oxidation state is more stable for Mo and W, with the greatest change in stability occurring between Cr and Mo. This pattern is repeated Trends in chemical properties 457 +3 +2 H2Cr2O7 +1 NE°/V MoO3 0 WO 3 –1 Cr2+ –2 Cr3+ –3 0 +1 +3 +5 +2 +4 Oxidation number, N +6 Figure 19.5 A Frost diagram for the chromium group in the d block (Group 6) in acidic solution (pH 0). +7 across the d block: for Groups 410 the highest oxidation state of an element becomes more stable on descending a group, with the greatest change in stability occurring between the first two series of the block. The increasing stability of high oxidation states for the heavier d metals can be seen in the formulas of their halides (Table 19.3), and the limiting formulas MnF4, TcF6, and ReF7 show a greater ease of oxidizing the 4d- and 5d-series metals than the 3d-series. The hexafluorides of the heavier d metals (as in PtF6) have been prepared from Group 6 through to Group 10 except for Pd. In keeping with the stability of high oxidation states for the heavier metals, WF6 is not a significant oxidizing agent. However, the oxidizing character of the hexafluorides increases to the right, and PtF6 is so potent that it can oxidize O2 to O2: O2(g) PtF6(s) → (O2)PtF6(s) Even Xe can be oxidized by PtF6 (Section 18.5). The ability to achieve the highest oxidation state does not correlate with the ease of oxidation of the bulk metal to an intermediate oxidation state. For example, although elemental iron is susceptible to oxidation by H(aq) under standard conditions, Fe(s) 2 H(aq) → Fe2(aq) H2(g) Ecell O 0.44 V no chemical oxidizing agent has been found that will take it to its group oxidation state of 8 in solution. By contrast, although the two heavier metals in Group 8 (Ru and Os) are not oxidized by H ions in acidic aqueous solution: Os(s) 2 H2O(aq) → OsO2(s) 2 H2(g) Ecell O 0.65 V they can be oxidized by oxygen to the 8 state, as RuO4 and OsO4: Os(s) 2 O2(g) → OsO4(s) Table 19.3 Highest oxidation states of the d-block binary halides Group 4 5 6 7 8 9 10 11 TiI4 VF5 CrF5† MnF4 FeBr3 CoF4 NiF4 CuBr2 ZrI4 NbI5 MoCl6 TcCl6 RuF6 RhF6 PdF4 AgF3 HfI4 TaI5 WBr6 ReF7 OsF6 IrF6 PtF6 AuF5 The formulas show the least electronegative halide that brings out the highest oxidation state of the d metal. † CrF6 exists for several days at room temperature in a passivated Monel chamber. 458 19 The d-block elements We see from Fig. 19.5 that the Frost diagrams for the positive oxidation states of Mo and (especially) W are quite flat. This flatness indicates that neither element exhibits the marked tendency to form M(III) that is so characteristic of chromium. Mononuclear complexes of Mo and W in oxidation states 3, 4, 5, and 6 are common. 19.5 Structural trends Key points: The 4d- and 5d-series elements often exhibit higher coordination numbers than their 3d-series congeners; compounds of d-metals in high oxidation states tend to have covalent structures. 90 Ionic radius, r/pm High spin 80 We might expect the ionic radii of d-metal ions to follow the same pattern as atomic radii and decrease across each period. However, in addition to this general trend, there are some subtle effects on the size of the ions caused by the order in which d orbitals are occupied and which will be explained more fully in Chapter 20. Figure 19.6 shows the variation in radius of the M2 ions for six-coordinate complexes of the 3d-series metals. To understand the two trends shown in the illustration we need to know that three of the 3d orbitals point between the ligands and that the remaining two point directly at them (this feature is explained more fully in Section 20.1). For the so-called ‘low-spin complexes’, in which electrons first individually occupy the three 3d orbitals that point between the ligands, there is a general decrease in radius across the series up to the d6 ion Fe2. After Fe2, the additional electrons occupy the two d orbitals that point towards the ligands, which they repel slightly and thus result in an effective increase in radius. The trend for the so-called ‘high-spin complexes’, in which the electrons occupy all five 3d orbitals singly before pairing with any already present, is more complicated. Initially, at Ti2 (d2) and V2 (d3), the electrons occupy the three 3d orbitals that point between the ligands and the radius decreases. The next two electrons occupy the two orbitals that point at the ligands, and the radii increase accordingly. Then the sequence starts again at Fe2 (d6), as the additional electrons pair with those already present, first in the ‘non-repelling’ set of three orbitals and then, finally, in the two ‘repelling’ orbitals. As might be anticipated from consideration of atomic and ionic radii, the 4d- and 5dseries elements often have higher coordination numbers than their smaller 3d-series congeners. Table 19.4 illustrates this trend for the fluorido and cyanido complexes of the early d metals. Note that with the small F ligand, these 3d-series metals tend to form sixcoordinate complexes but that the larger 4d- and 5d-series metals in the same oxidation state tend to form seven-, eight-, and nine-coordinate complexes. The octacyanidomolybdate complex, [Mo(CN)8]3, illustrates the tendency towards high coordination numbers with compact ligands. The same complex is readily reduced electrochemically or chemically without change of coordination number: [Mo(CN)8]3(aq) e → [Mo(CN)8]4(aq) 70 Low spin 60 Ti V Cr Mn Fe Co Ni Cu Zn Figure 19.6 The ionic radii of the M2 ions of the 3d metals. Where there are alternatives, red represents high-spin and blue low-spin complexes. In the orbital energy diagrams, the three lower levels are the d orbitals that point between ligands and the two upper levels are the d orbitals that point directly at them. E O 0.73 V Structural changes also result from changes in metal oxidation state. Compounds of d metals in low oxidation states often exist as ionic solids, whereas compounds of d metals in high oxidation states tend to take on covalent character: compare OsO2, which is an ionic solid with the rutile structure, and OsO4, which is a covalent molecular species (Section 19.8). We discussed the effect in Section 1.9e. Table 19.4 Coordination numbers of some early d-block fluorido and cyanido complexes Group 3 4 5 3d (NH4)3[ScF6] (6) Na2[TiF6] (6) K[VF6] (6); K2[V(CN)7].2H2O (7) 4d Na6[YF9] (9) Na3[ZrF7] (7) K2[NbF7] (7); K4[Nb(CN)8] (8) 5d Na6[LaF9] (9) Na3[HfF7] (7) K3[TaF8] (8) The number in parentheses is the coordination number of the d-metal atom in the complex anion enclosed in square brackets. Trends in chemical properties 19.6 Noble character Key point: Metals on the right of the d block tend to exist in low oxidation states and form compounds with soft ligands. With the exception of Group 12, the metals at the lower right of the d block are resistant to oxidation. This resistance is largely due to strong intermetallic bonding and high ionization energies. It is most evident for Ag, Au, and the 4d- and 5d-series metals in Groups 810 (Fig. 19.7). The latter are referred to as the platinum metals because they occur together in platinum-bearing ores. In recognition of their traditional use, Cu, Ag, and Au are referred to as the coinage metals. Gold occurs as the metal; silver, gold, and the platinum metals are also recovered in the electrolytic refining of copper. The prices of the individual platinum metals vary widely because they are recovered together but their consumption is not proportional to their abundance. Rhodium is by far the most expensive metal in this group because it is widely used in industrial catalytic processes and in automotive catalytic converters (Box 26.1). Copper, silver, and gold are not susceptible to oxidation by H under standard conditions, and this noble character accounts for their use, together with platinum, in jewellery and ornaments. Aqua regia, a 3:1 mixture of concentrated hydrochloric and nitric acids, is an old but effective reagent for the oxidation of gold and platinum. Its function is twofold: the NO3 ions provide the oxidizing power and the Cl ions act as complexing agents. The overall reaction is Au(s) 4 H(aq) NO3(aq) 4 Cl(aq) → AuCl4 NO(g) 2 H2O(l) The active species in solution are thought to be Cl2 and NOCl, which are generated in the reaction 3 HCl(aq) HNO3(aq) → Cl2(aq) NOCl(aq) 2 H2O(l) Oxidation state preferences are erratic in Group 11. For Cu, the 1 and 2 states are most common, but for Ag 1 is typical, and for Au 1 and 3 are common. The simple aqua ions Cu(aq) and Au(aq) undergo disproportionation in aqueous solution: 2 Cu(aq) → Cu(s) Cu2(aq) 3 Au(aq) → 2 Au(s) Au3(aq) Complexes of Cu(I), Ag(I), and Au(I) are often linear. For example, [H3NAgNH3] forms in aqueous solution and linear [XAgX] complexes have been identified by X-ray crystallography. The currently preferred explanation for the tendency towards linear coordination is the similarity in energy of the outer nd and (n1)s and (n1)p orbitals, which permits the formation of collinear spd hybrids (Fig. 19.8). The soft Lewis acid character of Cu, Ag, and Au is illustrated by their affinity order, which is I Br Cl. Complex formation, as in the formation of [Cu(NH3)2] and [AuI2], provides a means of stabilizing the 1 oxidation state of these metals in aqueous solution. Many tetrahedral complexes are also known for Cu(I), Ag(I), and Au(I) (Section 7.8). Square-planar complexes are common for the platinum metals and gold in oxidation states that yield the d8 electronic configuration, which include Rh(I), Ir(I), Pd(II), Pt(II), and Au(III) (Section 20.1f). An example is [Pt(NH3)4]2. Characteristic reactions for these Figure 19.8 The hybridization of s, pz, and dz 2 with the choice of phases shown here produces a pair of collinear orbitals that can be used to form strong bonds. 7 8 9 10 11 12 459 Al Mn Fe Co Ni Cu Zn Ga Tc Ru Rh Pd Ag Cd In Re Os Ir Platinum metals Pt Au Hg Tl Coinage metals Figure 19.7 The location of the platinum and coinage metals in the periodic table. 460 19 The d-block elements complexes are ligand substitution (Section 21.3) and, except for Au(III) complexes, oxidative addition (Section 22.22). The noble character that these elements developed across the d block is suddenly lost at Group 12 (Zn, Cd, Hg), where the metals once again become susceptible to atmospheric oxidation. The greater ease of oxidation of the Group 12 metals is due to a reduction in the extent of intermetallic bonding and an abrupt lowering of d-orbital energies at the end of the d block, with the higher energy (n1)s electrons participating in reactions. Representative compounds As a result of the convention of assigning a negative oxidation number to any nonmetal in combination with a metal, high formal oxidation states for d metals may be encountered in their compounds, such as Re(VII) in [ReH9]2 and W(VI) in W(CH3)6. However, these compounds are not oxidizing agents in the usual sense, and are best discussed together with other organometallic compounds (Chapter 22). The discussion here is confined to compounds containing electronegative ligands such as the halogens, oxygen, nitrogen, and sulfur. 19.7 Metal halides Key points: Binary halides of the d-block elements span all metals and most oxidation states; dihalides are typically ionic solids, with higher halides taking on covalent character. Binary metal halides of the d-block elements occur for all the elements with nearly all oxidation states represented. As we should expect, the more strongly oxidizing halogens bring out the higher oxidation states, with the corollary that the low oxidation state binary halides are more stable as iodides and bromides. Of all the groups, only the members of Group 11 (Cu, Ag, Au) have simple monohalides. For Cu, these salts are highly insoluble in water and dissolve only when complexed by other ligands (which include halides). Silver(I) halides are sparingly soluble and photosensitive, decomposing to the metal (a process that is exploited in photography). The only monohalide of Au that exists is the chloride; it is oxidized by water. More common are the dihalides which, as noted in Section 19.3b (Table 19.2), are typically ionic solids that dissolve in water to give M2(aq) ions; some of the earlier dihalides are reducing. Many of the intermediate halides exhibit metalmetal bonds and form clusters (Section 19.11). Higher halides exist for most of the d block, and covalent character becomes more prevalent with high oxidation state, especially for the lower halogens. For instance, in Group 4, whereas TiF4 is a solid with melting point 284ºC, TiCl4 melts at 24ºC and boils at 136ºC. In Group 6, not even the fluoride has ionic character and both MoF6 and WF6 are liquids at room temperature. 19.8 Metal oxides and oxido complexes Because oxygen is readily available in an aqueous environment, in the atmosphere, and as the donor atom in many organic molecules, it is not surprising that oxides and oxygencontaining ligands play a major role in the chemistry of the metallic elements. (a) Metal oxides Key point: Many different oxides of the d-block elements exist, with a wide variety of structures, varying from ionic lattices to covalent molecules. Many different oxides are known for the d-block elements, with a number of different structures. We have already noted the ability of oxygen to bring out the highest oxidation state for some elements, but oxides exist for some elements in very low oxidation states: in Cu2O, copper is present as Cu(I). Monoxides are known for all of the 3d-series metals, except Cr. The monoxides have the rock-salt structure characteristic of ionic solids but their properties, which are discussed in more detail in Chapter 24, indicate significant deviations from the simple ionic M2O2 model. For example, TiO has metallic conductivity 461 Representative compounds and FeO is always deficient in iron. The early d-block monoxides are strong reducing agents. Thus, TiO is easily oxidized by water or oxygen, and MnO is a convenient oxygen scavenger that is used in the laboratory to remove oxygen impurity in inert gases down to the parts-per-billion range. As we have already noted, very high oxidation state oxides can show covalent structures. For example ruthenium tetroxide and osmium tetroxide are low melting, highly volatile, toxic, molecular compounds that are used as selective oxidizing agents. Indeed, osmium tetroxide is used as the standard reagent to oxidize alkenes to cis-diols: OsO4 O O Os O O H2O OH OH (b) Mononuclear oxido complexes Key points: The conversion of an aqua ligand to an oxido ligand is favoured by a high pH and by a high oxidation state of the central metal atom; in vanadium complexes in oxidation state 4 or 5, the site trans to the oxido ligand may be vacant or occupied by a weakly coordinating ligand. 2+ N N N Ru N Cl OH2 3 cis-[RuLCl(OH2)]2+ .61 cis-[Ru(V)(Cl)(O)L]2+ cis(III) 2+ [Ru (Cl)(OH2)L] .21 E°/V Our focus in this section is on the oxido ligand (O2) and its ability to bring out the high oxidation states of the chemically hard metals on the left of the d block. Another important issue is the relation between oxido and aqua (H2O) ligands resulting from proton transfer equilibria. Elements in high oxidation states typically occur as oxoanions in aqueous solution, such as MnO4, which contains Mn(VII), and CrO42, which contains Cr(VI). The existence of these oxoanions contrasts with the existence of simple aqua ions for the same metals in lower oxidation states, such as [Mn(OH2)6]2 for manganese(II) and [Cr(OH2)6]3 for chromium(III). The formation of an oxido complex rather than an aqua complex is favoured by high pH because the OH ions in basic solution tend to remove protons from the aqua ligands. The occurrence of aqua complexes for low oxidation state metal cations is explained by noting the relatively small electron-withdrawing effect exerted by these cations on the O atoms of the OH2 ligand. As a result, the aqua ligands are only weak proton donors. A metal ion in a high oxidation state, however, depletes the electron density on the O atoms attached to it and thereby increases the Brønsted acidity of H2O and OH ligands. The influence of pH on the stabilities of different oxidation states is best summarized by a Pourbaix diagram (Section 5.14). The example shown in Fig. 19.9 is for a complex (3) containing a tetradentate ligand L, which results in a similar coordination geometry for a wide range of conditions. The aqua complex cis-[RuLCl(OH2)] in solution at pH 2 is stable up to 0.40 V. Just above this potential, simple oxidation (that is, electron removal) occurs to give cis-[RuLCl(OH2)]2. Under even more strongly oxidizing conditions (at 0.95 V) and pH 2, both oxidation and deprotonation occur and result in the formation of a Ru(IV) oxido species, cis-[RuLCl(O)]. At an even higher potential (about 1.4 V), further oxidation yields cis-[RuLCl(O)]2. As already remarked, the influence of more basic conditions is to deprotonate an OH2 ligand and thus to favour hydroxido or oxido complexes. For example, at pH 8, the Ru(III) species is deprotonated to the hydroxido complex cis-[RuLCl(OH)], which in turn is converted to the oxido-Ru(IV) complex at a lower potential than for the Ru(III)Ru(IV) transformation at pH 2. Table 19.5 gives a list of simple complexes containing oxido ligands. Many complexes are known that contain the vanadium(IV) group, VO2 (sometimes referred to as vanadyl), with V in its penultimate oxidation state (4). Vanadyl complexes generally contain four additional ligands and are square pyramidal (4). Many of these d1 complexes are blue (as a result of a dd transition) and may take on a weakly bound sixth ligand trans to the oxido ligand. The vanadyl VO bond length (158 pm) in [VO(acac)2] is short compared with the four VO bond lengths to the acac ligand (197 pm). The short bond lengths in vanadyl complexes, together with high VO stretching wavenumbers (9401000 cm1) provide strong evidence for VO multiple bonding in which lone pairs on the oxido ligand are donated to the central V atom. The V3dO2p -orbital overlap involved in a multiple bond is illustrated in Fig. 19.10; thus it can be seen that the O2 ligand is not only a donor but also a donor capable of making two bonds and resulting in a bond order of up to three (though by convention such interactions are depicted as M O, preserving electroneutrality). This strong .8 0 .4 0 cis-[Ru(IV)(Cl)(O)L]+ cis[Ru(III)(Cl)(OH)L]+ cis-[Ru(II)(Cl)(OH2)L]+ 0 0 2 6 4 8 0 1 H p Figure 19.9 A Pourbaix diagram for cis-[RuLCl(OH2)]2 (3) and related species. (Adapted from C.-K. Li, W.-T. Wang, C.-M. Chi, K.-Y. Wong, R.-J. Wang, and T.C.W. Mak, J. Chem. Soc., Dalton Trans. 1991, 1909.) 462 19 The d-block elements Table 19.5 Some common monoxido and dioxido complexes O 2– V Cl Group Element, configuration Structure Formula 5 V(IV), d1 Square pyramidal [V(O)(acac)2], [V(O)Cl4]2 cis-octahedral [V(O)2(OH2)4] Tetrahedral [M(O)2(Cl)2] cis-octahedral [Mo(O)2(acac)2] trans-octahedral [Re(O)2(CN)4]3 V(V), d 0 6 Mo(VI), d0; W(VI), d0 7, 8 Re(V), d2; Os(VI), d2 4 [VOCl4]2– [Os(O)2(Cl)2] [Os(O)2(Cl)4]2 z y x O V multiple bonding with the O atom appears to be responsible for the trans influence of the oxido ligand, which disfavours attachment of the ligand trans to O (Section 21.4). Vanadium in its highest (and group) oxidation state, 5, forms an extensive series of oxido compounds, many of which are the polyoxo species discussed later. The simplest oxido complex [V(O)2(OH2)4] exists in the acidic solution formed when the sparingly soluble vanadium(V) oxide, V2O5, dissolves in water. This pale yellow complex has a cis geometry (5). Here again we see the trans influence of an oxido ligand: the cis geometry minimizes competition between the oxido ligands for -bonding to the V atom. As shown in Fig. 19.11, the two O2 ligands in the trans configuration can -bond with the same two d orbitals; in the cis configuration the metal atom has only one d orbital in common with the orbitals on the O atoms. In contrast to the cis structure of the d0 complex [V(O)2(OH2)4], many trans-dioxido complexes are known for the d2 metal centres Os(VI) and Re(V) (6); Table 19.5 gives some examples. This geometry is probably favoured because the trans configuration leaves a vacant low-energy orbital that can be occupied by the two d electrons. According to this explanation, the avoidance of the destabilizing effect of the d2 electrons more than offsets the disadvantage of the ligands having to compete for the same d orbital in the trans-oxido geometry. (c) Polyoxometallates Key points: In their highest oxidation states, metals in Groups 5 and 6 readily form polyoxometallates and heteropolyoxometallates; pH is crucial in determining which compounds form. Figure 19.10 The dp -bonding between O and V in the vanadyl group VO2. It is convenient to think of the O as having oxidation number 2 and V as 4. Thus, both electron pairs for the two bonds are donated from the O2 ion into the metal dyz and dzx orbitals. A polyoxometallate is an oxoanion containing more than one metal atom. The H2O ligand created by protonation of an oxido ligand at low pH can be eliminated from the central metal atom and thus lead to the condensation of mononuclear oxometallates. A familiar example is the reaction of a basic chromate solution, which is yellow, with excess acid to form the oxido-bridged dichromate ion, which is orange: +2 Cr O O V O O O O O O O 2- O Cr Cr O O O H2O 2 CrO42 (aq) 2 H (aq) → Cr2O72(aq) H O(l) 2 V O (a ) 2- O 2 H+ (b ) Figure 19.11 Comparison of the competition between cis and trans oxygens attached to vanadium. (a) In the cis configuration the metal has only one d orbital in common with the orbitals on the O atoms. (b) In the trans configuration the metal atom has two d orbitals in common with the orbitals on the two O atoms. O V O e R y p OH2 5 [V(O)2(OH2)4]+ 6 [Re(O)2(py)4]+, py = C5H5N Representative compounds In highly acidic solution, oxido-bridged Cr(VI) species with longer chains are formed. The tendency for Cr(VI) to form polyoxo species is limited by the fact that the O tetrahedra link only through vertices: edge and face bridging would result in too close an approach of the metal atoms. By contrast, it is found that five- and six-coordinate metal oxido complexes, which are common with the larger 4d- and 5d-series metal atoms, can share oxido ligands between either vertices or edges. These structural possibilities lead to a richer variety of polyoxometallates than found with the 3dseries metals. Chromium’s neighbours in Groups 5 and 6 form six-coordinate polyoxo complexes (Fig. 19.12). In Group 5, the polyoxometallates are most numerous for vanadium, which forms many V(V) complexes and a few V(IV) or mixed oxidation state V(IV)V(V) polyoxido complexes. Polyoxometallate formation is most pronounced in Groups 5 and 6 for V(V), Mo(VI), and W(VI). It is often convenient to represent the structures of the polyoxometallate ions by polyhedra, with the metal atom understood to be in the centre and O atoms at the vertices. For example, the sharing of O atom vertices in the dichromate ion, Cr2O72, may be depicted in either the traditional way (7) or in the polyhedral representation (8). Similarly, the important M6O19 structure of [Nb6O19]8, [Ta6O19]8, [Mo6O19]2, and [W6O19]2 is depicted by the conventional or polyhedral structures shown in Fig. 19.13. The structures for this series of polyoxometallates contain terminal O atoms (those projecting out from a single metal atom) and two types of bridging O atoms: two-metal bridges, MOM, and one hypercoordinated O atom in the centre of the structure that is common to all six metal atoms. The structure consists of six MO6 octahedra, each sharing an edge with four neighbours. The overall symmetry of the M6O19 array is Oh. Another example of a polyoxometallate is [W12O40(OH)2]10 (9). As shown by this formula, the polyoxoanion is partially protonated. Polyoxometallate anions can be prepared by carefully adjusting pH and concentrations, for example polyoxomolybdates and polyoxotungstates are formed by acidification of solutions of the simple molybdate or tungstate: 5 4 Ti V +4 ,+5 6 463 7 Cr Mn +6 Zr Nb Mo Tc +5 +6 Hf Ta W Re +5 +6 Figure 19.12 Elements in the d block that form polyoxometallates. The elements coloured yellow form the greatest variety of polyoxometallates. 2– O Cr 7 Cr2O72– 2– 2 6 [MoO4]2(aq) 10 H(aq) → [Mo6O19]2(aq) 5 H O(l) 2 8 [MoO4]2(aq) 12 H(aq) → [Mo8O26]4(aq) 6 H O(l) 2 5 , and there is a large Mixed metal polyoxometallates are also common, as in MoV9O28 class of heteropolyoxometallates, such as the molybdates and tungstates, that also incorporate P, As, and other heteroatoms. For example, [PMo12O40]3 contains a PO43 tetrahedron that shares O atoms with surrounding octahedral MoO6 groups (10). Many different heteroatoms can be incorporated into this structure, and the general formulation is X(N)Mo12O40 where X(N) represents the oxidation state of the heteroatom X, which may be As(V), Si(IV), Ge(IV), or Ti(IV). An even broader range of heteroatoms is observed with the analogous tungsten heteropolyoxoanions. Heteropolyoxomolybdates and tungstates can undergo one-electron reduction with no change in structure but with the formation of a deep blue colour. The colour seems to arise from the excitation of the added electron from Mo(V) or W(V) to an adjacent Mo(VI) or W(VI) site. 8 Cr2O72– 10– 9 [W12O40(OH)2]10– 3– ) (a ) (b (c) Figure 19.13 (a) Conventional and (c) polyhedral representation of the six edge-shared octahedra as found in [M6O19]2. The intermediate structure (b) shows the process of constructing the polyhedral representation in progress. 10 [PMo12O40]3– 464 19 The d-block elements 19.9 Metal sulfides and sulfide complexes Sulfur is softer as a Lewis base and less electronegative than oxygen; it therefore has a broader range of oxidation states and a stronger affinity for the softer metals on the right of the d block. For example, zinc(II) sulfide is readily precipitated when Zn2(aq) is added to an aqueous solution containing H2S and NH3 (to adjust the pH) whereas the harder Lewis acid Sc3(aq) gives Sc(OH)3(s) instead. The covalent contribution to the lattice enthalpy of all the softer metal sulfides cannot be overcome in water, and in general the sulfides are only very sparingly soluble. Another striking feature of sulfur is its tendency to form catenated sulfide ions, Section 16.14. A broad set of compounds containing Sn2 ions can be prepared, and the smaller polysulfides can serve as chelating ligands towards d-metal ions. (a) Monosulfides Key point: Monosulfides with the nickel-arsenide structure are formed by most 3d metals. As with the d-metal monoxides, the monosulfides are most common in the 3d series (Table 19.6). In contrast to the monoxides, however, most of the monosulfides have the nickel-arsenide structure (Fig. 3.36). The different structures are consistent with the rock-salt structure of the monoxides being favoured by the ionic (harder) cationanion combination and the nickel-arsenide structure being favoured by more covalent (softer) combinations. The nickel-arsenide structure is expected when there is appreciable metalmetal bonding, resulting in the shorter metalmetal distances found for the hexagonal packing. Monosulfides are formed by most 3d-series metal ions. (b) Disulfides Key points: The 4d- and 5d-series metals often form disulfides with alternating layers of metal ions and sulfide ions; binary disulfides of the early d metals often have a layered structure, whereas Fe2 and ions. many of the later d-metal disulfides contain discrete S2 2 The disulfides of the d metals fall into two broad classes (Table 19.7). One class consists of layered compounds with either the CdI2 or the MoS2 structure; the other consists of compounds containing discrete S22 groups, the pyrites and marcasite structures. The layered disulfides are built from a sulfide layer, a metal layer, and then another sulfide layer (for example, Fig. 19.14). These sandwiches stack together in the crystal with sulfide layers in one slab adjacent to a sulfide layer in the next. Clearly, this crystal Table 19.6 Structures of d-block MS compounds Group 4 5 Nickel-arsenide structure (shaded) Ti V Rock-salt structure (unshaded) Zr Nb 6 7 8 9 10 Mn† Fe Co Ni Metal monosulfides of Group 6 are not shown; some of the heavier metals have more complex structures. † MnS has two polymorphs; one has a rock-salt structure, the other has a wurtzite structure. Table 19.7 Structures of d-block MS2 compounds Group 4 5 6 Layered (shaded) Ti Pyrite or marcasite Zr Nb Mo (unshaded) Hf Ta W 7 8 9 10 11 Mn Fe Co Ni Cu Ru Rh Os Ir Re Pt Metals not shown do not form disulfides or have disulfides with complex structures. Adapted from A.F. Wells, Structural inorganic chemistry. Oxford University Press (1984). Representative compounds structure is not consistent with a simple ionic model and its formation is a sign of covalence in the bonds between the soft sulfide ion and d-metal cations. The metal ion in these layered structures is surrounded by six S atoms. Its coordination environment is octahedral in some cases (such as PtS2, which adopts the CdI2 structure shown in Fig. 19.14) and trigonal prismatic in others (MoS2). The layered MoS2 structure is favoured by SS bonding as indicated by short SS distances within each of the MoS2 slabs. The common occurrence of the trigonal-prismatic structure in many of these compounds is in striking contrast to the isolated metal complexes, where the octahedral arrangement of ligands is by far the most common. Some of the layered metal sulfides readily undergo intercalation reactions in which ions or molecules penetrate between adjacent sulfide layers (Section 24.10). Compounds containing discrete S22 ions adopt the pyrite or marcasite structure (Fig. 19.15). The stability of the formal S22 ion in metal sulfides is much greater than that of the O22 ion in peroxides, and there are many more metal sulfides in which the anion is S22 than there are peroxides. 465 I Cd Figure 19.14 The CdI2 structure adopted by many disulfides. Fe E X A M PL E 19. 2 Contrasting the structures of two different d-block disulfides S Compare the structures of MoS2 and FeS2, and explain their existence in terms of the oxidation states of the metal ions. Answer We need to decide whether the metals are likely to be present as M(IV), in which case the two S atoms would be present as S2 ions, or whether the metal is present as M(II) with the S atoms present as the SS bonded species S2 . Because it is readily oxidized, S2 will be found only with a metal ion in 2 an oxidation state that is not easily reduced. As with many of the 4d- and 5d-series metals, Mo is easily oxidized to Mo(IV), therefore Mo(IV) can coexist with S2. Molybdenum(IV) sulfide (MoS2) has the layered structure typical of metal disulfides. In contrast, Fe is not readily oxidized to Fe(IV). Therefore, Fe(IV) cannot coexist with S2. The compound is therefore likely to contain Fe(II) and S2 . The mineral name for FeS2 is 2 pyrite; its common name, ‘fool’s gold’, indicates its misleading colour. Self-test 19.2 Suggest a use for molybdenum(IV) sulfide that makes use of its solid-state structure. Rationalize your suggestion. Figure 19.15 The structure of pyrite, FeS2. (c) Sulfide complexes Key point: Chelating polysulfide ligands are common in metalsulfur coordination compounds of the 4d- and 5d-series metals. The coordination chemistry of sulfur is quite different from that of oxygen. Much of this difference is connected with the ability of sulfur to catenate (form chains), the availability of low-lying empty 3d orbitals that allow an S atom to act as a acceptor, and the preference of sulfur for metal centres that are not highly oxidizing (Section 16.11). Simple thiometallate complexes such as [MoS4]2 can be synthesized easily by passing H2S gas through a strongly basic aqueous solution of molybdate or tungstate ions: [MoO4]2(aq) 4 H2S(g) → [MoS4]2(aq) 4 H2O(l) 2– S Mo These tetrathiometallate anions are building blocks for the synthesis of complexes containing more metal atoms. For example, they will coordinate to many dipositive metal ions, such as Co2 and Zn2: Co2(aq) 2 [MoS4]2(aq) → [S2MoS2CoS2MoS2]2(aq) The polysulfide ions, such as S22 and S32, which are formed by addition of elemental sulfur to a solution of ammonium sulfide, can also act as ligands. An example is [Mo2(S2)6]2 ; it contains side-bonded (11), which is formed from ammonium polysulfide and MoO2 4 S22 ligands. The larger polysulfides bond to metal atoms forming chelate rings, as in [MoS(S4)2]2 (12), which contains chelating S42 ligands. Clusters of Fe and S atoms (‘FeS clusters’) are cofactors in many enzymes, including nitrogenase, which converts N2 to NH3 under ambient conditions (Chapter 15). Synthetic analogues such as the cubane shown in (13) have been prepared and characterized. The cubane contains Fe and S atoms on alternate corners, so each S atom bridges three Fe 11 [Mo2(S2)6]2– 2– S Mo 12 [MoS(S4)2]2– 466 19 The d-block elements 2– R atoms; each of the Fe atoms has a thiolate group, RS, occupying a terminal position. These compounds can be obtained from simple starting materials in the absence of air using a polar aprotic solvent such as dimethylsulfoxide: 4 FeCl3 4 NaSH 6 RSH 10 NaOCH3 → [Fe4S4(SR)4]2 RSSR 14 Na 12 Cl 10 CH3OH Fe The HS ion provides the sulfide ligands for the cage, the RS ion serves as both a ligand and a reducing agent, and the CH3O ion acts as a base. Addition of the bulky cation from (Et4N)Cl results in the formation of black crystals of (Et4N)2[Fe4S4(SR)4]. The observation that this reaction is successful, in good yield, with many different R groups indicates that the Fe4S4 cage is thermodynamically more stable than other possibilities. The cage undergoes reversible one-electron reduction to [Fe4S4(SR)4]3, a reaction that is important in biological oxidation and reduction (Section 27.8). S 13 [Fe4S4(SR)4]2– 19.10 Nitrido and alkylidyne complexes Key points: Multiple ML bonds are common with high oxidation state metals of the early d-metal series; these bonds weaken the bonds to the trans ligands. N The nitrido ligand (N⬅) and alkylidyne ligand (RC⬅, previously known as carbido) are present formally as N3 and RC3. Thus, complexes containing them are usually of very high oxidation state. The highly reactive compound (Me3CO)3W⬅W(OCMe3)3 cleaves the C⬅N bond in benzonitrile to give both a nitrido and an alkylidyne ligand: – (Me3CO)3W⬅W(OCMe3)3 PhC⬅N → (Me3CO)3W⬅CPh (Me3CO)3W⬅N M l C 14 [M(N)Cl4]– 3– Br Ta N 15 [Br4Ta(µ-N)TaBr4]3– The remarkable feature of this reaction is that the C⬅N bond is broken despite its considerable strength (890 kJ mol1), so the W⬅C and W⬅N bond enthalpies must be substantial. Both nitrido and alkylidyne complexes are known to have short MN and MC bonds, which confirms the existence of multiple metalligand bonds. Like the O2 group, these ligands are strong donors and weaken the trans metalligand bond. The order of this influence is RC⬅ N⬅ O ; this weakening is attributed to competition for donation into a common set of metal d orbitals. Many nitrido complexes are known for the elements in Groups 58. They are most numerous for Mo and W in Group 6, Re in Group 7, and Ru and Os in Group 8. Square-pyramidal nitrido complexes of formula [MNX4] are known for M(VI), with M Mo, Re, Ru, and Os, and X F, Cl, Br, and (in some cases) I. These squarepyramidal structures (14) have short M⬅N bonds (157 to 166 pm). In a few cases, a ligand is attached in the sixth site, but the resulting bond is long and weak, like the oxido complexes discussed earlier. Examples of M N M species are known, such as [Ta2NBr8]3 (15), but nitrogen analogues of the polyoxometallates described earlier are unknown. 19.11 Metalmetal bonded compounds and clusters The traditional view of the chemical properties of the d metals is from the perspective of ionic solids and complexes that contain a single central metal ion surrounded by a set of ligands. However, with the development of improved techniques for the determination of structure, it has been recognized that there are also many d-metal compounds that have metalmetal (MM) bond distances comparable to or shorter than those in the elemental metal. These findings have stimulated additional X-ray structural investigations of compounds that might contain MM bonds and have inspired research into syntheses designed to broaden the range of such cluster compounds. Examples of these metal (and nonmetal) cluster compounds are now known in every block of the periodic table (Fig. 19.16), but they are most numerous in the d block. A rigorous definition of metal clusters restricts them to molecular complexes with metalmetal bonds that form triangular or larger structures. This definition, however, would exclude linear MM compounds, and is normally relaxed. We shall consider any MM bonded system to be a cluster. Representative compounds EnHn clusters Organometallic clusters i B L e a N g M K C a e H B C NO Mtalclusters e ith-donor w π ligands Metal clusters with π-acceptor ligands e F Sc T i V rC M n l Si A g A b R r N b o MT c u R R hP d Sr Y Z s C a B La H f aT R W e Os Ir t u P A rF a R Ac R f b D Sg B h s tM H s D F N e P S l A C r s A Se B r a G e G d C In Sn Sb eT g H l T b P o iN u C C Z n i B o P r K I e X t A n R Naked clusters Rg 467 Suboxide clusters Figure 19.16 The major classes of clusterforming elements. Note that carbon is in two classes (for example it occurs as C8H8 and as C60). (Adapted from D.M.P. Mingos and D.J. Wales, Introduction to cluster chemistry. Prentice Hall, Englewood Cliffs (1990).) (a) Metalmetal bonds Key point: Metalmetal bonds with bond orders up to five are formed by many d metals in low oxidation states. The first d-block metalmetal bonded species to be identified was the Hg22 ion of mercury(I) compounds, as occurs in Hg2Cl2, and examples of metalmetal bonded compounds and clusters are now known for most of the d metals. Some of their common structural motifs are an ethane-like structure (16), an edge-shared bioctahedron (17), a face-shared bioctahedron (18), and the tetragonal prism of [Re2Cl8]2 (19). If we consider the possible overlap between d orbitals on adjacent metal atoms, we can see that (Fig. 19.17): r a bond between two metal atoms can arise from the overlap of a dz2 orbital from each atom r two bonds can arise from the overlap of dzx or dyz orbitals σ r two bonds can be formed from the overlap of two face-to-face dxy or dx2-y2 orbitals. π W Cl W N y p Cl π CH3 16 [(Me2N)3WWCl(NMe2)2], Me = CH3 17 [(Cl)2(py)2W(µ-Cl)2W(Cl)2(py)2] δ Cl 3– 2– Cl 104° δ W Re 18 [(Cl)3W(µ-Cl)3W(Cl)3]3– 19 [Re2Cl8]2– Figure 19.17 The origin of , , and interactions between the d orbitals of two d-metal atoms situated along the z-axis. Only bonding combinations are shown. 468 19 The d-block elements σ Energy π δ δ π σ Figure 19.18 Approximate molecular orbital energy level scheme for MM interactions. Ar' Ar' Cr Cr Thus a quintuple bond could result if all the bonding orbitals are occupied to give the electron configuration 2 44 (Fig.19.18). Five d electrons would be needed from each metal for a quintuple bond, and this is exactly the case for the d5 Cr(I) centres in (20). In this molecule, the two Cr atoms are separated by a very short distance of 183.5 pm, unsupported by any additional bridging ligand interactions; compare this distance with the separation of Cr atoms in the bulk metal, which is 258 pm. The Re compound (19) also lacks bridging ligand interactions, but here the two d4 Re(III) centres can form only a quadruple ReRe bond. The four d electrons from each Re atom in (19) result in the configuration 2 42, and it is believed that the dx2y2 orbital is involved in bonding to the Cl ligands. Evidence for quadruple bonding comes from the observation that [Re2Cl8]2 has an eclipsed array of Cl ligands, which is sterically unfavourable. It is argued that the bond, which is formed only when the dxy orbitals are confacial, locks the complex in the eclipsed conformation. Many other species with multiple metalmetal bonds, where the dx2y2 orbital is involved in bonding to ligand species, are known. In all these complexes, the molecular orbital diagram shown in Figure 19.19 becomes appropriate and the maximum possible bond order is 4. A well-known example is the quadruply bonded compound molybdenum(II) acetate (21), which is prepared by heating Mo(CO)6 with acetic acid: 2 Mo(CO)6 4 CH3COOH → Mo2(O2CCH3)4 2 H2 12 CO The dimolybdenum complex is an excellent starting material for the preparation of other MoMo compounds. For example, the quadruply bonded chlorido complex is obtained when the acetato complex is treated with concentrated hydrochloric acid at below room temperature: Ar' Ar' 20 Mo2(O2CCH3)4(aq) 4 H(aq) 8 Cl(aq) → [Mo2Cl8]4(aq) 4 CH3COOH(aq) σ Energy π δ δ π As shown in Table 19.8, incomplete occupation of the bonding orbitals can result in a reduction of the formal bond order to 3.5 or to the triply bonded M⬅M systems. These complexes are more numerous than the quadruply bonded complexes and, because bonds are weak, M⬅M bond lengths are often similar to those of quadruply bonded systems. A decrease of bond order can also stem from the occupation of both the orbitals and, once these are fully occupied, successive occupation of the two higher lying orbitals leads to further decrease in the bond order from 2.5 to 1. As with carboncarbon multiple bonds, metalmetal multiple bonds are centres of reaction. However, the variety of structures resulting from the reactions of metalmetal multiple bonded compounds is more diverse than for organic compounds. For example: H σ Cp(OC)2Mo Mo(CO)2Cp HI Cp(OC)2Mo Mo(CO)2Cp I Figure 19.19 Approximate molecular orbital energy level scheme for the MM interactions in a quadruply bonded system, where only the dx2y2 is utilized in bonding to the ligands. In this reaction, HI adds across a triple bond but both the H and I bridge the metal atoms; the outcome is quite unlike the addition of HX to an alkyne, which results in a substituted alkene. The reaction product can be regarded as containing a 3c,2e MHM bridge and an iodide anion bonding by two conventional 2c,2e bonds, one to each Mo atom. Larger metal clusters can be synthesized by addition to a metalmetal multiple bond. For example, Pt(PPh3)4 loses two triphenylphosphine ligands when it adds to the Mo⬅Mo triple bond, resulting in a three-metal cluster: Ph3P PPh3 Pt o M O Cp(OC)2Mo Mo(CO)2Cp Pt(PPh3)4 Cp(OC)2Mo Mo(CO)2Cp 2PPh3 C CH3 21 [Mo2(µ-CH3CO2)4] (b) Clusters Key points: Metal clusters may be formed; for early d-block elements they are favoured by good donor ligands, whereas for later d-block elements -acceptor ligands are needed. Representative compounds Table 19.8 Examples of metalmetal bonded tetragonal prismatic complexes Complex O Configuration Bond order MM bond length/pm 2 42 4 211 2 41 3.5 217 2 4 3 222 2 42∗1 ∗2 2.5 227 2 42∗2 ∗2 2 238 2 42∗1 ∗4 1.5 232 2 42∗2 ∗4 1 239 4– O S O OO O Mo Mo O OO O O O 3– S O OO O Mo Mo O OO O O 2– OH P O OO O Mo Mo O OO O – Cl C O OO Cl O Ru Ru O Cl OO O Cl C O OO (H3C)2CO Ru O O Ru OO OC(CH3)2 O + CH3 O H2O C OO Rh Rh O OO O OH2 O CH3 O H2O C OO Rh Rh O OO O OH2 O When multiple bridging ligands are present, only one is shown in detail. 469 470 19 The d-block elements 22 23 24 Sc Cl Figure 19.16 showed how cluster compounds may be classified according to ligand type. Although this classification is not appropriate in detail for all cluster compounds, it provides information on the principal ligand types and some insight into the bonding. For example, alkyllithium compounds often have alkyl groups attached to a metal cluster by two-electron multicentre bonds. Clusters of the early d-block elements and the lanthanoids generally contain donor ligands such as Br. These ligands can fill some of the low-lying orbitals of these electron-poor metal atoms by - and -electron donation. In contrast, electron-rich metal clusters on the right of the d block generally contain -acceptor ligands such as CO, which remove some of the electron density from the metal. Many p-block elements do not require ligands to complete the valence shells of the individual atoms in clusters and can exist as naked clusters, which are clusters of the form En with no accompanying ligands. The ionic clusters Pb52 and Sn94 are two examples of naked clusters formed by metals. Clusters may involve MM bonds within a discrete molecular cluster or in extended solid-state compounds. When no bridging ligands are present in a cluster, as in [Re2Cl8]2 (19), the presence of a metalmetal bond is unambiguous. When bridging ligands are present, careful observations and measurement (typically of bond lengths and magnetic properties) are needed to identify direct metalmetal bonding. We shall see in Section 22.20 that there is an extensive range of organometallic metal cluster compounds of the middle-to-late d-block elements that are stabilized by -acceptor ligands, particularly CO. The bonding patterns in most metal cluster compounds are so intricate that metal metal bond strengths cannot be determined with great precision. Some evidence, however, such as the stability of compounds and the magnitudes of MM force constants, indicates that there is an increase in MM bond strength down a group, perhaps on account of the greater spatial extension of d orbitals in heavier atoms. This trend may be the reason why there are so many more metalmetal bonded compounds for the 4d- and 5d-series metals than for their 3d-series counterparts. Figure 19.1 indicates that for the bulk metals, the metalmetal bonds in the d block are strongest in the 4d and 5d series, and this feature carries over into their compounds. By contrast, elementelement bonds weaken down a group in the p block. Not all the early d-block metalmetal bonded compounds are discrete clusters. Many extended metalmetal bonded compounds exist, such as the multiple-chain scandium subhalides, Sc7Cl10 (Fig. 19.20) and Sc5Cl6, and layered compounds such as ZrCl (Fig. 19.21). In ZrCl, the Zr atoms are within bonding distance in two adjacent layers of metal atoms sandwiched between Cl layers. We have already seen that Sc and Zr adopt their group oxidation states (3 and 4, respectively) when exposed to air and moisture, so these metalmetal bonded compounds are prepared out of contact with air and moisture. For example, ZrCl is prepared by reducing zirconium tetrachloride with zirconium metal in a sealed tantalum tube at high temperatures: 600 −800 C 3 Zr(s) + ZrCl4 (g) ⎯⎯⎯⎯⎯ → 4 ZrCl(s) Figure 19.20 The structure of Sc7Cl10. Discrete clusters—as distinct from the extended MM bonded solid-state compounds just described—are often soluble, and can be manipulated in solution. The -donor ligands in these clusters typically occur in one of several locations, namely, in a terminal position (22), bridging two metal atoms (23), or bridging three metal atoms (24). Clusters may also be linked by bridging halides or chalcogenides in the solid state. For example, the solid compound of formula MoCl2 consists of octahedral Mo clusters linked by Cl bridges. Zr Figure 19.21 The structure of ZrCl consists of layers of metal atoms in graphite-like hexagonal nets. Cl Further reading The bridged structure of MoCl2 is in sharp contrast to CrCl2, which has discrete Cr atoms arranged in a distorted rutile structure. Samples of MoCl2 may be prepared in a sealed glass tube by the reaction of MoCl5 in a mixture of molten NaAlCl4 and AlCl3, together with aluminium metal as the reducing agent: 471 2– Cl NaAlCl /AlCl (l), 200 C 4 3 MoCl5 (s) + Al(s) ⎯⎯⎯⎯⎯⎯⎯⎯⎯ → MoCl2 (s) + AlCl3 (l) Mo The compound can withstand oxidation under mild conditions. When treated with hydrochloric acid it forms the anionic cluster [Mo6Cl14]2. This cluster contains an octahedral array of Mo atoms with a Cl atom bridging each triangular face and a terminal Cl atom on each Mo vertex (25). These terminal Cl atoms can be replaced by other halogens, alkoxides, and phosphines. An analogous series of tungsten cluster compounds is known. Once formed, these Mo and W compounds can be handled in air and water at room temperature on account of their kinetic barriers to decomposition. The oxidation number of the metal is 2, so the metal valence electron count is (6 2) 6 24. One-electron oxidation and reduction of the clusters is possible. Similar octahedral clusters are known for Nb and Ta in Group 5, and for Zr in Group 4. The cluster [Nb6Cl12L6]2 and its Ta analogue have an octahedral framework with edgebridging Cl atoms and six terminal ligands (26). The metal valence electron count in this case is 16, but these clusters can be oxidized in several steps. One example from Group 4 is [Zr6Cl18C]4, which has the same metal and Cl atom array together with a C atom in the centre of the octahedron. Rhenium trichloride, which consists of Re3Cl9 clusters linked by weak halide bridges (27), provides the starting material for the preparation of a series of M3 clusters that have been studied thoroughly. As with molybdenum dichloride, the intercluster bridges in the solid state can be broken by reaction with potential ligands. For example, treatment of Re3Cl9 with Cl ions produces the discrete complex [Re3Cl12]3. Neutral ligands, such as trialkylphosphines, can also occupy these coordination sites and result in clusters of the general formula Re3Cl9L3. 25 [Mo6Cl14]2– 2+ M X L 26 [M6X12L6]2+ l C e R E X A M PL E 19. 3 Metalmetal bonding and clusters Suggest which interactions might be responsible for, and thus the bond order of, the metalmetal bond in the Hg2 ion. 2 27 Re3Cl9 Answer We need to judge the types of bonds that can form from the available atomic orbitals on each metal atom and the bond order that results from their occupation. The oxidation state of mercury in Hg2 2 is Hg(I) and therefore its electron configuration is d10s1. Although overlap of the d orbitals in the manner depicted in Fig. 19.17 is possible, the 20 d electrons from the two Hg ions would result in complete filling of both the bonding and the antibonding orbitals, with no effective bonding. Therefore the bonding must come from the overlap of s orbitals on each ion and the two remaining s electrons: bonding and antibonding orbitals can be constructed and as only the former is occupied, it results in a single HgHg bond. Even if a degree of sd hybridization is invoked, the description still demands the involvement of s orbitals in the bonding and the bond order remains 1. Self-test 19.3 Describe the probable structure of the compound formed when Re3Cl9 is dissolved in a solvent containing PPh3. FURTHER READING D.M.P. Mingos, Essential trends in inorganic chemistry. Oxford University Press (1998). A survey of inorganic chemistry from the perspective of structure and bonding. R.B. King (ed.), Encyclopedia of inorganic chemistry. Wiley (2005). M.T. Pope, Polyoxoanions: synthesis and structure. In Comprehensive coordination chemistry II, Vol. 4 (ed. J.A. McCleverty and T.J. Meyer), Chapter 10. Elsevier (2004). A discussion of metal oxides in solution, focusing on their tendency to form polyoxoanions. M.H. Chisholm (ed.) Early transition metal clusters with -donor ligands. VCH, Weinheim (1995). 472 19 The d-block elements EXERCISES 19.1 Without reference to a periodic table, sketch the first series of the d block, including the symbols of the elements. Indicate those elements for which the group oxidation number is common by C, those for which the group oxidation number can be reached but is a powerful oxidizing agent by O, and those for which the group oxidation number is not achieved by N. 19.2 Explain why the enthalpy of sublimation of Re(s) is significantly greater than that of Mn(s). 19.3 State the trend in the stability of the group oxidation state on descending a group of metallic elements in the d block. Illustrate the trend using standard potentials in acidic solution for Groups 5 and 6. 19.4 For each part, give balanced chemical equations or NR (for no reaction) and rationalize your answer in terms of trends in oxidation states. (a) Cr 2+ (aq) + Fe3+ (aq) → (b) CrO24− (aq) + MoO2 (s) → (c) MnO4− (aq) + Cr 3+ (aq) → 19.5 (a) Which ion, Ni2(aq) or Mn2(aq), is more likely to form a sulfide in the presence of H2S? (b) Rationalize your answer with the trends in hard and soft character across Period 4. (c) Give a balanced chemical equation for the reaction. 19.6 Preferably without reference to the text (a) write out the d block of the periodic table, (b) indicate the metals that form difluorides with the rutile or fluorite structures, and (c) indicate the region of the periodic table in which metalmetal bonded halide compounds are formed, giving one example. 19.7 Write a balanced chemical equation for the reaction that occurs when cis-[RuLCl(OH2)] (see Fig. 19.9) in acidic solution at 0.2 V is made strongly basic at the same potential. Write a balanced equation for each of the successive reactions when this same complex at pH 6 and 0.2 V is exposed to progressively more oxidizing environments up to 1.0 V. Give other examples and a reason for the redox state of the metal centre affecting the extent of protonation of coordinated oxygen. 19.8 Give plausible balanced chemical reactions (or NR for no reaction) for the following combinations, and state the basis for your answer: (a) MnO4(aq) plus Fe2(aq) in acidic solution, (b) the preparation of [Mo6O19]2(aq) from K2MoO4(s), (c) ReCl5(s) plus KMnO4(aq), (d) MoCl2(s) plus warm HBr(aq), (e) TiO(s) with HCl(aq) under an inert atmosphere, (f) Cd(s) added to Hg2(aq). 19.9 Speculate on the structures of the following species and present bonding models to justify your answers: (a) [Re(O)2(py)4], (b) [V(O)2(ox)2]3, (c) [Mo(O)2(CN)4]4, (d) [VOCl4]2. 19.10 Which of the following are likely to have structures that are typical of (a) predominantly ionic, (b) significantly covalent, (c) metalmetal bonded compounds: NiI2, NbCl4, FeF2, PtS, and WCl2? Rationalize the differences and speculate on the structures. 19.11 Indicate the probable occupancy of , , and bonding and antibonding orbitals, and the bond order for the following tetragonal prismatic complexes: (a) [Mo2(O2CCH3)4], (b) [Cr2(O2CC2H5)4], (c) [Cu2(O2CCH3)4]. 19.12 Explain the differences in the following redox couples, measured at 25ºC: MnO4− MnO2 + 1.69 V TcO4− TcO2 + 0.74 V ReO4− ReO2 + 0.51V 19.13 Addition of sodium ethanoate to aqueous solutions of Cr(II) gives a red diamagnetic product. Draw the structure of the product, noting any features of interest. 19.14 Consider the two ruthenium complexes in Table 19.8. Using the bonding scheme depicted in Figure 19.19, confirm the bonding orders and electron configurations given in the table. PROBLEMS 19.1 An amateur chemist claimed the existence of a new metallic element, grubium (Gr), which has the following characteristics. Metallic Gr reacts with 1 m H(aq) in the absence of air to produce Gr3(aq) and H2(g). In the absence of air GrCl2(s) dissolves in 1 m H(aq) and very slowly yields H2(g) plus Gr3(aq). When Gr3(aq) is exposed to air GrO2(aq) is produced. From this information, estimate the range of potentials for (a) the reduction of Gr3(aq) to Gr(s), (b) the reduction of Gr3(aq) to GrCl2(s), and (c) the reduction of GrO2(aq) to Gr3(aq). Suggest a known element that fits the description of grubium. 19.3 Think about the mechanism of electrical conduction and then suggest how electrical conductivity might vary across the d block. 19.2 Compared with the p block, the variation in the properties of the elements of the d block across each period is rather modest. Provide evidence for this statement by reference to isostructural compounds. Speculate on reasons why this statement is true. 19.6 Discuss how the presence of a quintuple bond in (20) was confirmed, and how a sextuple bond between two metal atoms might arise (see T. Nguyen, A.D. Sutton, M. Brynda, J.C. Fettinger, G.J. Long, and P.P. Power, Science 2005, 310, 844). 19.4 Many metal salts can be vaporized to a small extent at high temperatures and their structures studied in the vapour phase by electron diffraction. Speculate on the structures that you might expect to find for the following gas-phase species and present your reasoning: (a) TaF5, (b) MoF6. 19.5 Discuss ways in which the triamidoamine ligands [(RNCH2CH2)3N]3 (where R is a bulky substituent) can be used to stabilize nitrido, phosphido, and arsenido groups (see R.R. Schrock, Acc. Chem. Res., 1997, 30, 9). d-Metal complexes: electronic structure and properties d-Metal complexes play an important role in inorganic chemistry. In this chapter, we discuss the nature of ligandmetal bonding in terms of two theoretical models. We start with the simple but useful crystal-field theory, which is based on an electrostatic model of the bonding, and then progress to the more sophisticated ligand-field theory. Both theories invoke a parameter, the ligand-field splitting parameter, to correlate spectroscopic and magnetic properties. We then examine the electronic spectra of complexes and see how ligand-field theory allows us to interpret the energies and intensities of electronic transitions. We now examine in detail the bonding, electronic structure, electronic spectra, and magnetic properties of the d-metal complexes introduced in Chapter 7. The striking colours of many d-metal complexes were a mystery to Werner when he elucidated their structures, and the origin of the colours was clarified only when the description of electronic structure in terms of orbitals was applied to the problem in the period from 1930 to 1960. Tetrahedral and octahedral complexes are the most important, and the discussion begins with them. Electronic structure There are two widely used models of the electronic structure of d-metal complexes. One (‘crystal-field theory’) emerged from an analysis of the spectra of d-metal ions in solids; the other (‘ligand-field theory’) arose from an application of molecular orbital theory. Crystalfield theory is more primitive, and strictly speaking it applies only to ions in crystals; however, it can be used to capture the essence of the electronic structure of complexes in a straightforward manner. Ligand-field theory builds on crystal-field theory: it gives a more complete description of the electronic structure of complexes and accounts for a wider range of properties. 20.1 Crystal-field theory In crystal-field theory, a ligand lone pair is modelled as a point negative charge (or as the partial negative charge of an electric dipole) that repels electrons in the d orbitals of the central metal ion. The theory concentrates on the resulting splitting of the d orbitals into groups with different energies, and uses that splitting to rationalize and correlate the optical spectra, thermodynamic stability, and magnetic properties of complexes. (a) Octahedral complexes Key points: In the presence of an octahedral crystal field, d orbitals are split into a lower-energy triply degenerate set (t2g) and a higher-energy doubly degenerate set (eg) separated by an energy O; the ligand-field splitting parameter increases along a spectrochemical series of ligands and varies with the identity and charge of the metal atom. In the model of an octahedral complex used in crystal-field theory, six point negative charges representing the ligands are placed in an octahedral array around the central metal ion. These charges (which we shall refer to as the ‘ligands’) interact strongly with the 20 Electronic structure 20.1 Crystal-field theory 20.2 Ligand-field theory Electronic spectra 20.3 Electronic spectra of atoms 20.4 Electronic spectra of complexes 20.5 Charge-transfer bands 20.6 Selection rules and intensities 20.7 Luminescence Magnetism 20.8 Cooperative magnetism 20.9 Spin crossover complexes FURTHER READING EXERCISES PROBLEMS 474 20 d-Metal complexes: electronic structure and properties z y x eg t2g Spherical environment Octahedral crystal field dz dzx dx –y 2 2 dyz 2 dxy Figure 20.1 The orientation of the five d orbitals with respect to the ligands of an octahedral complex: the degenerate (a) eg and (b) t2g orbitals. eg d 3 5 ∆O 2 5 ∆O ∆O t2g Figure 20.2 The energies of the d orbitals in an octahedral crystal field. Note that the mean energy remains unchanged relative to the energy of the d orbitals in a spherically symmetrical environment (such as in a free atom). central metal ion, and the stability of the complex stems in large part from this attractive interaction between opposite charges. However, there is a much smaller but very important secondary effect arising from the fact that electrons in different d orbitals interact with the ligands to different extents. Although this differential interaction is little more than about 10 per cent of the overall metalligand interaction energy, it has major consequences for the properties of the complex and is the principal focus of this section. Electrons in dz2 and dx2y2 orbitals (which are of symmetry type eg in Oh; Section 6.1) are concentrated close to the ligands, along the axes, whereas electrons in dxy, dyz, and dzx orbitals (which are of symmetry type t2g) are concentrated in regions that lie between the ligands (Fig. 20.1). As a result, the former are repelled more strongly by the negative charge on the ligands than the latter and lie at a higher energy. Group theory shows that the two eg orbitals have the same energy (although this is not readily apparent from drawings), and that the three t2g orbitals also have the same energy. This simple model leads to an energy-level diagram in which the three degenerate t2g orbitals lie below the two degenerate eg orbitals (Fig. 20.2). The separation of the two sets of orbitals is called the ligand-field splitting parameter, O (where the subscript O signifies an octahedral crystal field). 400 25 lmax = 493 nm n~max = 20 300 cm–1 Absorbance A note on good practice In the context of crystal-field theory, the ligand-field splitting parameter should be called the crystal-field splitting parameter, but we use ligand-field splitting parameter to avoid a proliferation of names. 500 l /nm 20 n~/(10 3 cm–1) Figure 20.3 The optical absorption spectrum of [Ti(OH2)6]3. 670 15 The energy level that corresponds to the hypothetical spherically symmetrical environment (in which the negative charge due to the ligands is evenly distributed over a sphere instead of being localized at six points) defines the barycentre of the array of levels, with the two eg orbitals lying at 35 ∆O above the barycentre and the three t2g orbitals lying at 25 ∆O below it. As in the representation of the configurations of atoms, a superscript is used to 2 . indicate the number of electrons in each set, for example t2g The simplest property that can be interpreted by crystal-field theory is the absorption spectrum of a one-electron complex. Figure 20.3 shows the optical absorption spectrum of the d1 hexaaquatitanium(III) ion, [Ti(OH2)6]3. Crystal-field theory assigns the first absorption maximum at 493 nm (20 300 cm1) to the transition eg ← t2g and identifies 20 300 cm1 with ∆O for the complex. It is not so straightforward to obtain values of ∆O for complexes with more than one d electron because the energy of a transition then depends not only on orbital energies but also on the electronelectron repulsion energies. This aspect is treated more fully in Section 20.4 and the results from the analyses described there have been used to obtain the values of ∆O in Table 20.1. Electronic structure Table 20.1 Ligand-field splitting parameters ∆O of ML6 complexes Ions d3 d 5 Cr3 Mn 2 Fe3 d 6 d8 Ligands Cl H2O NH3 en CN 13 700 17 400 21 500 21 900 26 600 7500 8500 10 100 30 000 11 000 14 300 2 (35 000) Fe 10 400 Co3 (20 700) (22 900) (32 800) (23 200) (34 800) (45 500) Rh3 (20 400) (27 000) (34 000) (34 600) Ni2 7500 8500 10 800 11 500 Values are in cm1; entries in parentheses are for low-spin complexes. Source: H.B. Gray, Electrons and chemical bonding. Benjamin, Menlo Park (1965). A note on good practice The convention in spectroscopic notation is to indicate transitions as [upper state] ← [lower state]. The ligand-field splitting parameter, ∆O, varies systematically with the identity of the ligand. For instance, in the series of complexes [CoX(NH3)5]n with X I, Br, Cl, H2O, and NH3, the colours range from purple (for X I) through pink (for Cl) to yellow (with NH3). This sequence indicates that the energy of the lowest energy electronic transition (and therefore ∆O) increases as the ligands are varied along the series. The same order is followed regardless of the identity of the metal ion. Thus ligands can be arranged in a spectrochemical series, in which the members are arranged in order of increasing energy of transitions that occur when they are present in a complex: I Br S2 SCN Cl NO2 N3 F OH C2O42 O2 H2O NCS CH3C⬅N py NH3 en bpy phen NO2 PPh3 CN CO (The donor atom in an ambidentate ligand is underlined.) Thus, the series indicates that, for the same metal, the optical absorption of the cyano complex will occur at higher energy than that of the corresponding chlorido complex. A ligand that gives rise to a highenergy transition (such as CO) is referred to as a strong-field ligand, whereas one that gives rise to a low-energy transition (such as Br) is referred to as a weak-field ligand. Crystalfield theory alone cannot explain these strengths, but ligand-field theory can, as we shall see in Section 20.2. The ligand-field strength also depends on the identity of the central metal ion, the order being approximately: Mn2 Ni2 Co2 Fe2 V2 Fe3 Co3 Mo3 Rh3 Ru3 Pd4 Ir3 Pt4 The value of ∆O increases with increasing oxidation state of the central metal ion (compare the two entries for Fe and Co) and also increases down a group (compare, for instance, the locations of Co, Rh, and Ir). The variation with oxidation state reflects the smaller size of more highly charged ions and the consequently shorter metalligand distances and stronger interaction energies. The increase down a group reflects the larger size of the 4d and 5d orbitals compared with the compact 3d orbitals and the consequent stronger interactions with the ligands. (b) Ligand-field stabilization energies Key point: The ground-state configuration of a complex reflects the relative values of the ligand-field splitting parameter and the pairing energy. For 3dn species with n 47, high-spin and low-spin complexes occur in the weak-field and strong-field cases, respectively. Complexes of 4d- and 5d-series metals are typically low-spin. Because the d orbitals in a complex do not all have the same energy, the ground-state electron configuration of a complex is no longer immediately obvious. To predict it, we use the 475 476 20 d-Metal complexes: electronic structure and properties d-orbital energy level diagram shown in Fig. 20.2 as a basis for applying the building-up principle. That is, we identify the lowest energy configuration subject to the Pauli exclusion principle (a maximum of two electrons in an orbital) and (if more than one degenerate orbital is available) to the requirement that electrons first occupy separate orbitals and do so with parallel spins. First, we consider complexes formed by the 3d-series elements. In an octahedral complex, the first three d electrons of a 3dn complex occupy separate t2g nonbonding orbitals, and do so with parallel spins. For example, the ions Ti2 and V2 have electron configurations 3d2 and 3d3, respectively. The d electrons occupy the lower t2g orbitals as shown in (1) and (2), respectively. The energy of a t2g orbital relative to the barycentre of an octahedral ion is 0.4∆O and the complexes are stabilized by 2 (0.4∆O) 0.8∆O (for Ti2) and 3 (0.4∆O) 1.2∆O (for V2). This additional stability, relative to the barycentre is called the ligand-field stabilization energy (LFSE). 1 A note on good practice The term crystal-field stabilization energy (CFSE) is widely used in place of LFSE, but strictly speaking the term is appropriate only for ions in crystals. 2 3 4 The next electron needed for the 3d4 ion Cr2 may enter one of the t2g orbitals and pair with the electron already there (3). However, if it does so, it experiences a strong Coulombic repulsion, which is called the pairing energy, P. Alternatively, the electron may occupy one of the eg orbitals (4). Although the pairing penalty is now avoided, the orbital energy is 4 ), there is a stabilisation of 1.6∆O, countered by the pairhigher by ∆O. In the first case (t2g 3 1 eg), the LFSE is ing energy of P, giving a net LFSE of 1.6∆O P. In the second case (t2g 3 (0.4∆O) 0.6∆O 0.6∆O, as there is no pairing energy to consider. Which configuration is adopted depends on which of 1.60∆O P and 0.60∆O is the larger. If ∆O P, which is called the weak-field case, a lower energy is achieved when the upper 3 1 eg. If ∆O P, which is called the strong-field orbital is occupied to give the configuration t2g case, a lower energy is achieved by occupying only the lower orbitals despite the cost of the 4 . For example, [Cr(OH2)6]2 has the pairing energy. The resulting configuration is now t2g 4 3 1 ground-state configuration t2g eg whereas [Cr(CN)6] , with relatively strong-field ligands 4 . In the weak-field case (as indicated by the spectrochemical series), has the configuration t2g all the electrons occupy different orbitals and have parallel spins. The resulting spin correlation effect (the tendency of electrons of the same spin to avoid each other) helps to offset the cost of occupying orbitals of higher energy. The ground-state electron configurations of 3d1, 3d2, and 3d3 complexes are unambiguous because there is no competition between the additional stabilization achieved 1 2 3 , t2g , and t2g , by occupying the t2g orbitals and the pairing energy: the configurations are t2g respectively, with each electron in a separate orbital. As remarked above, there are two possible configurations for 3d4 complexes; the same is true of 3dn complexes in which n 5, 6, or 7. In the strong-field case, the lower orbitals are occupied preferentially and in the weak-field case, electrons avoid the pairing energy by occupying the upper orbitals. When alternative configurations are possible, the species with the smaller number of parallel electron spins is called a low-spin complex, and the species with the greater number of parallel electron spins is called a high-spin complex. As we have noted, an octahedral 3d4 complex is likely to be low-spin if the ligand field is strong but high-spin if the field is weak (Fig. 20.4); the same applies to 3d5, 3d6, and 3d7 complexes: Weak-field ligands Strong-field ligands Configuration Unpaired electrons Configuration Unpaired electrons 3d4 3 1 t2g eg 4 4 t2g 2 3d5 3 2 t2g eg 5 5 t2g 1 6 3d 4 2 2g g t e 4 6 2g t 0 3d7 5 2 t2g eg 3 6 1 t2g eg 1 The ground-state electron configurations of 3d8, 3d9, and 3d10 complexes are unambiguous 6 2 6 3 6 4 eg, t2g eg, and t2g eg. and the configurations are t2g x y In general, the net energy of a t2g eg configuration relative to the barycentre, without taking the pairing energy into account, is (0.4x 0.6y)∆O. Pairing energies need to be taken Electronic structure into account only for pairing that is additional to the pairing that occurs in a spherical field. Figure 20.5 shows the case of a d6 ion. In both the free ion and the high-spin complex two electrons are paired, whereas in the low-spin case all six electrons occur as three pairs. Thus we do not need to consider the pairing energy in the high-spin case, as there is no additional pairing. There are two additional pairings in the low-spin case, so two pairing energy contributions must be taken into account. In general, high-spin complexes always have the same number of unpaired electrons as in a spherical field (free ion), and we therefore do not need to consider pairing energies for high-spin complexes. Table 20.2 lists the values for the LFSE of the various configurations of octahedral ions, with the appropriate pairing energies taken into account for the low-spin complexes. Remember that the LFSE is generally only a small fraction of the overall interaction between the metal atom and the ligands. The strength of the crystal field (as measured by the value of ∆O) and the spin-pairing energy (as measured by P) depend on the identity of both the metal and the ligand, so it is not possible to specify a universal point in the spectrochemical series at which a complex changes from high spin to low spin. For 3d-metal ions, low-spin complexes commonly occur for ligands that are high in the spectrochemical series (such as CN) and high-spin complexes are common for ligands that are low in the series (such as F). For octahedral dn complexes with n 13 and 810 there is no ambiguity about the configuration (see Table 20.2), and the designations high-spin and low-spin are not used. As we have seen, the values of ∆O for complexes of 4d- and 5d-series metals are typically higher than for the 3d-series metals. Pairing energies for the 4d- and 5d-series metals tend to be lower than for the 3d-series metals because the orbitals are less compact and electronelectron repulsions correspondingly weaker. Consequently, complexes of these metals generally have electron configurations that are characteristic of strong crystal fields and typically have low spin. An example is the 4d4 complex [RuCl6]2, which has 4 configuration, which is typical of a strong crystal field despite Cl being low in the a t2g 5 whereas spectrochemical series. Likewise, [Ru(ox)3]3 has the low-spin configuration t2g 3 3 2 [Fe(ox)3] has the high-spin configuration t2g eg. Weak field Free ion 477 Strong field High spin Low spin Figure 20.4 The effect of weak and strong ligand fields on the occupation of electrons for a d4 complex. The former results in a high-spin configuration and the latter in a low-spin configuration. Weak field Free ion Strong field High spin Low spin Table 20.2 Ligand-field stabilization energies for octahedral complexes dn Example d0 N (high spin) LFSE/O 0 0 d1 Ti3 1 0.4 d 2 V3 2 0.8 d 3 3 1.2 4 0.6 3 Cr , V d4 2 Cr2,Mn3 N (low spin) LFSE/O Figure 20.5 The effect of weak and strong ligand fields on the occupation of electrons for a d6 complex. The former results in a high-spin configuration and the latter in a low-spin configuration. 2 1.6 P Mn , Fe 5 0 1 2.0 2P d6 Fe2,Co3 4 0.4 0 2.4 2P d7 Co2 3 0.8 1 1.8 P 8 Ni2 2 1.2 Cu2 1 0.6 Cu, Zn2 0 0 d d 5 2 d9 d 10 3 N is the number of unpaired electrons. E X A M PL E 20.1 Calculating the LFSE Determine the LFSE for the following octahedral ions from first principles and confirm the value matches those in Table 20.2: (a) d3, (b) high-spin d5, (c) high-spin d6, (d) low-spin d6, (e) d9. Answer We need to consider the total orbital energy in each case and, when appropriate, the pairing energy. 3 (a) A d3 ion has configuration t2g (no pairing of electrons) and therefore LFSE 3 (0.4∆O) 1.2∆O. 5 3 2 eg (no pairing of electrons) therefore LFSE 3 (0.4∆O) 2 (b) A high spin d ion has configuration t2g 4 2 (0.6∆O) 0. (c) A high-spin d6 ion has configuration t2g eg with the pairing of two electrons. However, 478 20 d-Metal complexes: electronic structure and properties since those two electrons would be paired in a spherical field there is no additional pairing energy to be concerned with. Therefore LFSE 4 (0.4∆O) 2 (0.6∆O) 0.4∆O. (d) A low-spin d6 ion has 6 configuration t2g with the pairing of three pairs of electrons. However, since one pair of electrons would be paired in a spherical field the additional pairing energy is 2P. Therefore LFSE 6 (0.4∆O) 2P 6 3 2.4∆O 2P. (e) A d9 ion has configuration t2g eg with the pairing of four pairs of electrons. However, since all four pairs of electrons would be paired in a spherical field there is no additional pairing energy. Therefore LFSE 6 (0.4∆O) 3 (0.6∆O) 0.6∆O. Self-test 20.1 What is the LFSE for both high- and low-spin d7 configurations? (c) Magnetic measurements Key points: Magnetic measurements are used to determine the number of unpaired spins in a complex and hence to identify its ground-state configuration. A spin-only calculation may fail for low-spin d5 and for high-spin 3d6 and 3d7 complexes. The experimental distinction between high-spin and low-spin octahedral complexes is based on the determination of their magnetic properties. Compounds are classified as diamagnetic if they are repelled by a magnetic field and paramagnetic if they are attracted by a magnetic field. The two classes are distinguished experimentally by magnetometry (Chapter 8). The magnitude of the paramagnetism of a complex is commonly reported in terms of the magnetic dipole moment it possesses: the higher the magnetic dipole moment of the complex, the greater the paramagnetism of the sample. In a free atom or ion, both the orbital and the spin angular momenta give rise to a magnetic moment and contribute to the paramagnetism. When the atom or ion is part of a complex, any orbital angular momentum is normally quenched, or suppressed, as a result of the interactions of the electrons with their nonspherical environment. However, if any electrons are unpaired the net electron spin angular momentum survives and gives rise to spin-only paramagnetism, which is characteristic of many d-metal complexes. The spinonly magnetic moment, μ, of a complex with total spin quantum number S is 1 μ 2{S(S 1)} 2 μB (20.1) where μB is the Bohr magneton, μB e /2me with the value 9.274 10 J T . Because S 12 N, where N is the number of unpaired electrons, each with spin s 12 , 24 1 1 μ {N(N 2)} 2 μB (20.2) A measurement of the magnetic moment of a d-block complex can usually be interpreted in terms of the number of unpaired electrons it contains, and hence the measurement can be used to distinguish between high-spin and low-spin complexes. For example, magnetic 4 2 eg (N 4, S 2, measurements on a d6 complex easily distinguish between a high-spin t2g 6 μ 4.90μB) configuration and a low-spin t2g (N 0, S 0, μ 0) configuration. The spin-only magnetic moments for some electron configurations are listed in Table 20.3 and compared there with experimental values for a number of 3d complexes. For most 3d complexes (and some 4d complexes), experimental values lie reasonably close to spin-only predictions, so it becomes possible to identify correctly the number of unpaired electrons and assign the ground-state configuration. For instance, [Fe(OH2)6]3 is paramagnetic with a magnetic moment of 5.9μB. As shown in Table 20.3, this value is consistent with there be3 2 eg configuration. ing five unpaired electrons (N 5 and S 25 ), which implies a high-spin t2g Table 20.3 Calculated spin-only magnetic moments Ion Electron configuration S μ/μB Calculated Experimental 1 t2g 1 2 Ti3 V3 1.73 1.71.8 2 t2g 1 2.83 2.72.9 3 2g Cr t 3 2 3.87 3.8 Mn3 3 1 t2g eg 2 4.90 4.84.9 3 2 2g g 5 2 5.92 5.9 3 3 Fe t e 479 Electronic structure The interpretation of magnetic measurements is sometimes less straightforward than this example might suggest. For example, the potassium salt of [Fe(CN)6]3 has μ 2.3μB, which is between the spin-only values for one and two unpaired electrons (1.7μB and 2.8μB, respectively). In this case, the spin-only assumption has failed because the orbital contribution to the magnetic moment is substantial. For orbital angular momentum to contribute, and hence for the paramagnetism to differ significantly from the spin-only value, there must be one or more unfilled or half-filled orbitals similar in energy to the orbitals occupied by the unpaired spins and of the appropriate symmetry (one that is related to the occupied orbital by rotation round the direction of the applied field). If that is so, the applied magnetic field can force the electrons to circulate around the metal ion by using the low-lying orbitals and hence it generates orbital angular momentum and a corresponding orbital contribution to the total magnetic moment (Fig. 20.6). Departure from spin-only values is generally large for low-spin d5 and for high-spin 3d6 and 3d7 complexes. It is also possible for the electronic state of the metal ion to change (for example with temperature), leading to a change from high-spin to low-spin and a change in the magnetic moment. Such complexes are referred to as spin-crossover complexes and are discussed in more detail, together with the effects of cooperative magnetism, in Sections 20.8 and 20.9. dx –y 2 2 dxy Figure 20.6 If there is a low-lying orbital of the correct symmetry, the applied field may induce the circulation of the electrons in a complex and hence generate orbital angular momentum. This diagram shows the way in which circulation may arise when the field is applied perpendicular to the xy-plane (perpendicular to this page). E X A M PL E 20. 2 Inferring an electron configuration from a magnetic moment The magnetic moment of a certain octahedral Co(II) complex is 4.0B. What is its d-electron configuration? Answer We need to match the possible electron configurations of the complex with the observed magnetic 5 2 dipole moment. A Co(II) complex is d7. The two possible configurations are t2g eg (high spin, N 3, S 32 ) with 1 6 1 three unpaired electrons or t2g eg (low spin, N 1, S 2 ) with one unpaired electron. The spin-only magnetic moments are 3.87B and 1.73B, respectively (see Table 20.3). Therefore, the only consistent assignment is the 5 2 high-spin configuration t2g eg. Self-test 20.2 The magnetic moment of the complex [Mn(NCS)6]4 is 6.06B. What is its electron configuration? (d) Thermochemical correlations Key point: The experimental variation in hydration enthalpies reflects a combination of the variation in radii of the ions (the linear trend) and the variation in LFSE (the saw-tooth variation). 2989 3000 2986 2936 2904 –∆hydH°(M2+)/(kJ mol–1) 2800 2600 The concept of ligand-field stabilization energy helps to explain the double-humped variation in the hydration enthalpies of the high-spin octahedral 3d-metal M2 ions (Fig. 20.7). The nearly linear increase across a period shown by the filled circles represents the increasing strength of the bonding between H2O ligands and the central metal ion as the ionic radii decrease from left to right across the period. The deviation of hydration enthalpies from a straight line reflects the variation in the ligand-field stabilization energies. As Table 20.2 shows, the LFSE increases from d1 to d3, decreases again to d5, then rises to d8. The filled circles in Fig. 20.7 were calculated by subtracting the high-spin LFSE from ∆hydH by using the spectroscopic values of ∆O in Table 20.1. We see that the LFSE calculated from spectroscopic data accounts for the additional ligand binding energy for the complexes shown in the illustration. 2814 2843 2799 2743 E X A M PL E 20. 3 Using the LFSE to account for thermochemical properties 2468 The oxides of formula MO, which all have octahedral coordination of the metal ions in a rock-salt structure, have the following lattice enthalpies: CaO TiO VO MnO 3460 3878 3913 3810 kJ mol1 Account for the trends in terms of the LFSE. Answer We need to consider the simple trend that would be expected on the basis of trends in ionic radii and then deviations that can be traced to the LFSE. The general trend across the d block is the increase 2400 Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Figure 20.7 The hydration enthalpy of M2 ions of the first row of the d block. The straight line shows the trend when the ligand-field stabilization energy has been subtracted from the observed values. Note the general trend to greater hydration enthalpy (more exothermic hydration) on crossing the period from left to right. 480 20 d-Metal complexes: electronic structure and properties in lattice enthalpy from CaO (d0) to MnO (d5) as the ionic radii of the metals decrease (recall that lattice enthalpy is proportional to 1/(r r), Section 3.12). The Ca2 ion has an LFSE of zero as it has no d electrons and the Mn2 ion, being high spin (O2 is a weak field ligand), also has an LFSE of zero. For a linear increase in lattice enthalpy from calcium to manganese oxides we would expect the lattice enthalpies to increase by (3810 3460)/5 kJ mol1 from Ca2 to Sc2 to Ti2 to V2 to Mn2. We would therefore expect TiO and VO to have lattice enthalpies of 3600 and 3670 kJ mol1, respectively. In fact, TiO (d2) has a lattice enthalpy of 3878 kJ mol1 and we can ascribe this difference of 278 kJ mol1 to an LFSE of 0.8∆O. Likewise the actual lattice enthalpy of 3913 kJ mol1 for VO (d3) is 243 kJ mol1 greater than predicted, with this difference arising from an LFSE of 1.2∆O. Self-test 20.3 Account for the variation in lattice enthalpy of the solid fluorides in which each metal ion is surrounded by an octahedral array of F ions: MnF2 (2780 kJ mol1), FeF2 (2926 kJ mol1), CoF2 (2976 kJ mol1), NiF2 (3060 kJ mol1), and ZnF2 (2985 kJ mol1). (e) Tetrahedral complexes Key points: In a tetrahedral complex, the e orbitals lie below the t2 orbitals; only the high-spin case need be considered. Spherical environment Tetrahedral crystal field t2 d 2 5 ∆T 3 5 ∆T ∆T e Figure 20.8 The orbital energy level diagram used in the application of the building-up principle in a crystal-field analysis of a tetrahedral complex. Four-coordinate tetrahedral complexes are second only in abundance to octahedral complexes for the 3d metals. The same kind of arguments based on crystal-field theory can be applied to these species as we used for octahedral complexes. A tetrahedral crystal field splits d orbitals into two sets but with the two e orbitals (the dx2y2 and the dz2) lower in energy than the three t2 orbitals (the dxy, the dyz and the dzx) (Fig. 20.8).1 The fact that the e orbitals lie below the t2 orbitals can be understood from a consideration of the spatial arrangement of the orbitals: the e orbitals point between the positions of the ligands and their partial negative charges whereas the t2 orbitals point more directly towards the ligands (Fig. 20.9). A second difference is that the ligand-field splitting parameter in a tetrahedral complex, ∆T , is less than ∆O, as should be expected for complexes with fewer ligands, none of which is oriented directly at the d orbitals (in fact, ∆T ⬇ –94 ∆O). The pairing energy is invariably more unfavourable than ∆T , and normally only high-spin tetrahedral complexes are encountered. Ligand-field stabilization energies can be calculated in exactly the same way as for octahedral complexes. Since tetrahedral complexes are always high-spin, there is never any t2 dzx dyz dxy z e dz 2 dx –y 2 2 Figure 20.9 The effect of a tetrahedral crystal field on a set of d orbitals is to split them into two sets; the e pair (which point less directly at the ligands) lie lower in energy than the t2 triplet. 1 Because there is no centre of inversion in a tetrahedral complex, the orbital designation does not include the parity label g or u. Electronic structure Table 20.4 Ligand-field stabilization energies for tetrahedral complexes dn Configuration d0 d 1 d2 d3 N LFSE/DT 0 0 1 1 0.6 e2 2 1.2 e e2t21 3 e2t22 4 e2t23 5 d6 e3t23 d7 d4 Table 20.5 Values of ∆T for representative tetrahedral complexes Complex VCl4 DT /cm1 9010 2 0.8 [CoCl4] 3300 0.4 [CoBr4]2 2900 0 [CoI4] 2 2700 4 0.6 [Co(NCS)4]2 4700 e4t23 3 1.2 d8 e4t24 2 0.8 9 d 4 5 2 et 1 0.4 d10 e4t26 0 0 d5 481 N is the number of unpaired electrons. need to consider the pairing energy in the LFSE and the only differences compared with octahedral complexes are the order of occupation (e before t2) and the contribution of each orbital to the total energy ( 35 ∆T for an e orbital and 25 ∆T for a t2 orbital). Table 20.4 lists the configurations of tetrahedral dn complexes together with the calculated values of the LFSE and Table 20.5 lists some experimental values of ∆T for a number of complexes. (f) Square-planar complexes Key point: A d8 configuration, coupled with a strong ligand field, favours the formation of squareplanar complexes. This tendency is enhanced with the 4d and 5d metals due to their larger size and the greater ease of electron pairing. Although a tetrahedral arrangement of four ligands is the least sterically demanding arrangement, some complexes exist with four ligands in an apparently higher energy squareplanar arrangement. A square-planar arrangement gives the d-orbital splitting shown in Fig. 20.10, with dx2y2 raised above all the others. This arrangement may become energetically favourable when there are eight d electrons and the crystal field is strong enough to 2 2 dz22 dxy configuration. In this configuration the electronic stabilfavour the low-spin d2yz dzx ization energy can more than compensate for any unfavourable steric interactions. Thus, many square-planar complexes are found for complexes of the large 4d8 and 5d8 Rh(I), Ir(I), Pt(II), Pd(II), and Au(III) ions, in which unfavourable steric constraints have less effect and there is a large ligand-field splitting associated with the 4d- and 5d-series metals. By contrast, small 3d-series metal complexes such as [NiX4]2, with X a halogen, are generally tetrahedral because the ligand-field splitting parameter is generally quite small and will not compensate sufficiently for the unfavourable steric interactions. Only when the ligand is high in the spectrochemical series is the LFSE large enough to result in the formation of a square-planar complex, as, for example, with [Ni(CN)4]2. We have already noted that pairing energies for the 4d- and 5d-series metals tend to be lower than for the 3d-series metals, and this difference provides a further factor that favours the formation of low-spin square-planar complexes with these metals. The sum of the three distinct orbital splittings in Fig. 20.10 is denoted ∆SP. Simple theory predicts that ∆SP 1.3∆O for complexes of the same metal and ligands with the same ML bond lengths. 2 x –y ∆3 xy ∆SP ∆ 2 z (g) Tetragonally distorted complexes: the JahnTeller effect Key points: A tetragonal distortion can be expected when the ground electronic configuration of a complex is orbitally degenerate; the complex will distort so as to remove the degeneracy and achieve a lower energy. Six-coordinate d9 complexes of copper(II) usually depart considerably from octahedral geometry and show pronounced tetragonal distortions (Fig. 20.11). High-spin d4 (for instance, 2 2 ∆1 yz, zx Figure 20.10 The orbital splitting parameters for a square-planar complex. 482 20 d-Metal complexes: electronic structure and properties (a) (b) Figure 20.11 (a) A tetragonally distorted complex where two of the ligands have moved further away from the central ion. (b) A tetragonally distorted complex where two of the ligands have moved closer towards the central ion. Oh D4h 2 x –y 2 2 z Mn3) and low-spin d7 six-coordinate complexes (for instance, Ni3) may show a similar distortion, but they are less common. These distortions are manifestations of the Jahn–Teller effect: if the ground electronic configuration of a nonlinear complex is orbitally degenerate, and asymmetrically filled, then the complex distorts so as to remove the degeneracy and achieve a lower energy. The physical origin of the effect is quite easy to identify. Thus, a tetragonal distortion of a regular octahedron, corresponding to extension along the z-axis and compression on the x- and y-axes, lowers the energy of the eg(dz2) orbital and increases the energy of the eg(dx2y2) orbital (Fig. 20.12). Therefore, if one or three electrons occupy the eg orbitals (as in high-spin d4, low-spin d7, and d9 complexes) a tetragonal distortion may be energetical6 3 eg in ly advantageous. For example, in a d9 complex (with configuration that would be t2g Oh), such a distortion leaves two electrons in the dz2 orbital with a lower energy and one in the dx2y2 orbital with a higher energy. Similar distortions can occur with tetrahedral complexes. The JahnTeller effect identifies an unstable geometry (a nonlinear complex with an orbitally degenerate ground state); it does not predict the preferred distortion. For instance, with an octahedral complex, instead of axial elongation and equatorial compression the degeneracy can also be removed by axial compression and equatorial elongation. Which distortion occurs in practice is a matter of energetics, not symmetry. However, because axial elongation weakens two bonds but equatorial elongation weakens four, axial elongation is more common than axial compression. A JahnTeller distortion can hop from one orientation to another and give rise to the dynamic JahnTeller effect. For example, below 20 K the EPR spectrum of [Cu(OH2)6]2 shows a static distortion (more precisely, one that is effectively stationary on the timescale of the resonance experiment). However, above 20 K the distortion disappears because it hops more rapidly than the timescale of the EPR observation. A JahnTeller effect is possible for other electron configurations (for an octahedral complex the d1, d2, low-spin d4 and d5, high-spin d6, and d7 configurations, for a tetrahedral complex the d1, d3, d4, d6, d8, and d9 configurations). However, as neither the t2g orbitals in an octahedral complex nor any of the d orbitals in a tetrahedral complex point directly at the ligands, the effect is too small to induce a measurable distortion in the structure. (h) Octahedral versus tetrahedral coordination Key points: Consideration of the LFSE predicts that d3 and d8 ions strongly prefer an octahedral geometry over a tetrahedral one; for other configurations the preference is less pronounced, and LFSE has no bearing on the geometry of d0, high-spin d5, and d10 ions. xy yz, zx Octahedral Tetragonal Figure 20.12 The effect of tetragonal distortions (compression along x and y and extension along z) on the energies of d orbitals. The electron occupation is for a d9 complex. An octahedral complex has six ML bonding interactions and, in the absence of significant steric and electronic effects, this arrangement will have a lower energy than a tetrahedral complex with just four ML bonding interactions. We have already discussed the effects of steric bulk on a complex (Section 7.3), and have just seen the electronic reasons that favour a square-planar complex. We can now complete the discussion by considering the electronic effects that favour an octahedral complex over a tetrahedral one. Figure 20.13 illustrates the variation of the LFSE for tetrahedral and high-spin octahedral complexes for all electronic configurations. It is apparent that, in terms of LFSE, octahedral geometries are strongly preferred over tetrahedral for d3 and d8 complexes: chromium(III) (d3) and nickel(II) (d8) do indeed show an exceptional preference for octahedral geometries. Similarly, d4 and d9 configurations show a preference for octahedral complexes (for example Mn(III) and Cu(II); note that the JahnTeller effect enhances this preference), whereas tetrahedral complexes of d1, d2, d6, and d7 ions will not be too disfavoured; thus V(II) (d2) and Co(II) (d7) form tetrahedral complexes (MX42) with halide ligands. The geometry of complexes of ions with d0, d5, and d10 configurations will not be affected by the number of d electrons, as there is no LFSE for these species. Because the size of the d-orbital splitting, and hence the LFSE, depends on the ligand, it follows that a preference for octahedral coordination will be least pronounced for weak-field ligands. With strong-field ligands, low-spin complexes might be preferred and, although the situation is complicated by the pairing energy, the LFSE of a low-spin octahedral complex will be greater than that of a high-spin complex. There will thus be a correspondingly greater preference for octahedral over tetrahedral coordination when the octahedral complex is low-spin. Electronic structure 1.2 1.0 0.8 LFSE/∆ This preference for octahedral over tetrahedral coordination plays an important role in the solid state by influencing the structures that are adopted by d-metal compounds. This influence is demonstrated by the ways in which the different metal ions A and B in spinels (of formula AB2O4, Sections 3.9b and 24.7c) occupy the octahedral or tetrahedral sites. Thus, Co3O4 is a normal spinel because the low-spin d6 Co(III) ion strongly favours octahedral coordination, resulting in (Co2)t(2Co3)oO4, whereas Fe3O4 (magnetite) is an inverse spinel because Fe(II), but not Fe(III), can acquire greater LFSE by occupying an octahedral site. Thus magnetite is formulated as (Fe3)t(Fe2Fe3)oO4. 483 0.6 0.4 (i) The IrvingWilliams series Key point: The IrvingWilliams series summarizes the relative stabilities of complexes formed by M2 ions, and reflects a combination of electrostatic effects and LFSE. 0.2 Figure 20.14 shows log Kf values (Section 7.12) for complexes of the octahedral M2 ions of the 3d series. The variation in formation constants shown there is summarized by the IrvingWilliams series: 0 The order is relatively insensitive to the choice of ligands. In general, the increase in stability correlates with ionic radius, which suggests that the IrvingWilliams series reflects electrostatic effects. However, beyond Mn2 there is a sharp increase in the value of Kf for d6 Fe(II), d7 Co(II), d8 Ni(II), and d9 Cu(II), with strong-field ligands. These ions experience an additional stabilization proportional to the ligand-field stabilization energies (Table 20.2). There is one important exception: the stability of Cu(II) complexes is greater than that of Ni(II) even though Cu(II) has an additional antibonding eg electron. This anomaly is a consequence of the stabilizing influence of the JahnTeller effect, which results in strong binding of four of the ligands in the plane of the tetragonally distorted Cu(II) complex, and that stabilization enhances the value of Kf. 20.2 Ligand-field theory Crystal-field theory provides a simple conceptual model that can be used to interpret magnetic, spectroscopic, and thermochemical data by using empirical values of ∆O. However, the theory is defective because it treats ligands as point charges or dipoles and does not take into account the overlap of ligand and metal atom orbitals. One consequence of this oversimplification is that crystal-field theory cannot account for the ligand spectrochemical series. Ligand-field theory, which is an application of molecular orbital theory that concentrates on the d orbitals of the central metal atom, provides a more substantial framework for understanding the origins of ∆O. The strategy for describing the molecular orbitals of a d-metal complex follows procedures similar to those described in Chapter 2 for bonding in polyatomic molecules: the valence orbitals on the metal and ligand are used to form symmetry-adapted linear combinations (SALCs; Section 6.6), and then estimating the relative energies of the molecular orbitals by using empirical energy and overlap considerations. These relative energies can be verified and positioned more precisely by comparison with experimental data (particularly UV/visible absorption and photoelectron spectroscopy). We shall first consider octahedral complexes, initially taking into account only the metalligand bonding. We then consider the effect of π bonding, and see that it is essential for understanding ∆O (which is one reason why crystal-field theory cannot explain the spectrochemical series). Finally, we consider complexes with different symmetries, and see that similar arguments apply to them. Later in the chapter we shall see how information from optical spectroscopy is used to refine the discussion and provide quantitative data on the ligand-field splitting parameter and electronelectron repulsion energies. (a) Bonding Key point: In ligand-field theory, the building-up principle is used in conjunction with a molecular orbital energy level diagram constructed from metal atom orbitals and symmetry-adapted linear combinations of ligand orbitals. 2 3 4 5 6 7 8 9 10 Number of electrons, n Figure 20.13 The ligand-field stabilization energy for dn complexes in octahedral (high-spin, circles) and tetrahedral (squares) complexes. The LFSE is shown in terms of ∆O, by applying the relationship ∆T 94 ∆O. 10 NH2CH2CO2– 8 log Kf Ba2 Sr2 Ca2 Mg2 Mn2 Fe2 Co2 Ni2 Cu2 Zn2 0 1 6 4 O2CCO2– – 2 0 Ba Sr Ca Mg Mn Fe Co Ni Cu Zn Figure 20.14 The variation of formation constants for the M2 ions of the IrvingWilliams series. 484 20 d-Metal complexes: electronic structure and properties We begin by considering an octahedral complex in which each ligand (L) has a single valence orbital directed towards the central metal atom (M); each of these orbitals has local symmetry with respect to the ML axis. Examples of such ligands include the NH3 molecule and the F ion. In an octahedral (Oh) environment, the orbitals of the central metal atom divide by symmetry into four sets (Fig. 20.15 and Resource section 4): Metal orbital Symmetry label Degeneracy s a1g 1 px, py, pz t1u 3 dx2y2, dz2 eg 2 dxy, dyz, dzx t2g 3 Six symmetry-adapted linear combinations of the six ligand orbitals can also be formed. These combinations can be taken from Resource section 5 and are also shown in Fig. 20.15. One (unnormalized) SALC has symmetry a1g: a1g: 1 2 3 4 5 6 where i denotes a orbital on ligand i. There are three SALCs of symmetry t1u: t1u: 1 3, 2 4, 5 6 and two SALCs of symmetry eg: eg: 1 2 3 4, 2 6 2 5 1 2 3 4 These six SALCs account for all the ligand orbitals of symmetry: there is no combination of ligand orbitals that has the symmetry of the metal t2g orbitals, so the latter do not participate in bonding.2 Molecular orbitals are formed by combining SALCs and metal atomic orbitals of the same symmetry. For example, the (unnormalized) form of an a1g molecular orbital is cMMs cLLa1g, where Ms is the s orbital on the metal atom M and La1g is the ligand SALC of symmetry a1g. The metal s orbital and ligand a1g SALC overlap to give two molecular orbitals, one bonding and one antibonding. Similarly, the doubly degenerate metal eg orbitals and the ligand eg SALCs overlap to give four molecular orbitals (two degenerate bonding, two degenerate antibonding), and the triply degenerate metal t1u orbitals and the three t1u SALCs overlap to give six molecular orbitals (three degenerate bonding, three degenerate antibonding). There are therefore six bonding combinations in all and six antibonding combinations. The three triply degenerate metal t2g orbitals remain nonbonding and fully localized on the metal atom. Calculations of the resulting energies (adjusted to agree with a variety of spectroscopic data of the kind to be discussed in Section 20.4) result in the molecular orbital energy level diagram shown in Fig. 20.16. The greatest contribution to the molecular orbital of lowest energy is from atomic orbitals of lowest energy (Section 2.9). For NH3, F, and most other ligands, the ligand orbitals are derived from atomic orbitals with energies that lie well below those of the metal d orbitals. As a result, the six bonding molecular orbitals of the complex are mainly ligand-orbital in character (that is, cL2 cM2). These six bonding orbitals can accommodate the 12 electrons provided by the six ligand lone pairs. The electrons that we can regard as provided by the ligands are therefore largely confined to the ligands in the complex, just as the crystal-field theory presumes. However, because the coefficients cM are nonzero, the bonding molecular orbitals do have some d-orbital character and the ‘ligand electrons’ are partly delocalized on to the central metal atom. The total number of electrons to accommodate, in addition to those supplied by the ligands, now depends on the number of d electrons, n, supplied by the metal atom. These additional electrons enter the orbitals next in line for occupation, which are the nonbonding d orbitals (the t2g orbitals) and the antibonding combination (the upper eg orbitals) of the d orbitals and ligand orbitals. The t2g orbitals are wholly confined (in the present approximation) to the metal atom and the antibonding eg orbitals are largely metal atom in 1 1 The normalization constants (with overlap neglected) are N(a1g) ( 61 ) 2 , N(t1u) ( 12 ) 2 for all three orbitals, 1 and N(eg) 12 and ( 121 ) 2 , respectively. 2 Electronic structure Metal atom Ligands Metal atom Metal Ligands Complex 485 Ligands t1u 4p 4s a1g t1u a1g a1g eg a1g 3d eg + t2g ∆O t2g eg t1u eg eg eg a1g t1u a1g Figure 20.16 Molecular orbital energy levels of a typical octahedral complex. The frontier orbitals are inside the tinted box. t1u t1u t2g Figure 20.15 Symmetry-adapted combinations of ligand orbitals (represented here by spheres) in an octahedral complex. For symmetry-adapted orbitals in other point groups, see Resource section 5. character too, so the n electrons supplied by the central atom remain largely on that atom. The frontier orbitals of the complex are therefore the nonbonding entirely metal t2g orbitals and the antibonding, mainly metal eg orbitals. Thus, we have arrived at an arrangement that is qualitatively the same as in crystal-field theory. In the ligand-field approach the octahedral ligand-field splitting parameter, ∆O, is the separation between the molecular orbitals largely, but not completely, confined to the metal atom, Fig. 20.16. With the molecular orbital energy level diagram established, we use the building-up principle to construct the ground-state electron configuration of the complex. For a dn complex, there are 12 n electrons to accommodate. The six bonding molecular orbitals accommodate the 12 electrons supplied by the ligands. The remaining n electrons are accommodated in the nonbonding t2g orbitals and the antibonding eg orbitals. Now the story is essentially the same as for crystal-field theory, the types of complexes that are obtained (high-spin or low-spin, for instance) depending on the relative values of ∆O and the pairing energy P. The principal difference from the crystal-field discussion is that ligand-field theory gives deeper insight into the origin of the ligand-field splitting, and we can begin to understand why some ligands are strong and others are weak. For instance, a good -donor ligand should result in strong metalligand overlap, hence a more strongly antibonding eg set and consequently a larger value of ∆O. However, before drawing further conclusions, we must go on to consider what crystal-field theory ignores completely, the role of π bonding. 486 20 d-Metal complexes: electronic structure and properties E X A M PL E 20. 4 Using a photoelectron spectrum to obtain information about a complex Mo–CO The photoelectron spectrum of gas-phase [Mo(CO)6] is shown in Fig. 20.17. Use the spectrum to infer the energies of the molecular orbitals of the complex. Signal intensity CO t2g 8 16 20 12 Ionization energy/eV Figure 20.17 The He(II) (30.4 nm) photoelectron spectrum of Mo(CO)6. With six electrons from Mo and twelve from :CO, the ground-state configuration of the 2 6 4 6 t1u eg t2g . (From B.R. Higginson, complex is a1g D.R. Lloyd, P. Burroughs, D.M. Gibson, and A.F. Orchard, J. Chem. Soc., Faraday II, 1973, 69, 1659.) Signal intensity Mg(Cp)2 Fe(Cp)2 6 10 18 14 Ionization energy/eV Figure 20.18 Photoelectron spectra of ferrocene and magnesocene. dxy(M) y py(L) x Figure 20.19 The π overlap that may occur between a ligand p orbital perpendicular to the ML axis and a metal dxy orbital. Answer We need to identify the electron configuration of the complex, and then match the order of ionization energies to the order of the orbitals from which the electrons are likely to come. Twelve electrons are provided by the six CO ligands (treated as :CO); they enter the bonding orbitals and result in the 2 6 4 configuration a1g t1u eg . The oxidation number of molybdenum, Group 6, is 0, so Mo provides a further six valence electrons. The ligand and metal valence electrons are distributed over the orbitals shown in the box in Fig. 20.15 and, as CO is a strong-field ligand, the ground-state electron configuration of the complex is 2 6 4 6 expected to be low-spin a1g t1u eg t2g . The HOMOs are the three t2g orbitals largely confined to the Mo atom, and their energy can be identified by ascribing the peak of lowest ionization energy (close to 8 eV) to them. The group of ionization energies around 14 eV are probably due to the MoCO -bonding orbitals. The value of 14 eV is close to the ionization energy of CO itself, so the variety of peaks at that energy also arise from bonding orbitals in CO. Self-test 20.4 Suggest an interpretation of the photoelectron spectra of [Fe(C5H5)2] and [Mg(C5H5)2] shown in Fig. 20.18. (b) π Bonding Key points: π-Donor ligands decrease O whereas π-acceptor ligands increase O; the spectrochemical series is largely a consequence of the effects of π bonding when such bonding is feasible. If the ligands in a complex have orbitals with local π symmetry with respect to the ML axis (as two of the p orbitals of a halide ligand have), they may form bonding and antibonding π orbitals with the metal orbitals (Fig. 20.19). For an octahedral complex the combinations that can be formed from the ligand π orbitals include SALCs of t2g symmetry. These ligand combinations have net overlap with the metal t2g orbitals, which are therefore no longer purely nonbonding on the metal atom. Depending on the relative energies of the ligand and metal orbitals, the energies of the now molecular t2g orbitals lie above or below the energies they had as nonbonding atomic orbitals, so ∆O is decreased or increased, respectively. To explore the role of π bonding in more detail, we need two of the general principles described in Chapter 2. First, we shall make use of the idea that, when atomic orbitals overlap strongly, they mix strongly: the resulting bonding molecular orbitals are significantly lower in energy and the antibonding molecular orbitals are significantly higher in energy than the atomic orbitals. Second, we note that atomic orbitals with similar energies interact strongly, whereas those of very different energies mix only slightly even if their overlap is large. A π-donor ligand is a ligand that, before any bonding is considered, has filled orbitals of π symmetry around the ML axis. Such ligands include Cl, Br, OH, O2 and even H2O. In Lewis acidbase terminology (Section 4.9), a π-donor ligand is a π base. The energies of the full π orbitals on the ligands will not normally be higher than their -donor orbitals (HOMO) and must therefore also be lower in energy than the metal d orbitals. Because the full π orbitals of π donor ligands lie lower in energy than the partially filled d orbitals of the metal, when they form molecular orbitals with the metal t2g orbitals, the bonding combination lies lower than the ligand orbitals and the antibonding combination lies above the energy of the d orbitals of the free metal atom (Fig. 20.20). The electrons supplied by the ligand π orbitals occupy and fill the bonding combinations, leaving the electrons originally in the d orbitals of the central metal atom to occupy the antibonding t2g orbitals. The net effect is that the previously nonbonding metal t2g orbitals become antibonding and hence are raised closer in energy to the antibonding eg orbitals. It follows that π-donor ligands decrease ∆O. A π-acceptor ligand is a ligand that has empty π orbitals that are available for occupation. In Lewis acidbase terminology, a π-acceptor ligand is a π acid. Typically, the π-acceptor orbitals are vacant antibonding orbitals on the ligand (usually the LUMO), as in CO and N2, which are higher in energy than the metal d orbitals. The two π orbitals of CO, for instance, have their largest amplitude on the C atom and have the correct symmetry for overlap with the metal t2g orbitals, so CO can act as a π-acceptor ligand Electronic spectra (Section 22.5). Phosphines (PR3) are also able to accept π-electron density and also act as π acceptors (Section 22.6). Because the π-acceptor orbitals on most ligands are higher in energy than the metal d orbitals, they form molecular orbitals in which the bonding t2g combinations are largely of metal d-orbital character (Fig. 20.21). These bonding combinations lie lower in energy than the d orbitals themselves. The net result is that π-acceptors increase ∆O. We can now put the role of π bonding in perspective. The order of ligands in the spectrochemical series is partly that of the strengths with which they can participate in ML bonding. For example, both CH3 and H are very high in the spectrochemical series because they are very strong donors. However, when π bonding is significant, it has a strong influence on ∆O: π-donor ligands decrease ∆O and π-acceptor ligands increase ∆O. This effect is responsible for CO (a strong π acceptor) being high on the spectrochemical series and for OH (a strong π donor) being low in the series. The overall order of the spectrochemical series may be interpreted in broad terms as dominated by π effects (with a few important exceptions), and in general the series can be interpreted as follows: — increasing O → π donor weak π donor no π effects π acceptor Representative ligands that match these classes are π donor I , Br , Cl , F weak π donor no π effects π acceptor H2O NH3 PR3, CO Notable examples of where the effect of bonding dominates include amines (NR3), CH3, and H, none of which has orbitals of π symmetry of an appropriate energy and thus are neither π-donor nor π-acceptor ligands. It is important to note that the classification of a ligand as strong-field or weak-field does not give any guide as to the strength of the M—L bond. M σ only 20.3 Electronic spectra of atoms Key point: Electronelectron repulsions result in multiple absorptions in the electronic spectrum. Figure 20.22 sets the stage for our discussion by showing the electronic absorption spectrum of the d3 complex [Cr(NH3)6]3 in aqueous solution. The band at lowest energy (longest wavelength) is very weak; later we shall see that it is an example of a ‘spin-forbidden’ transition. Next are two bands with intermediate intensities; these are ‘spin-allowed’ transitions between the t2g and eg orbitals of the complex, which are mainly derived from the Ligands eg ∆O t2g π Figure 20.20 The effect of π bonding on the ligand-field splitting parameter. Ligands that act as π donors decrease ∆O. Only the π orbitals of the ligand are shown. M σ only ML6 σ+π Ligands π eg ∆O Electronic spectra Now that we have considered the electronic structure of d-metal complexes, we are in a position to understand their electronic spectra and to use the data they provide to refine the discussion of structure. The magnitudes of ligand-field splittings are such that the energy of electronic transitions corresponds to an absorption of ultraviolet radiation and visible light. However, the presence of electronelectron repulsions within the metal orbitals means that the absorption frequencies are not in general a direct portrayal of the ligand-field splitting. The role of electronelectron repulsion was originally determined by the analysis of atoms and ions in the gas phase, and much of that information can be used in the analysis of the spectra of metal complexes provided we take into account the lower symmetry of a complex. Keep in mind that the purpose of the following development is to find a way to extract the value of the ligand-field splitting parameter from the electronic absorption spectrum of a complex with more than one d electron, when electronelectron repulsions are important. First, we discuss the spectra of free atoms and see how to take electronelectron repulsions into account. Then we see what energy states atoms adopt when they are embedded in an octahedral ligand field. Finally, we see how to represent the energies of these states for various field strengths and electronelectron repulsion energies (in the TanabeSugano diagrams of Section 20.4e), and how to use these diagrams to extract the value of the ligand-field splitting parameter. ML6 σ+π 487 t2g Figure 20.21 Ligands that act as π acceptors increase O. Only the π orbitals of the ligand are shown. 20 d-Metal complexes: electronic structure and properties 4 log(e /(dm3 mol–1 cm–1)) 488 2 2 1 3 t 2geg ← t2g CT 4 4 T1g← A2g 4 4 T2g← A2g 2 4 Eg ← A2g Magnified absorption 0 200 (50 000 cm–1) 400 600 (25 000 cm–1) (17 000 cm–1) l /nm 800 (12 500 cm–1) Figure 20.22 The spectrum of the d3 complex [Cr(NH3)6]3, which illustrates the features studied in this section, and the assignments of the transitions as explained in the text. metal d orbitals. The third feature in the spectrum is an intense charge-transfer band at short wavelength (labelled CT, denoting ‘charge transfer’), of which only the low-energy tail is evident in the illustration. One problem that immediately confronts us is why two absorptions can be ascribed to 2 3 eg1 ← t2g . This splitting of a single transition into two the apparently single transition t2g bands is in fact an outcome of the electronelectron repulsions mentioned above. To understand how it arises, and to extract the information it contains, we need to consider the spectra of free atoms and ions. (a) Spectroscopic terms Key points: Different microstates exist for the same electronic configuration; for light atoms, RussellSaunders coupling is used to describe the terms, which are specified by symbols in which the value of L is indicated by one of the letters S, P, D, . . . , and the value of 2S 1 is given as a left superscript. In Chapter 1 we expressed the electronic structures of atoms by giving their electronic configurations, the designation of the number of electrons in each orbital (as in 1s22s1 for Li). However, a configuration is an incomplete description of the arrangement of electrons in atoms. In the configuration 2p2, for instance, the two electrons might occupy orbitals with different orientations of their orbital angular momenta (that is, with different values of ml from among the possibilities 1, 0, and 1 that are available when l 1). Similarly, the designation 2p2 tells us nothing about the spin orientations of the two electrons ms 12 or 12 . The atom may in fact have several different states of total orbital and spin angular momenta, each one corresponding to the occupation of orbitals with different values of ml by electrons with different values of ms. The different ways in which the electrons can occupy the orbitals specified in the configuration are called the microstates of the configuration. For example, one microstate of a 2p2 configuration is (1,1); this notation signifies that both electrons occupy an orbital with ml 1 but do so with opposite spins, the superscript indicating ms 12 and indicating ms 12 . Another microstate of the same configuration is (1,0). In this microstate, both electrons have ms 12 but one occupies the 2p orbital with ml 1 and the other occupies the orbital with ml 0. The microstates of a given configuration have the same energy only if electronelectron repulsions on the atom are negligible. However, because atoms and most molecules are compact, interelectronic repulsions are strong and cannot always be ignored. As a result, microstates that correspond to different relative spatial distributions of electrons have different energies. If we group together the microstates that have the same energy when electronelectron repulsions are taken into account, we obtain the spectroscopically distinguishable energy levels called terms. For light atoms and the 3d series, it turns out that the most important property of a microstate for helping us to decide its energy is the relative orientation of the spins of the electrons. Electronic spectra Next in importance is the relative orientation of the orbital angular momenta of the electrons. It follows that we can identify the terms of light atoms and put them in order of energy by sorting the microstates according to their total spin quantum number S (which is determined by the relative orientation of the individual spins) and then according to their total orbital angular momentum quantum number L (which is determined by the relative orientation of the individual orbital angular momenta of the electrons). For heavy atoms, such as those of the 4d and 5d series, the relative orientations of orbital momenta or of spin momenta are less important. In these atoms the spin and orbital angular momenta of individual electrons are strongly coupled together by spinorbit coupling, so the relative orientation of the spin and orbital angular momenta of each electron is the most important feature for determining the energy. The terms of heavy atoms are therefore sorted on the basis of the values of the total angular momentum quantum number j for an electron in each microstate. The process of combining electron angular momenta by summing first the spins, then the orbital momenta, and finally combining the two resultants is called RussellSaunders coupling. This coupling scheme is used to identify the terms of light atoms (i.e. the 3d metals), and we consider it in detail here. The coupling scheme most appropriate to heavy atoms (that is, atoms of the 4d and 5d series of elements) is called jj-coupling, but we shall not consider it further. Our first task is to identify the values of L and S that can arise from the orbital and spin angular momenta of individual electrons. Suppose we have two electrons with quantum numbers l1, s1 and l2, s2. Then, according to the ClebschGordan series, the possible values of L and S are L l1 l2, l1 l2 1, … , \|l1 l2\| S s1 s2, s1 s2 1, … , \|s1 s2\| (20.3) (The modulus signs appear because neither L nor S can, by definition, be negative.) For example, an atom with configuration d2 (l1 2, l2 2) can have the following values of L: L 2 2, 2 2 1, … , \|2 2\| 4, 3, 2, 1, 0 The total spin (because s1 S 12 , 1 2 1 2 1 2 1 2 , s2 12 ) can be 1, … , \| 12 12 \| 1, 0 To find the values of L and S for atoms with three electrons, we continue the process by combining l3 with the value of L just obtained, and likewise for s3. Once L and S have been found, we can write down the allowed values of the quantum numbers ML and MS, ML L, L 1, … , L MS S, S 1, … , S These quantum numbers give the orientation of the angular momentum relative to an arbitrary axis: there are 2L 1 values of ML for a given value of L and 2S 1 values of MS for a given value of S. The values of ML and MS for a given microstate can be found very easily by adding together the values of ml or ms for the individual electrons. Therefore, if one electron has the quantum number ml1 and the other has ml2, then ML ml1 ml2 A similar expression applies to the total spin: MS ms1 ms2 Thus, for example, (0,1) is a microstate with ML 0 1 1 and MS 12 ( 12 ) 0 and may contribute to any term for which these two quantum numbers apply. By analogy with the notation s, p, d, … for orbitals with l 0, 1, 2,…, the total orbital angular momentum of an atomic term is denoted by the equivalent uppercase letter: L 0 1 2 3 4 S P D F G then alphabetical (omitting J) The total spin is normally reported as the value of 2S 1, which is called the multiplicity of the term: S 0 1 2 1 3 2 2 2S 1 1 2 3 4 5 489 490 20 d-Metal complexes: electronic structure and properties The multiplicity is written as a left superscript on the letter representing the value of L, and the entire label of a term is called a term symbol. Thus, the term symbol 3P denotes a term (a collection of nearly degenerate states) with L 1 and S 1, and is called a triplet term. E X A M PL E 20. 5 Deriving term symbols Give the term symbols for an atom with the configurations (a) s1, (b) p1, and (c) s1p1. Answer We need to use the ClebschGordan series to couple any angular momenta, identify the letter for the term symbol from the table above, and then attach the multiplicity as a left superscript. (a) The single s electron has l 0 and s 21 . Because there is only one electron, L 0 (an S term), S s 21 , and 2S 1 2 (a doublet term). The term symbol is therefore 2S. (b) For a single p electron, l 1 so L 1 and the term is 2P. (These terms arise in the spectrum of an alkali metal atom, such as Na.) (c) With one s and one p electron, L 0 1 1, a P term. The electrons may be paired (S 0) or parallel (S 1). Hence both 1P and 3P terms are possible. Self-test 20.5 What terms arise from a p1d1 configuration? (b) The classification of microstates Key point: The allowed terms of a configuration are found by identifying the values of L and S to which the microstates of an atom can contribute. The Pauli principle restricts the microstates that can occur in a configuration and consequently affects the terms that can occur. For example, two electrons cannot both have the same spin and be in a d orbital with ml 2. Therefore, the microstate (2,2) is forbidden and so are the values of L and S to which such a microstate might contribute. We shall illustrate how to determine what terms are allowed by considering a d2 configuration, as the outcome will be useful in the discussion of the complexes encountered later in the chapter. An example of a species with a d2 configuration is a Ti2 ion. We start the analysis by setting up a table of microstates of the d2 configuration (Table 20.6); only the microstates allowed by the Pauli principle have been included. We then use a process of elimination to classify all the microstates. First, we note the largest value of ML, which for a d2 configuration is 4. This state must belong to a term with L 4 (a G term). Table 20.6 shows that the only value of MS that occurs for this term is MS 0, so the G term must be a singlet. Moreover, as there are nine values of ML when L 4, one of the microstates in each of the boxes in the column below (2,2) must belong to this term.3 We can therefore strike out one microstate from each row in the central column of Table 20.6, which leaves 36 microstates. Table 20.6 Microstates of the d2 configuration ML MS 0 1 4 (2,2) 3 (2 ,1 ) 2 (2,0) 1 1 (2,1)(2,1) (2,1) (2,0)(2,0)(1,1) (2 ,1 )(1 ,0 ) (2 ,1 )(2 ,1 ) (2,0) (2,1)(1,0) (1,0)(1,0) 0 (1 ,1 )(2 ,2 ) (1,1)(1,1) (1,1)(2,2) (2,2)(2,2) (0,0) 1 to 4 The lower half of the table is a reflection of the upper half. 3 In fact, it is unlikely that one of the microstates itself will correspond to one of these states: in general, a state is a linear combination of microstates. However, as N linear combinations can be formed from N microstates, each time we cross off one microstate, we are taking one linear combination into account, so the bookkeeping is correct even though the detail may be wrong. Electronic spectra The next largest value is ML 3, which must stem from L 3 and hence belong to an F term. That row contains one microstate in each column (that is, each box contains one unassigned combination for MS 1, 0, and 1), which signifies S 1 and therefore a triplet term. Hence the microstates belong to 3F. The same is true for one microstate in each of the rows down to ML 3, which accounts for a further 3 7 21 microstates. If we strike out one state in each of the 21 boxes, we are left with 15 to be assigned. There is one unassigned microstate in the row with ML 2 (which must arise from L 2) and the column under MS 0 (S 0), which must therefore belong to a 1D term. This term has five values of ML, which removes one microstate from each row in the column headed MS 0 down to ML 2, leaving 10 microstates unassigned. Because these unassigned microstates include one with ML 1 and MS 1, nine of these microstates must belong to a 3P term. There now remains only one microstate in the central box of the table, with ML 0 and MS 0. This microstate must be the one and only state of a 1S term (which has L 0 and S 0). At this point we can conclude that the terms of a 3d2 configuration are 1G, 3F, 1D, 3P, and 1S. These terms account for all 45 permitted states (see table in the margin). (c) The energies of the terms Key point: Hund’s rules indicate the likely ground term of a gas-phase atom or ion. Once the values of L and S that can arise from a given configuration are known, it is possible to identify the term of lowest energy by using Hund’s rules. The first of these empirical rules was introduced in Section 1.7, where it was expressed as ‘the lowest energy configuration is achieved if the electron spins are parallel’. Because a high value of S stems from parallel electron spins, an alternative statement is 1. For a given configuration, the term with the greatest multiplicity lies lowest in energy. The rule implies that a triplet term of a configuration (if one is permitted) has a lower energy than a singlet term of the same configuration. For the d2 configuration, this rule predicts that the ground state will be either 3F or 3P. By inspecting spectroscopic data, Hund also identified a second rule for the relative energies of the terms of a given multiplicity: 2. For a term of given multiplicity, the term with the greatest value of L lies lowest in energy. The physical justification for this rule is that when L is high, the electrons can stay clear of one another and hence experience a lower repulsion. If L is low, the electrons are more likely to be closer to each other, and hence repel one another more strongly. The second rule implies that, of the two triplet terms of a d2 configuration, the 3F term is lower in energy than the 3P term. It follows that the ground term of a d2 species such as Ti2 is expected to be 3F. The spin multiplicity rule is fairly reliable for predicting the ordering of terms, but the ‘greatest L’ rule is reliable only for predicting the ground term, the term of lowest energy; there is generally little correlation of L with the order of the higher terms. Thus, for d2 the rules predict the order F 3P 1G 1D 1S 3 but the order observed for Ti2 from spectroscopy is F 1D 3P 1G 1S 3 Normally, all we want to know is the identity of the ground term of an atom or ion. The procedure may then be simplified and summarized as follows: 1. Identify the microstate that has the highest value of MS. This step tells us the highest multiplicity of the configuration. 2. Identify the highest permitted value of ML for that multiplicity. This step tells us the highest value of L consistent with the highest multiplicity. Term Number of states 1 91 9 3 7 3 21 1 51 5 3 33 9 1 11 1 G F D P S Total: 491 45 492 20 d-Metal complexes: electronic structure and properties 2 2 d 1 D d What is the ground term of the configurations (a) 3d5 of Mn2 and (b) 3d3 of Cr3? S 2 F 2 D 2 P H 2 G 4 P 2 1 G 3 P D 1 3 F E X A M PL E 20.6 Identifying the ground term of a configuration 3 4 F Figure 20.23 The relative energies of the terms arising from d2 (left) and d3 (right) configurations of a free atom. Answer First we need to identify the term with maximum multiplicity, as this will be the ground term. Then we need to identify the L value for any terms that have the maximum multiplicity, for the term with the highest L value will be the ground term. (a) Because the d5 configuration permits occupation of each d orbital singly with parallel spins, the maximum value of S is 52 , giving a multiplicity of 2 52 1 6, a sextet term. If each of the electrons is to have the same spin quantum number, all must occupy different orbitals and hence have different ML values. Thus, the ML values of the occupied orbitals will be 2, 1, 0, 1, and 2. This configuration is the only one possible for a sextet term. Because the sum of the ML is 0, it follows that L 0 and the term is 6S. (b) For the configuration d3, the maximum multiplicity corresponds to all three electrons having the same spin quantum number, so S 32 . The multiplicity is therefore 2 32 1 4, a quartet. Again, the three ML values must be different if the electrons are all parallel. There are several possible arrangements that give quartet terms, but the one that gives a maximum value of ML has the three electrons with ML 2, 1 and 0, giving a total of 3, which must arise from a term with L 3, an F term. Hence, the ground term of d3 is 4F. Self-test 20.6 Identify the ground terms of (a) 2p2 and (b) 3d9. (Hint: Because d9 is one electron short of a closed shell with L 0 and S 0, treat it on the same footing as a d1 configuration.) Figure 20.23 shows the relative energies of the terms for the d2 and d3 configurations of free atoms. Later, we shall see how to extend these diagrams to include the effect of a crystal field (Section 20.4). (d) Racah parameters Key points: The Racah parameters summarize the effects of electronelectron repulsion on the energies of the terms that arise from a single configuration; the parameters are the quantitative expression of the ideas underlying Hund’s rules and account for deviations from them. Different terms of a configuration have different energies on account of the repulsion between electrons. To calculate the energies of the terms we must evaluate these electron electron repulsion energies as complicated integrals over the orbitals occupied by the electrons. Mercifully, however, all the integrals for a given configuration can be collected together in three specific combinations and the repulsion energy of any term of a configuration can be expressed as a sum of these three quantities. The three combinations of integrals are called the Racah parameters and denoted A, B, and C. The parameter A corresponds to an average of the total interelectron repulsion, and B and C relate to the repulsion energies between individual d electrons. We do not even need to know the theoretical values of the parameters or the theoretical expressions for them because it is more reliable to use A, B, and C as empirical quantities obtained from gas-phase atomic spectroscopy. Each term stemming from a given configuration has an energy that may be expressed as a linear combination of all three Racah parameters. For a d2 configuration a detailed analysis shows that E(1S) A 14B 7C E(1G) A 4B 2C E(3P) A 7B E(3F) A 8B E(1D) A 3B 2C The values of A, B, and C can be determined by fitting these expressions to the observed energies of the terms. Note that A is common to all the terms (as remarked above it is the average of the total interelectron repulsion energy); therefore, if we are interested only in their relative energies, we do not need to know its value. All three Racah parameters are positive as they represent electronelectron repulsions. Therefore, provided C 5B, the energies of the terms of the d2 configuration lie in the order 3 F 3P 1D 1G 1S This order is nearly the same as obtained by using Hund’s rules. However, if C 5B, the advantage of having an occupation of orbitals that corresponds to a high orbital angular momentum is greater than the advantage of having a high multiplicity, and the 3P term lies above 1D (as is in fact the case for Ti2). Table 20.7 shows some experimental values of Electronic spectra 493 Table 20.7 Racah parameters for some d-block ions 1 2 Ti 720 (3.7) V 765 (3.9) 3 860 (4.8) Cr 830 (4.1) 1030 (3.7) Mn 960 (3.5) 1130 (3.2) Fe 1060 (4.1) Co 1120 (3.9) Ni Cu 4 1040 (4.1) 1080 (4.5) 1220 (4.0) 1240 (3.8) The table gives the B parameter in cm1 with the value of C/B in parentheses. B and C. The values in parentheses indicate that C ≈ 4B so the ions listed there are in the region where Hund’s rules are not reliable for predicting anything more than the ground term of a configuration. The parameter C appears only in the expressions for the energies of states that differ in multiplicity from the ground state. Hence if, as is usual, we are interested only in the relative energies of terms of the same multiplicity as the ground state (that is excitation without a change in spin state), we do not need to know the value of C. The parameter B is of the most interest, and we return to factors that affect its value in Section 20.4f. 20.4 Electronic spectra of complexes The preceding discussion related only to free atoms, and we will now expand our discussion to encompass complex ions. The spectrum of [Cr(NH3)6]3 in Fig. 20.22 has two central bands with intermediate intensities and with energies that differ on account of the electronelectron repulsions (as we explain soon). Because both the transitions are between orbitals that are predominantly metal d orbital in character, with a separation characterized by the strength of the ligand-field splitting parameter ∆O, these two transitions are called dd transitions or ligand-field transitions. (a) Ligand-field transitions Key point: Electronelectron repulsion splits ligand-field transitions into components with different energies. According to the discussion in Section 20.1, we expect the octahedral d3 complex 3 . The absorption near 25 000 cm1 [Cr(NH3)6]3 to have the ground-state configuration t2g 2 1 3 can be identified as arising from the excitation t2g eg ← t2g because the corresponding energy (close to 3 eV) is typical of ligand-field splittings in complexes. Before we embark on a Racah-like analysis of the transition, it will be helpful to see qualitatively from the viewpoint of molecular orbital theory why the transition gives rise to two bands. First, note that a dz2 ← dxy transition, which is one way of achieving eg ← t2g, promotes an electron from the xy-plane into the already electron-rich z-direction: that axis is electron-rich because both dyz and dzx are occupied (Fig. 20.24). However, a dz2 ← dzx transition, which is another way of achieving eg ← t2g, merely relocates an electron that is already largely concentrated along the z-axis. In the former case, but not in the latter, there is a distinct increase in electron repulsion and, as a result, the two eg ← t2g transitions lie at different 2 1 3 eg ← t2g transitions, and all resemble one or other of these energies. There are six possible t2g two cases: three of them fall into one group and the other three fall into the second group. (b) The spectroscopic terms Key points: The terms of an octahedral complex are labelled by the symmetry species of the overall orbital state; a superscript prefix shows the multiplicity of the term. The two bands we are discussing in Fig. 20.22 are labelled 4T2g ← 4A2g (at 21 550 cm1) and 4T1g ← 4A2g (at 28 500 cm1). The labels are molecular term symbols and serve a (a) (b) Figure 20.24 The shifts in electron density that accompany the two transitions discussed in the text. There is a considerable relocation of electron density towards the ligands on the z-axis in (a), but a much less substantial relocation in (b). 494 20 d-Metal complexes: electronic structure and properties C4 (a) C4 (b) Figure 20.25 The changes in sign that occur under C4 rotations about the z-axis: (a) a dxy orbital is rotated into the negative of itself; (b) a dyz orbital is rotated into a dzx orbital. purpose similar to that of atomic term symbols. The left superscript denotes the multiplicity, so the superscript 4 denotes a quartet state with S 32 , as expected when there are three unpaired electrons. The rest of the term symbol is the symmetry label of the overall electronic orbital state of the complex. For example, the nearly totally symmetrical ground state of a d3 complex (with an electron in each of the three t2g orbitals) is denoted A2g. We say nearly totally symmetrical because close inspection of the behaviour of the three occupied t2g orbitals shows that the C3 rotation of the Oh point group transforms the product t2g t2g t2g into itself, which identifies the complex as an A symmetry species (see the character table in Resource section 4). Moreover, because each orbital has even parity (g), the overall parity is also g. However, each C4 rotation transforms one t2g orbital into the negative of itself and the other two t2g orbitals into each other (Fig. 20.25), so overall there is a change of sign under this operation and its character is 1. The term is therefore A2g rather than the totally symmetrical A1g of a closed shell. It is more difficult to establish that the term symbols that can arise from the quartet 2 1 eg excited configuration are 4T2g and 4T1g, and we shall not consider this aspect here. The t2g superscript 4 implies that the upper configuration continues to have the same number of unpaired spins as in the ground state, and the subscript g stems from the even parity of all the contributing orbitals. (c) Correlating the terms Key point: In the ligand field of an octahedral complex the free atom terms split and are then labelled by their symmetry species as enumerated in Table 20.8. Table 20.8 The correlation of spectroscopic terms for d electrons in Oh complexes Atomic term Number of states Terms in Oh symmetry S 1 A1g P 3 T1g D 5 T2g Eg F 7 T1g T2g A2g G 9 A1g Eg T1g T2g In a free atom, where all five d orbitals in a shell are degenerate, we needed to consider only the electronelectron repulsions to arrive at the relative ordering of the terms of a given dn configuration. In a complex, the d orbitals are not all degenerate and it is necessary to take into account the difference in energy between the t2g and eg orbitals as well as the electronelectron repulsions. Consider the simplest case of an atom or ion with a single valence electron. Because a totally symmetrical orbital in one environment becomes a totally symmetrical orbital in another environment, an s orbital in a free atom becomes an a1g orbital in an octahedral field. We express the change by saying that the s orbital of the atom ‘correlates’ with the a1g orbital of the complex. Similarly, the five d orbitals of a free atom correlate with the triply degenerate t2g and doubly degenerate eg sets in an octahedral complex. Now consider a many-electron atom. In exactly the same way as for a single electron, the totally symmetrical overall S term of a many-electron atom correlates with the totally symmetrical A1g term of an octahedral complex. Likewise, an atomic D term splits into a T2g term and an Eg term in Oh symmetry. The same kind of analysis can be applied to other states, and Table 20.8 summarizes the correlations between free atom terms and terms in an octahedral complex. E X A M PL E 20.7 Identifying correlations between terms What terms in a complex with Oh symmetry correlate with the 3P term of a free atom with a d2 configuration? Answer We argue by analogy: if we know how p orbitals correlate with orbitals in a complex, then we can use that information to express how the overall states correlate, simply by changing to uppercase letters. The three p orbitals of a free atom become the triply degenerate t1u orbitals of an octahedral complex. Therefore, if we disregard parity for the moment, a P term of a many-electron atom becomes a T1 term in the point group Oh. Because d orbitals have even parity, the term overall must be g, and specifically T1g. The multiplicity is unchanged in the correlation, so the 3P term becomes a 3T1g term. Self-test 20.7 What terms in a d2 complex of Oh symmetry correlate with the 3F and 1D terms of a free atom? (d) The energies of the terms: weak- and strong-field limits Key points: For a given metal ion, the energies of the individual terms respond differently to ligands of increasing field strength, and the correlation between free atom terms and terms of a complex can be displayed on an Orgel diagram. Electronelectron repulsions are difficult to take into account, but the discussion is simplified by considering two extreme cases. In the weak-field limit the ligand field, as 495 Electronic spectra 2 1 1 t2g eg eg2 t2g In an octahedral field, these configurations have different energies; that is, as we noted earlier, the 3F term splits into three terms. From the information in Fig. 20.2, we can write their energies as 2 5 2 5 Energy 2 D 2 T2g Ligand field strength→ Figure 20.26 Correlation diagram for a free ion (left) and the strong-field terms (right) of a d1 configuration. 3 A2g 3 3 P 15B T1g ∆O 1 1 e g t 2g 3 0 2∆O 2 eg T2g 3 2 t 2g E(t e ) ( )∆O 0.2∆O 1 1 2g g E2g F E(t ) 2( ∆O) 0.8∆O 2 2g 2 Energy measured by ∆O, is so weak that only electronelectron repulsions are important. As the Racah parameters B and C fully describe the interelectron repulsions, these are the only parameters we need at this limit. In the strong-field limit the ligand field is so strong that electronelectron repulsions can be ignored and the energies of the terms can be expressed solely in terms of ∆O. Then, with the two extremes established, we can consider intermediate cases by drawing a correlation diagram between the two. We shall illustrate what is involved by considering two simple cases, namely, d1 and d2. Then we show how the same ideas are used to treat more complicated cases. The only term arising from the d1 configuration of a free atom is 2D. In an octahedral 1 , which gives rise to a 2T2g term, or eg1, which gives complex the configuration is either t2g 2 rise to a Eg term. Because there is only one electron, there are no electronelectron repulsions to worry about, and the separation of the 2T2g and 2Eg terms is the same as the separation of the t2g and eg orbitals, which is ∆O. The correlation diagram for the d1 configuration will therefore resemble that shown in Fig. 20.26. We saw earlier that for a d2 configuration the lowest energy term in the free atom is the triplet 3F. We need consider only electronic transitions that start from the ground state, and, in this section, will discuss only those in which there is no change in spin. There is an additional triplet term (3P); relative to the lower term (3F), the energies of the terms are E(3F) 0 and E(3P) 15B. These two energies are marked on the left of Fig. 20.27. Now consider the very strong field limit. A d2 atom has the configurations 3 5 3 0 T1g Ligand field strength→ E(eg2) 2( 35 )∆O 1.2∆O Therefore, relative to the energy of the lowest term, their energies are 2 , T1g) 0 E(t2g 1 1 E(t2g eg , T2g) ∆O E(eg2, A2g) 2∆O These energies are marked on the right in Fig. 20.27. Our problem now is to account for the energies when neither the ligand-field nor the electron repulsion terms is dominant. To do so, we correlate the terms in the two extreme 2 configuration gives rise to a 3T1g term, and this correlates with the 3F cases. The triplet t2g term of the free atom. The remaining correlations can be established similarly, and we 1 1 eg configuration gives rise to a T2g term and that the eg2 configuration gives see that the t2g rise to a A2g term; both terms correlate with the 3F term of the free atom. Note that some terms, such as the 3T1g term that correlates with 3P, are independent of the ligand-field strength. All the correlations are shown in Fig. 20.27, which is a simplified version of an Orgel diagram. An Orgel diagram can be constructed for any d-electron configuration, and several electronic configurations can be combined on the same diagram. Orgel diagrams are of considerable value for simple discussions of the electronic spectra of complexes; however, they consider only some of the possible transitions (the spin-allowed transitions, which is why we considered only the triplet terms) and cannot be used to extract a value for the ligand-field splitting parameter, ∆O. (e) TanabeSugano diagrams Key point: TanabeSugano diagrams are correlation diagrams that depict the energies of electronic states of complexes as a function of the strength of the ligand field. Diagrams showing the correlation of all terms can be constructed for any electron configuration and strength of ligand field. The most widely used versions are called TanabeSugano diagrams, after the scientists who devised them. Figure 20.28 shows the diagram for d2 and we can see splittings for all the atomic terms that split; thus the 3F splits into three, the 1 D into two, and the 1G into four. In these diagrams the term energies, E, are expressed as E/B and plotted against ∆O/B, where B is the Racah parameter. The relative energies of the Figure 20.27 Correlation diagram for a free ion (left) and the strong-field terms (right) of a d2 configuration. 496 20 d-Metal complexes: electronic structure and properties 1 3 1 A1 70 E A2 1 60 T1 1 1 S T2 E /B 50 40 3 T1 3 T2 1 A1 30 1 G 20 3 P 1 D 10 1 E 1 T2 3 3 F 0 10 20 ∆O/B T1 30 Figure 20.28 The TanabeSugano diagram for the d2 configuration. Note that the left-hand axis corresponds to Fig. 20.23 (left). A complete collection of diagrams for dn configurations is given in Resource section 6. The parity subscript g has been omitted from the term symbols for clarity. terms arising from a given configuration are independent of A, and by choosing a value of C (typically setting C ≈ 4B), terms of all energies can be plotted on the same diagrams. Some lines in TanabeSugano diagrams are curved because of the mixing of terms of the same symmetry type. Terms of the same symmetry obey the noncrossing rule, which states that, if the increasing ligand field causes two weak-field terms of the same symmetry to approach, then they do not cross but bend apart from each other (Fig. 20.29). The effect of the noncrossing rule can be seen for the two 1E terms, the two 1T2 terms, and the two 1 A1 terms in Fig. 20.28. TanabeSugano diagrams for Oh complexes with configurations d2 to d8 are given in Resource section 6. The zero of energy in a TanabeSugano diagram is always taken as that of the lowest term. Hence the lines in the diagrams have abrupt changes of slope when there is a change in the identity of the ground term brought about by the change from high-spin to low-spin with increasing field strength (see the diagram for d4, for instance). The purpose of the preceding discussion has been to find a way to extract the value of the ligand-field splitting parameter from the electronic absorption spectrum of a complex with more than one d electron, when electronelectron repulsions are important, such as that in Fig. 20.22. The strategy involves fitting the observed transitions to the correlation lines in a TanabeSugano diagram and identifying the values of ∆O and B where the observed transition energies match the pattern. This procedure is illustrated in Example 20.8. As we shall see in Section 20.6, certain transitions are allowed and certain transitions are forbidden. In particular, those that correspond to a change of spin state are forbidden, whereas those that do not are allowed. In general, spin-allowed transitions will dominate the UV/visible absorption spectrum, thus we might only expect to see three transitions for a d2 ion, specifically 3T2g ← 3T1g, 3T1g ← 3T1g, and 3A2g ← 3T1g. However, some complexes (for instance high-spin d5 ions such as Mn2) do not have any spin-allowed transitions and none of the 11 possible transitions dominates. E X A M PL E 20. 8 Calculating O and B using a Tanabe–Sugano diagram Deduce the values of ∆O and B for [Cr(NH3)6]3 from the spectrum in Fig. 20.22 and a TanabeSugano diagram. En y rg e Sta 2 te Sta 1 te t e rm ip g n a h C Figure 20.29 The noncrossing rule states that if two states of the same symmetry are likely to cross as a parameter is changed (as shown by the blue lines), they will in fact mix together and avoid the crossing (as shown by the brown lines). Answer We need to identify the relevant TanabeSugano diagram and then locate the position on the diagram where the observed ratio of transition energies (as wavenumbers) matches the theoretical ratio. The relevant diagram (for d3) is shown in Fig. 20.30. We need concern ourselves only with the spin-allowed transitions, of which there are three for a d3 ion (a 4T2g ← 4A2g and two 4T1g ← 4A2g transitions). We have seen that the spectrum in Fig. 20.22 exhibits two low-energy ligand-field transitions at 21 550 and 28 500 cm1, which correspond to the two lowest energy transitions (4T2g ← 4A2g and 4T1g ← 4A2g). The ratio of the energies of the transitions is 1.32, and the only point in Fig. 20.30 where this energy ratio is satisfied is on the far right. Hence, we can read off the value of ∆O /B 33.0 from the location of this point. The tip of the arrow representing the lower energy transition lies at 32.8B vertically, so equating 32.8B and 21 550 cm1 gives B 657 cm1 and therefore ∆O 21 700 cm1. Self-test 20.8 Use the same TanabeSugano diagram to predict the energy of the first two spin-allowed quartet bands in the spectrum of [Cr(OH2)6]3 for which ∆O 17 600 cm1 and B 700 cm1. A TanabeSugano diagram also provides some understanding of the widths of some absorption lines. Consider Fig. 20.28 and the 3T2g ← 3T1g and 3A2g ← 3T1g transitions. The line representing 3T2g is not parallel to that representing the ground state 3T1g, and so any variation in the value of ∆O (such as those caused by molecular vibration) results in a change in the energy of the transition and thus a broadening of the absorption band. The line representing 3 A2g is even less parallel to the line representing 3T1g; the energy of this transition will be affected even more by variations in ∆O and will thus be even broader. By contrast, the line representing the lower 1T2g term is almost parallel to that representing 3T1g and thus the energy of this transition is largely unaffected by variations in ∆O and the absorption is consequently very sharp (albeit weak, because it is forbidden). (f) The nephelauxetic series Key points: Electronelectron repulsions are lower in complexes than in free ions because of electron delocalization; the nephelauxetic parameter is a measure of the extent of d-electron delocalization on to the ligands of a complex; the softer the ligand, the smaller the nephelauxetic parameter. 497 Electronic spectra A small value of indicates a large measure of d-electron delocalization on to the ligands and hence a significant covalent character in the complex. Thus the series shows that a Br ligand results in a greater reduction in electron repulsions in the ion than an F ion, which is consistent with a greater covalent character in bromido complexes than in analogous fluorido complexes. As an example, compare [NiF6]4, for which B 843 cm1, with [NiBr4]2, for which B 600 cm1. Another way of expressing the trend represented by the nephelauxetic series is: the softer the ligand, the smaller the nephelauxetic parameter. 20.5 Charge-transfer bands Key points: Charge-transfer bands arise from the movement of electrons between orbitals that are predominantly ligand in character and orbitals that are predominantly metal in character; such transitions are identified by their high intensity and the sensitivity of their energies to solvent polarity. Another feature in the spectrum of [Cr(NH3)6]3 in Fig. 20.22 that remains to be explained is the very intense shoulder of an absorption that appears to have a maximum at well above 50 000 cm1. The high intensity suggests that this transition is not a simple ligandfield transition, but is consistent with a charge-transfer transition (CT transition). In a CT transition, an electron migrates between orbitals that are predominantly ligand in character and orbitals that are predominantly metal in character. The transition is classified as a ligand-to-metal charge-transfer transition (LMCT transition) if the migration of the electron is from the ligand to the metal, and as a metal-to-ligand charge-transfer transition (MLCT transition) if the charge migration occurs in the opposite direction. An example of an MLCT transition is the one responsible for the red colour of tris(bipyridyl)iron(II), the complex used for the colorimetric analysis of Fe(II). In this case, an electron makes a transition from a d orbital of the central metal into a π orbital of the ligand. Figure 20.31 summarizes the transitions we classify as charge transfer. Several lines of evidence are used to identify a band as due to a CT transition. The high intensity of the band, which is evident in Fig. 20.22, is one strong indication. Another indication is if such a band appears following the replacement of one ligand with another, as this implies that the band is strongly dependent on the ligand. The CT character is most often identified (and distinguished from π ← π transitions on ligands) by demonstrating solvatochromism, the variation of the transition frequency with changes in solvent permittivity. Solvatochromism indicates that there is a large shift in electron density as a result of the transition, which is more consistent with a metalligand transition than a ligandligand or metalmetal transition. Figure 20.32 shows another example of a CT transition in the visible and UV spectrum of [CrCl(NH3)5]2 (5). If we compare this spectrum with that of [Cr(NH3)6]3 in Fig. 20.22, then we can recognize the two ligand-field bands in the visible region. The replacement of one NH3 ligand by a weaker field Cl ligand moves the lowest energy ligand-field bands to lower energy than those of [Cr(NH3)6]3. Also, a shoulder appears on the high-energy side of one of the ligand-field bands, indicating an additional transition that is the result of the reduction in symmetry from Oh to C4v. The major new feature in the spectrum is the strong absorption maximum in the ultraviolet, near 42 000 cm1. This band is at lower energy than the corresponding band in the spectrum of [Cr(NH3)6]3 and is due to an 4 The name is from the Greek words for ‘cloud expanding’. 4 T1 40 4 T2 2 F 2 30 2 20 2 G T2 T1 2 E 4 P 10 4 A2 4 F 10 20 ∆O/B 30 Figure 20.30 The TanabeSugano diagram for the d3 configuration. Note that the left-hand axis corresponds to Fig. 20.22 (left). A complete collection of diagrams for dn configurations is given in Resource section 6. The parity subscript g has been omitted from the term symbols for clarity. L π MLCT 50 A1 eg MLCT 2 t 2g Md LMCT Br CN , Cl NH3 H2O F 2 D 2 P 2 H T1 60 LMCT The values of depend on the identity of the metal ion and the ligand, and a list of ligands ordered by the value of gives the nephelauxetic series: 4 2 D LMCT (20.4) A2 LMCT B(complex)/B(free ion) 2 70 E/B In Example 20.8 we found that B 657 cm1 for [Cr(NH3)6]3, which is only 64 per cent of the value for a Cr3 ion in the gas phase. This reduction is a general observation and indicates that electron repulsions are weaker in complexes than in the free atoms and ions. The weakening occurs because the occupied molecular orbitals are delocalized over the ligands and away from the metal. The delocalization increases the average separation of the electrons and hence reduces their mutual repulsion. The reduction of B from its free ion value is normally reported in terms of the nephelauxetic parameter, :4 Lπ Lσ Figure 20.31 A summary of the chargetransfer transitions in an octahedral complex. 498 20 d-Metal complexes: electronic structure and properties LMCT transition from the Cl ligand to the metal. The LMCT character of similar bands in [CoX(NH3)5]2 is confirmed by the decrease in energy in steps of about 8000 cm1 as X is varied from Cl to Br to I. In this LMCT transition, a lone-pair electron of the halide ligand is promoted into a predominantly metal orbital. 4 Key points: Ligand-to-metal charge-transfer transitions are observed in the visible region of the spectrum when the metal is in a high oxidation state and ligands contain nonbonding electrons; the variation in the position of LMCT bands can be parameterized in terms of optical electronegativities. d–d (17 000 cm–1) 200 d–d (25 000 cm–1) 0 (a) LMCT transitions (50 000 cm–1) log(e /(dm3 mol–1 cm–1)) LMCT 400 l /nm 600 Figure 20.32 The absorption spectrum of [CrCl(NH3)5]2 in water in the visible and ultraviolet regions. The peak corresponding to the transition 2E ← 4A is not visible on this magnification. 2+ NH3 Cr Cl 5 [CrCl(NH3)5]2+ Charge-transfer bands in the visible region of the spectrum (and hence contributing to the intense colours of many complexes) may occur if the ligands have lone pairs of relatively high energy (as in sulfur and selenium) or if the metal atom has low-lying empty orbitals. The tetraoxidoanions of metals with high oxidation numbers (such as MnO4) provide what are probably the most familiar examples of LMCT bands. In these, an O lone-pair electron is promoted into a low-lying empty metal e orbital. High metal oxidation numbers correspond to a low d-orbital population (many are formally d0), so the acceptor level is available and low in energy. The trend in LMCT energies is: Oxidation number 7 MnO4 TcO4 ReO4 6 CrO42 MoO42 WO42 5 VO43 NbO43 TaO43 The UV/visible spectra of the tetraoxido anions of the Group 6 metals, CrO42, MoO42, and WO42 are shown in Fig. 20.33. The energies of the transitions correlate with the order of the electrochemical series, with the lowest energy transitions taking place to the most easily reduced metal ions. This correlation is consistent with the transition being the transfer of an electron from the ligands to the metal ion, corresponding, in effect, to the reduction of the metal ion by the ligands. Polymeric and monomeric oxidoanions follow the same trends, with the oxidation state of the metal the determining factor. The similarity suggests that these LMCT transitions are localized processes that take place on discrete molecular fragments. The variation in the position of LMCT bands can be expressed in terms of the optical electronegativities of the metal, metal, and the ligands, ligand. The wavenumber of the transition is then written as the difference between the two electronegativities: Absorbance = ligand − metal o 2– MoO4 2– WO4 2– CrO4 Optical electronegativities have values comparable to Pauling electronegativities (Table 1.7) if we set ␯ o 3.0 104 cm1 (Table 20.9), and can be used in a similar fashion. If the LMCT transition terminates in an eg orbital, ∆O must be added to the energy predicted by this equation. Electron pairing energies must also be taken into account if the transition results in the population of an orbital that already contains an electron. The values for metals are different in complexes of different symmetry, and the ligand values are different if the transition originates from a π orbital rather than a orbital. Table 20.9 Optical electronegativities 250 350 l /nm 450 Figure 20.33 Optical absorption spectra of the ions CrO42, WO42, and MoO42. On descending the group, the absorption maximum moves to shorter wavelengths, indicating an increase in the energy of the LMCT band. Ligand 1.81.9 F 3.9 4.4 2.3 Cl 3.0 3.4 2.8 3.3 2.5 3.0 Metal Oh Cr(III) Co(III) Td Ni(II) 2.02.1 Br Co(II) 1.81.9 I Rh(III) 2.3 H 2O 3.5 Mo(VI) 2.1 NH3 3.3 Low-spin complexes. Electronic spectra N (b) MLCT transitions 499 N Key point: Charge-transfer transitions from metal to ligand are observed when the metal is in a low oxidation state and the ligands have low-lying acceptor orbitals. Charge-transfer transitions from metal to ligand are most commonly observed in complexes with ligands that have low-lying π orbitals, especially aromatic ligands. The transition occurs at low energy and appears in the visible spectrum if the metal ion is in a low oxidation state, as its d orbitals are then relatively close in energy to the empty ligand orbitals. The family of ligands most commonly involved in MLCT transitions are the diimines, which have two N donor atoms: two important examples are 2,2′-bipyridine (bpy, 6) and 1,10-phenanthroline (phen, 7). Complexes of diimines with strong MLCT bands include tris(diimine) species such as tris(2,2′-bipyridyl)ruthenium(II) (8), which is orange. A diimine ligand may also be easily substituted into a complex with other ligands that favour a low oxidation state. Two examples are [W(CO)4(phen)] and [Fe(CO)3(bpy)]. However, the occurrence of MLCT transitions is by no means limited to diimine ligands. Another important ligand type that shows typical MLCT transitions is dithiolene, S2C2R22 (9). Resonance Raman spectroscopy (Section 8.4) is a powerful technique for the study of MLCT transitions. The MLCT excitation of tris(2,2′-bipyridyl)ruthenium(II) has been the subject of intense research efforts because the excited state that results from the charge transfer has a lifetime of microseconds, and the complex is a versatile photochemical redox reagent. The photochemical behaviour of a number of related complexes has also been studied on account of their relatively long excited-state lifetimes. 6 2,2’-Bipyridine (bpy) N N 7 1,10-Phenanthroline (phen) 2+ N N Ru N N N N 8 Tris(2,2’-bipyridyl)ruthenium(II) 20.6 Selection rules and intensities S – S– Key point: The strength of an electronic transition is determined by the transition dipole moment. The contrast in intensity between typical charge-transfer bands and typical ligand-field bands raises the question of the factors that control the intensities of absorption bands. In an octahedral, nearly octahedral, or square-planar complex, the maximum molar absorption coefficient εmax (which measures the strength of the absorption)5 is typically less than or close to 100 dm3 mol1 cm1 for ligand-field transitions. In tetrahedral complexes, which have no centre of symmetry, εmax for ligand-field transitions might exceed 250 dm3 mol1 cm1. By contrast, charge-transfer bands usually have an εmax in the range 100050 000 dm3 mol1 cm1. To understand the intensities of transitions in complexes we have to explore the strength with which the complex couples with the electromagnetic field. Intense transitions indicate strong coupling; weak transitions indicate feeble coupling. The strength of coupling when an electron makes a transition from a state with wavefunction i to one with wavefunction f is measured by the transition dipole moment, which is defined as the integral ␮fi ∫f ␮i dτ (20.5) where ␮ is the electric dipole moment operator, er. The transition dipole moment can be regarded as a measure of the impulse that a transition imparts to the electromagnetic field: a large impulse corresponds to an intense transition; zero impulse corresponds to a forbidden transition. The intensity of a transition is proportional to the square of its transition dipole moment. A spectroscopic selection rule is a statement about which transitions are allowed and which are forbidden. An allowed transition is a transition with a nonzero transition dipole moment, and hence nonzero intensity. A forbidden transition is a transition for which the transition dipole moment is calculated as zero. Formally forbidden transitions may occur in a spectrum if the assumptions on which the transition dipole moment were calculated are invalid, such as the complex having a lower symmetry than assumed. 5 The molar absorption coefficient is the constant in the Beer–Lambert law for the transmittance T If /Ii when light passes through a length L of solution of molar concentration [X] and is attenuated from an intensity Ii to an intensity If: log T ε[X]L (the logarithm is a common logarithm, to the base 10). Its older but still widely used name is the ‘extinction coefficient’. R R 9 Dithiolene 500 20 d-Metal complexes: electronic structure and properties (a) Spin selection rules Key points: Electronic transitions with a change of multiplicity are forbidden; intensities of spin-forbidden transitions are greater for 4d- or 5d-series metal complexes than for comparable 3d-series complexes. (a) (b) Figure 20.34 (a) A spin-allowed transition does not change the multiplicity. (b) A spinforbidden transition results in a change in the multiplicity. The electromagnetic field of the incident radiation cannot change the relative orientations of the spins of the electrons in a complex. For example, an initially antiparallel pair of electrons cannot be converted to a parallel pair, so a singlet (S 0) cannot undergo a transition to a triplet (S 1). This restriction is summarized by the rule ∆S 0 for spinallowed transitions, Fig. 20.34. The coupling of spin and orbital angular momenta can relax the spin selection rule, but such spin-forbidden, ∆S ≠ 0, transitions are generally much weaker than spin-allowed transitions. The intensity of spin-forbidden bands increases as the atomic number increases because the strength of the spinorbit coupling is greater for heavy atoms than for light atoms. The breakdown of the spin selection rule by spinorbit coupling is often called the heavy-atom effect. In the 3d series, in which spinorbit coupling is weak, spin-forbidden bands have εmax less than about 1 dm3 mol1 cm1; however, spin-forbidden bands are a significant feature in the spectra of heavy d-metal complexes. The very weak transition labelled 2Eg ← 4A2g in Fig. 20.22 is an example of a spin-forbidden transition. Some metal ions, such as the d5 Mn2 ion, have no spin-allowed transitions, and hence are only weakly coloured. (b) The Laporte selection rule Key points: Transitions between d orbitals are forbidden in octahedral complexes; asymmetric vibrations relax this restriction. The Laporte selection rule states that in a centrosymmetric molecule or ion, the only allowed transitions are those accompanied by a change in parity. That is, transitions between g and u terms are permitted, but a g term cannot undergo a transition to another g term and a u term cannot undergo a transition to another u term: g↔u g↔g u↔u In many cases it is enough to note that in a centrosymmetric complex, if there is no change in quantum number l, then there can be no change in parity. Thus, ss, pp, dd , and ff transitions are forbidden. Since s and d orbitals are g, whereas p and f orbitals are u it follows that sp, pd, and df transitions are allowed whereas sd and pf transitions are forbidden. A more formal treatment of the Laporte selection rule is based on the properties of the transition dipole moment, which is proportional to r. Because r changes sign under inversion (and is therefore u), the entire integral in eqn 20.5 also changes sign under inversion if i and f have the same parity because g u g u and u u u u. Therefore, because the value of an integral cannot depend on the choice of coordinates used to evaluate it,6 it vanishes if i and f have the same parity. However, if they have opposite parity, the integral does not change sign under inversion of the coordinates because g u u g and therefore need not vanish. In a centrosymmetric complex, dd ligand-field transitions are g ↔ g and are therefore forbidden. Their forbidden character accounts for the relative weakness of these transitions in octahedral complexes (which are centrosymmetric) compared with those in tetrahedral complexes, on which the Laporte rule is silent (they are noncentrosymmetric, and have no g or u as a subscript). The question remains why dd ligand-field transitions in octahedral complexes occur at all, even weakly. The Laporte selection rule may be relaxed in two ways. First, a complex may depart slightly from perfect centrosymmetry in its ground state, perhaps on account of the intrinsic asymmetry in the structure of polyatomic ligands or a distortion imposed by the environment of a complex packed into a crystal. Alternatively, the complex might undergo an asymmetrical vibration, which also destroys its centre of inversion. In either case, a Laporte-forbidden dd ligand-field band tends to be much more intense than a spin-forbidden transition. 6 An integral is an area, and areas are independent of the coordinates used for their evaluation. Electronic spectra Table 20.10 summarizes typical intensities of electronic transitions of complexes of the 3d-series elements. The width of spectroscopic absorption bands is due principally to the simultaneous excitation of vibration when the electron is promoted from one distribution to another. According to the FranckCondon principle, the electronic transition takes place within a stationary nuclear framework. As a result, after the transition has occurred, the nuclei experience a new force field and the molecule begins to vibrate anew. 501 Table 20.10 Intensities of spectroscopic bands in 3d complexes Band type dmax / (dm3 mol1 cm1) Spin-forbidden 1 20100 E X A M PL E 20. 9 Assigning a spectrum using selection rules Laporte-forbidden dd Laporte-allowed dd c. 250 Assign the bands in the spectrum in Fig. 20.32 by considering their intensities. Symmetry-allowed (e.g. CT) 100050 000 Answer If we assume that the complex is approximately octahedral, examination of the TanabeSugano diagram for a d3 ion reveals that the ground term is 4A2g. Transitions to the higher terms 2Eg, 2T1g, and 2T2g are spin-forbidden and will have εmax 1 dm3 mol1 cm1. Thus, very weak bands for these transitions are predicted, and will be difficult to distinguish. The next two higher terms of the same multiplicity are 4T2g and 4T1g. These terms are reached by spin-allowed but Laporte-forbidden ligand-field transitions, and have εmax ≈ 100 dm3 mol1 cm1: these are the two bands at 360 and 510 nm. In the near UV, the band with εmax ≈ 10 000 dm3 mol1 cm1 corresponds to the LMCT transitions in which an electron from a chlorine π lone pair is promoted into a molecular orbital that is principally metal d orbital in character. Self-test 20.9 The spectrum of [Cr(NCS)6]3 has a very weak band near 16 000 cm1, a band at 17 700 cm1 with εmax 160 dm3 mol1 cm1, a band at 23 800 cm1 with εmax 130 dm3 mol1 cm1, and a very strong band at 32 400 cm1. Assign these transitions using the d3 TanabeSugano diagram and selection rule considerations. (Hint: NCS has low-lying π orbitals.) 20.7 Luminescence Key points: A luminescent complex is one that re-emits radiation after it has been electronically excited. Fluorescence occurs when there is no change in multiplicity, whereas phosphorescence occurs when an excited state undergoes intersystem crossing to a state of different multiplicity and then undergoes radiative decay. A complex is luminescent if it emits radiation after it has been electronically excited by the absorption of radiation. Luminescence competes with nonradiative decay by thermal degradation of energy to the surroundings. Relatively fast radiative decay is not especially common at room temperature for d-metal complexes, so strongly luminescent systems are comparatively rare. Nevertheless, they do occur, and we can distinguish two types of process. Traditionally, rapidly decaying luminescence was called ‘fluorescence’ and luminescence that persists after the exciting illumination is extinguished was called ‘phosphorescence’. However, because the lifetime criterion is not reliable, the modern definitions of the two kinds of luminescence are based on the distinctive mechanisms of the processes. Fluorescence is radiative decay from an excited state of the same multiplicity as the ground state. The transition is spin-allowed and is fast; fluorescence half-lives are a matter of nanoseconds. Phosphorescence is radiative decay from a state of different multiplicity from the ground state. It is a spin-forbidden process, and hence is often slow. The initial excitation of a phosphorescent complex usually populates a state by a spinallowed transition, so the mechanism of phosphorescence involves intersystem crossing, the nonradiative conversion of the initial excited state into another excited state of different multiplicity. This second state acts as an energy reservoir because radiative decay to the ground state is spin-forbidden. However, just as spinorbit coupling allows the intersystem crossing to occur, it also breaks down the spin selection rule, so the radiative decay can occur. Radiative decay back to the ground state is slow, so a phosphorescent state of a d-metal complex may survive for microseconds or even longer. An important example of phosphorescence is provided by ruby, which consists of a low concentration of Cr3 ions in place of Al3 in alumina. Each Cr3 ion is surrounded octahedrally by six O2 ions and the initial excitations are the spin-allowed processes 2 1 3 eg ← t2g : t2g T2g ← 4A2g and 4T1g ← 4A2g 4 These absorptions occur in the green and violet regions of the spectrum and are respon3 sible for the red colour of the gem (Fig. 20.35). Intersystem crossing to a 2E term of the t2g configuration occurs in a few picoseconds or less, and red 627 nm phosphorescence occurs 502 20 d-Metal complexes: electronic structure and properties 4 T1g 2.3 Internal conversion 4 T2g Intersystem crossing 2 Eg Red emission Green absorption Violet absorption Energy/eV 2.0 as this doublet decays back into the quartet ground state. This red emission adds to the red perceived by the subtraction of green and violet light from white light, and adds lustre to the gem’s appearance. This effect was utilised in the first laser to be constructed (in 1960). A similar 2E → 4A phosphorescence can be observed from a number of Cr(III) complexes 3 configuration, which is the same as the ground in solution. The 2E term arises from the t2g state, and thus the strength of the ligand field is not important. Hence the emission is always in the red (and close to the wavelength of ruby emission). If the ligands are rigid, as in [Cr(bpy)3]3, the 2E term may live for several microseconds in solution. Another interesting example of a phosphorescent state is found in [Ru(bpy)3]2. The excited singlet term produced by a spin-allowed MLCT transition of this d6 complex under5 1 π . goes intersystem crossing to the lower energy triplet term of the same configuration, t2g Bright orange emission then occurs with a lifetime of about 1 s (Fig. 20.36). The effects of other molecules (quenchers) on the lifetime of the emission may be used to monitor the rate of electron transfer from the excited state. 4 A2g 0 Magnetism 1 The diamagnetic and paramagnetic properties of complexes were introduced in Section 20.1c, but the discussion was restricted to magnetically dilute species, where the individual paramagnetic centres, the atoms with unpaired d-electrons, are separate from each other. We now consider two further aspects of magnetism, one where magnetic centres can interact with one another and one where the spin-state may change. MLCT absorption 3 500 l /nm 600 –1 (17 000 cm–1) 400 (20 000 cm ) MLCT emission (25 000 cm–1) Intentsity of absorption or emission Figure 20.35 The transitions responsible for the absorption and luminescence of Cr3 ions in ruby. Figure 20.36 The absorption and phosphorescence spectra of [Ru(bpy)3]2. 20.8 Cooperative magnetism Key point: In solids, the spins on neighbouring metal centres may interact to produce magnetic behaviour, such as ferromagnetism and anitferromagnetism, that are representative of the whole solid. In the solid state the individual magnetic centres are often close together and separated by only a single atom, typically O. In such arrays cooperative properties can arise from interactions between electron spins on different atoms. The magnetic susceptibility, , of a material is a measure of how easy it is to align electron spins with the applied magnetic field in the sense that the induced magnetic moment is proportional to the applied field, with the constant of proportionality. A paramagnetic material has a positive susceptibility and a diamagnetic material has a negative susceptibility. Magnetic effects arising from cooperative phenomena can be very much larger than those arising from individual atoms and ions. The susceptibility and its variation with temperature are different for different types of magnetic materials and are summarized in Table 20.11 and Fig. 20.37. The application of a magnetic field to a paramagnetic material results in the partial alignment of the spins parallel to the field. As a paramagnetic material is cooled, the disordering effect of thermal motion is reduced, more spins become aligned, and the magnetic susceptibility increases. In a ferromagnetic substance, which is one example of a cooperative magnetic property, the spins on different metal centres are coupled into a parallel alignment that is sustained over thousands of atoms to form a magnetic domain (Fig. 20.38). The net magnetic moment, and hence the magnetic susceptibility, may be Table 20.11 Magnetic behaviour of materials Magnetic behaviour Typical value of Variation of with temperature Field dependence Diamagnetism (no unpaired spins) 8 106 for Cu None No Paramagnetism 4 103 for FeSO4 Decreases No Ferromagnetism 5 103 for Fe Decreases Yes Antiferromagnetism 0102 Increases (Yes) 503 Magnetism Paired spins Paramagnetic Magnetic susceptibility, c very large because the magnetic moments of individual spins add to each other. Moreover, once established and with the temperature maintained below the Curie temperature (TC), the magnetization persists after the applied field is removed because the spins are locked together. Ferromagnetism is exhibited by materials containing unpaired electrons in d or, more rarely, f orbitals that couple with unpaired electrons in similar orbitals on surrounding atoms. The key feature is that this interaction is strong enough to align spins but not so strong as to form covalent bonds, in which the electrons would be paired. At temperatures above TC the disordering effect of thermal motion overcomes the ordering effect of the interaction and the material becomes paramagnetic (Fig. 20.37). The magnetization, M, of a ferromagnet, its bulk magnetic moment, is not proportional to the applied field strength H. Instead, a ‘hysteresis loop’ is observed like that shown in Fig. 20.39. For hard ferromagnets the loop is broad and M remains large when the applied field has been reduced to zero. Hard ferromagnets are used for permanent magnets where the direction of the magnetization does not need to be reversed. A soft ferromagnet has a narrower hysteresis loop and is therefore much more responsive to the applied field. Soft ferromagnets are used in transformers, where they must respond to a rapidly oscillating field. In an antiferromagnetic material, neighbouring spins are locked into an antiparallel alignment (Fig. 20.40). As a result, the collection of individual magnetic moments cancel and the sample has a low magnetic moment and magnetic susceptibility (tending, in fact, to zero). Antiferromagnetism is often observed when a paramagnetic material is cooled to a low temperature and is indicated by a sharp decrease in magnetic susceptibility at the Néel temperature, TN (Fig. 24.37). Above TN the magnetic susceptibility is that of a paramagnetic material, and decreases as the temperature is raised. The spin coupling responsible for antiferromagnetism generally occurs through intervening ligands by a mechanism called superexchange. As indicated in Fig. 20.41, the spin on one metal atom induces a small spin polarization on an occupied orbital of a ligand, and this spin polarization results in an antiparallel alignment of the spin on the adjacent metal atom. This alternating …↑↓↑↓… alignment of spins then propagates throughout the material. Many d-metal oxides exhibit antiferromagnetic behaviour that can be ascribed to a superexchange mechanism involving O atoms, for example MnO is antiferromagnetic below 122 K and Cr2O3 is antiferromagnetic below 310 K. Coupling of spins through intervening ligands is frequently observed in molecular complexes containing two ligandbridged metal ions but it is weaker than with a simple O2 link between metal sites and as a result the ordering temperatures are much lower, typically below 100 K. In ferrimagnetism, a net magnetic ordering of ions with different individual magnetic moments is observed below the Curie temperature. These ions can order with opposed spins, as in antiferromagnetism, but because the individual spin moments are different, there is incomplete cancellation and the sample has a net overall moment. As with antiferro-magnetism, these interactions are generally transmitted through the ligands; an example is magnetite Fe3O4. There are a large number of molecular systems where magnetic coupling is observed. Typical systems have two or more metal atoms bridged by ligands that mediate the coupling. Simple examples include copper acetate (10), which exists as a dimer with antiferromagnetic coupling between the two d9 centres. Many metalloenzymes (Sections 27.9 to 27.14) have multiple metal centres that show magnetic coupling. Ferromagnetic Antiferromagnetic TN TC Temperature, T Figure 20.37 The temperature dependence of the susceptibilities of paramagnetic, ferromagnetic, and antiferromagnetic substances. Figure 20.38 The parallel alignment of individual magnetic moments in a ferromagnetic material. M H Figure 20.39 Magnetization curves for ferromagnetic materials. A hysteresis loop results because the magnetization of the sample with increasing field (→) is not retraced as the field is decreased (←). Blue line: hard ferromagnet; red line: soft ferromagnet. O H2O OO Cu Figure 20.40 The antiparallel arrangement of individual magnetic moments in an antiferromagnetic material. Polarization (favourable alignment) Figure 20.41 Antiferromagnetic coupling between two metal centres created by spin polarization of a bridging ligand. O Cu O O O 10 OH2 O 504 20 d-Metal complexes: electronic structure and properties 100 100 (b) (c) 50 0 Temperature, T Figure 20.42 The change to high-spin might be (a) abrupt, (b) gradual, or (c) stepped. Percentage of high spin/% Percentage of high spin/% (a) 50 0 Temperature, T Figure 20.43 A hysteresis loop that can occur with some spin crossover systems. 20.9 Spin crossover complexes Key point: When the factors that determine the spin state of a d-metal centre are closely matched, complexes that change spin state in response to external stimuli are possible. Ph N N N Ph 11 We have seen how a number of factors, such as oxidation state and ligand type, determine whether a complex is high- or low-spin. With some complexes, normally of the 3d-series metals, there is only a very small energy difference between the two states, leading to the possibility of spin crossover complexes. Such complexes change their spin state in response to an external stimulus (such as heat or pressure), which in turn leads to a change in their bulk magnetic properties. An example is the d6 iron complex of two diphenylterpyridine ligands (11), which is low-spin (S 0) below 300 K, but high-spin (S 2) above 323 K. The transition from one spin state to another can be abrupt, gradual, or even stepped (Fig 20.42). In the solid state, a further feature of spin crossover complexes is the existence of cooperativity between the magnetic centres, which can lead to hysteresis like that shown in Fig. 20.43. A preference normally exists for the low-spin state under high pressure and low temperature. This preference can be understood on the basis that the eg orbitals, which are more extensively occupied in the high-spin state, have significant metalligand antibonding character. Thus the high-spin form occupies a larger volume, which is favoured by low pressure or high temperature. Spin crossover complexes occur in many geological systems, are implicated in the binding of O2 to haemoglobin, and have the potential to be exploited in both practical magnetic information storage and pressure-sensitive devices. FURTHER READING E.I. Solomon and A.B.P. Lever, Inorganic electronic structure and spectroscopy. Wiley, Chichester (2006). A thorough account of the material covered in this chapter, including a useful discussion of solvatochromism. E.U. Condon and G.H. Shortley, The theory of atomic spectra. Cambridge University Press (1935); revised as E.U. Condon and H. Odabas¸i, Atomic structure. Cambridge University Press (1980). The standard reference text on atomic spectra. S.F.A. Kettle, Physical inorganic chemistry: a co-ordination chemistry approach. Oxford University Press (1998). A.F. Orchard, Magnetochemistry. Oxford University Press (2003). This book provides detailed, modern explanations, based on ligandfield theory, of the origins and interpretations of magnetic effects in complexes and materials. B.N. Figgis and M.A. Hitchman, Ligand field theory and its applications. Wiley, Chichester (2000). Problems 505 EXERCISES x y 20.1 Determine the configuration (in the form t2g eg or ext2y, as appropriate), the number of unpaired electrons, and the ligand-field stabilization energy in terms of ∆O or ∆T and P for each of the following complexes using the spectrochemical series to decide, where relevant, which are likely to be high-spin and which low-spin. (a) [Co(NH3)6]3, (b) [Fe(OH2)6]2, (c) [Fe(CN)6]3, (d) [Cr(NH3)6]3, (e) [W(CO)6], (f) tetrahedral [FeCl4]2, (g) tetrahedral [Ni(CO)4]. 20.2 Both H and P(C6H5)3 are ligands of similar field strength, high in the spectrochemical series. Recalling that phosphines act as π acceptors, is π-acceptor character required for strong-field behaviour? What orbital factors account for the strength of each ligand? 20.3 Estimate the spin-only contribution to the magnetic moment for each complex in Exercise 20.1. 2 2 20.4 Solutions of the complexes [Co(NH3)6] , [Co(OH2)6] (both Oh), and [CoCl4]2 are coloured. One is pink, another is yellow, and the third is blue. Considering the spectrochemical series and the relative magnitudes of ∆T and ∆O, assign each colour to one of the complexes. 20.5 For each of the following pairs of complexes, identify the one that has the larger LFSE: (a) [Cr(OH2)6]2 or [Mn(OH2)6]2 (b) [Mn(OH2)6]2 or [Fe(OH2)6]3 (c) [Fe(OH2)6]3 or [Fe(CN)6]3 (d) [Fe(CN)6]3 or [Ru(CN)6]3 (e) tetrahedral [FeCl4]2 or tetrahedral [CoCl4]2 20.6 Interpret the variation, including the overall trend across the 3d series, of the following values of oxide lattice enthalpies (in kJ mol1). All the compounds have the rock-salt structure: CaO (3460), TiO (3878), VO (3913), MnO (3810), FeO (3921), CoO (3988), NiO (4071). 20.7 A neutral macrocyclic ligand with four donor atoms produces a red diamagnetic low-spin d8 complex of Ni(II) if the anion is the weakly coordinating perchlorate ion. When perchlorate is replaced by two thiocyanate ions, SCN, the complex turns violet and is high-spin with two unpaired electrons. Interpret the change in terms of structure. 20.8 Bearing in mind the JahnTeller theorem, predict the structure of [Cr(OH2)6]2. 20.9 The spectrum of d1 Ti3(aq) is attributed to a single electronic transition eg ← t2g. The band shown in Fig. 20.3 is not symmetrical and suggests that more than one state is involved. Suggest how to explain this observation using the JahnTeller theorem. 20.10 Write the RussellSaunders term symbols for states with the angular momentum quantum numbers (L,S): (a) (0, 25 ), (b) (3, 32 ), (c) (2, 12 ), (d) (1,1). 20.11 Identify the ground term from each set of terms: (a) 1P, 3P, 3F, 1G, (b) 3P, 5D, 3H, 1I, 1G, (c) 6S, 4P, 4G, 2I. 20.12 Give the RussellSaunders terms of the configurations: (a) 4s1, (b) 3p2. Identify the ground term. 20.13 The gas-phase ion V3 has a 3F ground term. The 1D and 3P terms lie, respectively, 10 642 and 12 920 cm1 above it. The energies of the terms are given in terms of Racah parameters as E(3F) A 8B, E(3P) A 7B, E(1D) A 3B 2C. Calculate the values of B and C for V3. 20.14 Write the d-orbital configurations and use the TanabeSugano diagrams (Resource section 6) to identify the ground term of (a) low-spin [Rh(NH3)6]3, (b) [Ti(OH2)6]3, (c) high-spin [Fe(OH2)6]3. 20.15 Using the TanabeSugano diagrams in Resource section 6, estimate ∆O and B for (a) [Ni(OH2)6]2 (absorptions at 8500, 15 400, and 26 000 cm1) and (b) [Ni(NH3)6]2 (absorptions at 10 750, 17 500, and 28 200 cm1). 20.16 The spectrum of [Co(NH3)6]3 has a very weak band in the red and two moderate intensity bands in the visible to near-UV. How should these transitions be assigned? 20.17 Explain why [FeF6]3 is colourless whereas [CoF6]3 is coloured but exhibits only a single band in the visible region of the spectrum. 20.18 The Racah parameter B is 460 cm1 in [Co(CN)6]3 and 615 cm1 in [Co(NH3)6]3. Consider the nature of bonding with the two ligands and explain the difference in nephelauxetic effect. 20.19 An approximately ‘octahedral’ complex of Co(III) with ammine and chloro ligands gives two bands with εmax between 60 and 80 dm3 mol1 cm1, one weak peak with εmax 2 dm3 mol1 cm1, and a strong band at higher energy with εmax 2 104 dm3 mol1 cm1. What do you suggest for the origins of these transitions? 20.20 Ordinary bottle glass appears nearly colourless when viewed through the wall of the bottle but green when viewed from the end so that the light has a long path through the glass. The colour is associated with the presence of Fe3 in the silicate matrix. Suggest which transitions are responsible for the colour. 20.21 Solutions of [Cr(OH2)6]3 ions are pale bluegreen but the chromate ion, CrO42, is an intense yellow. Characterize the origins of the transitions and explain the relative intensities. 20.22 Classify the symmetry type of the d orbitals in a tetragonal C4v symmetry complex, such as [CoCl(NH3)52, where the Cl lies on the z-axis. (a) Which orbitals will be displaced from their position in the octahedral molecular orbital diagram by π interactions with the lone pairs of the Cl ligand? (b) Which orbital will move because the Cl ligand is not as strong a base as NH3? (c) Sketch the qualitative molecular orbital diagram for the C4v complex. 20.23 Consider the molecular orbital diagram for a tetrahedral complex (based on Fig. 20.8) and the relevant d-orbital configuration and show that the purple colour of MnO4 ions cannot arise from a ligand-field transition. Given that the wavenumbers of the two transitions in MnO4 are 18 500 and 32 200 cm1, explain how to estimate ∆T from an assignment of the two charge-transfer transitions, even though ∆T cannot be observed directly. 20.24 The lowest energy band in the spectrum of [Fe(OH2)]3 (in 1M HClO4) occurs at lower energy than the equivalent transition in the spectrum of [Mn(OH2)]2. Explain why this is. PROBLEMS 20.1 In a fused magma liquid from which silicate minerals crystallize, the metal ions can be four-coordinate. In olivine crystals, the M(II) coordination sites are octahedral. Partition coefficients, which are defined as Kp [M(II)]olivine/[M(II)]melt, follow the order Ni(II) Co(II) Fe(II) Mn(II). Account for this in terms of ligand-field theory. (See I.M. Dale and P. Henderson, 24th Int. Geol. Congress, Sect. 1972, 10, 105.) 506 20 d-Metal complexes: electronic structure and properties 20.2 In Problem 7.11 we looked at the successive formation constants for ethylenediamine complexes of three different metals. Using the same data, discuss the effect of the metal on the formation constant. How might the Irving–Williams series provide insight into these formation constants? 20.3 By considering the splitting of the octahedral orbitals as the symmetry is lowered, draw the symmetry-adapted linear combinations and the molecular orbital energy level diagram for bonding in a trans-[ML4X2] complex. Assume that the ligand X is lower in the spectrochemical series than L. 20.4 By referring to Resource section 5, draw the appropriate symmetry-adapted linear combinations and the molecular orbital diagram for bonding in a square-planar complex. The point group is D4h. Take note of the small overlap of the ligand with the dz2 orbital. What is the effect of π bonding? 20.5 Figures 20.12 and 20.10 show the relationship between the frontier orbitals of octahedral and square-planar complexes. Construct similar diagrams for two-coordinate linear complexes. 20.6 Consider a trigonal prismatic six-coordinate ML6 complex with D3h symmetry. Use the D3h character table to divide the d orbitals of the metal into sets of defined symmetry type. Assume that the ligands are at the same angle relative to the xy-plane as in a tetrahedral complex. 20.9 The compound Ph3SnRe(CO)3(tBu-DAB) (tBu-DAB tBuN CH NtBu) has a metal-to-diimine ligand MLCT band as the lowest energy transition in its spectrum. Irradiation of this band gives an Re(CO)3(tBu-DAB) radical as a photochemical product. The EPR spectrum shows extensive hyperfine splitting by 14N and 1H. The radical is assigned as a complex of the DAB radical anion with Re(I). Explain the argument (see D.J. Stufkens, Coord. Chem. Rev., 1990, 104, 39). 20.10 MLCT bands can be recognized by the fact that the energy is a sensitive function of the polarity of the solvent (because the excited state is more polar than the ground state). Two simplified molecular orbital diagrams are shown in Fig. 20.44. In (a) is a case with a ligand π level higher than the metal d orbital. In (b) is a case in which the metal d orbital and the ligand level are at the same energy. Which of the two MLCT bands should be more solvent sensitive? These two cases are realized by [W(CO)4(phen)] and [W(CO)4(iPr-DAB)], where DAB 1,4-diaza-1,3-butadiene, respectively. (See P.C. Servas, H.K. van Dijk, T.L. Snoeck, D.J. Stufkens, and A. Oskam, Inorg. Chem., 1985, 24, 4494.) Comment on the CT character of the transition as a function of the extent of backdonation by the metal atom. LUMO 20.7 In a trigonal bipyramidal complex ligand the axial and equatorial sites have different steric and electronic interactions with the central metal ion. Consider a range of some common ligands and decide which coordination site in a trigonal bipyramidal complex they would favour. (See A.R. Rossi and R. Hoffmann, Inorg. Chem., 1975, 14, 365.) 20.8 Vanadium(IV) species that have the V O group have quite distinct spectra. What is the d-electron configuration of V(IV)? The most symmetrical of such complexes are VOL5 with C4v symmetry with the O atom on the z-axis. What are the symmetry species of the five d orbitals in VOL5 complexes? How many dd bands are expected in the spectra of these complexes? A band near 24 000 cm1 in these complexes shows vibrational progressions of the V O vibration, implicating an orbital involving V O bonding. Which dd transition is a candidate? (See C.J. Ballhausen and H.B. Gray, Inorg. Chem., 1962, 1, 111.) Ligand π LUMO Ligand π Metal d Metal d HOMO HOMO Figure 20.44 Representation of the orbitals involved in MLCT transitions for cases in which the energy of the ligand π orbital varies with respect to the energy of the metal d orbital. See Problem 20.10. 20.11 Consider spin crossover complexes and identify the features that a complex would need for it to be used in (a) a practical pressure sensor and (b) a practical information storage device (see P. Gütlich, Y. Garcia, and H.A. Goodwin, Chem. Soc. Rev., 2000, 29, 419). Coordination chemistry: reactions of complexes We now look at the evidence and experiments that are used in the analysis of the reaction pathways of metal complexes and so develop a deeper understanding of their mechanisms. Because a mechanism is rarely known definitively, the nature of the evidence for it should always be kept in mind in order to recognize that there might be other consistent possibilities. In the first part of this chapter we consider ligand exchange reactions and describe how reaction mechanisms are classified. We consider the steps by which the reactions take place and the details of the formation of the transition state. These concepts are then used to describe the mechanisms of the redox reactions of complexes. 21 Ligand substitution reactions 21.1 Rates of ligand substitution 21.2 The classification of mechanisms Ligand substitution in square-planar complexes 21.3 The nucleophilicity of the entering group 21.4 The shape of the transition state Coordination chemistry is not the sole preserve of d metals. Whereas Chapter 20 dealt exclusively with d metals, this chapter builds on the introduction to coordination chemistry in Chapter 7 and applies it to all metals regardless of the block to which they belong. However, there are special features of each block, and we shall point them out. Ligand substitution in octahedral complexes 21.5 Rate laws and their interpretation 21.6 The activation of octahedral complexes 21.7 Base hydrolysis Ligand substitution reactions 21.8 Stereochemistry 21.9 Isomerization reactions The most fundamental reaction a complex can undergo is ligand substitution, a reaction in which one Lewis base displaces another from a Lewis acid: Y MX → MY X This class of reaction includes complex formation reactions, in which the leaving group, the displaced base X, is a solvent molecule and the entering group, the displacing base Y, is some other ligand. An example is the replacement of a water ligand by Cl: [Co(OH2)6]2(aq) Cl−(aq) → CoCl(OH2)5 H2O(l) The thermodynamic aspects of complex formation are discussed in Sections 7.12 to 7.15. 21.1 Rates of ligand substitution Key points: The rates of substitution reactions span a very wide range and correlate with the structures of the complexes; complexes that react quickly are called labile, those that react slowly are called inert or nonlabile. Rates of reaction are as important as equilibria in coordination chemistry. The numerous isomers of the ammines of Co(III) and Pt(II), which were so important to the development of the subject, could not have been isolated if ligand substitutions and interconversion of the isomers had been fast. But what determines whether one complex will survive for long periods whereas another will undergo rapid reaction? The rate at which one complex converts into another is governed by the height of the activation energy barrier that lies between them. Thermodynamically unstable complexes that survive for long periods (by convention, at least a minute) are commonly called ‘inert’, but nonlabile is more appropriate and is the term we shall use. Complexes that undergo more rapid equilibration are called labile. An example of each type is the labile complex [Ni(OH2)6]2, which has a half-life of the order of milliseconds before the H2O is replaced Redox reactions 21.10 The classification of redox reactions 21.11 The inner-sphere mechanism 21.12 The outer-sphere mechanism Photochemical reactions 21.13 Prompt and delayed reactions 21.14 dd and charge-transfer reactions 21.15 Transitions in metalmetal bonded systems FURTHER READING EXERCISES PROBLEMS 508 21 Coordination chemistry: reactions of complexes by another H2O or a stronger base, and the nonlabile complex [Co(NH3)5(OH2)]3, in which H2O survives for several minutes as a ligand before it is replaced by a stronger base. Figure 21.1 shows the characteristic lifetimes of the important aqua metal ion complexes. We see a range of lifetimes starting at about 1 ns, which is approximately the time it takes for a molecule to diffuse one molecular diameter in solution. At the other end of the scale are lifetimes in years. Even so, the illustration does not show the longest times that could be considered, which are comparable to geological eras. We shall examine the lability of complexes in greater detail when we discuss the mechanism of reactions later in this section, but we can make two broad generalizations now. The first is that complexes of metals that have no additional factor to provide extra stability (for instance, the LFSE and chelate effects) are among the most labile. Any additional stability of a complex results in an increase in activation energy for a ligand replacement reaction and hence decreases the lability of the complex. A second generalization is that very small ions are often less labile because they have greater ML bond strengths and it is sterically very difficult for incoming ligands to approach the metal atom closely. Some further generalizations are as follows: 1. All complexes of s-block ions except the smallest (Be2 and Mg2) are very labile. 2. Complexes of the M(III) ions of the f block are all very labile. 3. Complexes of the d10 ions (Zn2, Cd2, and Hg2) are normally very labile. 4. Across the 3d series, complexes of d-block M(II) ions are generally moderately labile, with distorted Cu(II) complexes among the most labile. 5. Complexes of d-block M(III) ions are distinctly less labile than d-block M(II) ions. 6. d-Metal complexes with d3 and low-spin d6 configurations (for example Cr(III), Fe(II), and Co(III)) are generally nonlabile as they have large LFSEs. Chelate complexes with the same configuration, such as [Fe(phen)3]2, are particularly inert. 7. Nonlability is common among the complexes of the 4d and 5d series, which reflects the high LFSE and strength of the metalligand bonding. Table 21.1 illustrates the range of timescales for a number of reactions. The natures of the ligands in the complex also affect the rates of reactions. The identity of the incoming ligand has the greatest effect, and equilibrium constants of displacement reactions can be used to rank ligands in order of their strength as Lewis bases. However, a different order may be found if bases are ranked according to the rates at which they displace a ligand from the central metal ion. Therefore, for kinetic considerations, we replace the equilibrium concept of basicity by the kinetic concept of nucleophilicity, the rate of attack on a complex by a given Lewis base relative to the rate of attack by a reference Lewis base. The shift from equilibrium to kinetic considerations is emphasized by referring to ligand displacement as nucleophilic substitution. Ligands other than the entering and leaving groups may play a significant role in controlling the rates of reactions; these ligands are referred to as spectator ligands. Li Na K Rb Cs 2+ V2+ Be2+ Ir3+ Rh3+ 1010 108 Cr3+ 104 Al3+ 102 Fe3+ 1 10–2 Lifetime, t /s Ca2+ Sr2+ Ba2+ Hg Mg2+ 2+ 2+ 2+ 2+ Ni2+ Co Fe Mn2+ Zn2+Cd Cu 3+ V3+ Ti3+ In3+ La 10–4 10–6 Figure 21.1 Characteristic lifetimes for exchange of water molecules in aqua complexes. 10–8 10–10 Ligand substitution reactions Table 21.1 Representative timescales of chemical and physical processes Timescale Process Example 108 s Ligand exchange (inert complex) [Cr(OH2)6]3 H2O (c. 6 days) 60 s Ligand exchange (nonlabile complex) [V(OH2)6]3 H2O (50 s) 1 ms Ligand exchange (labile complex) [Pt(OH2)4]2 H2O (0.4 ms) 1 μs Intervalence charge transfer 1 ns Ligand exchange (labile complex) [Ni(OH2)5(py)]2 H2O (1 ns) 10 ps Ligand association Cr(CO)5 THF (10 ps) 1 ps Rotation time in liquid CH3CN (1 ps) 1 fs Molecular vibration SnCl stretch (300 fs) (H 3N) 5Ru II N N Ru III(NH3) (0.5 μs) Approximate time at room temperature. For instance, it is observed for square-planar complexes that the ligand trans to the leaving group X has a great effect on the rate of substitution of X by the entering group Y. 21.2 The classification of mechanisms The mechanism of a reaction is the sequence of elementary steps by which the reaction takes place. Once the mechanism has been identified, attention turns to the details of the activation process of the rate-determining step. In some cases the overall mechanism is not fully resolved, and the only information available is the rate-determining step. (a) Association, dissociation, and interchange Key points: The mechanism of a nucleophilic substitution reaction is the sequence of elementary steps by which the reaction takes place and is classified as associative, dissociative, or interchange; an associative mechanism is distinguished from an interchange mechanism by demonstrating that the intermediate has a relatively long life. The first stage in the kinetic analysis of a reaction is to study how its rate changes as the concentrations of reactants are varied. This type of investigation leads to the identification of rate laws, the differential equations governing the rate of change of the concentrations of reactants and products. For example, the observation that the rate of formation of [Ni(NH3)(OH2)5]2 from [Ni(OH2)6]2 is proportional to the concentration of both NH3 and [Ni(OH2)6]2 implies that the reaction is first order in each of these two reactants, and that the overall rate law is: rate kr[Ni(OH2)62][NH3] (21.1) A note on good practice In rate equations, as in expressions for equilibrium constants, we omit the brackets that are part of the chemical formula of the complex; the surviving brackets denote molar concentration. We denote rate constants by kr, to avoid possible confusion with Boltzmann’s constant. In simple sequential reaction schemes, the slowest elementary step of the reaction dominates the overall reaction rate and the overall rate law, and is called the rate-determining step. However, in general, all the steps in the reaction may contribute to the rate law and affect its rate. Therefore, in conjunction with stereochemical and isotopic labelling studies, the determination of the rate law is the route to the elucidation of the mechanism of the reaction. Three main classes of reaction mechanism have been identified. A dissociative mechanism, denoted D, is a reaction sequence in which an intermediate of reduced coordination number is formed by the departure of the leaving group: MLnX → MLn X MLn Y → MLnY 509 Potential energy 510 21 Coordination chemistry: reactions of complexes Here MLn (the metal atom and any spectator ligands) is a true intermediate that can, in principle, be detected (or even isolated). The typical form of the corresponding reaction profile is shown in Fig. 21.2. {X,M,Y} ■ A brief illustration. The substitution of hexacarbonyltungsten by phosphine takes place by dissociation of CO from the complex X–M + Y W(CO)6 → W(CO)5 CO followed by coordination of phosphine: X + M–Y Reaction coordinate Figure 21.2 The typical form of the reaction profile of a reaction with a dissociative mechanism. W(CO)5 PPh3 → W(CO)5(PPh3) Under the conditions in which this reaction is usually performed in the laboratory, the intermediate W(CO)5 is rapidly captured by the solvent, such as tetrahydrofuran, to form [W(CO)5(thf)]. This complex in turn is converted to the phosphine product, presumably by a second dissociative process. ■ An associative mechanism, denoted A, involves a step in which an intermediate is formed with a higher coordination number than the original complex: Potential energy MLnX Y → MLnXY MLnXY → MLnY X X–M–Y Once again, the intermediate MLnXY can, in principle at least, be detected. This mechanism plays a role in many reactions of square-planar Au(III), Pt(II), Pd(II), Ni(II), and Ir(I) d8 complexes. The typical form of the reaction profile is similar to that of the dissociative mechanism, and is shown in Fig. 21.3. X–M + Y ■ A brief illustration. The first step in the exchange of 14CN with the ligands in the square-planar X + M–Y complex [Ni(CN)4]2 is the coordination of a ligand to the complex: [Ni(CN)4]2 14CN− → [Ni(CN)4(14CN)]3− Reaction coordinate Figure 21.3 The typical form of the reaction profile of a reaction with an associative mechanism. A ligand is then discarded: [Ni(CN)4(14CN)]3− → [Ni(CN)3(14CN)]2− CN− The radioactivity of carbon-14 provides a means of monitoring this reaction, and the intermediate [Ni(CN)5]3 has been detected and isolated. ■ An interchange mechanism, denoted I, takes place in one step: MLnX Y → X MLn Y → MLnY X Potential energy X...M...Y X–M + Y X + M–Y Reaction coordinate Figure 21.4 The typical form of the reaction profile of a reaction with an interchange mechanism. The leaving and entering groups exchange in a single step by forming a transition state but not a true intermediate. The interchange mechanism is common for many reactions of six-coordinate complexes. The typical form of the reaction profile is shown in Fig. 21.4. The distinction between the A and I mechanisms hinges on whether or not the intermediate persists long enough to be detectable. One type of evidence is the isolation of an intermediate in another related reaction or under different conditions. If an argument by extrapolation to the actual reaction conditions suggests that a moderately long-lived intermediate might exist during the reaction in question, then the A path is indicated. For example, the synthesis of the first trigonal-bipyramidal Pt(II) complex, [Pt(SnCl3)5]3, indicates that a five-coordinate platinum complex may be plausible in substitution reactions of square-planar Pt(II) ammine complexes. Similarly, the fact that [Ni(CN)5]3 is observed spectroscopically in solution, and that it has been isolated in the crystalline state, provides support for the view that it is involved when CN exchanges with the square-planar tetracyanidonickelate(II) ion. A second indication of the persistence of an intermediate is the observation of a stereochemical change, which implies that the intermediate has lived long enough to undergo rearrangement. Cis to trans isomerization is observed in the substitution reactions of certain square-planar phosphine Pt(II) complexes, which is in contrast to the retention of configuration usually observed. This difference implies that the trigonal-bipyramidal intermediate lives long enough for an exchange between the axial and equatorial ligand positions to occur. Ligand substitution reactions 511 (b) The rate-determining step Key point: The rate-determining step is classified as associative or dissociative according to the dependence of its rate on the identity of the entering group. Now we consider the rate-determining step of a reaction and the details of its formation. The step is called associative and denoted a if its rate depends strongly on the identity of the incoming group. Examples are found among reactions of the d8 square-planar complexes of Pt(II), Pd(II), and Au(III), including [Ni(OH2)6]2(aq) NH3(aq) → [Ni(NH3)(OH2)5]2(aq) H2O(l) It is found that replacement of NH3 by pyridine in this reaction changes the rate by at most a few per cent. The weak dependence on Y of a dissociatively activated process indicates that the rate of formation of the transition state is determined largely by the rate at which the bond to the leaving group X can break. A reaction with an associative mechanism (A) will be dissociatively activated (d) provided the loss of X from the intermediate YMLnX is the ratedetermining step; such a reaction is designated Ad. A reaction with a dissociative mechanism (D) is dissociatively activated (d) if the initial loss of X from the reactant MLnX is the rate-determining step, such a reaction is designated Dd. In this case, the intermediate MLn would not be detected. Figure 21.6 shows the reaction profiles for dissociatively activated A and D mechanisms. A reaction that has an interchange mechanism (I) can be either associatively or dissociatively activated, and is designated either Ia or Id, respectively. In an Ia mechanism, the rate of reaction depends on the rate at which the M … Y bond forms, whereas in an Id reaction the rate of reaction depends on the rate at which the M … X bond breaks (Fig. 21.7). The distinction between these possibilities may be summarized as follows, where MLnX denotes the initial complex: A a I d D a d a d Ratedetermining step Y attaching to MLnX Loss of X from YMLnX Y attaching to MLnX Loss of X from YMLnX Y attaching to MLn Loss of X from MLnX Detect intermediate? no MLnXY detectable no no MLn detectable no Potential energy where dien is diethylenetriamine (NH2CH2CH2NHCH2CH2NH2). It is found, for instance, that use of I instead of Br increases the rate constant by an order of magnitude. Experimental observations on the substitution reactions of square-planar complexes support the view that the rate-determining step is associative. The strong dependence of the rate-determining step on entering group Y indicates that the transition state must involve significant bonding to Y. A reaction with an associative mechanism (A) will be associatively activated (a) if the attachment of Y to the initial reactant MLnX is the rate-determining step; such a reaction is designated Aa, and in this case the intermediate MLnXY would not be detected. A reaction with a dissociative mechanism (D) is associatively activated (a) if the attachment of Y to the intermediate MLn is the rate-determining step; such a reaction is designated Da. Figure 21.5 shows the reaction profiles for associatively activated A and D mechanisms. For the reactions to proceed, it is necessary to have established a population of an encounter complex XM, Y in a preequilibrium step. The rate-determining step is called dissociative and denoted d if its rate is largely independent of the identity of Y. This category includes some of the classic examples of ligand substitution in octahedral d-metal complexes, including X–M–Y X–M + Y X + M–Y (a) PtCl(dien) I−(aq) → PtI(dien) Cl−(aq) Mechanism: Activation: Potential energy Direct spectroscopic detection of the intermediate, and hence an indication of A rather than I, may be possible if a sufficient amount accumulates. Such direct evidence, however, requires an unusually stable intermediate with favourable spectroscopic characteristics. Reaction coordinate {X,M,Y} X–M + Y X + M–Y (b) Reaction coordinate Figure 21.5 The typical form of the reaction profile of reactions with an associatively activated step: (a) associative mechanism, Aa; (b) a dissociative mechanism, Da. 21 Coordination chemistry: reactions of complexes Potential energy Potential energy X...M...Y X–M–Y X–M + Y X–M + Y X + M–Y X + M–Y (a) Potential energy (a) Reaction coordinate {X,M,Y} Potential energy 512 X–M + Y Reaction coordinate X...M...Y X–M + Y X + M–Y (b) X + M–Y Reaction coordinate Figure 21.6 The typical form of the reaction profile of reactions with a dissociatively activated step: (a) an associative mechanism, Ad; (b) a dissociative mechanism, Dd. (b) Reaction coordinate Figure 21.7 The typical form of the reaction profile of reactions with an interchange mechanism: (a) associatively activated, Ia; (b) dissociatively activated, Id. Ligand substitution in square-planar complexes The mechanism of ligand exchange in square-planar Pt complexes has been studied extensively, largely because the reactions occur on a timescale that is very amenable to investigation. We might expect an associative mechanism of ligand exchange because square-planar complexes are sterically uncrowded—they can be considered as octahedral complexes with two ligands missing—but it is rarely that simple. The elucidation of the mechanism of the substitution of square-planar complexes is often complicated by the occurrence of alternative pathways. For instance, if a reaction such as PtCl(dien) I(aq) → PtI(dien) Cl(aq) is first order in the complex and independent of the concentration of I, then the rate of reaction will be equal to kr,1[PtCl(dien)]. However, if there is a pathway in which the rate law is first order in the complex and first order in the incoming group (that is, overall second order) then the rate would be given by kr,2[PtCl(dien)][I]. If both reaction pathways occur at comparable rates, the rate law has the form rate (kr,1 kr,2 [I])[PtCl(dien)] (21.2) A reaction like this is usually studied under the conditions [I] [complex] so that [I] does not change significantly during the reaction. This simplifies the treatment of the data as kr,1 kr,2[I] is effectively constant and the rate law is now pseudo-first order: rate kr,obs[PtCl(dien)] kr,obs kr,1 kr,2[I] (21.3) A plot of the observed pseudo-first-order rate constant against [I ] gives kr,2 as the slope and kr,1 as the intercept. 513 Ligand substitution in square-planar complexes In the following sections, we examine the factors that affect the second-order reaction and then consider the first-order process. 21.3 The nucleophilicity of the entering group Key points: The nucleophilicity of an entering group is expressed in terms of the nucleophilicity parameter defined in terms of the substitution reactions of a specific square-planar platinum complex; the sensitivity of other platinum complexes to changes in the entering group is expressed in terms of the nucleophilic discrimination factor. We start by considering the variation of the rate of the reaction as the entering group Y is varied. The reactivity of Y (for instance, I in the reaction above) can be expressed in terms of a nucleophilicity parameter, nPt: nPt = log kr,2 (Y ) o kr,2 (21.4) Table 21.2 A selection of nPt values for a range of nucleophiles Nucleophile Donor atom nPt CH3OH where kr,2(Y) is the second-order rate constant for the reaction trans-[PtCl2(py)2] Y → trans-[PtClY(py)2] Cl o and kr,2 is the rate constant for the same reaction with the reference nucleophile methanol. The entering group is highly nucleophilic, or has a high nucleophilicity, if nPt is large. Table 21.2 gives some values of nPt. One striking feature of the data is that, although the entering groups in the table are all quite simple, the rate constants span nearly nine orders of magnitude. Another feature is that the nucleophilicity of the entering group towards Pt appears to correlate with soft Lewis basicity (Section 4.12), with Cl I, O S, and NH3 PR3. The nucleophilicity parameter is defined in terms of the reaction rates of a specific platinum complex. When the complex itself is varied we find that the reaction rates show a range of different sensitivities towards changes in the entering group. To express this range of sensitivities we rearrange eqn 21.4 into log kr,2(Y) nPt(Y) C O 0 Cl 3.04 Br Br 4.18 I 5.42 C 7.14 Cl I CN S 5.75 N3 N 3.58 SCN C6H5SH S 4.15 NH3 N 3.07 (C6H5)3P P 8.93 (21.5) where C log k . Now consider the analogous substitution reactions for the general complex [PtL3X]: o r,2 2 SeCN– [PtL3X] Y → [PtL3Y] X – SCN 0 I– log kr,2(Y) SnPt(Y) C (21.6) The parameter S, which characterizes the sensitivity of the rate constant to the nucleophilicity parameter, is called the nucleophilic discrimination factor. We see that the straight line obtained by plotting log kr,2(Y) against nPt for reactions of Y with trans-[PtCl2(PEt3)2], the red circles in Fig. 21.8, is steeper than that for reactions with cis-[PtCl2(en)], the blue squares in Fig. 21.8. Hence, S is larger for the former reaction, which indicates that the rate of the reaction is more sensitive to changes in the nucleophilicity of the entering group. Some values of S are given in Table 21.3. Note that S is close to 1 in all cases, so all the complexes are quite sensitive to nPt. This sensitivity is what we expect for associatively activated reactions. Another feature to note is that larger values of S are found for complexes of platinum with softer base ligands. E X A M PL E 21.1 Using the nucleophilicity parameter The second-order rate constant for the reaction of I with trans-[Pt(CH3)Cl(PEt3)2] in methanol at 30C is 40 dm3 mol1 s1. The corresponding reaction with N3 has k2 7.0 dm3 mol1 s1. Estimate S and C for the reaction given the nPt values of 5.42 and 3.58, respectively, for the two nucleophiles. log kr,2(Y) The relative rates of these reactions can be expressed in terms of the same nucleophilicity parameter nPt provided we replace eqn 21.5 by –2 NH3 –4 N3– Br– H2O I – Tu SCN– – Br N3– Cl– NO2– –6 CH3OH –8 0 1 2 3 4 nPt(Y) 5 Figure 21.8 The slope of the straight line obtained by plotting log kr,2(Y) against the nucleophilicity parameter nPt(Y) for a series of ligands is a measure of the responsiveness of the complex to the nucleophilicity of the entering group (data from U. Belluso, L. Cattaini, F. Basolo, R.G. Pearson, and A. Turco; J. Am. Chem. Soc., 1965, 87, 241). 6 514 21 Coordination chemistry: reactions of complexes Table 21.3 Nucleophilic discrimination factors S trans-[PtCl2(PEt3)2] 1.43 trans-[PtCl2(py)2] 1.00 [PtCl2(en)] 0.64 trans-[PtCl(dien)] 0.65 Answer To determine S and C, we need to use the two pieces of information to set up and solve two simultaneous equations based on eqn 21.6. Substituting the two values of nPt into eqn 21.6 gives 1.60 5.42S C (for I) 0.85 3.58S C (for N3 ) Solving these two simultaneous equations gives S 0.41 and C 0.62. The value of S is fairly small, showing that the discrimination of this complex among different nucleophiles is not great. This lack of sensitivity is related to the fairly large value of C, which corresponds to the rate constant being large and hence to the complex being reactive. It is commonly found that high reactivity correlates with low selectivity. Self-test 21.1 Calculate the second-order rate constant for the reaction of the same complex with NO2 , for which nPt 3.22. 21.4 The shape of the transition state Careful studies of the variation of the reaction rates of square-planar complexes with changes in the composition of the reactant complex and the conditions of the reaction shed light on the general shape of the transition state. They also confirm that substitution almost invariably has an associative rate-determining stage; hence intermediates are rarely detected. (a) The trans effect Key point: A strong -donor ligand or π-acceptor ligand greatly accelerates substitution of a ligand that lies in the trans position. X T Y The spectator ligands T that are trans to the leaving group in square-planar complexes influence the rate of substitution. This phenomenon is called the trans effect. It is generally accepted that the trans effect arises from two separate influences: one arising in the ground state and the other in the transition state itself. The trans influence is the extent to which the ligand T weakens the bond trans to itself in the ground state of the complex. The trans influence correlates with the -donor ability of the ligand T because, broadly speaking, ligands trans to each other use the same orbitals on the metal for bonding. Thus if one ligand is a strong donor, then the ligand trans to it cannot donate electrons to the metal so well, and thus has a weaker interaction with the metal. The trans influence is assessed quantitatively by measuring bond lengths, stretching frequencies, and metal-to-ligand NMR coupling constants (Section 8.5). The transition state effect correlates with the π-acceptor ability of the ligand. Its origin is thought to be the increase in electron density on the metal atom arising due to the incoming ligand: any ligand that can accept this increased electron density will stabilize the transition state (1). The trans effect is the combination of both effects; it should be noted that the same factors contribute to a large ligand-field splitting. Trans effects are listed in Table 21.4 and follow the order: For a T -donor: OH NH3 Cl Br CN, CH3 I SCN PR3, H For a T π-acceptor: Br I NCS NO2 CN CO, C2H4 1 Table 21.4 The effect of the trans ligand in reactions of trans-[PtCl(PEt3)2L] L kr,1/s1 kr,2 /(dm3 mol1 s1) CH3 1.7 104 6.7 102 C6H5 3.3 105 1.6 102 Cl 1.0 106 4.0 104 H 1.8 102 4.2 PEt3 1.7 10 3.8 2 E X A M PL E 21. 2 Using the trans effect synthetically Use the trans effect series to suggest synthetic routes to cis- and trans-[PtCl2(NH3)2] from [Pt(NH3)4]2 and [PtCl4]2. Answer If we consider the reaction of [Pt(NH3)4]2 with HCl we can see it leads to [PtCl(NH3)3]. Now, because the trans effect of Cl is greater than that of NH3, substitution reactions will occur preferentially trans to Cl, and further action of HCl gives trans-[PtCl2(NH3)2]: [Pt(NH3)4]2 Cl → [PtCl(NH3)3] → trans-[PtCl2(NH3)2] 515 Ligand substitution in square-planar complexes However, when the starting complex is [PtCl4]2, reaction with NH3 leads first to [PtCl3(NH3)]. A second step should substitute one of the two mutually trans Cl ligands with NH3 to give cis-[PtCl2(NH3)2]. [PtCl4]2 NH3 → [PtCl3(NH3)] → cis-[PtCl2(NH3)2] Self-test 21.2 Given the reactants PPh3, NH3, and [PtCl4]2, propose efficient routes to both cis- and trans[PtCl2(NH3)(PPh3)]. CH3 PEt3 M (b) Steric effects Key point: Steric crowding at the reaction centre usually inhibits associative reactions and facilitates dissociative reactions. Cl Steric crowding at the reaction centre by bulky groups that can block the approach of attacking nucleophiles will inhibit associative reactions. The rate constants for the replacement of Cl by H2O in cis-[PtClL(PEt3)2] complexes at 25ºC illustrate the point: L pyridine 2-methylpyridine 2,6-dimethylpyridine kr/s1 8 102 2.0 104 1.0 106 2 CH3 The methyl groups adjacent to the N donor atom greatly decrease the rate. In the 2-methylpyridine complex they block positions either above or below the plane. In the 2,6-dimethylpyridine complex they block positions both above and below the plane (2). Thus, along the series, the methyl groups increasingly hinder attack by H2O. The effect is smaller if L is trans to Cl. This difference is explained by the methyl groups then being further from the entering and leaving groups in the trigonal-bipyramidal transition state if the pyridine ligand is in the trigonal plane (3). Conversely, the decrease in coordination number that occurs in a dissociative reaction can relieve the steric overcrowding and thus increase the rate of the dissociative reaction. PEt3 Y M Cl 3 (c) Stereochemistry Key point: Substitution of a square-planar complex preserves the original geometry, which suggests a trigonal-pyramidal transition state. A Further insight into the nature of the transition state is obtained from the observation that substitution of a square-planar complex preserves the original geometry. That is, a cis complex gives a cis product and a trans complex gives a trans product. This behaviour is explained by the formation of an approximately trigonal-bipyramidal transition state with the entering, leaving, and trans groups in the trigonal plane (4).1 Trigonal-bipyramidal intermediates of this type account for the relatively small influence that the two cis spectator ligands have on the rate of substitution, as their bonding orbitals will be largely unaffected by the course of the reaction. The steric course of the reaction is shown in Fig. 21.9. We can expect a cis ligand to exchange places with the T ligand in the trigonal plane only if the intermediate lives long enough to be stereomobile. That is, it must be a long-lived associative (A) intermediate, with release of the ligand from the five-coordinate intermediate being the rate-determining step. (d) Temperature and pressure dependence Key point: Negative volumes and entropies of activation support the view that the rate-determining step of square-planar Pt(II) complexes is associative. Another clue to the nature of the transition state comes from the entropies and volumes of activation for reactions of Pt(II) and Au(III) complexes (Table 21.5). The entropy of activation is obtained from the temperature dependence of the rate constant, and indicates the change in disorder (of reactants and solvent) when the transition state forms. Likewise, the volume of activation, which is obtained (with considerable difficulty) from the pressure dependence of the rate constant, is the change in volume that occurs on formation of the transition state. 1 Note that the stereochemistry is quite different from that of p-block central atoms, such as Si(IV) and P(V), where the leaving group departs from the more crowded axial position. X Y X A Y 4 516 21 Coordination chemistry: reactions of complexes Figure 21.9 The stereochemistry of substitution in a square-planar complex. The normal path (resulting in retention) is from (a) to (c). However, if intermediate (b) is sufficiently long-lived, it can undergo pseudorotation to (d), which leads to isomer (e). (a) trans (d) (e) cis (b) (c) trans Table 21.5 Activation parameters for substitution in square-planar complexes (in methanol) Reaction kr,1 kr,2 H S V H 50 100 38 trans-[PtBrP2(mes)] SC(NH2)2 71 84 46 46 138 54 cis-[PtBrP2(mes)] I 84 59 67 63 121 63 cis-[PtBrP2(mes)] SC(NH2)2 79 71 71 59 121 54 54 17 ‡ ‡ ‡ trans-[PtCl(NO2)(py)2] py [AuCl(dien)] 2 Br ‡ S ‡ V ‡ [PtBrP2(mes)] is [PtBr(PEt3)2(2,4,6-Me3C6H2)]. Enthalpy in kilojoules per mole (kJ mol1); entropy in J K1 mol1 and volume in cm3 mol1. The limiting cases for the volume of activation in ligand substitution reactions correspond to the increase in molar volume of the outgoing ligand (for a dissociative reaction) and the decrease in molar volume of the incoming ligand (for an associative reaction). The two striking aspects of the data in the table are the consistently strongly negative values of both quantities. The simplest explanation of the decrease in disorder and the decrease in volume is that the entering ligand is being incorporated into the transition state without release of the leaving group. That is, we can conclude that the rate-determining step is associative. (e) The first-order pathway Key point: The first-order contribution to the rate law is a pseudo-first-order process in which the solvent participates. Having considered factors that affect the second-order pathway, we can now consider the first-order pathway for the substitution of square-planar complexes. The first issue we must address is the first-order pathway in the rate equation and decide whether kr,1 in the rate law in eqn 21.2 and its generalization rate (kr,1 kr,2[Y])[PtL4] (21.7) does indeed represent the operation of an entirely different reaction mechanism. It turns out that it does not, and kr,1 represents an associative reaction involving the solvent. In this pathway the substitution of Cl by pyridine in methanol as solvent proceeds in two steps, with the first rate-determining: [PtCl(dien)] CH3OH → [Pt(CH3OH)(dien)]2 Cl (slow) [Pt(CH3OH)(dien)]2 py → [Pt(py)(dien)]2 CH3OH (fast) The evidence for this two-step mechanism comes from a correlation of the rates of these reactions with the nucleophilicity parameters of the solvent molecules and the observation that reactions of entering groups with solvent complexes are rapid when compared to the Ligand substitution in octahedral complexes step in which the solvent displaces a ligand. Thus, the substitution of ligands at a squareplanar platinum complex results from two competing associative reactions. Ligand substitution in octahedral complexes Octahedral complexes occur for a wide variety of metals in a wide range of oxidation states and with a great diversity of bonding modes. We might therefore expect a wide variety of mechanisms of substitution; however, almost all octahedral complexes react by the interchange mechanism. The only real question is whether the rate-determining step is associative or dissociative. The analysis of rate laws for reactions that take place by such a mechanism helps to formulate the precise conditions for distinguishing these two possibilities and identifying the substitution as Ia (interchange with an associative ratedetermining stage) or Id (interchange with a dissociative rate-determining stage). The difference between the two classes of reaction hinges on whether the rate-determining step is the formation of the new Y⋅⋅⋅M bond or the breaking of the old M⋅⋅⋅X bond. 21.5 Rate laws and their interpretation Rate laws provide an insight into the detailed mechanism of a reaction in the sense that any proposed mechanism must be consistent with the observed rate law. In the following section we see how the rate laws found experimentally for ligand substitution are interpreted. (a) The EigenWilkins mechanism Key points: In the EigenWilkins mechanism, an encounter complex is formed in a pre-equilibrium step and the encounter complex forms products in a subsequent rate-determining step. As an example of a ligand substitution reaction, we consider [Ni(OH2)6]2 NH3 → [Ni(NH3)(OH2)5]2 H2O The first step in the EigenWilkins mechanism is an encounter in which the complex ML6, in this case [Ni(OH2)6]2, and the entering group Y, in this case NH3, diffuse together and come into contact: [Ni(OH2)6]2 NH3 → {[Ni(OH2)6]2,NH3} The two components of the encounter pair, the entity {A,B}, may also separate at a rate governed by their ability to migrate by diffusion through the solvent: {[Ni(OH2)6]2,NH3} → [Ni(OH2)6]2 NH3 Because in aqueous solution the lifetime of an encounter pair is approximately 1 ns, the formation of the pair can be treated as a pre-equilibrium in all reactions that take longer than a few nanoseconds. Consequently, we can express the concentrations in terms of a pre-equilibrium constant KE: ML6 Y {ML6,Y} KE = [{ML 6 ,Y}] [ML 6 ][Y] The second step in the mechanism is the rate-determining reaction of the encounter complex to give products: {[Ni(OH2)6]2,NH3} → [Ni(NH3)(OH2)5]2 H2O and in general {ML6,Y} → ML5Y L rate kr[{ML6,Y}] We cannot simply substitute [{ML6,Y}] KE[ML6][Y] into this expression because the concentration of ML6 must take into account the fact that some of it is present as the encounter pair; that is, [M]tot [{ML6,Y}] [ML6], the total concentration of the complex. It follows that rate = kr KE [M]tot [Y] 1 + KE [Y] (21.8) 517 518 21 Coordination chemistry: reactions of complexes It is rarely possible to conduct experiments over a range of concentrations wide enough to test eqn 21.8 exhaustively. However, at such low concentrations of the entering group that KE[Y] 1, the rate law reduces to rate kr,obs[M]tot[Y] kr,obs krKE (21.9) Because kr,obs can be measured and KE can be either measured or estimated as we describe below, the rate constant kr can be found from kr,obs/KE. The results for reactions of Ni(II) hexaaqua complexes with various nucleophiles are shown in Table 21.6. The very small variation in kr indicates a model Id reaction with very slight sensitivity to the nucleophilicity of the entering group. When Y is a solvent molecule the encounter equilibrium is ‘saturated’ in the sense that, because the complex is always surrounded by solvent, a solvent molecule is always available to take the place of one that leaves the complex. In such a case KE[Y] 1 and kr,obs kr. Thus, reactions with the solvent can be directly compared to reactions with other entering ligands without needing to estimate the value of KE. (b) The FuossEigen equation Key point: The FuossEigen equation provides an estimate of the pre-equilibrium constant based on the strength of the Coulombic interaction between the reactants and their distance of closest approach. The equilibrium constant KE for the encounter pair can be estimated by using a simple equation proposed independently by R.M. Fuoss and M. Eigen. Both sought to take the complex size and charge into account, expecting larger, oppositely charged ions to meet more frequently than small ions of the same charge. Fuoss used an approach based on statistical thermodynamics and Eigen used one based on kinetics. Their result, which is called the FuossEigen equation, is KE 4/3 a3NAeV/kT (21.10) In this expression a is the distance of closest approach of ions of charge numbers z1 and z2 in a medium of permittivity ε, V is the Coulombic potential energy (z1z2e2/4πεa) of the ions at that distance, and NA is Avogadro’s constant. Although the value predicted by this equation depends strongly on the details of the charges and radii of the ions, typically it clearly favours the encounter if the reactants are large (so a is large) or oppositely charged (V negative). ■ A brief illustration. If one of the reactants is uncharged (as in the case of substitution by NH3), then V 0 and KE 34 a3NA. For an encounter distance of 200 pm for neutral species, we find KE = 4π 3 × (2.00 × 1010 m) × (6.022 × 1023 mol1) = 2.02 × 105 m3 mol1 3 or 2.02 × 10−2 dm3 moll. For two singly charged ions of opposite charge in water at 298 K (when εr 78), other factors being equal, the value of KE is increased by a factor of 2 /4πε akT eV / kT = ee = 36 ■ Table 21.6 Complex formation by the [Ni(OH2)6]2 ion Ligand kr,obs /(dm3 mol1 s1) KE /(dm3 mol1) (kr,obs /KE)/s1 CH3CO2 1 105 3 3 104 F 8 105 1 8 103 HF 3 10 0.15 2 104 3 3 103 H2O NH3 [NH2(CH2)2NH3] SCN 3 5 10 0.15 3 104 4 102 0.02 2 104 6 10 1 6 103 3 The solvent is always in encounter with the ion so that KE is undefined and all rates are inherently first order. 519 Ligand substitution in octahedral complexes 21.6 The activation of octahedral complexes Many studies of substitution in octahedral complexes support the view that the ratedetermining step is dissociative, and we summarize these studies first. However, the reactions of octahedral complexes can acquire a distinct associative character in the case of large central ions (as in the 4d and 5d series) or where the d-electron population at the metal is low (the early members of the d block). More room for attack or lower π electron density appears to facilitate nucleophilic attack and hence permit association. (a) Leaving-group effects Key points: A large effect of the leaving group X is expected in Id reactions; a linear relation is found between the logarithms of the rate constants and equilibrium constants. –4 We can expect the identity of the leaving group X to have a large effect in dissociatively activated reactions because their rates depend on the scission of the M…X bond. When X is the only variable, as in the reaction –5 it is found that the rate constant and equilibrium constant of the reaction are related by (21.11) F– –8 [NiXL5] H2O → [NiL5(OH2)] X are much faster when L is NH3 than when it is H2O. This difference can be explained on the grounds that, as NH3 is a stronger donor than H2O, it increases the electron density at the metal atom and hence facilitates the scission of the MX bond and the formation of X. In the transition state, the stronger donor stabilizes the reduced coordination number. (c) Steric effects Key point: Steric crowding favours dissociative activation because formation of the transition state can relieve strain. Steric effects on reactions with dissociative rate-determining steps can be illustrated by considering the rate of hydrolysis of the first Cl ligand in two complexes of the type [CoCl2(bn)2]: [CoCl2(bn)2] H2O → [CoCl(OH2)(bn)2]2 Cl 1 Figure 21.10 The straight line obtained when the logarithm of a rate constant is plotted against the logarithm of an equilibrium constant shows the existence of a linear free energy relation. The graph is for the reaction [Co(NH3)5X]2 H2O → [Co(NH3)5(OH2)]3 X− with different leaving groups X. Co ...X Potential energy In Co(III), Cr(III), and related octahedral complexes, both cis and trans ligands affect rates of substitution in proportion to the strength of the bonds they form with the metal atom. For instance, hydrolysis reactions such as 0 log K –1 (b) The effects of spectator ligands Key points: In octahedral complexes, spectator ligands affect rates of substitution; the effect is related to the strength of the metalligand interaction, with stronger donor ligands increasing the reaction rate by stabilizing the transition state. H2PO4– –7 (21.12) with p and b constants (and p ≈ 1). The existence of an LFER of unit slope, as for the reaction of [CoX(NH3)5]2, shows that changing X has the same effect on ∆‡G for the conversion of CoX to the transition state as it has on ∆rG O for the complete elimination of X (Fig. 21.11). This observation in turn suggests that in a reaction with an interchange mechanism and a dissociative rate-determining step (Id), the leaving group (an anionic ligand) has already become a solvated ion in the transition state. An LFER with a slope of less than 1, indicating some associative character, is observed for the corresponding complexes of Rh(III). For Co(III), the reaction rates are in the order I Br Cl whereas for Rh(III) they are reversed, and are in the order I Br Cl. This difference should be expected as the softer Rh(III) centre forms more stable complexes with I, compared with Br and Cl, whereas the harder Co(III) centre forms more stable complexes with Cl. Cl– Br– –6 This correlation is illustrated in Fig. 21.10. Because both logarithms are proportional to Gibbs energies (ln kr is approximately proportional to the activation Gibbs energy, ∆‡G, and ln K is proportional to the standard reaction Gibbs energy, ∆rG O ), we can write the following linear free energy relation (LFER): ∆‡G p∆rG O b I– log (kr /s–1) [CoX(NH3)5]2 H2O → [Co(NH3)5(OH2)]3 X ln kr ln K c NO3– Co ...X‘ =\| ∆G =\| ∆G‘ Co–X’ Co–X ∆rG° CoY + X ∆rG°‘ CoY + X’ Reaction coordinate Figure 21.11 The existence of a linear free energy relation with unit slope shows that changing X has the same effect on ∆‡G for the conversion of MX to the transition state as it has on ∆rG O for the complete elimination of X. The reaction profile shows the effect of changing the leaving group from X to X´. 520 21 Coordination chemistry: reactions of complexes Me Cl N Co bn 5 [Co(Cl)2(bn)2]+ Cl Me N Co bn 6 [Co(Cl)2(bn)2]+ The ligand bn is 2,3-butanediamine, and may be either chiral (5) or achiral (6). The important observation is that the complex formed with the chiral form of the ligand hydrolyses 30 times more slowly than the complex of the achiral form. The two ligands have very similar electronic effects, but the CH3 groups are on the opposite sides of the chelate ring in (5) but adjacent and crowded in (6). The latter arrangement is more reactive because the strain is relieved in the dissociative transition state with its lowered coordination number. In general, steric crowding favours an Id process because the five-coordinate transition state can relieve strain. Quantitative treatments of the steric effects of ligands have been developed using molecular modelling computer software that takes into account van der Waals interactions. However, a more pictorial semiquantitative approach was introduced by C.A. Tolman. In this approach, the extent to which various ligands (especially phosphines) crowd each other is assessed by approximating the ligand by a cone with an angle determined from a space-filling model and, for phosphine ligands, an MP bond length of 228 pm (Fig. 21.12 and Table 21.7).2 The ligand CO is small in the sense of having a small cone angle; P(tBu)3 is regarded as bulky because it has a large cone angle. Bulky ligands have considerable steric repulsion with each other when packed around a metal centre. They favour dissociative activation and inhibit associative activation. As an illustration, the rate of the reaction of [Ru(CO)3(PR3)(SiCl3)2] (7) with Y to give [Ru(CO)2Y(PR3)(SiCl3)2] is independent of the identity of Y, which suggests that the ratedetermining step is dissociative. Furthermore, it has been found that there is only a small variation in rate for the substituents Y with similar cone angles but significantly different values of pKa. This observation supports the assignment of the rate changes to steric effects because changes in pKa should correlate with changes in electron distributions in the ligands. (d) Activation energetics Key point: A significant loss of LFSE on going from the starting complex to the transition state results in nonlabile complexes. One factor that has a strong bearing on the activation of complexes is the difference between ligand field stabilization energy (LFSE, Section 20.1) of the reacting complex and that of the transition state (the LFSE‡). This difference is known as the ligand field activation energy (LFAE): q LFAE LFSE‡ LFSE 228 pm Figure 21.12 The determination of the ligand cone angles from space-filling molecular models of the ligand and an assumed MP bond length of 228 pm. (21.13) Table 21.8 gives calculated values of LFAE for the replacement of H2O at a hexaaqua ion, assuming a square-pyramidal transition state (that is, a dissociatively activated reaction), and shows there is correlation between a large ∆‡H and a large LFAE. Thus, we can begin to see why Ni2 and V2 complexes are not very labile: they have large activation energies which arise, in part, from a significant loss of LFSE on moving from an octahedral complex to a transition state. Table 21.7 Tolman cone angles for various ligands Cl Si Ru CO PR3 7 [Ru(CO)3(PR3)(SiCl3)2] Ligand / Ligand / CH3 90 P(OC6H5)3 127 CO 95 PBu3 130 Cl, Et 102 PEt3 132 PF3 104 5 -C5H5(Cp) 136 Br, Ph 105 PPh3 145 I, P(OCH3)3 107 -C5Me5 (Cp) 165 PMe3 118 2,4-Me2C5H3 180 t-Butyl 126 P(t-Bu)3 182 5 2 Tolman’s studies were on nickel complexes, so strictly speaking we are talking about a NiP distance of 228 pm. Ligand substitution in octahedral complexes Table 21.8 Activation parameters for the H2O exchange reactions [M(OH2)6]2 H17 O → [M(OH2)5(17OH2)]2 H2O 2 ‡H/(kJ mol1) Ti2(d2) 2 3 V (d ) 68.6 Cr2(d4, hs) LFSE/O (LFSE)‡/O† LFAE/O 0.8 0.91 0.11 1.2 1 0.2 0.6 0.91 0.31 ‡V/(cm3 mol1) 4.1 Mn2(d5, hs) 33.9 0 0 0 5.4 Fe2(d6, hs) 31.2 0.4 0.46 0.06 3.8 Co2(d7, hs) 43.5 0.8 0.91 0.11 6.1 Ni2(d8) 58.1 1.2 1 0.2 7.2 Octahedral. † Square pyramidal. hs, high spin. (e) Associative activation Key point: A negative volume of activation indicates association of the entering group into the transition state. As we have seen, the activation volume reflects the change in compactness (including that of the surrounding solvent) when the transition state forms from the reactants. The last column in Table 21.8 gives ∆‡V for some H2O ligand-exchange reactions. We see that ∆‡V becomes more positive, from 4.1 cm3 mol1 for V2 to 7.2 cm3 mol1 for Ni2. Because a negative volume of activation can be interpreted as the result of shrinkage when an H2O molecule becomes part of the transition state, we can infer that the activation has significant associative character.3 The increase in ∆‡V follows the increase in the number of nonbonding d electrons from d3 to d8 across the 3d series. In the earlier part of the d block, associative reaction appears to be favoured by a low population of d electrons. Negative volumes of activation are also observed in the 4d and 5d series, such as for Rh(III), and indicate an associative interaction of the entering group in the transition state of the reaction. Associative activation begins to dominate when the metal centre is more accessible to nucleophilic attack, either because it is large or has a low (nonbonding or π) d-electron population, and the mechanism shifts from Id towards Ia. Table 21.9 shows some data for the formation of Br, Cl, and NCS complexes from [Cr(NH3)5(OH2)]3 and [Cr(OH2)6]3. In contrast to the strong dependence of the hexaaqua complex, the pentaammine complex shows only a weak dependence on the identity of the nucleophile. The two complexes probably mark a transition from Id to Ia. In addition, the rate constants for the replacement of H2O in [Cr(OH2)6]3 by Cl, Br, or NCS are smaller by a factor of about 104 than those for the analogous reactions of [Cr(NH3)5(OH2)]3. This difference suggests that the NH3 ligands, which are stronger donors than H2O, promote dissociation of the sixth ligand more effectively. As we saw above, this behaviour is to be expected in dissociatively activated reactions. Table 21.9 Kinetic parameters for anion attack on Cr(III) L H2O X kr / ∆‡H/ 8 3 1 1 (10 dm mol s ) (kJ mol1) ∆‡S/ (J K1 mol1) L NH3 kr / (104 dm3 mol1 s1) Br 0.46 122 8 3.7 Cl 1.15 126 38 0.7 48.7 105 4 4.2 NCS Reaction is [CrL5(OH2)]3 X → [CrL5X]2 H2O ‡ 3 The limiting value for ∆ V is approximately ±18 cm3 mol–1, the molar volume of water, with A reactions having negative and D reactions positive values. 521 522 21 Coordination chemistry: reactions of complexes E X A M PL E 21. 3 Interpreting kinetic data in terms of a mechanism The second-order rate constants for formation of [VX(OH2)5] from [V(OH2)6]2 and X for X Cl, NCS, and N3 are in the ratio 1:2:10. What do the data suggest about the rate-determining step for the substitution reaction? Answer We need to consider the factors that might affect the rate of the reaction. All three ligands are singly charged anions of similar size, so we can expect the encounter equilibrium constants to be similar. Therefore, the second-order rate constants are proportional to first-order rate constants for substitution in the encounter complex. The second-order rate constant is equal to KEkr,2, where KE is the pre-equilibrium constant and kr,2 is the first-order rate constant for substitution of the encounter complex. The greater rate constants for NCS than for Cl, and especially the five-fold difference of NCS from its close structural analogue N3, suggest some contribution from nucleophilic attack and an associative reaction. By contrast, there is no such systematic pattern for the same anions reacting with Ni(II), for which the reaction is believed to be dissociative. Self-test 21.3 Use the data in Table 21.8 to estimate an appropriate value for KE and calculate kr,2 for the reactions of V(II) with Cl if the observed second-order rate constant is 1.2 102 dm3 mol1 s1. 21.7 Base hydrolysis Key points: Octahedral substitution can be greatly accelerated by OH ions when ligands with acidic hydrogens are present as a result of the decrease in charge of the reactive species and the increased ability of the deprotonated ligand to stabilize the transition state. Consider a substitution reaction in which the ligands possess acidic protons, such as [CoCl(NH3)5]2 OH → [Co(OH)(NH3)5]2 Cl An extended series of studies has shown that whereas the rate law is overall second order, with rate kr[CoCl(NH3)52][OH], the mechanism is not a simple bimolecular attack by OH on the complex. For instance, whereas the replacement of Cl by OH is fast, the replacement of Cl by F is slow, even though F resembles OH more closely in terms of size and nucleophilicity. There is a considerable body of indirect evidence relating to the problem but one elegant experiment makes the essential point. This conclusive evidence comes from a study of the 18O/16O isotope distribution in the product [Co(OH)(NH3)5]2. It is known that the 18O/16O ratio differs between H2O and OH at equilibrium, and this fact can be used to establish whether the incoming group is H2O or OH. The 18O/16O isotope ratio in the cobalt product matches that for H2O, not that for the OH ions, proving that it is an H2O molecule that is the entering group. The mechanism that takes these observations into account supposes that the role of OH is to act as a Brønsted base, not an entering group: [CoCl(NH3)5]2 OH → [CoCl(NH2)(NH3)4] H2O [CoCl(NH2)(NH3)4] → [Co(NH2)(NH3)4]2 Cl (slow) Table 21.10 Stereochemical course of hydrolysis reactions of [CoAX(en)2] A cis X Percentage cis in product OH Cl 100 Cl 100 NCS Cl 100 Cl Br 100 Cl trans 2 NO Cl 0 NCS Cl 5070 Cl Cl 35 75 OH Cl X is the leaving group. [Co(NH2)(NH3)4]2 H2O → [Co(OH)(NH3)5]2 (fast) In the first step, an NH3 ligand acts as a Brønsted acid, resulting in the formation of its conjugate base, the NH2 ion, as a ligand. Because the deprotonated form of the complex has a lower charge, it will be able to lose a Cl ion more readily than the protonated form, thus accelerating the reaction. In addition, according to the conjugate base mechanism, loss of a proton from an NH3 ligand changes it from a pure -donor ligand to a strong and π donor (as NH2) and so helps to stabilize the five-coordinate transition state, greatly accelerating the loss of the Cl ion (see the next brief illustration). 21.8 Stereochemistry Key point: Reaction through a square-pyramidal intermediate results in retention of the original geometry but reaction through a trigonal-bipyramidal intermediate can lead to isomerization. Classic examples of octahedral substitution stereochemistry are provided by Co(III) complexes. Table 21.10 shows some data for the hydrolysis of cis- and trans-[CoAX(en)2] Ligand substitution in octahedral complexes (8) and (9), respectively, where X is the leaving group (either Cl or Br) and A is OH, NCS, or Cl. The stereochemical consequences of substitution of octahedral complexes are much more intricate than those of square-planar complexes. The cis complexes do not undergo isomerization when substitution occurs, whereas the trans forms show a tendency to isomerize in the order A NO2 Cl NCS OH. The data can be understood in terms of an Id mechanism and by recognizing that the five-coordinate metal centre in the transition state may resemble either of the two stable geometries for five-coordination, namely square-pyramidal or trigonal-bipyramidal. As can be seen from Fig. 21.13, reaction through the square-pyramidal complex results in retention of the original geometry but reaction through the trigonal-bipyramidal complex can lead to isomerization. The cis complex gives rise to a square-pyramidal intermediate but the trans isomer gives a trigonal-bipyramidal intermediate. For d metals, trigonal-bipyramidal complexes are favoured when the ligands in the equatorial positions are good π donors, and a good π-donor ligand trans to the leaving group Cl favours isomerization (10). 523 A X Co en 8 cis-[CoAX(en)2]+ ■ A brief illustration. Substitution of Co(III) complexes of the type [CoAX(en)2] results in trans to cis isomerization, but only when the reaction is catalysed by a base. In a base hydrolysis reaction, one of the NH2R groups of the en ligands loses a proton and becomes its conjugate base, :NHR. The :NHR ligand group is a strong π-donor ligand and favours a trigonal bipyramid of the type shown in Fig. 21.13 and it may be attacked in the way shown there. If the direction of attack of the incoming ligands were random, we would expect 33 per cent trans and 67 per cent cis product. ■ A 21.9 Isomerization reactions Co en Key points: Isomerization of a complex can take place by mechanisms that involve substitution, bond cleavage, and reformation, or twisting. Isomerization reactions are closely related to substitution reactions; indeed, a major pathway for isomerization is often via substitution. The square-planar Pt(II) and octahedral Co(III) complexes we have discussed can form five-coordinate trigonal-bipyramidal transition states. The interchange of the axial and equatorial ligands in a trigonal-bipyramidal complex can be pictured as occurring by a Berry pseudorotation through a squarepyramidal conformation (Section 7.4 and Fig. 21.14). As we have seen, when a trigonalbipyramidal complex adds a ligand to produce a six-coordinate complex, a new direction of attack of the entering group can result in isomerization. If a chelate ligand is present, isomerization can occur as a consequence of metalligand bond breaking, and substitution need not occur. An example is the exchange of the ‘outer’ CD3 group with the ‘inner’ CH3 group during the isomerization of a substituted tris(acetylacetonato)cobalt(III) complex, (11) → (12). An octahedral complex can also undergo isomerization by an intramolecular twist without loss of a ligand or breaking of a bond. There is evidence, for example, that racemization of [Ni(en)3]2 occurs by such an internal twist. Two possible paths are the Bailar twist and the RayDutt twist (Fig. 21.15). A Y + X 9 trans-[CoAX(en)2]+ T Co 10 A Y –X A Y X A A –X ) (3 ) (2 ) (1 A A ) (1 (2 ) )( 3 Figure 21.13 Reaction through a square-pyramidal complex (top path) results in retention of the original geometry, but reaction through a trigonal-bipyramidal complex (bottom path) can lead to isomerization. Figure 21.14 The exchange of axial and equatorial ligands by a twist through a square-pyramidal conformation of the complex. 524 21 Coordination chemistry: reactions of complexes H 3C O O D 3C O C H3 (a) Co D3 C H 3C O O H 3C Co O 11 (b) O C H3 O H3 C O C H3 Co O C H3 Co C D3 C D3 O O Figure 21.15 (a) The Bailar twist and (b) the RayDutt twist by which an octahedral complex can undergo isomerization without losing a ligand or breaking a bond. 12 Redox reactions As remarked in Chapter 5, redox reactions can occur by the direct transfer of electrons (as in some electrochemical cells and in many solution reactions) or by the transfer of atoms and ions, as in the transfer of O atoms in reactions of oxidoanions. Because redox reactions in solution involve both an oxidizing and a reducing agent, they are usually bimolecular in character. The exceptions are reactions in which one molecule has both oxidizing and reducing centres. 21.10 The classification of redox reactions Key points: In an inner-sphere redox reaction a ligand is shared to form a transition state; in an outersphere redox reaction there is no bridging ligand between the reacting species. In the 1950s, Henry Taube identified two mechanisms of redox reactions for metal complexes. One is the inner-sphere mechanism, which includes atom-transfer processes. In an inner-sphere mechanism, the coordination spheres of the reactants share a ligand transitorily and form a bridged transition state. The other is an outer-sphere mechanism, which includes many simple electron transfers. In an outer-sphere mechanism, the complexes come into contact without sharing a bridging ligand and the electron tunnels from one metal atom to the other. The mechanisms of some redox reactions have been definitively assigned as inner- or outer-sphere. However, the mechanisms of a vast number of reactions are unknown because it is difficult to make unambiguous assignments when complexes are labile. Much of the study of well-defined examples is directed towards the identification of the parameters that differentiate the two paths with the aim of being able to making correct assignments in more difficult cases. 21.11 The inner-sphere mechanism Key point: The rate-determining step of an inner-sphere redox reaction may be any one of the component processes, but a common one is electron transfer. 4+ Co Cl Cr NH3 OH2 13 The inner-sphere mechanism was first confirmed for the reduction of the nonlabile complex [CoCl(NH3)5]2 by Cr2(aq). The products of the reaction included both Co2(aq) and [CrCl(OH2)5]2, and addition of 36Cl to the solution did not lead to the incorporation of any of the isotope into the Cr(III) product. Furthermore, the reaction is much faster than reactions that remove Cl from nonlabile Co(III) or introduce Cl into the nonlabile [Cr(OH2)6]3 complex. These observations suggest that Cl has moved directly from the coordination sphere of one complex to that of the other during the reaction. The Cl attached to Co(III) can easily enter into the labile coordination sphere of [Cr(OH2)6]2 to produce a bridged intermediate (13). Inner-sphere reactions, though involving more steps than outer-sphere reactions, can be fast. Figure 21.16 summarizes the steps necessary for such a reaction to occur. Redox reactions Inner sphere Outer sphere Reactants Association Precursor complex Bridge formation Electron transfer Successor complex Diffusion apart Products Figure 21.16 The different pathways followed by inner- and outer-sphere mechanisms. The first two steps of an inner-sphere reaction are the formation of a precursor complex and the formation of the bridged binuclear intermediate. These two steps are identical to the first two steps in the EigenWilkins mechanism (Section 21.5). The final steps are electron transfer through the bridging ligand to give the successor complex, followed by dissociation to give the products. The rate-determining step of the overall reaction may be any one of these processes, but the most common one is the electron-transfer step. However, if both metal ions have a nonlabile electron configuration after electron transfer, then the break-up of the bridged complex is rate determining. An example is the reduction of [RuCl(NH3)5]2 by [Cr(OH2)6]2, in which the rate-determining step is the dissociation of the Cl-bridged complex [RuII(NH3)5(μ-Cl)CrIII(OH2)5]4. Reactions in which the formation of the bridged complex is rate determining tend to have similar rate constants for a series of partners of a given species. For example, the oxidation of V2(aq) has similar rate constants for a long series of Co(III) oxidants with different bridging ligands. The explanation is that the ratedetermining step is the substitution of an H2O molecule from the coordination sphere of V(II), which is quite slow (Table 21.8). 525 526 21 Coordination chemistry: reactions of complexes Table 21.11 Second-order rate constants for selected inner-sphere reactions with variable bridging ligands Oxidant Reductant [Co(NH3)6]3 Bridging ligand kr /(dm3 mol1 s1) 8 105 [Cr(OH2)6]2 [CoF(NH3)5] [Cr(OH2)6] F 2.5 105 [CoCl(NH3)5]2 [Cr(OH2)6]2 Cl 6.0 105 3.0 106 2 2 2 2 [Col(NH3)5] [Cr(OH2)6] I [Co(NCS)(NH3)5]2 [Cr(OH2)6]2 NCS 1.9 101 2 [Co(SCN)(NH3)5] 2 [Cr(OH2)6] SCN 1.9 105 [Co(OH2)(NH3)5]2 [Cr(OH2)6]2 H2O 1.0 101 7.4 103 2 [CrF(OH2)5] 2 [Cr(OH2)6] F The numerous reactions in which electron transfer is rate determining do not display such simple regularities. Rates vary over a wide range as metal ions and bridging ligands are varied.4 The data in Table 21.11 show some typical variations as bridging ligand, oxidizing metal, and reducing metal are changed. All the reactions in Table 21.11 result in the change of oxidation number by ±1. Such reactions are still often called one-equivalent processes, the name reflecting the largely outmoded term ‘chemical equivalent’. Similarly, reactions that result in the change of oxidation number by ±2 are often called two-equivalent processes and may resemble nucleophilic substitutions. This resemblance can be seen by considering the reaction [PtIICl4]2 [PtIVCl6]2 → [PtIVCl6]2 [PtIICl4]2 Cl 5– Pt(IV) Cl Pt(II) 14 N N Mo(IV) NO3 → MoONO2 → Mo O NO2 15 Pyrazine N which occurs through a Cl bridge (14). The reaction depends on the transfer of a Cl ion in the break-up of the successor complex. There is no difficulty in assigning an inner-sphere mechanism when the reaction involves ligand transfer from an initially nonlabile reactant to a nonlabile product. With more labile complexes, inner-sphere reactions should always be suspected when ligand transfer occurs as well as electron transfer, and if good bridging groups such as Cl, Br, I, N3, CN, SCN, pyrazine (15), 4,4-bipyridine (16), and 4-dimethylaminopyridine (17) are present. Although all these ligands have lone pairs to form the bridge, this may not be an essential requirement. For instance, just as the carbon atom of a methyl group can act as a bridge between OH and I in the hydrolysis of iodomethane, so it can act as a bridge between Cr(II) and Co(III) in the reduction of methylcobalt species by Cr(II). The oxidation of a metal centre by oxidoanions is also an example of an inner-sphere process. For example, in the oxidation of Mo(IV) by NO3 ions, an O atom of the nitrate ion binds to the Mo atom, facilitating the electron transfer from Mo to N, and then remains bound to the Mo(VI) product: ■ A brief illustration. The rate constant for the oxidation of the Ru(II) centre by the Co(III) centre N in the bimetallic complex (18) is 1.0 102 dm3 mol1 s1, whereas the rate constant for complex (19) is 1.6 102 dm3 mol1 s1. In both complexes there is a pyridine carboxylic acid group bridging the two metal centres. These groups are bound to both metal atoms and could facilitate an 16 4,4’-Bipyridine H3 N N NMe2 H3 N NH3 C o I N R u O H O NH NH 3 3 NH 3 NH3 17 4-Dimethylaminopyridine I 18 H 3 N NH 3 4+ H 3 N NH 3 O 2 H3 N H 3N N R uII NH 3 OH 2 NH3 NH 3 C oIII O NH NH 3 3 4+ O 19 4 Some bridged intermediates have been isolated with the electron clearly located on the bridge, but we shall not consider these here. 527 Redox reactions electron-transfer process through the bridge, suggesting an inner-sphere process. The fact that the rate constant changes between the two complexes, when the only substantive difference between them is in the substitution pattern of the pyridine ring, confirms that the bridge must be playing a role in the electron transfer process. ■ 2+ 3+ 21.12 The outer-sphere mechanism (a) Key points: An outer-sphere redox reaction involves electron tunnelling between two reactants without any major disturbance of their covalent bonding or inner coordination spheres; the rate constant depends on the electronic and geometrical structures of the reacting species and on the Gibbs energy of reaction. e– A conceptual starting point for understanding the principles of outer-sphere electron transfer is the deceptively simple reaction called electron self-exchange. A typical example is the exchange of an electron between [Fe(OH2)6]3 and [Fe(OH2)6]2 ions in water. (b) [Fe(OH2)6]3 [Fe(OH2)6]2 → [Fe(OH2)6]2 [Fe(OH2)6]3 kET = Nee− ‡ G / RT (21.14) in which kET is the rate constant for electron transfer and ∆‡G is given by 2 ⎛ GO ⎞ (21.15) ‡G = 41 ⎜ 1 + r ⎟ ⎠ ⎝ with ∆rG O the standard reaction Gibbs energy (which is obtained from the difference in standard potentials of the redox partners) and the reorganization energy, the energy required to move the nuclei associated with the reactant to the positions they adopt in the product immediately before the transfer of the electron. This energy depends on the changes in metalligand bond lengths (the so-called inner-sphere reorganization energy) and alterations in solvent polarization, principally the orientation of the solvent molecules around the complex (the outer-sphere reorganization energy). The pre-exponential factor in eqn 21.14 has two components, the nuclear frequency factor vN and the electronic factor e. The former is the frequency at which the two complexes, having already encountered each other in the solution, attain the transition state. The electronic factor gives the probability on a scale from 0 to 1 that an electron will transfer when the transition state is reached; its precise value depends on the extent of overlap of the donor and acceptor orbitals. A small reorganization energy and a value of e close to 1 corresponds to a redox couple capable of fast electron self-exchange. The first requirement is achieved if the transferred 5 Tunnelling refers to a process where, according to classical physics, the electrons do not have sufficient energy to overcome the barrier but penetrate into or through it. 3+ 2+ (c) Figure 21.17 Electron transfer between two metal ions in a precursor complex is not productive until their coordination shells have reorganized to be of equal size. (a) Reactants; (b) reactant complexes having distorted into the same geometry; (c) products. ‘Reactant’ form ‘Product’ form Potential energy Self-exchange reactions can be studied over a wide dynamic range with techniques ranging from isotopic labelling to NMR, with EPR being useful for even faster reactions. The rate constant of the Fe3/Fe2 reaction is about 1 dm3 mol1 s1 at 25°C. To set up a scheme for the mechanism, we suppose that Fe3 and Fe2 come together to form a weak outer-sphere complex (Fig. 21.17). We need to consider, by assuming that the overlap of their respective acceptor and donor orbitals is sufficient to give a reasonable tunnelling probability,5 how rapidly an electron transfers between the two metal ions. To explore this problem, we invoke the FranckCondon principle introduced originally to account for the vibrational structure of electronic transitions in spectroscopy, which states that electronic transitions are so fast that they take place in a stationary nuclear framework. In Fig. 21.18 the nuclear motions associated with the ‘reactant’ Fe3 and its ‘conjugate product’ Fe2 are represented as displacements along a reaction coordinate. If [Fe(OH2)6]3 lies at its energy minimum, then an instantaneous electron transfer would give a compressed state of [Fe(OH2)6]2. Likewise, the removal of an electron from Fe2 at its energy minimum would give an expanded state of [Fe(OH2)6]3. The only instant at which the electron can transfer within the precursor complex is when both [Fe(OH2)6]3 and [Fe(OH2)6]2 have achieved the same nuclear configuration by thermally induced fluctuations. That configuration corresponds to the point of intersection of the two curves, and the energy required to reach this position is the Gibbs energy of activation, ∆‡G. If [Fe(OH2)6]3 and [Fe(OH2)6]2 differ in their nuclear configurations, ∆‡G is larger and electron exchange is slower. The difference in rates is expressed quantitatively by the Marcus equation l = ∆\|G Nuclear coordinates Figure 21.18 The potential energy curves for electron self-exchange. The nuclear motions of both the oxidized and reduced species (shown displaced along the reaction coordinate) and the surrounding solvent are represented by potential wells. Electron transfer to oxidized metal ion (left) occurs once fluctuations of its inner and outer coordination shell bring it to a point (denoted ) on its energy surface that coincides with the energy surface of its reduced state (right). This point is at the intersection of the two curves. The activation energy depends on the horizontal displacement of the two curves (representing the difference in sizes of the oxidized and reduced forms). 21 Coordination chemistry: reactions of complexes Potential energy 528 l ∆rG = 0 = ∆\|G = ␭ /4 Nuclear coordinates Potential energy (a) (b) \| ∆=G = 0 ∆rG = – ␭ Nuclear coordinates electron is removed from or added to a nonbonding orbital, as the change in metalligand bond length is then least. It is also likely if the metal ion is shielded from the solvent, in the sense of it being sterically difficult for solvent molecules to approach close to the metal ion, because the polarization of the solvent is normally a major component of the reorganization energy. Simple metal ions such as aqua species typically have well in excess of 1 eV, whereas buried redox centres in enzymes, which are very well shielded from the solvent, can have values as low as 0.25 eV. A value of e close to 1 is achieved if there is good orbital overlap between the two components of the precursor complex. For a self-exchange reaction, ∆rG O 0 and therefore, from eqn 21.15, ∆‡G 41 and the rate of electron transfer is controlled by the reorganization energy (Fig. 21.19a). To a considerable extent, the rates of self-exchange can be interpreted in terms of the types of orbitals involved in the transfer (Table 21.12). In the [Cr(OH2)6]3/2 self-exchange reaction, an electron is transferred between antibonding orbitals, and the consequent extensive change in metalligand bond lengths results in a large inner-sphere reorganization energy and therefore a slow reaction. The [Co(NH3)6]3/2 couple has an even greater reorganization energy because two electrons are moved into the orbital as rearrangement occurs and the reaction is even slower. With the other hexaaqua and hexaammine complexes in the table, the electron is transferred between weakly antibonding or nonbonding π orbitals, the inner-sphere reorganization is less extensive, and the reactions are faster. The bulky, hydrophobic chelating ligand bipyridyl acts as a solvent shield, thus decreasing the outersphere reorganization energy. Bipyridyl and other π-acceptor ligands allow electrons in an orbital with π symmetry on the metal ion to delocalize on to the ligand. This delocalization effectively lowers the reorganization energy when the electron is transferred between π orbitals, as occurs with Fe and Ru, where the electron transfer is between t2g orbitals (which, as explained in Section 20.2, can participate in π bonding), but not with Ni, where the electron transfer is between eg orbitals. Delocalization can also increase the electronic factor. Self-exchange reactions are helpful for pointing out the concepts that are involved in electron transfer, but chemically useful redox reactions occur between different species and involve net electron transfer. For the latter reactions, ∆rG O is nonzero and contributes to the rate through eqns 21.14 and 21.15. Provided \|∆rG O \| \|\| eqn 21.15 becomes 2 G⎞ 2rG ⎞ ⎛ ⎛ † ∆ †G = 41 ⎜ 1 + r ⎟ ≈ 41 ⎜ 1 + = ⎠ ⎟⎠ ⎝ ⎝ and then according to eqn 21.14 Potential energy ∆=G kET ≈ Nee− (+2 rG ∆rG < – ␭ (\|∆rG \| > ␭ ) O 1 4 ( + 2 G) r )/ 4 RT Because 0 and ∆rG O 0 for thermodynamically feasible reactions, provided \|∆rG O \| \|\| the rate constant increases exponentially as ∆rG O becomes increasingly favourable (that is, more negative). However, as \|∆rG O \| becomes comparable to \|\|, this equation breaks down and we see that the reaction rate peaks before declining as \|∆rG O \| \|\|. Equation 21.15 shows that ∆‡G 0 when ∆rG O . That is, the reaction becomes ‘activationless’ when the standard reaction Gibbs energy and the reorganization energy Table 21.12 Correlations between rate constants for electron self-exchange reactions Reaction (c) Electron configuration d/pm k11 / (dm3 mol1 s1) 3 3 1 t2g /t2g eg 20 1 105 t /t 13 1 105 Nuclear coordinates Figure 21.19 Variation of the activation Gibbs energy (∆‡G) with reaction Gibbs energy (∆rG O ). (a) In a self-exchange reaction, ∆rG O 0 and ∆‡G /4. (b) A reaction is ‘activationless’ when ∆rG O . (c) ∆‡G increases (the rate diminishes) as ∆rG O becomes more negative beyond ∆rG O . [Cr(OH2)6]3/2 3/2 [V(OH2)6] [Fe(OH2)6]3/2 2 2g 3 2g 3 2 4 2 t2g eg /t2g eg 13 1.1 3/2 5 6 t2g /t2g 9 20 [Ru(NH3)6]3/2 5 6 t2g /t2g 4 9.6 102 [Ru(OH2)6] [Co(NH3)6] t /t e 22 2 108 [Fe(bpy)3]3/2 5 6 t2g /t2g 0 3 108 t /t 0 4 108 6 1 6 2 t2g e g /t2g eg 12 1.5 103 3/2 [Ru(bpy)3] 3/2 [Ni(bpy)3]3/2 6 2g 5 2g 5 2 2g g 6 2g Redox reactions N N O P h Ir N N Ir C O P P P h C O 4 2 0 –2 0 50 100 150 –∆rG ° 200 Figure 21.20 The theoretical dependence of the log10 of the reaction rate (in arbitrary units) on ∆rG O for a redox reaction with 1.0 eV (96.5 kJ mol1). N R 6 log kr cancel (Fig. 21.19b). The activation energy now increases as ∆rG O becomes more negative and the reaction rate decreases. This slowing of the reaction as the standard Gibbs energy of the reaction becomes more exergonic is called inverted behaviour (Fig. 21.19c). Inverted behaviour has important consequences, a notable one relating to the long-range electron transfer involved in photosynthesis. Photosystems are complex proteins containing lightexcitable pigments such as chlorophyll and a chain of redox centres having low reorganization energies. In this chain, one highly exergonic recombination of the photoelectron with oxidized chlorophyll is sufficiently retarded (to 30 ns) to allow the electron to escape (in 200 ps) and proceed down the photosynthetic electron-transport chain, ultimately to produce reduced carbon compounds (Section 27.10). The theoretical dependence of the reaction rate of a reaction on the standard Gibbs energy is plotted in Fig. 21.20, and Fig. 21.21 shows the observed variation of reaction rate with ∆rG O for the iridium complex (20). The results plotted in Fig. 21.21 represent the first unambiguous experimental observation of an inverted region for a synthetic complex. 529 P h O N R P h 12 20 Ox1 Red2 → Red1 Ox2 8 If we suppose that the reorganization energy for this reaction is the average of the values for the two self-exchange processes, we can write 12 12 (11 22) and then manipulation of eqns 21.14 and 21.15 gives the Marcus cross-relation k12 (k11k22K12f12) 1/2 (21.16) in which k12 is the rate constant, K12 is the equilibrium constant obtained from ∆rG O and k11 and k22 are the respective self-exchange rate constants for the two reaction partners. For reactions between simple ions in solution, when the standard Gibbs reaction energy is not too large, an LFER of the kind expressed by eqn 21.12 exists between ∆‡G and ∆rG O and f12 can normally be set to 1. However, for reactions that are highly favourable thermodynamically (that is, ∆rG O large and negative) the LFER breaks down. The term f12 takes into consideration the nonlinearity of the relationship between ∆‡G and ∆rG O and is given by log f12 = (log K12 )2 4 log (k11k22 / Z 2 ) 10 log kr The Marcus equation can be used to predict the rate constants for outer-sphere electron transfer reactions between different species. Consider the electron-transfer reaction between an oxidant Ox1 and reductant Red2: (21.17) where Z is the constant of proportionality between the encounter density in solution (in moles of encounters per cubic decimetre per second) and the molar concentrations of the reactants; it is often taken to be 1011 mol1 dm3 s1. ■ A brief illustration. The rate constants for k11 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → [Co(bpy)3]3 [Co(bpy)3]2 [Co(bpy)3]2 [Co(bpy)3]3 ⎯ k 22 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → [Co(terpy)2]3 [Co(terpy)2]2 [Co(terpy)2]2 [Co(terpy)2]3 ⎯ (where bpy is bipyridyl and terpy is tripyridyl) are k11 9.0 dm3 mol1 s1 and k22 48 dm3 mol1 s1, and K12 3.57. Then for the outer-sphere reduction of [Co(bpy)3]3 by [Co(terpy)2]2, eqn 21.16 with f12 1 (as noted above) gives k12 (9.0 48 3.57)½ dm3 mol1 s1 39 dm3 mol1 s1 This result compares reasonably well with the experimental value, which is 64 dm3 mol1 s1. ■ 6 0 50 100 150 –∆rG ° 200 Figure 21.21 Plot of log10 kr against ∆rG O for the iridium complex (18) in acetonitrile solution at room temperature. The rates are for electron transfer from the photoexcited Ir2 unit to the outlying pyridinium functionalities. (Data from L.S. Fox, M. Kozik, J.R. Winkler, and H.B. Gray, Science 1990, 247, 1069). 530 21 Coordination chemistry: reactions of complexes Photochemical reactions The absorption of a photon of ultraviolet radiation or visible light increases the energy of a complex by between 170 and 600 kJ mol1. Because these energies are larger than typical activation energies it should not be surprising that new reaction channels are opened. However, when the high energy of a photon is used to provide the energy of the primary forward reaction, the back reaction is almost always very favourable, and much of the design of efficient photochemical systems lies in trying to avoid the back reaction. 21.13 Prompt and delayed reactions Key point: Reactions of electronically excited species are classified as prompt or delayed. In some cases, the excited state formed after absorption of a photon dissociates almost immediately after it is formed. Examples include formation of the pentacarbonyl intermediates that initiate ligand substitution in metal carbonyl compounds: hν Cr(CO)6 ⎯⎯→ Cr(CO)5 CO and the scission of CoCl bonds: hν ( λ = 350 nm) → [CoII(NH ) ]2 Cl. [CoIIICl(NH3)5]2 ⎯⎯⎯⎯⎯⎯ 3 5 Both processes occur in less than 10 ps and hence are called prompt reactions. In the second reaction, the quantum yield, the amount of reaction per mole of photons absorbed, increases as the wavelength of the radiation is decreased (and the photon energy correspondingly increased, Ephoton hc/). The energy in excess of the bond energy is available to the newly formed fragments and increases the probability that they will escape from each other through the solution before they have an opportunity to recombine. Some excited states have long lifetimes. They may be regarded as energetic isomers of the ground state that can participate in delayed reactions. The excited state of [RuII(bpy)3]2 created by photon absorption in the metal-to-ligand charge-transfer band (Section 20.5) may be regarded as a Ru(III) cation complexed to a radical anion of the ligand. Its redox reactions can be explained by adding the excitation energy (expressed as a potential by using FE ∆rG and equating ∆rG to the molar excitation energy) to the ground-state reduction potential (Fig. 21.22). 21.14 dd and charge-transfer reactions [Ru(III)(bpy)(bpy)2]2+ +202 kJ mol–1 hν (590 nm) E° = –0.84 V (+80 kJ mol–1) [Ru(III)(bpy)3]3+ E° = –1.26 V (+122 kJ mol–1) [Ru(II)(bpy)3]2+ Figure 21.22 The photoexcitation of [RuII(bpy)3]2 can be treated as if the excited state is a Ru(III) cation complexed to a radical anion of the ligand. Key point: A useful first approximation is to associate photosubstitution and photoisomerization with dd transitions and photoredox reactions with charge-transfer transitions, but the rule is not absolute. There are two main types of spectroscopically observable electron promotion in d-metal complexes, namely dd transitions and charge-transfer transitions (Sections 20.4 and 20.5). A dd transition corresponds to the essentially angular redistribution of electrons within a d shell. In octahedral complexes, this redistribution often corresponds to the 2 3 eg1 ← t2g ) occupation of ML antibonding eg orbitals. An example is the 4T1g ← 4A2g (t2g transition in [Cr(NH3)6]3. The occupation of the antibonding eg orbital results in a quantum yield close to 1 (specifically 0.6) for the photosubstitution hν [Cr(NH3)6]3 H2O ⎯⎯→ [Cr(NH3)5(OH2)]3 NH3 This is a prompt reaction, as it occurs in less than 5 ps. Charge-transfer transitions correspond to the radial redistribution of electron density. They correspond to the promotion of electrons into predominantly ligand orbitals if the transition is metal-to-ligand or into orbitals of predominantly metal character if the transition is ligand-to-metal. The former process corresponds to oxidation of the metal centre and the latter to its reduction. These excitations commonly initiate photoredox reactions of the kind already mentioned in connection with Co(III) and Ru(II). 531 Photochemical reactions Although a useful first approximation is to associate photosubstitution and photoisomerization with dd transitions and photoredox with charge-transfer transitions, the rule is not absolute. For example, it is not uncommon for a charge-transfer transition to result in photosubstitution by an indirect path: hν [CoIIICl(NH3)5]2 H2O ⎯⎯→ [CoII(NH3)5(OH2)]2 Cl. [CoII(NH3)5(OH2)]2 Cl. → [CoIII(NH3)5(OH2)]3 Cl In this case, the aqua complex formed after the homolytic fission of the CoCl bond is reoxidized by the Cl atom. The net result leaves the Co substituted. Conversely, some excited states show no differences in substitutional reactivity compared with the ground state: the long-lived excited 2E state of [Cr(bpy)3]3 results from a pure dd transition and its lifetime of several microseconds allows the excess energy to enhance its redox reactions. The standard potential (1.3 V), calculated by adding the excitation energy to the ground-state value, accounts for its function as a good oxidizing agent, in which it undergoes reduction to [Cr(bpy)3]2. 21.15 Transitions in metalmetal bonded systems Key points: Population of a metalmetal antibonding orbital can sometimes initiate photodissociation; such excited states have been shown to initiate multielectron redox photochemistry. We might expect the ← transition in metalmetal bonded systems to initiate photodissociation as it results in the population of an antibonding orbital of the metalmetal system. It is more interesting that such excited states have also been shown to initiate multi-electron redox photochemistry. One of the best characterized systems is the dinuclear platinum complex [Pt2(μ-P2O5H2)4]4-, called informally ‘PtPOP’ (21). There is no metalmetal bonding in the ground state of this Pt(II)Pt(II) d8d8 species. The HOMOLUMO pattern indicates that excitation populates a bonding orbital between the two metal atoms (Fig. 21.23). The lowest-lying excited state has a lifetime of 9 μs and is a powerful reducing agent, reacting by both electron and halogen-atom transfer. The most interesting oxidation products are Pt(III)––Pt(III), which contain X ligands (where X is halogen or pseudohalogen) at both ends and a metalmetal single bond. Irradiation in the presence of (Bu)3SnH gives a dihydrido product that can eliminate H2. Irradiation of the quadruply bonded dinuclear cluster [Mo2(O2P(OC6H5)2)4] (22) at 500 nm in the presence of ClCH2CH2Cl results in production of ethene and the addition of two Cl atoms to the two Mo atoms, with a two-electron oxidation. The reaction proceeds in one-electron steps, and requires a complex with the metal atoms shielded by sterically crowding ligands. If smaller ligands are present, the reaction that occurs instead is a photochemical oxidative addition of the organic molecule. Pt Pt–Pt Pt pσ pz 2 × dx2–y2 pz pσ dσ OPh dz2 dz2 P H O o M Pt P O 21 [Pt2(µ-P2O5H2)4]4–, PtPOP 22 [Mo2(O2P(OPh)2)4] 2 × (dxy,dyz,dzx) dσ Figure 21.23 The dinuclear complex [Pt2(-P2O5H2)4]4− consists of two faceto-face square-planar complexes held together by a bridging pyrophosphito ligand. The metal pz and dz2 orbitals interact along the PtPt axis. The other p and d orbitals are considered to be nonbonding. Photoexcitation results in an electron in the antibonding orbital moving into the bonding orbital. 532 21 Coordination chemistry: reactions of complexes FURTHER READING G.J. Leigh and N. Winterbottom (ed.), Modern coordination chemistry: the legacy of Joseph Chatt. Royal Society of Chemistry, Cambridge (2002). A readable historical discussion of this area. M.L. Tobe and J. Burgess, Inorganic reaction mechanisms. Longman, Harlow (1999). R.G. Wilkins, Kinetics and mechanism of reactions of transition metal complexes. VCH, Weinheim (1991). A special issue of Coordination Chemistry Reviews has been dedicated to the work of Henry Taube. See Coord. Chem. Rev., 2005, 249. Two readable discussions of redox processes are to be found in Taube’s 1983 Nobel Prize lecture reprinted in Science, 1984, 226, 1028 and in Marcus’s 1992 Nobel Prize lecture published in Nobel Lectures: Chemistry 19911995, World Scientific, Singapore (1997). EXERCISES 21.1 The rate constants for the formation of [CoX(NH3)5]2 from [Co(NH3)5OH2]3 for X Cl, Br, N3, and SCN differ by no more than a factor of two. What is the mechanism of the substitution? with OH. Explain the anomaly. What is the implication of your explanation for the behaviour of a complex lacking Brønsted acidity on the ligands? 21.2 If a substitution process is associative, why may it be difficult to characterize an aqua ion as labile or inert? 21.14 Predict the products of the following reactions: 21.3 The reactions of Ni(CO)4 in which phosphines or phosphites replace CO to give Ni(CO)3L all occur at the same rate regardless of which phosphine or phosphite is being used. Is the reaction d or a? (b) [PtCl4]2 2 PR3 21.4 Write the rate law for formation of [MnX(OH2)5] from the aqua ion and X. How would you undertake to determine if the reaction is d or a? 21.15 Put in order of increasing rate of substitution by H2O the complexes (a) [Co(NH3)6]3, (b) [Rh(NH3)6]3, (c) [Ir(NH3)6]3, (d) [Mn(OH2)6]2, (e) [Ni(OH2)6]2. 21.5 Octahedral complexes of metal centres with high oxidation numbers or of d metals of the second and third series are less labile than those of low oxidation number and d metals of the first series of the block. Account for this observation on the basis of a dissociative rate-determining step. 21.16 State the effect on the rate of dissociatively activated reactions of Rh(III) complexes of (a) an increase in the overall charge on the complex, (b) changing the leaving group from NO3 to Cl, (c) changing the entering group from Cl to I, (d) changing the cis ligands from NH3 to H2O. (a) [Pt(PR3)4]2 2 Cl (c) cis-[Pt(NH3)2(py)2]2 2 Cl 21.17 Write out the inner- and outer-sphere pathways for reduction of azidopentaamminecobalt(III) ion with V2(aq). What experimental data might be used to distinguish between the two pathways? 21.6 A Pt(II) complex of tetramethyldiethylenetriamine is attacked by Cl 105 times less rapidly than the diethylenetriamine analogue. Explain this observation in terms of an associative rate-determining step. 21.7 The rate of loss of chlorobenzene, PhCl, from [W(CO)4L(PhCl)] increases with increase in the cone angle of L. What does this observation suggest about the mechanism? 21.8 The pressure dependence of the replacement of chlorobenzene (PhCl) by piperidine in the complex [W(CO)4(PPh3)(PhCl)] has been studied. The volume of activation is found to be 11.3 cm3 mol1. What does this value suggest about the mechanism? 21.18 The compound [Fe(SCN)(OH2)5]2 can be detected in the reaction of [Co(NCS)(NH3)5]2 with Fe2(aq) to give Fe3(aq) and Co2(aq). What does this observation suggest about the mechanism? 21.19 Calculate the rate constants for electron transfer in the oxidation of [V(OH2)6]2 (E O (V3/V2) 0.255 V) and the oxidants (a) [Ru(NH3)6]3 (E O (Ru3/Ru2) 0.07 V), (b) [Co(NH3)6]3 (E O (Co3/Co2) 0.10 V). Comment on the relative sizes of the rate constants. 21.20 Calculate the rate constants for electron transfer in the oxidation of [Cr(OH2)6]2 (E O (Cr3/Cr2) 0.41 V) and each of the oxidants [Ru(NH3)6]3 (E O (Ru3/Ru2) 0.07 V), [Fe(OH2)6]3 (E O (Fe3/Fe2) 0.77 V) and [Ru(bpy)3]3 (E O (Ru3/Ru2) 1.26 V). Comment on the relative sizes of the rate constants. 21.9 Does the fact that [Ni(CN)5]3 can be isolated help to explain why substitution reactions of [Ni(CN)4]2 are very rapid? 21.10 Reactions of [Pt(Ph)2(SMe2)2] with the bidentate ligand 1,10-phenanthroline (phen) give [Pt(Ph)2phen]. There is a kinetic pathway with activation parameters ∆‡H 101 kJ mol1 and ∆‡S 42 J K1 mol1. Propose a mechanism. 21.11 Design two-step syntheses of cis- and trans-[PtCl2(NO2)(NH3)] starting from [PtCl4]2. 21.12 How does each of the following modifications affect the rate of a square-planar complex substitution reaction? (a) Changing a trans ligand from H to Cl. (b) Changing the leaving group from Cl to I. (c) Adding a bulky substituent to a cis ligand. (d) Increasing the positive charge on the complex. 21.13 The rate of attack on Co(III) by an entering group Y is nearly independent of Y with the spectacular exception of the rapid reaction 21.21 The photochemical substitution of [W(CO)5(py)] (py pyridine) with triphenylphosphine gives W(CO)5(P(C6H5)3). In the presence of excess phosphine, the quantum yield is approximately 0.4. A flash photolysis study reveals a spectrum that can be assigned to the intermediate W(CO)5. What product and quantum yield do you predict for substitution of [W(CO)5(py)] in the presence of excess triethylamine? Is this reaction expected to be initiated from the ligand field or MLCT excited state of the complex? 21.22 From the spectrum of [CrCl(NH3)5]2 shown in Fig. 20.32, propose a wavelength for photoinitiation of reduction of Cr(III) to Cr(II) accompanied by oxidation of a ligand. Problems 533 PROBLEMS O 21.1 Given the following mechanism for the formation of a chelate complex [Ni(OH2)6]2 LL [Ni(OH2)6]2,LL KE, rapid [Ni(OH2)6]2,LL [Ni(OH2)5LL]2 H2O ka, ka [Ni(OH2)4LL] H2O kb, kb 2 [Ni(OH2)5LL] 2 X O kobs/(10–3 s–1) 12 8 4 Y kr /(dm3 mol1 s1) H2O H2O 105107 CH3OH CH3OH 2 106 CH3CN CH3CN 1.1 105 PPh3 PPh3 1.5 105 CH3CN PR3 107109 PR3 CH3CN 101102 N-donor H2O 102103 21.6 The activation enthalpy for the reduction of cis-[CoCl2(en)2] by Cr2(aq) is 24 kJ mol1. Explain the negative value. (See R.C. Patel, R.E. Ball, J.F. Endicott, and R.G. Hughes, Inorg. Chem., 1970, 9, 23.) 21.7 The rate of reduction of [Co(NH3)5(OH2)]3 by Cr(II) is seven orders of magnitude slower than reduction of its conjugate base, [Co(NH3)5(OH)]2, by Cr(II). For the corresponding reductions with [Ru(NH3)6]2, the two differ by less than a factor of 10. What do these observations suggest about mechanisms? 21.8 Consider the complexes (18) and (19) discussed in the illustration on page 526. Think about the potential routes of the electron transfer in the two complexes and suggest why there is such a difference in the rates of electron transfer between the two complexes. 21.9 Calculate the rate constants for outer-sphere reactions from the following data. Compare your results to the measured values in the last column. SO32– + HSO3– I– N3– NO2– SCN– OH– S2O32– Reaction k11/(dm3 mol1 s1) k22 /(dm3 mol1 s1) E O /V kobs /(dm3 mol1 s1) Cr2 Fe2 2 105 4.0 1.18 2.3 103 4 104 4.4 0.90 108 7.4 10 3 10 0.20 1.7 105 3 107 4.4 0.36 1.4 105 4 [Fe(CN)6] 0 10 20 30 40 [Reactant]/(mmol dm–3) O X [W(CN)8]4 Ce(IV) 0 Y O 23 21.2 The complex [PtH(PEt3)3] was studied in deuterated acetone in the presence of excess PEt3. In the absence of excess ligand the 1H-NMR spectrum in the hydride region exhibits a doublet of triplets. As excess PEt3 ligand is added the hydride signal begins to change, the line shape depending on the ligand concentration. Suggest a mechanism to account for the effects of excess PEt3. 21.4 Figure 21.24 (which is based on J.B. Goddard and F. Basolo, Inorg. Chem., 1968, 7, 936) shows the observed first-order rate constants for the reaction of [PdBrL] with various Y to give [PdYL], where L is Et2NCH2CH2NHCH2CH2NEt2. Note the large slope for S2O32 and zero slopes for Y N3, I, NO2, and SCN. Propose a mechanism. O Rh O derive the rate law for the formation of the chelate. Discuss the step that is different from that for two monodentate ligands. The formation of chelates with strongly bound ligands occurs at the rate of formation of the analogous monodentate complex but the formation of chelates of weakly bound ligands is often significantly slower. Assuming an Id mechanism, explain this observation. (See R.G. Wilkins, Acc. Chem. Res., 1970, 3, 408.) 21.3 Solutions of [PtH2(PMe3)2] exist as a mixture of cis and trans isomers. Addition of excess PMe3 led to formation of [PtH2(PMe3)3] at a concentration that could be detected using NMR. This complex exchanged phosphine ligands rapidly with the trans isomer but not the cis. Propose a pathway. What are the implications for the trans effect of H versus PMe3? (See D.L. Packett and W.G. Trogler, Inorg. Chem., 1988, 27, 1768.) OO Rh 50 Figure 21.24 The data required for Problem 21.4. 21.5 The substitution reactions of the bridged dinuclear Rh(II) complex [Rh2(μ-O2CCH3)4XY] (23) have been studied by M.A.S. Aquino and D.H. Macartney (Inorg. Chem., 1987, 26, 2696). Reaction rates show little dependence on the choice of the entering group. The table below shows the dependence on the leaving group, X, and the ligand on the opposite Rh, (trans) Y, at 298K. What conclusions can you draw concerning the mechanism? Note that the complex has a d7 configuration at each Rh and a single RhRh bond. MnO 4 [Fe(phen)3]2 Ce(IV) 2 3 21.10 In the presence of catalytic amounts of [Pt(P2O5H2)4]4 (21) and light, 2-propanol produces H2 and acetone (E.L. Harley, A.E. Stiegman, A. Vlcek, Jr., and H.B. Gray, J. Am. Chem. Soc., 1987, 109, 5233; D.C. Smith and H.B. Gray, Coord. Chem. Rev., 1990, 100, 169). (a) Give the equation for the overall reaction. (b) Give a plausible molecular orbital scheme for the metalmetal bonding in this tetragonal-prismatic complex and indicate the nature of the excited state that is thought to be responsible for the photochemistry. (c) Indicate the metal complex intermediates and the evidence for their existence. 22 Bonding 22.1 Stable electron configurations 22.2 Electron count preference 22.3 Electron counting and oxidation states 22.4 Nomenclature Ligands 22.5 Carbon monoxide d-Metal organometallic chemistry Organometallic chemistry is the chemistry of compounds containing metalcarbon bonds. Much of the basic organometallic chemistry of the s- and p-block metals was understood by the early part of the twentieth century and has been discussed in Chapters 1116. The organometallic chemistry of the d and f blocks has been developed much more recently. Since the mid-1950s this field has grown into a thriving area that spans new types of reactions, unusual structures, and practical applications in organic synthesis and industrial catalysis. We discuss the organometallic chemistry of the d and f blocks separately, covering d metals in this chapter and f metals in the next. The widespread use of organometallic compounds in synthesis is covered in Chapter 26 (on catalysis). 22.6 Phosphines 22.7 Hydrides and dihydrogen complexes 22.8 -Alkyl, -alkenyl, -alkynyl, and -aryl ligands 1 22.9 2-Alkene and -alkyne ligands 22.10 Nonconjugated diene and polyene ligands 22.11 Butadiene, cyclobutadiene, and cyclooctatetraene 22.12 Benzene and other arenes 22.13 The allyl ligand 22.14 Cyclopentadiene and cycloheptatriene 22.15 Carbenes 22.16 Alkanes, agostic hydrogens, and noble gases 22.17 Dinitrogen and nitrogen monoxide Compounds 22.18 d-Block carbonyls 22.19 Metallocenes 22.20 Metalmetal bonding and metal clusters Reactions 22.21 Ligand substitution 22.22 Oxidative addition and reductive elimination 22.23 -Bond metathesis 22.24 1,1-Migratory insertion reactions 22.25 1,2-Insertions and -hydride elimination 22.26 -, -, and -Hydride eliminations and cyclometallations FURTHER READING EXERCISES PROBLEMS A few d-block organometallic compounds were synthesized and partially characterized in the nineteenth century. The first of them (1), an ethene complex of platinum(II), was prepared by W.C. Zeise in 1827, with the first metal carbonyls, [PtCl2(CO)2] and [PtCl2(CO)]2, being reported by P. Schützenberger in 1868. The next major discovery was tetracarbonylnickel (2), which was synthesized by L. Mond, C. Langer, and F. Quinke in 1890. Beginning in the 1930s, W. Hieber synthesized a wide variety of metal carbonyl cluster compounds, many of which are anionic, including [Fe4(CO)13]2 (3). It was clear from this work that metal carbonyl chemistry was potentially a very rich field. However, as the structures of these and other d- and f-block organometallic compounds are difficult or impossible to deduce by chemical means alone, fundamental advances had to await the development of X-ray diffraction for precise structural data on solid samples and of IR and NMR spectroscopy for structural information in solution. The discovery of the remarkably stable organometallic compound ferrocene, Fe(C5H5)2 (4), occurred at a time (in 1951) when these techniques were becoming widely available. The ‘sandwich’ structure of ferrocene was soon correctly inferred from its IR spectrum and then determined in detail by X-ray crystallography. The stability, structure, and bonding of ferrocene defied the classical Lewis description and therefore captured the imagination of chemists. This puzzle in turn set off a train of synthesizing, characterizing, and theorizing that led to the rapid development of d-block organometallic chemistry. Two highly productive research workers in the formative stage of the subject, Ernst-Otto Fischer in Munich and Geoffrey Wilkinson in London, were awarded the Nobel Prize in 1973 for their contributions. Similarly, f-block organometallic chemistry blossomed soon after the discovery in the late 1970s that the pentamethylcyclopentadienyl ligand, C5Me5, forms stable f-block compounds (5). We adhere to the convention that an organometallic compound contains at least one metal–carbon (MC) bond. Thus, compounds (1) to (5) clearly qualify as organometallic, whereas a complex such as [Co(en)3]3, which contains carbon but has no MC bonds, does not. Cyano complexes, such as hexacyanidoferrate(II) ions, do have MC bonds, but as their properties are more akin to those of conventional coordination complexes they are generally not considered as organometallic. In contrast, complexes of the isoelectronic ligand CO are considered to be organometallic. The justification for this somewhat arbitrary distinction is that many metal carbonyls are significantly different from coordination complexes both chemically and physically. In general, the distinctions between the two classes of compounds are clear: coordination complexes normally are charged, with variable d-electron count, and are soluble in water; organometallic compounds are often neutral, with fixed d-electron count, and are Bonding 535 2– CO – CO t P Fe i N Cl CH2= CH 2 2 Ni(CO)4 1 [PtCl3(C2H4)]– 3 [Fe4(CO)13]2– soluble in organic solvents such as tetrahydrofuran. Most organometallic compounds have properties that are much closer to organic compounds than inorganic salts, with many of them having low melting points (some are liquid at room temperature). Cp Bonding Although there are many organometallic compounds of the s and p blocks, the bonding in these compounds is often relatively simple and normally adequately described solely by bonds. The d metals, in contrast, form a large number of organometallic compounds with many different bonding modes. For instance, to describe fully the bonding of a cyclopentadienyl group to iron in ferrocene (and in general to any d metal), we need to invoke , π, and bonds. Unlike coordination compounds, d-metal organometallic compounds normally have relatively few stable electron configurations and often have a total of 16 or 18 valence electrons around the metal atom. This restriction to a limited number of electronic configurations is due to the strength of the π (and , where appropriate) bonding interactions between the metal atom and the carbon-containing ligands. e F 4 FeCp2, Cp = C5H5– Me Me Me 22.1 Stable electron configurations We start by examining the bonding patterns so that we can appreciate the importance of π bonds and understand the origin of the restriction of the d-metal organometallic compounds to certain electron configurations. (a) 18-Electron compounds Key points: Six -bonding interactions are possible in an octahedral complex and, when π-acceptor ligands are present, bonding combinations can be made with the three orbitals of the t2g set, leading to nine bonding MOs, and space for a total of 18 electrons. In the 1920s, N.V. Sidgwick recognized that the metal atom in a simple metal carbonyl, such as Ni(CO)4, has the same valence electron count (18) as the noble gas that terminates the long period to which the metal belongs. Sidgwick coined the term ‘inert gas rule’ for this indication of stability, but it is now usually referred to as the 18-electron rule.1 It becomes readily apparent, however, that the 18-electron rule is not as uniformly obeyed for d-block organometallic compounds as the octet rule is obeyed for compounds of Period 2 elements, and we need to look more closely at the bonding to establish the reasons for the stability of both the compounds that have the 18-electron configurations and those that do not. Figure 22.1 shows the energy levels that arise when a strong-field ligand such as carbon monoxide bonds to a d-metal atom (Section 20.2). Carbon monoxide is a strong-field ligand, even though it is a poor donor, because it can use its empty π orbitals to act as 1 The 18-electron rule is sometimes referred to as the effective atomic number or EAN rule. Me Me Me Th H Me Me OR Me Me 5 [Th(Cp)2H(OR)], Cp = C5(CH3)5– 536 22 d-Metal organometallic chemistry Metal 4s Antibonding 4p Complex t1u a1g Ligands t1u a1g t2g eg ∆O eg + t2g 3d t2g eg t1u a1g Bonding eg t1u a1g Figure 22.1 The energy levels of the d orbitals of an octahedral complex with strong-field ligands. Metal Complex 4p Ligands eu a1g (b) 16-Electron square-planar compounds a1u Key point: With strong-field ligands, a square-planar complex has only eight bonding MOs, thus a 16-electron configuration is the most energetically favourable configuration. 4s b1gdx2–y2 b2g dxy 3d a good π acceptor. In this picture of the bonding, the t2g orbitals of the metal atom are no longer nonbonding, as they would be in the absence of π interactions, but are bonding. The energy level diagram shows six bonding MOs that result from the ligandmetal interactions, and three bonding MOs that result from π interactions. Thus up to 18 electrons can be accommodated in the nine bonding MOs. Compounds that have this configuration are remarkably stable, for instance the 18-electron Cr(CO)6 is a colourless air-stable compound. An indication of the size of the HOMO-LUMO gap (∆O) can be gained from a consideration of its lack of colour, which results from a lack of any electronic transitions in the visible region of the spectrum, that is ∆O is so large that such transitions are shifted to the UV. The only way to accommodate more than 18 valence electrons in an octahedral complex with strong-field ligands is to use an antibonding orbital. As a result, such complexes are unstable, being particularly prone to electron loss and acting as reducing agents. Compounds with fewer than 18 electrons will not necessarily be very unstable, but such complexes will find it energetically favourable to acquire extra electrons by reaction and so populate their bonding MOs fully. As we shall see later, compounds with fewer than 18 electrons often occur as intermediates in reaction pathways. The bonding characteristic of the carbonyl ligand is replicated with other ligands, which are often poor donors but good π acceptors. Hence, octahedral organometallic compounds are most stable when they have a total of 18 valence electrons around their central metal ion. Similar arguments can be used to rationalize the stability of the 18-electron configuration for other geometries, such as tetrahedral and trigonal bipyramidal, although in practice relatively few tetrahedral organometallic compounds are known. The steric requirements of most ligands normally preclude coordination numbers of greater than six for d-metal organometallic compounds. a1g dz2 eudyz,dzx b1g eu a1g ∆SP b1g eu a1g Figure 22.2 The energy levels of the molecular orbitals of a square-planar complex with strong-field ligands. The eight lowest MOs correspond to bonding interactions, with the higher MOs corresponding to antibonding interactions; the MOs are labelled with the d orbitals from which they are derived. PPh3 CO One other geometry already discussed in the context of coordination chemistry is the square-planar arrangement of four ligands (Section 20.1), where we noted that it occurred only for strong-field ligands and a d8 metal ion. Because organometallic ligands often produce a strong field, many square-planar organometallic compounds exist. Stable squareplanar complexes are normally found with a total of 16 valence electrons, which results in the population of all the bonding and none of the antibonding MOs (Fig. 22.2). The ligands in square-planar complexes can normally provide only two electrons each, for a total of eight electrons. Therefore, to reach 16 electrons, the metal ion must provide an additional eight electrons. As a result, organometallic compounds with 16 valence electrons are common only on the right of the d block, particularly in Groups 9 and 10 (Table 22.1). Examples of such complexes include [Ir(CO)Cl(PPh3)2] (6) and the anion of Zeise’s salt, [Pt(C2H4)Cl3] (1). Square-planar 16-electron complexes are particularly common for the heavier elements in Groups 9 and 10, especially for Rh(I), Ir(I), Pd(II), and Pt(II), because the ligand-field splitting is large and the ligand-field stabilization energy of these complexes favours the square-planar configuration. 22.2 Electron count preference Key point: On the left of the d block, steric requirements can mean that it is not possible to assemble enough ligands around a metal atom to achieve a total of 16 or 18 valence electrons. The preference of a metal atom for one particular geometry and electron count is not normally so strong that other geometries do not occur. For instance, although the chemistry of Table 22.1 Validity of the 16/18-electron rule for d-metal organometallic compounds Usually fewer than 18 electrons Cl Ir 6 trans-[IrCl(CO)(PPh3)2] Usually 18 electrons 16 or 18 electrons Sc Ti V Cr Mn Fe Co Ni Y Zr Nb Mo Tc Ru Rh Pd La Hf Ta W Re Os Ir Pt Bonding both Pd(II) and Rh(I) is dominated by 16-electron square-planar complexes, the cyclopentadienyl palladium complex (7) and the rhodium complex (8) are both 18-electron compounds. Steric factors may restrict the number of ligands that can bond to a metal atom and stabilize compounds with lower electron counts than might have been expected. For instance, the tricyclohexylphosphine ligands (9) in the trigonal Pt(0) compound Pt(PCy3)3 are so large that only three can be fitted around the metal atom, which consequently has only 16 valence electrons. Steric stabilization of the metal centre is partially a kinetic effect, in which the large groups protect the metal centre from reaction. Many complexes readily lose or gain ligands, forming other configurations transitorily during reactions—indeed, the accessibility of these other configurations is precisely why the organometallic chemistry of the d metals is so interesting. Unusual electron configurations are common on the left of the d block, where the metal atoms have fewer electrons, and it is often not possible to crowd enough ligands around the atom to bring the electron count up to 16 or 18. For example, the simplest carbonyl in Group 5, [V(CO)6], is a 17-electron complex. Other examples include [W(CH3)6], which has 12 valence electrons, and [Cr(η5-Cp)(CO)2(PPh3)] with 17. The latter compound provides another good example of the role of steric crowding. When the compact CO ligand is present in place of bulky triphenylphosphine (10), a dimeric compound with a long but definite CrCr bond is observed in the solid state and in solution. The formation of the CrCr bond in [Cr(η5-Cp)(CO)3]2 raises the electron count on each metal to 18. 537 OMe Pd N 7 CH3 PMe3 Rh 22.3 Electron counting and oxidation states The dominance of 16- and 18-electron configurations in organometallic chemistry makes it imperative to be able to count the number of valence electrons on a central metal atom because knowing that number allows us to predict the stabilities of compounds and to suggest patterns of reactivity. Although the concept of ‘oxidation state’ for organometallic compounds is regarded by many as tenuous at best, the vast majority of the research community uses it as a convenient shorthand for describing electron configurations. Oxidation states (and the corresponding oxidation number) help to systematize reactions such as oxidative addition (Section 22.22), and also bring out analogies between the chemical properties of organometallic and coordination complexes. Fortunately, the business of counting electrons and assigning oxidation numbers can be combined. Two models are routinely used to count electrons, the so-called neutral-ligand method (sometimes called the covalent method) and the donor-pair method (sometimes known as the ionic method). We introduce both briefly—they give identical results for electron counting—but in what follows we use the donor-pair method as it may easily be used to assign oxidation numbers too. (a) Neutral-ligand method Key point: All ligands are treated as neutral and are categorized according to how many electrons they are considered to donate. For the sake of counting electrons, each metal atom and ligand is treated as neutral. We include in the count all valence electrons of the metal atom and all the electrons donated by the ligands. If the complex is charged, we simply add or subtract the appropriate number of electrons to the total. Ligands are defined as L type if they are neutral two-electron donors (like CO, PMe3) and X type if, when they are considered to be neutral, they are one-electron radical donors (like halogen atoms, H, CH3). For example, Fe(CO)5 acquires 18 electrons from the eight valence electrons on the Fe atom and the 10 electrons donated by the five CO ligands. Some ligands are considered combinations of these types, for instance cyclopentadienyl is considered as a five-electron L2X donor; see Table 22.2. E X A M PL E 22 .1 Counting electrons using the neutral-ligand method Do (a) [IrBr2(CH3)(CO)(PPh3)2] and (b) [Cr(η5-C5H5)(η6-C6H6)] obey the 18-electron rule? Answer (a) We start with the Ir atom (Group 9) which has nine valence electrons, then add in the electrons from the two Br atoms and the CH3 group (each is a one-electron donor) and finally add in the electrons 8 [Rh(Me)(PMe3)4], Me = CH3 P 9 PCy3, Cy = cyclo-C6H11 Ph Ph Ph P 10 PPh3, Ph = C6H5 538 22 d-Metal organometallic chemistry from the CO and PPh3 (both are two-electron donors). Thus, the number of valence electrons on the metal atom is 9 (3 1) (3 2) 18. (b) In a similar fashion, the Cr atom (Group 6) has six valence electrons, the η5-C5H5 ligand donates five electrons, and the η6-C6H6 ligand donates six, so the number of metal valence electrons is 6 5 6 17. This complex does not obey the 18-electron rule and is not stable. A related but stable 18-electron compound is [Cr(η6-C6H6)2]. Self-test 22.1 Is [Mo(CO)7] likely to be stable? The advantage of the neutral-ligand method is that, given the information in Table 22.2, it is trivial to establish the electron count. The disadvantage, however, is that the method Table 22.2 Typical ligands and their electron counts (a) Neutral-ligand method Ligand Formula Designation Electrons donated Carbonyl CO L 2 Phosphine PR3 L 2 Hydride H X 1 Chloride Cl X 1 Dihydrogen H2 L 2 -Alkyl, -alkenyl, -alkynyl, and -aryl groups R X 1 2-Alkene CH2 CH2 L 2 2-Alkyne RCCR L 2 1 Dinitrogen N2 L 2 Butadiene CH2 CH—CH CH2 L2 4 Benzene C6H6 L3 6 3-Allyl CH2CHCH2 LX 3 -Cyclopentadienyl C5H5 L2X 5 5 (b) Donor-pair method Ligand Formula Carbonyl CO 2 Phosphine PR3 2 Hydride H 2 Chloride Cl 2 Dihydrogen H2 2 1-Alkyl, -alkenyl, -alkynyl, and -aryl groups R 2 2-Alkene CH2 CH2 2 -Alkyne RCCR 2 Dinitrogen N2 2 Butadiene CH2 CH—CH CH2 4 Benzene C6H6 6 2 Electrons donated -Allyl CH2CHCH 4 5-Cyclopentadienyl C5H5 6 3 We use this method throughout this book. 2 Bonding overestimates the degree of covalence and thus underestimates the charge at the metal. Moreover, it becomes confusing to assign an oxidation number to a metal, and meaningful information on some ligands is lost. (b) Donor-pair method Key point: Ligands are considered to donate electrons in pairs, resulting in the need to treat some ligands as neutral and others as charged. The donor-pair method requires a calculation of the oxidation number. The rules for calculating the oxidation number of an element in an organometallic compound are the same as for conventional coordination compounds. Neutral ligands, such as CO and phosphine, are considered to be two-electron donors and are formally assigned an oxidation number of 0. Ligands such as halides, H, and CH3 are formally considered to take an electron from the metal atom, and are treated as Cl, H, and CH3 (and hence are assigned oxidation number 1); in this anionic state they are considered to be two-electron donors. The cyclopentadienyl ligand, C5H5 (Cp), is treated as C5H5 (it is assigned an oxidation number of 1); in this anionic state it is considered to be a six-electron donor. Then: The oxidation number of the metal atom is the total charge of the complex minus the charges of any ligands. The number of electrons the metal provides is its group number minus its oxidation number. The total electron count is the sum of the number of electrons on the metal atom and the number of electrons provided by the ligands. The main advantage of this method is that with a little practice both the electron count and the oxidation number may be determined in a straightforward manner. The main disadvantage is that it overestimates the charge on the metal atom and can suggest reactivity that might be incorrect (see Section 22.7 on hydrides). Table 22.2 lists the maximum number of electrons available for donation to a metal for most common ligands. E X A M PL E 22 . 2 Assigning oxidation numbers and counting the electrons using the donor-pair method Assign the oxidation number and count the valence electrons on the metal atom in (a) [IrBr2(CH3)(CO) (PPh3)2], (b) [Cr(η5-C5H5)(η6-C6H6)], and (c) [Mn(CO)5]. Answer (a) We treat the two Br groups and the CH3 as three singly negatively charged two-electron donors and the CO and the two PPh3 ligands as three two-electron donors, providing 12 electrons in all. Because the complex is neutral overall, the Group 9 Ir atom must have a charge of 3 (that is, have oxidation number 3) to balance the charge of the three anionic ligands, and thus contributes 9 3 6 electrons. This analysis gives a total of 18 electrons for the Ir(III) complex. (b) We treat the η5-C5H5 ligand as C5H5 and thus it donates six electrons, with the η6-C6H6 ligand donating a further six. To maintain neutrality, the Group 6 Cr atom must have a charge of 1 (and an oxidation number of 1) and contributes 6 1 5 electrons. The total number of metal electrons is 12 5 17 for a Cr(I) complex. As noted before, this complex does not obey the 18-electron rule and is unlikely to be stable. (c) We treat each CO ligand as neutral and contributing two electrons, giving 10 electrons. The overall charge of the complex is 1; because all the ligands are neutral, we consider this charge to reside formally on the metal atom, giving it an oxidation number 1. The Group 7 Mn atom thus contributes 7 1 electrons, giving a total of 18 for a Mn(1) complex. Self-test 22.2 What is the electron count for and oxidation number of platinum in the anion of Zeise’s salt, [Pt(CH2 CH2)Cl3]? Treat CH2 CH2 as a neutral two-electron donor. 22.4 Nomenclature Key point: The naming of organometallic compounds is similar to the naming of coordination compounds, but certain ligands have multiple bonding modes, which is reported as the hapticity. According to the recommended convention, we use the same system of nomenclature for organometallic compounds as set out for coordination complexes in Section 7.2. 539 540 22 d-Metal organometallic chemistry C6H6 Mo CO 11 [Mo(η6-C6H6)(CO)3] M 12 η1-Cyclopentadienyl M 13 η3-Cyclopentadienyl Thus, ligands are listed in alphabetical order followed by the name of the metal, all of which is written as one word. The name of the metal should be followed by its oxidation number in parentheses. The nomenclature used in research journals, however, does not always obey these rules, and it is common to find the name of the metal buried in the middle of the name of the compound and the oxidation number omitted. For example, (11) is sometimes referred to as benzenemolybdenum-tricarbonyl, rather than the preferred name benzene(tricarbonyl)molybdenum(0). The IUPAC recommendation for the formula of an organometallic compound is to write it in the same form as for a coordination complex: the symbol for the metal is written first, followed by the ligands, listed in alphabetical order based on their chemical symbol. We shall follow these conventions unless a different order of ligands helps to clarify a particular point. Often a ligand with carbon donor atoms can exhibit multiple bonding modes—for instance, the cyclopentadienyl group can commonly bond to a d-metal atom in three different ways—thus, we need some additional nomenclature. Without going into the intimate details of the bonding of the various ligands (we do that later in this chapter), the extra information we need to describe a bonding mode is the number of points of attachment. This procedure gives rise to the notion of hapticity, the number of ligand atoms that are considered formally to be bonded to the metal atom. The hapticity is denoted ηn, where n is the number of atoms (and η is eta). For example, a CH3 group attached by a single MC bond is monohapto, η1, and if the two C atoms of an ethene ligand are both within bonding distance of the metal, the ligand is dihapto, η2. Thus, three cyclopentadienyl complexes might be described as having η1 (12), η3 (13), or η5 (14) cyclopentadienyl groups. Some ligands (including the simplest of them all, the hydride ligand, H) can bond to more than one metal atom in the same complex, and are then referred to as bridging ligands. We do not require any new concepts to understand bridging ligands other than those introduced in Section 2.11. Recall from Section 7.2 that the Greek letter (mu) is used to indicate how many atoms the ligand bridges. Thus a 2-CO is a carbonyl group that bridges two metal atoms and a 3-CO bridges three. M 14 η5-Cyclopentadienyl E X A M PL E 22 . 3 Naming organometallic compounds Give the formal names of (a) ferrocene (4) and (b) [RhMe(PMe3)4] (8). Answer (a) Ferrocene contains two cyclopentadienyl groups that are both bound to the metal atom through all five carbon atoms, thus both groups are designated η5. The full name for ferrocene is thus bis(η5-cyclopentadienyl)iron(II). (b) The rhodium compound contains one formally anionic methyl group and four neutral trimethylphosphine ligands, therefore the formal name is methyltetrakis-(trimethylphosphine) rhodium(I). Self-test 22.3 What is the formal name of [Ir(Br)2(CH3)(CO)(PPh3)2]? Ligands A large number of ligands are found in organometallic complexes, with many different bonding modes. Because the reactivity of the metal atom and the ligands is affected by the ML bonding, it is important to look at each ligand in some detail. 22.5 Carbon monoxide Key point: The 3 orbital of CO serves as a very weak donor and the π orbitals act as acceptors. Carbon monoxide is a very common ligand in organometallic chemistry, where it is known as the carbonyl group. Carbon monoxide is particularly good at stabilizing very low oxidation states, with many compounds (such as Fe(CO)5) having the metal in its zero oxidation state. We described the molecular orbital structure of CO in Section 2.9, and it would be sensible to review that section. A simple picture of the bonding of CO to a metal atom is to treat the lone pair on the carbon atom as a Lewis σ base (an electron-pair donor) and the empty CO antibonding orbital as a Lewis π acid (an electron-pair acceptor), which accepts π-electron density from the filled d orbitals on the metal atom. In this picture, the bonding can be considered to 541 Ligands be made up of two parts: a σ bond from the ligand to the metal atom (15) and a π bond from the metal atom to the ligand (16). This type of π bonding is sometimes referred to as backbonding. Carbon monoxide is not appreciably nucleophilic, which suggests that σ bonding to a d-metal atom is weak. As many d-metal carbonyl compounds are very stable, we can also infer that π backbonding is strong and the stability of carbonyl complexes arises mainly from the π-acceptor properties of CO. Further evidence for this view comes from the observation that stable carbonyl complexes exist only for metals that have filled d orbitals of an energy suitable for donation to the CO antibonding orbital. For instance, elements in the s and p blocks do not form stable carbonyl complexes. However, the bonding of CO to a d metal atom is best regarded as a synergistic (that is, mutually enhancing) outcome of both σ and π bonding: the π backbonding from the metal to the CO increases the electron density on the CO, which in turn increases the ability of the CO to form a σ bond to the metal atom. A more formal description of the bonding can be derived from the molecular orbital scheme for CO (Fig. 22.3), which shows that the HOMO has σ symmetry and is essentially a lobe that projects away from the C atom. When CO acts as a ligand, this 3σ orbital serves as a very weak donor to a metal atom, and forms a σ bond with the central metal atom. The LUMOs of CO are the π orbitals. These two orbitals play a crucial role because they can overlap with metal d orbitals that have local π symmetry (such as the t2g orbitals in an Oh complex). The π interaction leads to the delocalization of electrons from filled d orbitals on the metal atom into the empty π orbitals on the CO ligands, so the ligand also acts as a π acceptor. One important consequence of this bonding scheme is the effect on the strength of the CO triple bond: the stronger the metalcarbon bond becomes through pushing electron density from the metal atom into the π bond, the weaker the CO bond becomes, as this electron density enters a CO antibonding orbital. In the extreme case, when two electrons are fully donated by the metal atom, a formal metalcarbon double bond is formed; because the two electrons occupy a CO antibonding orbital, this donation results in a decrease in the bond order of the CO to 2. In practice, the bonding is somewhere between MC⬅O, with no backbonding, and M C O, with complete backbonding. Infrared spectroscopy is a very convenient method of assessing the extent of π bonding; the CO stretch is clearly identifiable as it is both strong and normally clear of all other absorptions. In CO gas, the absorption for the triple bond is at 2143 cm1, whereas a typical metal carbonyl complex has a stretching mode in the range 21001700 cm1 (Table 22.3). The number of IR absorptions that occur in a particular carbonyl compound is discussed in Section 22.18g. Carbonyl stretching frequencies are often used to determine the order of acceptor or donor strengths for the other ligands present in a complex. The basis of the approach is that the CO stretching frequency is decreased when it serves as a π acceptor. However, as other π acceptors in the same complex compete for the d electrons of the metal atom, they cause the CO frequency to increase. This behaviour is opposite to that observed with donor ligands, which cause the CO stretching frequency to decrease as they supply electrons to the metal atom and hence, indirectly, to the CO π orbitals. Thus, strong σ-donor ligands attached to a metal carbonyl and a formal negative charge on a metal carbonyl anion both result in slightly greater CO bond lengths and significantly lower CO stretching frequencies. Carbon monoxide is versatile as a ligand because, as well as the bonding mode we have described so far (often referred to as ‘terminal’), it can bridge two (17) or three (18) metal atoms. Although the description of the bonding is now more complicated, the concepts of σ-donor and π-acceptor ligands remain useful. The CO stretching frequencies generally follow the order MCO M2CO M3CO, which suggests an increasing occupation of the π orbital as the CO molecule bonds to more metal atoms and more electron density from the metals enters the CO π orbitals. As a rule of thumb, carbonyls bridging two metal atoms typically have stretching bands in the range 19001750 cm1, and those that bridge three atoms have stretching bands in the range 18001600 cm1 (Fig. 22.4). We consider carbon monoxide to be a two-electron neutral ligand when terminal or bridging (thus a carbonyl bridging two metals can be considered to give one electron to each). A further bonding mode for CO that is sometimes observed is when the CO is terminally bound to one metal atom and the CO triple bond binds side-on to another metal (19). C O M 15 O C M 16 C O CO 4σ 2π 2p 2p 3σ 1π 2s 2σ 2s 1σ Figure 22.3 The molecular orbital scheme for CO shows that the HOMO has σ symmetry and is essentially a lobe that projects away from the C atom. The LUMO has π symmetry. Table 22.3 The influence of coordination and charge on CO stretching bands Compound v / cm−1 CO 2143 [Mn(CO)6] 2090 Cr(CO)6 2000 [V(CO)6] 2 [Ti(CO)6] 1860 1750 542 22 d-Metal organometallic chemistry This description is best considered as two separate bonding interactions, with the terminal interaction being the same as that described above and the side-on bonding being essentially identical to that of other side-on π donors such as alkynes and N2, which are discussed later. The synthesis, properties, and reactivities of compounds containing the carbonyl ligand are discussed in more detail in Section 22.18. CO M 22.6 Phosphines 17 Key point: Phosphines bond to metals by a combination of σ donation from the P atom and π backbonding from the metal atom. CO M 18 1600 1700 1800 1900 2000 2100 O C M Wavenumber, n~ /cm–1 Figure 22.4 Approximate ranges for CO stretching bands in neutral metal carbonyls. Note that high wavenumbers (and hence high frequencies) are on the left, in keeping with the way infrared spectra are generally plotted. M’ CO M 19 Although phosphine complexes are not organometallic because the ligands do not bond to the metal atom through a carbon atom, they are best discussed here as their bonding has many similarities to that of carbon monoxide. Phosphine, PH3 (formally, phosphane), is a reactive, noxious, poisonous, and flammable gas (Section 15.10). Like ammonia, phosphine can behave as a Lewis base and use its lone pair to donate electron density to a Lewis acid, and so act as a ligand. However, given the problems associated with handling phosphine, it is rarely used as a ligand. Substituted phosphines, on the other hand, such as trialkylphosphines (for example, PMe3, PEt3), or triarylphosphines (for example, PPh3, 10), or trialkyl- or triarylphosphites (for example, P(OMe)3, P(OPh)3, 20) and a whole host of bridged multidentate di- and triphosphines (for example, Ph2PCH2CH2PPh2 dppe, 21) are easy to handle (indeed some are air-stable odourless solids with no appreciable toxicity) and are widely used as ligands; all are colloquially referred to as ‘phosphines’. Phosphines have a lone pair on the P atom that is appreciably basic and nucleophilic, and can serve as a σ donor. Phosphines also have empty orbitals on the P atom that can overlap with filled d orbitals on 3d-metal ions and behave as π acceptors (22). The bonding of phosphines to a d-metal atom, made up of a σ bond from the ligand to the metal and a π bond from the metal back to the ligand, is completely analogous to the bonding of CO to a d-metal atom. Quite which orbitals on the P atom behave as π acceptors has been the subject of considerable debate in the past with some groups claiming a role for unoccupied 3d orbitals on P, and some claiming a role for the PR σ orbitals; current consensus favours the σ orbitals. In any case, each phosphine provides an additional two electrons to the valence electron count. As we have remarked, a huge variety of phosphines are both possible and widely available, including chiral systems such as 2,2-bis(diphenylphosphino)-1,1-binaphthyl (BINAP, 23), in which steric constraints result in compounds that can be resolved into diastereomers. Generally there are two properties of phosphine ligands that are considered important in discussions of the reactivity of their complexes: their steric bulk and their electron donating (and accepting) ability. We described in Section 21.6 how the steric bulk of phosphines can be expressed in terms of the notional cone occupied by the bonded ligand, and Table 22.4 lists some of the derived cone angles. The bonding of phosphines to d-metal atoms is, as we have seen, a composite of σ bonding from the ligand to the metal atom, and π backbonding from the metal atom to the ligand. The σ-donating ability and π-acceptor ability of phosphines are inversely PPh2 O O O P P P F PPh2 P M 20 P(OPh)3 21 Ph2PCH2CH2PPh2, dppe 22 23 2,2’-bis(diphenylphosphino)1,1’-binaphthyl, BINAP Ligands correlated in the sense that electron-rich phosphines, such as PMe3, are good σ donors and poor π acceptors, whereas electron-poor phosphines, such as PF3, are poor σ donors and good π acceptors. Thus Lewis basicity can normally be used as a single scale to indicate their donor/acceptor ability. The generally accepted order of basicity of phosphines is: PCy3 PEt3 PMe3 PPh3 P(OMe)3 P(OPh)3 PCl3 PF3 and is easily understood in terms of the electronegativity of the substituents on the P atom. The basicity of a phosphine is not simply related to the strength of the MP bond in a complex, for instance an electron-poor metal atom forms a stronger bond with an electron-rich (basic) phosphine, whereas an electron-rich metal atom will form a stronger bond with an electron-poor phosphine. If there are carbonyl ligands present in a metalphosphine complex, then the carbonyl stretching frequency can be used to assess the basicity of the phosphine ligand: this method allows us to conclude that PF3 is a π acceptor comparable to CO. The vast range of phosphines that are commonly used in organometallic chemistry is a testament to their versatility as ligands: judicious choice allows control over both steric and electronic properties of the metal atom in a complex. Phosphorus-31 (which occurs in 100 per cent natural abundance) is easy to observe by NMR and both the 31P chemical shift and the coupling constant to the metal atom (where appropriate) give considerable insight into the bonding and reactivity of a complex. Like carbonyls, phosphines can bridge either two or three metal atoms, providing additional variety in bonding modes. E X A M PL E 22 . 4 Interpreting carbonyl stretching frequencies and phosphine complexes (a) Which of the two isoelectronic compounds Cr(CO)6 and [V(CO)6] will have the higher CO stretching frequency? (b) Which of the two chromium compounds [Cr(CO)5(PEt3)] and [Cr(CO)5(PPh3)] will have the lower CO stretching frequency? Which will have the shorter MC bond? Answer We need to think about whether backbonding to the CO ligands is enhanced or diminished: more backbonding results in a weaker carbonoxygen bond. (a) The negative charge on the V complex will result in greater π backbonding to the CO π orbitals, compared to the Cr complex. This backbonding results in a weakening of the CO bond, with a corresponding decrease in stretching frequency. Thus the Cr complex has the higher CO stretching frequency. (b) PEt3 is more basic than PPh3 and thus the PEt3 complex will have greater electron density on the metal atom than the PPh3 complex. The greater electron density will result in greater backbonding and thus both a lower CO stretching frequency and a shorter MC bond. Self-test 22.4 Which of the two iron compounds Fe(CO)5 and [Fe(CO)4(PEt3)] will have the higher CO stretching frequency? Which will have the longer MC bond? 22.7 Hydrides and dihydrogen complexes Key points: The bonding of a hydrogen atom to a metal atom is a σ interaction, whereas the bonding of a dihydrogen ligand involves π backbonding. A hydrogen atom directly bonded to a metal is commonly found in organometallic complexes and is referred to as a hydride ligand. The name ‘hydride’ can be misleading as it implies a H ligand. Although the formulation H might be appropriate for most hydrides, such as [CoH(PMe3)4], some hydrides are appreciably acidic and behave as though they contain H, for example [CoH(CO)4] is an acid with pKa 8.3 (in acetonitrile). The acidity of organometallic carbonyls is described in Section 22.18. In the donor-pair method of electron counting, we consider the hydride ligand to contribute two electrons and to have a single negative charge (that is, to be H). The bonding of a hydrogen atom to a metal atom is simple because the only orbital of appropriate energy for bonding on the hydrogen is H1s and the MH bond can be considered as a σ interaction between the two atoms. Hydrides are readily identified by NMR spectroscopy as their chemical shift is rather unusual, typically occurring in the range 50 0. Infrared spectroscopy can also be useful in identifying metal hydrides as they normally have a stretching band in the range 28502250 cm1. X-ray diffraction, normally so valuable for identifying the structure of crystalline materials, is of little use in identifying hydrides because the diffraction is related to electron density, and the hydride ligand will have at most two electrons around it, compared with, for instance, 78 for a platinum. 543 Table 22.4 Tolman cone angles (in degrees) for selected phosphines PF3 104 P(OMe)3 107 PMe3 118 PCl3 125 P(OPh)3 127 PEt3 132 PPh3 145 PCy3 169 PtBu3 182 P(o-tolyl)3 193 544 22 d-Metal organometallic chemistry Neutron diffraction is of more use in locating hydride ligands, especially if the hydrogen atom is replaced by a deuterium atom, because deuterium has a large neutron scattering cross-section. An MH bond can sometimes be produced by protonation of an organometallic compound, such as neutral and anionic metal carbonyls (Section 22.18e). For example, ferrocene can be protonated in strong acid to produce an FeH bond: H H M Cp2Fe HBF4 → [Cp2FeH][BF4] 24 Bridging hydrides exist, where an H atom bridges either two or three metal atoms: here the bonding can be treated in exactly the same way we considered bridging hydrides in diborane, B2H6 (Section 2.11). Although the first organometallic metal hydride was reported in 1931, complexes of hydrogen gas, H2, were identified only in 1984. In such compounds, the dihydrogen molecule, H2, bonds side-on to the metal atom (in the older literature, such compounds were sometimes called non-classical hydrides). The bonding of dihydrogen to the metal atom is considered to be made up of two components: a σ donation of the two electrons in the H2 bond to the metal atom (24) and a π backdonation from the metal to the σ antibonding orbital of H2 (25). This picture of the bonding raises a number of interesting issues. In particular, as the π backbonding from the metal atom increases, the strength of the HH bond decreases and the structure tends to that of a dihydride: H H M 25 PiPr3 H H M CO H M H H M A dihydrogen molecule is treated as a neutral two-electron donor. Thus the transformation of the dihydrogen into two hydrides (each considered to have a single negative charge and to contribute two electrons) requires the formal charge on the metal atom to increase by two. That is, the metal is oxidized by two units and the dihydrogen is reduced. Although it might seem that this oxidation of the metal is just an anomaly thrown up by our method of counting electrons, two of the electrons on the metal atom have been used to backbond to the dihydrogen, and these two electrons are no longer available to the metal atom for further bonding. This transformation of the dihydrogen molecule to a dihydride is an example of oxidative addition, and is discussed more fully later in this chapter (Section 22.22). It is now recognized that complexes exist with structures at all points between these two extremes, and in some cases an equilibrium can be identified between the two. Work by G. Kubas on tungsten complexes used the HD coupling constant to show that it is possible to detect both the dihydrogen complex (26, 1JHD 34 Hz) and the dihydride (27, 2JHD 2 Hz), and to follow the conversion from one to the other. Certain microbes contain enzymes known as hydrogenases that use Fe and Ni at their catalytic centres to catalyse the rapid oxidation of H2 and reduction of H, via intermediate metal dihydrogen and hydride species (Section 27.14). W D H H 26 [W(HD)(PiPr3)2(CO)3], i Pr = CH(CH3)2 PiPr3 CO W D 22.8 η1-Alkyl, -alkenyl, -alkynyl, and -aryl ligands Key point: The metalligand bonding of η1-hydrocarbon ligands is a σ interaction. H 27 [W(H)(D)(PiPr3)2(CO)3] R M M M 28 29 30 Alkyl groups are often found as ligands in d-metal organometallic chemistry and their bonding presents no new features: it is best considered a simple covalent interaction between the metal atom and the carbon atom of the organic fragment. Alkyl groups with a hydrogen atom on a carbon atom adjacent to the one that bonds to the metal are prone to decompose by a process known as -hydrogen elimination (Section 22.25) and hence those alkyl groups that cannot react in this fashion, such as methyl, benzyl, (CH2C6H5), neopentyl (CH2CMe3), and trimethylsilylmethyl (CH2SiMe3), are more stable than those that can, such as ethyl. Alkenyl (28), alkynyl (29), and aryl (30) groups can bond to a metal atom in a similar fashion, binding to the metal atom through a single carbon atom, and hence are described as monohapto (η1). Although there is potential for each of these three groups to accept π electron density into antibonding orbitals, there is little evidence that this happens. For instance, even though an η1-alkynyl group might be considered analogous to a CO group, the stretching frequency of the triple bond in alkynyl complexes changes little on attachment to a metal. Bridging alkyl and aryl groups also exist, and the bonding can be considered in the same way as we have considered other bridging ligands, with 3c,2e bonds. Ligands 545 Alkyl, alkenyl, alkynyl, and aryl groups are commonly introduced into organometallic complexes by the displacement of a halide at a metal centre with a lithium or Grignard reagent. For example: Cl Ph3P Pd PPh 3 2 PhLi Ph Pd Ph3P Cl PPh 3 2 LiCl Ph We consider alkyl, alkenyl, alkynyl, and aryl ligands to be two-electron donors with a single negative charge (for example, Me, Ph) in the donor-pair scheme of electron counting. 22.9 η2-Alkene and -alkyne ligands Key point: The bonding of an alkene or an alkyne to a metal atom is best described as a σ interaction from the multiple bond to the metal atom, with a π backbonding interaction from the metal atom to the π antibonding orbital on the alkene or alkyne. Alkenes are routinely found bound to metal centres: the first organometallic compound isolated, Zeise’s salt (1), was a complex of ethene. Alkenes normally bond side-on to a metal atom with both carbon atoms of the double bond equidistant from the metal with the other groups on the alkene approximately perpendicular to the plane of the metal atom and the two carbon atoms (31). In this arrangement, the electron density of the C C π bond can be donated to an empty orbital on the metal atom to form a σ bond. In parallel with this interaction, a filled metal d orbital can donate electron density back to the empty π orbitals of the alkene to form a π bond. This description is called the DewarChattDuncanson model (Fig. 22.5) and η2-alkenes are considered to be two-electron neutral ligands. Electron donor and acceptor character appear to be fairly evenly balanced in most ethene complexes of the d metals, but the degree of donation and backdonation can be altered by substituents on the metal atom and on the alkene. When the π backbonding from the metal atom increases, the strength of the C C bond decreases as the electron density is located in the C C antibonding orbital and the structure tends to that of a CC singly bonded structure, a metallocyclopropane: M M M CH2=CH2 31 H H M Dihaptoalkenes with only a small degree of electron donation from the metal have their substituents bent slightly away from the metal atom, and the C C bond length is only slightly greater than in the free alkene (134 pm). When the degree of backdonation is greater, substituents on the alkene are bent away more from the metal atom and the CC bond length approaches that characteristic of a single bond. Steric constraints can also force the other groups on the alkene to bend away from the metal atom. Alkynes have two π bonds and hence the potential to be four-electron donors. When side-on to a single metal atom, the η2-carboncarbon triple bond is best considered as a two-electron donor, with the π orbitals accepting electron density from a metal atom in the same way as for alkenes. When strongly electron-withdrawing groups are attached to an alkyne, the ligand can become an excellent π acceptor and displace other ligands such as phosphines; the compound commonly known as dimethylacetylenedicarboxylate, CH3OCOC⬅CCO2CH3, is a good example. Substituted alkynes can form very stable polymetallic complexes in which the alkyne can be regarded as a four-electron donor. An example is η2-diphenylethyne-(hexacarbonyl) dicobalt(0), in which we can view one π bond as donating to one of the Co atoms and the second π bond as overlapping with the other Co atom (32). In this example, the alkyl or aryl groups present on the alkyne impart stability by lowering the tendency towards secondary reactions of the coordinated ethyne, such as loss of the slightly acidic ethynic H atom to the metal atom. M (a) Nonconjugated diene (C CXC C) and polyene ligands can also bond to metal atoms. It is simplest to consider them as linked alkenes, and hence they present no new bonding concepts. As with the chelate effect in coordination complexes (Section 7.14), the H M (b) H H π Figure 22.5 The interaction of ethene with a metal atom. (a) Donation of electron density from the filled π molecular orbital of ethene to a vacant metal orbital. (b) Acceptance of electron density from a filled dπ orbital into the vacant π orbital of ethene. 22.10 Nonconjugated diene and polyene ligands Key point: The bonding of nonconjugated alkenes to a metal atom is best described as independent multiple alkenes bonding to a metal centre. π H Ph Ph OC CO OC CO CO CO 32 [Co2(PhC=CPh)(CO)6] 546 22 d-Metal organometallic chemistry i N 33 Ni(cod)2 resulting polyene complexes are usually more stable than the equivalent complex with individual ligands because the entropy of dissociation of the complex is much smaller than when the liberated ligands can move independently. For example, bis(η4-cycloocta-1,5diene)nickel(0) (33) is more stable than the corresponding complex containing four ethene ligands. Cycloocta-1,5-diene (34) is a fairly common ligand in organometallic chemistry, where it is referred to engagingly as ‘cod’, and is normally introduced into the metal coordination sphere by simple ligand displacement reactions. An example is: Cl NCPh Pd Cl 34 Cycloocta-1,5-diene, cod NCPh cod Cl Pd 2 PhCN Cl Metalcod complexes are commonly used as starting materials because they often have intermediate stability. Many of them are sufficiently stable to be isolated and handled, but cod can be displaced by many other ligands. For example, if the highly toxic Ni(CO)4 molecule is needed in a reaction, then it may be generated from Ni(cod)2 directly in the reaction flask: Ni(cod)2(soln) 4 CO(g) → Ni(CO)4(soln) 2 cod(soln) 22.11 Butadiene, cyclobutadiene, and cyclooctatetraene Key points: Some insight into the bonding of butadiene and cyclobutadiene can be gained by treating them as containing two alkene units, but a full understanding needs a consideration of the molecular orbitals. Cyclooctatetraene bonds in a number of different ways; the most common mode in d-metal chemistry is as an 4-donor, analogous to butadiene. The temptation with both butadiene and cyclobutadiene is to treat them like two isolated double bonds. However, a proper molecular orbital approach is necessary to understand the bonding fully because the ligand–metal atom interactions are different in the two cases. Figure 22.6 shows the molecular orbitals for the π system in butadiene. The two occupied lower energy MOs can behave as donors to the metal, the lowest a σ donor and the next a π donor. The next higher unoccupied MO, the LUMO, can act as a π acceptor from the metal atom. Thus, attachment of a butadiene molecule to a metal atom results in population of an MO that is bonding between the two central C atoms (which are already nominally singly bonded) and antibonding between the nominally doubly bonded C atoms. The resulting modification of electron density results in a shortening of the central CC bond and a lengthening of the double CC bonds; in some complexes the central CC bond is found to be even shorter than the other two CC bonds. In theory, a δ-bonding interaction is possible between the dxy orbital of the metal atom and the most antibonding of the butadiene MOs, but there is no definitive evidence that it occurs. Butadiene is therefore considered to be a four-electron neutral ligand in electron counting schemes. 4π dxy 3π dyz 2π dzx 1π dz 2 Figure 22.6 The molecular orbitals of the π system in butadiene; also shown are metal d orbitals of appropriate symmetry to form bonding interactions. 547 Ligands Cyclobutadiene is rectangular (D2h) and unstable as a free molecule, as a result of bond angle constraints and its four-electron anti-aromatic configuration. However, stable complexes are known, including [Ru(η4-C4H4)(CO)3] (35). This species is one of many in which coordination to a metal atom stabilizes an otherwise unstable molecule. As a result of a distortion from the square arrangement,2 cyclobutadiene has an MO diagram similar to that of butadiene (Fig. 22.7). Population of the LUMO by backbonding now leads to greater bonding on the long sides of the rectangular cyclobutadiene molecule, tending towards a square (D4h) arrangement. If the cyclobutadiene formally accepts two electrons from the metal atom, it will have six π electrons with three MOs occupied and no incentive to distort from square; two of the MOs are degenerate in this configuration. All cyclobutadienemetal complexes are found to be square, suggesting that a sixelectron aromatic configuration more accurately describes the bonding within the carbon ring than a four-electron configuration does. This view of the bonding has led some to regard cyclobutadiene complexes as complexes of the R4C42 dianion (a six-electron donor), although for convenience most treat cyclobutadiene as a four-electron neutral ligand. Once again, a δ-bonding interaction is possible between the dxy orbital of the metal atom and the most antibonding of the butadiene MOs, but again there seems to be no definitive evidence that it occurs. Because cyclobutadiene is unstable, the ligand must be generated in the presence of the metal to which it is to be coordinated. This synthesis can be accomplished in a variety of ways. One method is the dehalogenation of a halogenated cyclobutene: C4H4 u R CO 35 [Ru(C4H4)(CO)3] Cl + Fe 2(CO) 9 Fe OC Cl CO CO FeCl 2 + 6 CO Another procedure is the dimerization of a substituted ethyne: PhC PhC CPh Co Ph Ph CPh Co Cp PPh 3 4π Ph Ph Ph Ph Cp PPh 3 Ph Ph Co Cp PPh 3 4π dxy 3π 3π 2π 2π 1π dyz dzx 1π dz 2 We can regard this distortion as an organic example of the Jahn–Teller effect (Section 20.1g). 2 Figure 22.7 The molecular orbitals of the π system in cyclobutadiene; also shown are metal d orbitals of appropriate symmetry to form bonding interactions. 548 22 d-Metal organometallic chemistry 36 Cyclooctatetraene Ru OC OC CO 37 [Ru(η4-C8H8)(CO)3] Cyclooctatetraene (36) is a large ligand that is found in a wide variety of bonding arrangements. Like cyclobutadiene, it is anti-aromatic as the free molecule. Cyclooctatetraene can bond to a metal atom in an octahapto arrangement, in which it is planar and all CC bond lengths are equal. In a similar fashion to cyclobutadiene, in this arrangement cyclooctatetraene is considered to extract two electrons and to become (formally) the aromatic dinegative ligand [C8H8]2 (that is, a 10-electron donor). Cyclooctatetraenes bound in this fashion are rarely found in d-metal compounds and are normally found only with lanthanoids and actinoids (Section 23.8). For example, two such dinegative ligands are found in bis(η8-cyclooctatetraenyl)-uranium(IV), [U(η8-C8H8)2], commonly referred to as uranocene. The more usual bonding modes for cyclooctatetraenes in d-metal compounds are as puckered η4-C8H8 ligands such as (37), where the bonding part of the ligand can be treated as a butadiene. Bridging modes, (38) and (39), are also possible. 22.12 Benzene and other arenes (CO ) u R u R (CO ) 3 3 38 [(CO)3Ru(C8H8)Ru(CO)3] Cp Co Co Key point: A consideration of the MOs of benzene leads to a picture of bonding of benzene to a metal atom that includes a significant δ backbonding interaction. If benzene is considered to have three localized double bonds, each double bond can behave as a ligand and the molecule could behave as a tridentate η6-ligand. A compound such as bis(η6-benzene)chromium (40) could then be considered to be made up of six coordinated double bonds, each donating two electrons, bound to a d6 metal atom, giving a total of 18 valence electrons for the octahedral complex. Bis(η6-benzene)chromium does exist, and is remarkably stable: it can be handled in air and sublimes with no decomposition. Although this description of the bonding is a first step towards understanding its structure, the true picture needs a deeper consideration of the molecular orbitals involved. In the molecular orbital picture of the π bonding in benzene there are three bonding and three antibonding orbitals. If we consider a single benzene molecule bonding to a single metal, and consider only the d orbitals, the strongest interaction is a σ interaction between the most strongly bonding a1 benzene MO and the dz2 orbital of the metal atom; π bonds are possible between the two other bonding benzene MOs and the dzx and dyz orbitals. Backbonding from the metal atom to the benzene is possible as a δ interaction between the dx2y2 and dxy orbitals and the empty antibonding e2 orbitals of benzene (Fig. 22.8). η6-Arenes are considered to be neutral ligands that donate six electrons and are normally considered to take up three coordination sites at a metal. Cp 39 [CpCo(C8H8)CoCp] C6H6 Cr dxy e2 dx2–y2 dzx 40 Cr(C6H6)2 e1 dyz a1 dz2 Figure 22.8 The molecular orbitals of the π system in benzene; also shown are metal d orbitals of appropriate symmetry to form bonding interactions. 549 Ligands Hexahapto (η6) arene complexes are very easy to make, often simply by dissolving a compound that has three replaceable ligands in the arene and refluxing the solution: OMe Mo(CO)6 PhOMe Mo OC CO CO One commonly invoked reaction intermediate of η6-arene complexes is a ‘slipping’ to an η4 complex, which donates only four electrons to the metal, and therefore allows a substitution reaction to proceed without an initial ligand loss: M 41 η -(CH2CH=CH2) 1 OMe P + R Mo OC CO OMe 3 R 3P CO OC η6-arene CO – Mo CO OMe Mo CO R 3P η4-arene CO CO M η6-arene M 42 43 η2-Arenes are also known and are analogous to η2-alkenes; they have an important role in the activation of arenes by metal complexes. M 22.13 The allyl ligand 44 η3-(CH2CH=CH2) Key points: A consideration of the MOs of η3-allyl complexes leads to a picture that gives two identical CC bond lengths; because the type of bonding of the allyl ligand is so variable, η3-allyl complexes are often highly reactive. The allyl ligand, CH2 CHCH2, can bind to a metal atom in either of two configurations. As an η1-ligand (41) it should be considered just like an η1-alkyl group (that is, as a two-electron donor with a single negative charge). However, the allyl ligand can also use its double bond as an additional two-electron donor and act as an η3-ligand (42); in this arrangement it acts as a four-electron donor with a single negative charge. The η3-allyl ligand can be thought of as a resonance between two forms (43), and because all evidence points towards a symmetrical structure, it is often depicted with a curved line representing all the bonding electrons (44). As with benzene, a more detailed understanding of the bonding of an allyl group needs a consideration of the molecular orbitals of the organic fragment (Fig. 22.9), whereupon it becomes apparent why a symmetrical arrangement is the correct description of the η3bonding mode. The filled 1π orbital on the allyl group behaves as a σ donor (into the dz2 orbital), the 2π orbital behaves as a π donor (into the dzx orbital) and the 3π orbital behaves as a π acceptor (from the dyz orbital). Thus the interactions of the metal atom with each of the terminal carbon atoms are identical and a symmetrical arrangement results. The terminal substituents of an η3-allyl group are bent slightly out of the plane of the three-carbon backbone and are either syn (45) or anti (46) relative to the central hydrogen. It is common to observe anti and syn group exchange, which in some cases is fast on an NMR timescale. A mechanism that involves the transformation η3 to η1 to η3 is often invoked to explain this exchange. 45 syn 46 anti 3π dyz R syn R R anti R M 2π dzx M M M Because of this flexibility in the bonding, η3-allyl complexes are often highly reactive as transformation to the η1-form allows them to bind readily to another ligand. There are many synthetic routes to allyl complexes. One is the nucleophilic attack of an allyl Grignard reagent on a metal halide: 2 C3H5MgBr NiCl2 → [Ni(3-C3H5)2] 2 MgBrCl 1π dz 2 Figure 22.9 The molecular orbitals of the π system of the allyl group; also shown are metal d orbitals of appropriate symmetry to form bonding interactions. 550 22 d-Metal organometallic chemistry Nucleophilic attack on a haloalkane by a metal atom in a low oxidation state also yields allyl complexes: CO − [Mn(CO) 5] + CH 2=CHCH 2Cl OC CO OC Mn OC Mn OC CO CO CO − Cl CO In complexes where the metal centre is not protonated directly, the protonation of a butadiene ligand can lead to an η3-allyl complex: Fe OC CO CO HCl Cl OC Fe CO CO 22.14 Cyclopentadiene and cycloheptatriene Key points: The common η5-bonding mode of a cyclopentadienyl ligand can be understood on the basis of both σ and π donation from the organic fragment to the metal in conjunction with δ backbonding; cycloheptatriene commonly forms either η6-complexes or η7-complexes of the aromatic tropilium cation (C7H7). Cyclopentadiene, C5H6, is a mildly acidic hydrocarbon that can be deprotonated to form the cyclopentadienyl anion, C5H5. The stability of the cyclopentadienyl anion can be understood when it is realized that the six electrons in its π system make it aromatic. The delocalization of these six electrons results in a ring structure with five equal bond lengths. As a ligand, the cyclopentadienyl group, commonly denoted Cp, has played a major role in the development of organometallic chemistry and continues to be the archetype of cyclic polyene ligands. We have already alluded to the role ferrocene (4) had in the development of organometallic chemistry. A huge number of metal cyclopentadienyl and substituted cyclopentadienyl compounds are known. Some compounds have C5H5 as a monohapto ligand (12), in which case it is treated like an η1-alkyl group; others contain C5H5 as a trihapto ligand (13), in which case it is treated like an η3-allyl group. Usually, though, C5H5 is present as a pentahapto ligand, bound through all five carbons of its ring. We treat the 5-C5H5 group as a six-electron donor. Formally, the electron donation to the metal now comes from the filled 1π ( bonding) and 2π (π bonding) MOs (Fig. 22.10) with backbonding to the dxy and dx2y2 orbitals on the metal atom. As we shall see in Section 22.19, coordinated Cp ligands behave as though they maintain their six-electron aromatic structure. 3π dx2–y2 dxy dyz dzx 2π 1π dz 2 Figure 22.10 The molecular orbitals for the π systems of the cyclopentadienyl group; also shown are metal d orbitals of appropriate symmetry to form bonding interactions. 551 Ligands The electronic and steric properties of cyclopentadiene can easily be tuned: electron withdrawing and donating groups can be attached to the five-membered ring, and steric bulk can be enhanced by additional substitution. The pentamethylcyclopentadienyl ligand (Cp) is commonly used to provide greater electron density on, and greater steric protection of, a metal atom. Chiral groups are often added to Cp groups so that complexes can be used in stereoselective reactions: the neo-menthyl group (47) is commonly used. The synthesis, properties, and reactivities of compounds containing the cyclopentadienyl ligand are discussed in more detail in Section 22.19. Cycloheptatriene, C7H8 (48), can form η6-complexes such as (49), which may be treated as having three η2-alkene molecules bound to the metal atom. Hydride abstraction from these complexes results in formation of η7-complexes of the six-electron aromatic cation C7H7 (50), for example (51). In η7-cycloheptatrienyl complexes, all carboncarbon bond lengths are equal; bonding to the metal atom and backbonding from the metal are similar to those found in arene and cyclopentadienyl complexes. 47 neo-Menthylcyclpentadienyl 48 Cycloheptatriene 22.15 Carbenes Key points: Fischer- and Schrock-type carbene complexes are considered to have a metalcarbon double bond; N-heterocyclic carbenes are considered to have a metalcarbon single bond, together with π backbonding. Carbene, CH2, has only six electrons around its C atom and is consequently highly reactive. Other substituted carbenes exist and are substantially less reactive and can behave as ligands towards metals. In principle, carbenes can exist in one of two electronic configurations: with a linear arrangement of the two groups bound to the carbon atom and the two remaining electrons unpaired in two p orbitals (52), or with the two groups bent, the two remaining electrons paired, and an empty p orbital (53). Carbenes with a linear arrangement of groups are referred to as ‘triplet carbenes’ (because the two unpaired electrons are unpaired and S 1) and are favoured when sterically very bulky groups are attached to the carbene carbon. Carbenes with the bent arrangement are known as ‘singlet carbenes’ (the two electrons are paired and S 0) and are the normal form for carbenes. The electron pair on the carbon atom of a singlet carbene is suitable to bond to a metal atom, resulting in a ligand-to-metal bond. The empty p orbital on the C atom can then accept electron density from the metal atom, thus stabilizing the electron-poor carbon atom (54). For historical reasons, carbenes bonded to metal atoms in this fashion are known as Fischer carbenes and are represented by a metalcarbon double bond. Fischer carbenes are electron deficient at the C atom and consequently are easily attacked by nucleophiles. When the backbonding to the C atom is very strong, the carbene can become electron rich and is thus prone to attack by electrophiles. Carbenes of this type are known as Schrock carbenes, after their discoverer. The term alkylidene technically refers only to carbenes with alkyl substituents (CR2), but is sometimes used to mean both Fischer and Schrock carbenes. More recently, a large number of derivatives of what are known as N-heterocyclic carbenes (NHCs) have been used as ligands. In most NHCs, two nitrogen atoms are adjacent to the carbene carbon atom, and if the lone pair on the nitrogen is considered to be largely p-orbital based, then strong π-donor interactions from the two nitrogen atoms can help to stabilize the carbene (55). Tying the carbene C atom and the two N atoms into a ring helps to stabilize the carbene, and five-membered rings are common (56). Additional stability can be achieved by a double bond in the ring, which provides an additional two electrons that may be considered part of a six-electron aromatic resonance structure (57). o M C O CO C O 49 [Mo(η6-C7H8)(CO)3] 50 C7H7+ B F Mo C O C O – 4 O C 51 52 C N M 53 54 N N N C C 55 56 552 22 d-Metal organometallic chemistry NHC ligands are considered to be two-electron σ donors and initial descriptions of the bonding suggested only minimal π backbonding from the metal atom. However, the current view is that there is actually significant π backbonding from the metal atom to the NHC. N N C Key point: Alkanes can donate the electron density from CH single bonds to a metal atom and, in the absence of other donors, even the electron density of a noble gas atom can enable it to behave as a ligand. 57 C H M 58 C H M 59 Re OC OC H C H Me Me H Me Ph3P Pd Me PPh3 61 i Pr Re OC F3P Xe 62 M N=N 63 Highly reactive metal intermediates can be generated by photolysis and, in the absence of any other ligands, alkanes and noble gases have been observed to coordinate to the metal atom. Such species were first identified in the 1970s in solid methane and noble gas matrices, and were initially regarded as mere curiosities. However, both species have recently been fully characterized in solution and are now accepted as important intermediates in some reactions. Alkanes are considered to donate electron density from a CH σ bond to the metal atom (58), and accept π-electron density back from the metal atom into the corresponding σ orbital (59), just like dihydrogen (Section 22.7). Although most alkane complexes are short lived, with the alkane being readily displaced, in 1998 the cyclopentane complex (60) was unambiguously identified in solution by NMR. Interactions between the CH bond of an already coordinated ligand and the metal atom have also been observed. These species are referred to as having agostic CH interactions, from the Greek for ‘to hold on to oneself’, and are thought to have additional stability due to the chelate effect (Section 7.14). Many examples of compounds with agostic interactions are now known; an example is (61). Though weakly bonding, each CH to metal atom interaction, whether it be agostic or not, is considered formally to donate two electrons to the metal. Unlikely as it might seem, noble gas atoms can behave as ligands towards metal centres, and a number of complexes of Kr and Xe have been identified by IR spectroscopy, with the relatively long-lived Xe complex (62) characterized in solution by NMR in 2005. These complexes are stable only in the absence of better ligands (such as alkanes). The noble gas is formally considered to be neutral and donate two electrons. 22.17 Dinitrogen and nitrogen monoxide 60 Br 22.16 Alkanes, agostic hydrogens, and noble gases Key points: The bonding of dinitrogen to a metal atom is weak but contains both a σ-donating and a π-accepting component; nitrogen monoxide can bind to a metal in two different ways—either bent or straight. Neither dinitrogen, N2, nor nitrogen monoxide, NO, is strictly an organometallic ligand, although they are sometimes found in organometallic compounds. Dinitrogen is a much sought after ligand as complexes can potentially take part in a catalytic reduction of nitrogen to more useful species. Dinitrogen can bind to metals in a number of different ways. The majority of complexes have a terminal monohapto link, η1-N2, in which the bonding can be considered to be like that of the isoelectronic CO ligand (63). Dinitrogen is both a weaker σ donor and a weaker π acceptor than CO, and hence is bound less strongly; in fact only good π-donor metal atoms bind N2. Like CO, the N2 ligand has a distinctive IR stretching band lying in the range 21501900 cm1. A dinitrogen molecule can participate in two bonding interactions and bridge two metal atoms (64). If the backdonation to the nitrogen is extensive in this kind of complex, it can formally be considered to have been reduced to a hydrazine (65). Occasionally, dinitrogen ligands are found bound in a dihapto (η2) side-on fashion (66). In these complexes the ligand is best considered analogous to an η2-alkyne. The side-on bonding mode seems to be particularly common in complexes of the f metals (Chapter 23). Nitrogen monoxide (nitric oxide) is a radical with 11 valence electrons. When bound, NO is referred to as the nitrosyl ligand and can bond in one of two modes to d-metal atoms, in either a bent or a linear fashion. In the linear arrangement (67), the ligand is considered to be the NO cation. The NO cation is isoelectronic with CO, and the bonding can be considered in a similar fashion (a two-electron σ donor, with strong π-acceptor ability). Compounds In the bent arrangement (68), NO is considered to behave as NO, again donating two electrons. In many complexes, NO can change its coordination mode; in effect, moving from the linear to the bent mode reduces the number of electrons on the metal by 2. E X A M PL E 22 . 5 Counting electrons in complexes M N N M N M 64 M N Which of the following compounds have 18 electrons: (a) the agostic Pd compound shown as (61), (b) [Re(iPr-Cp)(CO)(PF3)Xe] (62)? 65 Answer (a) We consider both the η1-alkenyl and the bromide ligand as two-electron singly negatively charged donors, with PPh3 ligands being considered as two-electron neutral donors. Thus the compound is a complex of Pd(II), which provides eight further electrons. The total number of electrons, before we consider the agostic interaction, is therefore (4 2) 8 16. The agostic interaction can be considered to donate a further two electrons, leading to an 18-electron compound. (b) The iPr-Cp ligand is considered a six-electron donor with a single negative charge, the CO, PF3, and Xe ligands are each considered to be two-electron neutral donors, implying that the Re must have oxidation number 1, so providing a further six electrons. The total electron count is therefore 6 2 2 2 6 18. N=N M 66 M Self-test 22.5 Show that both (a) [Mo(η6-C7H8)(CO)3] (49) and (b) [Mo(η7-C7H7)(CO)3] (51) are 18-electron species. N N O Compounds 22.18 d-Block carbonyls d-Block carbonyls have been studied extensively since the discovery of tetracarbonylnickel in 1890. Interest in carbonyl compounds has not waned, with many important industrial processes relying on carbonyl intermediates. (a) Homoleptic carbonyls Key points: The carbonyls of the Period 4 elements of Groups 6 to 10 obey the 18-electron rule; they have alternately one and two metal atoms and a decreasing number of CO ligands. A homoleptic complex is a complex with only one kind of ligand. Simple homoleptic metal carbonyls can be prepared for most of the d metals, but those of Pd and Pt are so unstable that they exist only at low temperatures. No simple neutral metal carbonyls are known for Cu, Ag, and Au or for the members of Group 12. The metal carbonyls are useful synthetic precursors for other organometallic compounds and are used in organic syntheses and as industrial catalysts. The 18-electron rule helps to systematize the formulas of metal carbonyls. As shown in Table 22.5, the carbonyls of the Period 4 elements of Groups 6 to 10 have alternately one and two metal atoms and a decreasing number of CO ligands. The binuclear carbonyls are formed by elements of the odd-numbered groups, which have an odd number of valence electrons and therefore dimerize by forming metalmetal (MM) bonds (each MM bond effectively increases the electron count on the metal by 1). The decrease in the number of CO ligands from left to right across a period matches the need for fewer CO ligands to achieve 18 valence electrons. The simple vanadium carbonyl V(CO)6 is an exception as it only has 17 valence electrons and is too sterically crowded to dimerize; it is, however, readily reduced to the 18-electron V(CO)6 anion. O 67 M The preceding discussion of ligands and their bonding modes suggests that there are likely to be a large number of organometallic compounds that have either 16 or 18 valence electrons. A detailed discussion of all these compounds is well beyond the scope of this book. However, we shall examine a number of different classes of compounds because they provide insight into the structures and properties of many other compounds that can be derived from them. Thus, we shall now consider the structures, bonding, and reactions of metal carbonyls, which historically formed the foundation of much of d-block organometallic chemistry. Then we consider some sandwich compounds, before describing the structures and reactions of metal cluster compounds. 553 68 554 22 d-Metal organometallic chemistry Table 22.5 Formulas and electron count for some 3d-series carbonyls Group formula Valence electrons 6 Cr Cr(CO)6 Structure 6 CO 6(CO) 12 OC Total 18 OC CO Cr CO CO 7 8 Mn2(CO)10 Mn Fe(CO)5 7 5(CO) 10 M—M 1 Total 18 Fe OC OC 10 Total 18 Mn CO Mn CO OC OC 8 5(CO) CO OC CO CO CO OC CO Fe CO CO 9 8 Co2(CO)8 Ni(CO)4 Co 9 4(CO) 8 M—M 1 Total 18 Ni 10 4(CO) 8 Total 18 OC CO CO Co Co OC C O C O CO CO CO Ni OC CO CO Simple metal carbonyl molecules often have well-defined, simple, symmetrical shapes that correspond to the CO ligands taking up the most distant locations, like regions of enhanced electron density in the VSEPR model. Thus, the Group 6 hexacarbonyls are octahedral, pentacarbonyliron(0) is trigonal bipyramidal, tetracarbonylnickel(0) is tetrahedral, and decacarbonyldimanganese(0) consists of two square-pyramidal Mn(CO)5 groups joined by a metalmetal bond. Bridging carbonyls are also found, for example one isomer of octacarbonyldicobalt(0) has its metalmetal bond bridged by two CO ligands. (b) Synthesis of homoleptic carbonyls Key points: Some metal carbonyls are formed by direct reaction, but of those that can be formed in this way most require high pressures and temperatures; metal carbonyls are commonly formed by reductive carbonylation. The two principal methods for the synthesis of monometallic metal carbonyls are direct combination of carbon monoxide with a finely divided metal and the reduction of a metal salt in the presence of carbon monoxide under pressure. Many polymetallic carbonyls are synthesized from monometallic carbonyls. In 1890 Mond, Langer, and Quinke discovered that the direct combination of nickel and carbon monoxide produced tetracarbonylnickel(0), Ni(CO)4, a reaction that is used in the Mond process for purifying nickel (Box 22.1): 50 C, 1 atm CO → Ni(CO)4(g) Ni(s) 4 CO(g) ⎯⎯⎯⎯⎯⎯ Tetracarbonylnickel(0) is in fact the metal carbonyl that is most readily synthesized in this way, with other metal carbonyls, such as Fe(CO)5, being formed more slowly. They are therefore synthesized at high pressures and temperatures (Fig. 22.11): 200 C, 200 atm Fe(s) 5 CO(g) ⎯⎯⎯⎯⎯⎯ → Fe(CO)5(l) 150 C, 35 atm 2 Co(s) 8 CO(g) ⎯⎯⎯⎯⎯⎯ → Co2(CO)8(s) Compounds Direct reaction is impractical for most of the remaining d metals, and reductive carbonylation, the reduction of a salt or metal complex in the presence of CO, is normally employed instead. Reducing agents vary from active metals such as aluminium and sodium, to alkylaluminium compounds, H2, and CO itself: 555 High-pressure gas supply Thermocouple AlCl3 , benzene CrCl3(s) Al(s) 6 CO(g) ⎯⎯⎯⎯⎯⎯ → AlCl3(soln) Cr(CO)6(soln) 150 C, 200 atm, CH3OH 3 Ru(acac)3(soln) H2(g) 12 CO(g) ⎯⎯⎯⎯⎯⎯⎯⎯ → Ru3(CO)12 (soln) … 250 C, 350 atm Re2O7(s) 17 CO(g) ⎯⎯⎯⎯⎯⎯ → Re2(CO)10(s) 7 CO2(g) (c) Properties of homoleptic carbonyls Key points: All the mononuclear carbonyls are volatile; all the mononuclear and many of the polynuclear carbonyls are soluble in hydrocarbon solvents; polynuclear carbonyls are coloured. Iron and nickel carbonyls are liquids at room temperature and pressure but all other common carbonyls are solids. All the mononuclear carbonyls are volatile; their vapour pressures at room temperature range from approximately 50 kPa for tetracarbonylnickel(0) to approximately 10 Pa for hexacarbonyltungsten(0). The high volatility of Ni(CO)4, coupled with its extremely high toxicity, means that unusual care is required in its handling. Although the other carbonyls appear to be less toxic, they too must not be inhaled or allowed to touch the skin. Because they are nonpolar, all the mononuclear and many of the polynuclear carbonyls are soluble in hydrocarbon solvents. The most striking exception among the common carbonyls is nonacarbonyldiiron(0), [Fe2(CO)9], which has a very low vapour pressure and is insoluble in solvents with which it does not react. By contrast, [Mn2(CO)10] and [Co2(CO)8] are soluble in hydrocarbon solvents and sublime readily. Most of the mononuclear carbonyls are colourless or lightly coloured. Polynuclear carbonyls are coloured, the intensity of the colour increasing with the number of metal atoms. For example, pentacarbonyliron(0) is a light straw-coloured liquid, nonacarbonyldiiron(0) forms golden-yellow flakes, and dodecacarbonyltriiron(0) is a deep green compound that looks black in the solid state. The colours of polynuclear carbonyls arise from electronic transitions between orbitals that are largely localized on the metal framework. The principal reactions of the metal centre of simple metal carbonyls are substitution (Section 22.21), oxidation, reduction, and condensation into clusters (Section 22.20). In certain cases, the CO ligand itself is also subject to attack by nucleophiles or electrophiles. Metal thermocouple well Glass container Stainless steel pressure vessel Figure 22.11 A high-pressure reaction vessel. The reaction mixture is in a glass container. (d) Oxidation and reduction of carbonyls Key points: Most metal carbonyls can be reduced to metal carbonylates; some metal carbonyls disproportionate in the presence of a strongly basic ligand, producing the ligated cation and a carbonylate anion; metal carbonyls are susceptible to oxidation by air; metalmetal bonds undergo oxidative cleavage. B OX 22 .1 The Mond process Ludwig Mond, Carl Langer, and Friederich Quinke discovered Ni(CO)4 in the course of studying the corrosion of nickel valves in process gas containing CO in 1890. They were not able to characterize the new compound fully (calling it ‘nickel-carbon-oxide’) commenting, ‘We have at present no suggestion to offer as to the constitution of this remarkable compound’. They were, however, able to assign the formula ‘Ni(CO)4’ to their new compound and were quick to apply their discovery to develop a new industrial process (the Mond process) for the purification of nickel. This process was such a success that it brought nickel all the way from Canada to Mond’s factory in Wales. The Mond process relies on the ease of synthesis of Ni(CO)4. At a pressure of 1 atm of carbon monoxide, nickel metal will react at around 50°C to give Ni(CO)4: Ni 4 CO → Ni(CO)4 At this temperature Ni(CO)4 (b.p. 34°C) is a gas and is easily separated from the residues and from the impure nickel. Tetracarbonylnickel decomposes to give pure nickel at about 220°C, liberating carbon monoxide, which can then be reused. Typically the impure nickel is generated from the reduction of nickel oxide ores with a mixture of hydrogen and carbon monoxide. Mond, Langer, and Quinke also attempted the synthesis of analogous compounds with other metals, but were unable to isolate anything new. They did, however, succeed in extracting nickel contaminants from samples of cobalt, suggesting a way to purify cobalt too. The centenary of the discovery of tetracarbonylnickel is celebrated in a special volume of J. Organomet. Chem., 1990, 383, which is devoted to metal carbonyl chemistry. 556 22 d-Metal organometallic chemistry Most neutral metal carbonyl complexes can be reduced to an anionic form known as a metal carbonylate. In monometallic carbonyls, two-electron reduction is generally accompanied by loss of the two-electron donor CO ligand, thus preserving the electron count at 18: THF 2 Na Fe(CO)5 ⎯⎯⎯ → (Na)2[Fe(CO)4]2 CO The metal carbonylate contains Fe with oxidation number 2, and it is rapidly oxidized by air. That much of the negative charge is delocalized over the CO ligands is confirmed by the observation of a low CO stretching band in the IR spectrum at about 1730 cm1. Polynuclear carbonyls, which obey the 18-electron rule through the formation of MM bonds, are generally cleaved by strong reducing agents. The 18-electron rule is obeyed in the product and a mononegative mononuclear carbonylate results: THF 2 Na (OC)5MnMn(CO)5 ⎯⎯⎯ → 2 Na[Mn(CO)5] Some metal carbonyls disproportionate in the presence of a strongly basic ligand, producing the ligated cation and a carbonylate. Much of the driving force for this reaction is the stability of the metal cation when it is surrounded by strongly basic ligands. Octacarbonyldicobalt(0) is highly susceptible to this type of reaction when exposed to a good Lewis base such as pyridine (py): 3 [Co2(0)(CO)8] 12 py → 2 [Co(2)(py)6][Co(−1)(CO)4]2 8 CO It is also possible for the CO ligand to be oxidized in the presence of the strongly basic ligand OH, the net outcome being the reduction of a metal centre: 2 3 3 [Fe(0)(CO)5] 4 OH → [Fe3( –)(CO)11]2 CO32 2 H2O 3 CO Carbonyl compounds that have only 17 electrons are particularly prone to reduction to give 18-electron carbonylates. Metal carbonyls are susceptible to oxidation by air. Although uncontrolled oxidation produces the metal oxide and CO or CO2, of more interest in organometallic chemistry are the controlled reactions that give rise to organometallic halides. One of the simplest of these is the oxidative cleavage of an MM bond: [(OC)5Mn(0)Mn(0)(CO)5] Br2 → 2 [Mn(1)Br(CO)5] In keeping with the loss of electron density from the metal when a halogen atom is attached, the CO stretching frequencies of the product are significantly higher than those of [Mn2(CO)10]. Table 22.6 Acidity constants of d-metal hydrides in acetonitrile at 25°C Hydride pKa [CoH(CO)4] 8.3 [CoH(CO)3P(OPh)3] 11.3 [Fe(H)2(CO)4] 11.4 [CrH(Cp)(CO)3] 13.3 [MoH(Cp)(CO)3] 13.9 [MnH(CO)5] 15.1 [CoH(CO)3PPh3] 15.4 [WH(Cp)(CO)3] 16.1 [MoH(Cp)(CO)3] 17.1 [Ru(H)2(CO)4] 18.7 [FeH(Cp)(CO)2] 19.4 [RuH(Cp)(CO)2] 20.2 [Os(H)2(CO)4] 20.8 [ReH(CO)5] 21.1 [FeH(Cp)(CO)2] 26.3 [WH(Cp)(CO)2PMe3] 26.6 (e) Metal carbonyl basicity Key points: Most organometallic carbonyl compounds can be protonated at the metal centre; the acidity of the protonated form depends on the other ligands on the metal. Many organometallic compounds can be protonated at the metal centre. Metal carbonylates provide many examples of this basicity: [Mn(CO)5] H → [MnH(CO)5] The affinity of metal carbonylates for the proton varies widely (Table 22.6). It is observed that, the greater the electron density on the metal centre of the anion, the higher its Brønsted basicity and hence the lower the acidity of its conjugate acid (the metal carbonyl hydride). As we noted in Section 22.7, d-block MH complexes are commonly referred to as ‘hydrides’, which reflects the assignment of oxidation number 1 to an H atom attached to a metal atom. Nevertheless, most of the carbonyl hydrides of metals to the right of the d block are Brønsted acids. The Brønsted acidity of a metal carbonyl hydride is a reflection of the π-acceptor strength of a CO ligand, which stabilizes the conjugate base. Thus, [CoH(CO)4] is acidic whereas [CoH(PMe3)4] is strongly hydridic. In striking contrast to p-block hydrogen compounds, the Brønsted acidity of d-block MH compounds decreases on descending a group. Neutral metal carbonyls (such as pentacarbonyliron, [Fe(CO)5]) can be protonated in air-free concentrated acid; the Brønsted basicity of a metal atom with oxidation number zero is associated with the presence of nonbonding d electrons. Compounds having metalmetal bonds, such as clusters (Section 22.20), are even more easily protonated; here the Compounds Brønsted basicity is associated with the ready protonation of MM bonds to produce a formal 3c,2e bond like that in diborane: [Fe3(CO)11]2 H → [Fe3H(CO)11] The MHM bridge is by far the most common bonding mode of hydrogen in clusters. Metal basicity is turned to good use in the synthesis of a wide variety of organometallic compounds. For example, alkyl and acyl groups can be attached to metal atoms by the reaction of an alkyl or acyl halide with an anionic metal carbonyl: [Mn(CO)5] CH3I → [Mn(CH3)(CO)5] I [Co(CO)4] CH3COI → [Co(COCH3)(CO)4] I A similar reaction with organometallic halides may be used to form MM bonds: [Mn(CO)5] [ReBr(CO)5] → [(OC)5MnRe(CO)5] Br (f) Reactions of the CO ligand Key points: The C atom of CO is susceptible to attack by nucleophiles if it is attached to a metal atom that is electron poor; the O atom of CO is susceptible to attack by electrophiles in electron-rich carbonyls. The C atom of CO is susceptible to attack by nucleophiles if it is attached to a metal atom that is not electron rich. Thus, terminal carbonyls with high CO stretching frequencies are liable to attack by nucleophiles. The d electrons in these neutral or cationic metal carbonyls are not extensively delocalized on to the carbonyl C atom and so that atom can be attacked by electron-rich reagents. For example, strong nucleophiles (such as methyllithium, Section 11.17) attack the CO in many neutral metal carbonyl compounds: 1 4 [Li4(CH3)4] [Mo(CO)6] → Li[Mo(COCH3)(CO)5] The resulting anionic acyl compound reacts with carbocation reagents to produce a stable and easily handled neutral product: CO O Li+ OC CO OEt Et3O+ Mo OC – BF 4– OC CO LiBF4 + Et2O Mo OC CO CO CO The product of this reaction, with a direct M C bond, is a Fischer carbene (Section 22.15). The attack of a nucleophile on the C atom is also important for the mechanism of the hydroxide-induced dissociation of metal carbonyls: [(OC)nM(CO)] OH → [(OC)nM(COOH)] [(OC)nM(COOH)] 3 OH → [M(CO)n]2 CO32 2 H2O In electron-rich metal carbonyls, considerable electron density is delocalized on the CO ligand. As a result, in some cases the O atom of a CO ligand is susceptible to attack by electrophiles. Once again, IR data provide an indication of when this type of reaction should be expected, as a low CO stretching frequency indicates significant backdonation to the CO ligand and hence appreciable electron density on the O atom. Thus a bridging carbonyl is particularly susceptible to attack at the O atom: AlEt3 O C Fe OC O C 2 + AlEt Fe C O CO Fe 3 OC Fe C O CO Et3Al The attachment of an electrophile to the oxygen of a CO ligand, as in the structure on the right of this equation, promotes migratory insertion reactions (Section 22.24) and CO cleavage reactions. The ability of some alkyl-substituted metal carbonyls to undergo a migratory insertion reaction to give acyl ligands, (CO)R, is discussed in detail in Section 22.24. 557 558 22 d-Metal organometallic chemistry E X A M PL E 22 .6 Converting CO to carbene and acyl ligands Propose a set of reactions for the formation of [W(C(OCH3)Ph)(CO)5] starting with hexacarbonyltungsten(0) and other reagents of your choice. Answer We know that CO ligands in hexacarbonyltungsten(0) are susceptible to attack by nucleophiles, and therefore that the reaction with phenyllithium should give a C-phenyl intermediate: – CO O OC Li+ W(CO)6 + PhLi W OC Ph CO CO This anion can then react with a carbon electrophile to attach an alkyl group to the O atom of the CO ligand: CO O Li+ OC W OC CO Ph CO – CO OMe Me3O+ BF4– OC W OC Ph CO LiBF 4 + Me2O CO Self-test 22.6 Propose a synthesis for [Mn(COCH3)(CO)4(PPh3)] starting with [Mn2(CO)10], PPh3, Na, and CH3I. Resultant 69 (g) Spectroscopic properties of carbonyl compounds Key points: The CO stretching frequency is decreased when it serves as a π acceptor; donor ligands cause the CO stretching frequency to decrease as they supply electrons to the metal; 13C-NMR is of less use as many carbonyl compounds are fluxional on the NMR timescale. Resultant 70 2000 1900 ~ ν/cm–1 1800 1700 O C Fe OC Fe C O CO Figure 22.12 The infrared spectrum of [Fe2(Cp)2(CO)4]. Note the two high-frequency terminal CO stretches and the lower frequency absorption of the bridging CO ligands. Although two bridging CO bands would be expected on account of the low symmetry of the complex, a single band is observed because the two bridging CO groups are nearly collinear. Infrared and 13C-NMR spectroscopy are widely used to determine the arrangement of atoms in metal carbonyl compounds, as separate signals are observed for inequivalent CO ligands. NMR spectra generally contain more detailed structural information than IR spectra provided the molecule is not fluxional (the timescales of the NMR and IR transition are different, Sections 8.4, 8.5). However, IR spectra are often simpler to obtain, and are particularly useful for following reactions. Most CO stretching bands occur in the range 21001700 cm1, a region that is generally free of bands due to organic groups. Both the range of CO stretching frequencies (see Fig. 22.4) and the number of CO bands (Table 22.7) are important for making structural inferences. Group theory allows us to predict the number of active CO stretches in both the IR and Raman spectra (Section 6.5). If the CO ligands are not related by a centre of inversion or a threefold or higher axis of symmetry, a molecule with N CO ligands will have N CO stretching absorption bands. Thus a bent OCMCO group (with only a twofold symmetry axis) will have two infrared absorptions because both the symmetric (69) and antisymmetric (70) stretches cause the electric dipole moment to change and are IR active. Highly symmetrical molecules have fewer bands than CO ligands. Thus, in a linear OCMCO group, only one IR band (corresponding to the out-of-phase stretching of the two CO ligands) is observed in the CO stretching region because the symmetrical stretch leaves the overall electric dipole moment unchanged. As shown in Fig. 22.12, the positions of the CO ligands in a metal carbonyl may be more symmetrical than the point group of the whole compound suggests and then fewer bands will be observed than are predicted on the basis of the overall point group. Raman spectroscopy can be very useful in assigning structures because the selection rules complement those of IR (Sections 6.5 and 8.4). Thus, for a linear OCMCO group, the symmetrical stretching of the two CO ligands is observed in the Raman spectrum. As we noted in Section 22.5, infrared spectroscopy is also useful for distinguishing terminal CO (MCO) from bridging CO (μ2-CO) and face-bridging CO (μ3-CO). It can also be used to determine the order of π-acceptor strengths for the other ligands present in a complex. A motionally averaged NMR signal is observed when a molecule undergoes changes in structure more rapidly than the technique can resolve (Section 8.5). Although this phenomenon is quite common in the NMR spectra of organometallic compounds, it is not normally observed in their IR or Raman spectra. An example of this difference is Fe(CO)5, 559 Compounds Table 22.7 Relation between the structure of a carbonyl complex and the number of CO stretching bands in its IR spectrum Complex Isomer Structure Point group Number of bands Complex Isomer Structure CO OC M(CO)6 M OC CO Oh 1 M(CO)5 ax OC M OC C4v 3† M(CO)4L eq L trans OC CO OC 1 CO M OC M(CO)3L2 trans OC OC L L M L 4‡ M(CO)3L2 cis L CO M fac OC L C2v M M(CO)4 OC L L C3v M CO CO 1 C3v 2 L 2 M(CO)3L3 fac OC M OC L L CO CO OC Td CO 3‡ CO M(CO)5 3 CO CO M Cs CO L M(CO)3L3 1 L C2v CO OC D3h CO L mer CO M L CO M(CO)3L3 4 L D4h CO cis C2v CO CO L M(CO)4L2 CO M L M(CO)4L2 3§ CO CO CO M C3v CO CO OC CO M CO L M(CO)5L Number of bands L CO CO OC Point group CO D3h 2 CO CO The number of IR bands expected in the CO stretching region is based on formal selection rules, and in some cases fewer bands are observed, as explained above. † If the fourfold array of CO ligands lies in the same plane as the metal atom, two bands will be observed. ‡ If the trans-CO ligands are nearly collinear, one fewer band will be observed. § If the threefold array of CO ligands is nearly planar, only two bands will be observed.. for which the 13C-NMR signal shows a single line at 210, whereas IR and Raman spectra are consistent with a trigonal-bipyramidal structure. E X A M PL E 22 .7 Determining the structure of a carbonyl from IR data The complex [Cr(CO)4(PPh3)2] has one very strong IR absorption band at 1889 cm1 and two other very weak bands in the CO stretching region. What is the probable structure of this compound? (The CO stretching frequencies are lower than in the corresponding hexacarbonyl because the phosphine ligands are better donors and poorer π acceptors than CO.) Answer If we consider the possible isomers, we can see that a disubstituted hexacarbonyl may exist with either cis or trans configurations. In the cis isomer the four CO ligands are in a low-symmetry (C2v) environment and therefore four IR bands should be observed, as indicated in Table 22.7. The trans isomer has a square-planar array of four CO ligands (D4h), for which only one band in the CO stretching region is expected (Table 22.7). The trans CO arrangement is indicated by the data because it is reasonable to assume that the weak bands reflect a small departure from D4h symmetry imposed by the PPh3 ligands. 560 22 d-Metal organometallic chemistry Self-test 22.7 The IR spectrum of [Ni2(5-Cp)2(CO)2] has a pair of CO stretching bands at 1857 cm1 (strong) and 1897 cm1 (weak). Does this complex contain bridging or terminal CO ligands, or both? (Substitution of 5-C5H5 ligands for CO ligands leads to small shifts in the CO stretching frequencies for a terminal CO ligand.) 22.19 Metallocenes As we have already noted, cyclopentadienyl compounds are often remarkably stable and the discovery of (Cp)2Fe, ferrocene, in 1951 sparked renewed interest in the whole field of d-block organometallic compounds. Many Cp complexes have two ring systems, with the metal sandwiched between the two rings, and it was for work on so-called ‘sandwich compounds’, formally metallocenes, that Wilkinson and Fischer were awarded the Nobel Prize in 1972. In keeping with the picture of a metallocene as a metal sandwiched between two planar carbon rings, we can consider compounds of 4-cyclobutadiene, 5-cyclopentadienyl, 6-arenes, 7-cycloheptatrienyl (C7H7), 8-cyclooctatriene, and even 3-cyclopropenium to be metallocenes. As all the carboncarbon bond lengths in each of these bound ligands are identical, it makes sense to treat each ligand as having an aromatic configuration; that is, 4-cyclobutadiene2 3-cyclopropenium 8-cyclooctatetraenyl2 5-cyclopentadienyl 6- arenes 7-cycloheptatrienyl π electrons: 2 6 10 This picture of the structure of metallocenes is not wholly in keeping with either system of electron counting, but is closer to the donor-pair method. We touched on the structures and reactivities of some metallocenes when we discussed the mode of bonding of ligands such as the cyclopentadienyl group, and here we look at some further aspects of their bonding and reactivity. (a) Synthesis and reactivity of cyclopentadienyl compounds Key points: Deprotonation of cyclopentadiene gives a convenient precursor to many metal cyclopentadienyl compounds; bound cyclopentadienyl rings behave as aromatic compounds and will undergo FriedelCrafts electrophilic reactions. Sodium cyclopentadienide, NaCp, is a common starting material for the preparation of cyclopentadienyl compounds. It can conveniently be prepared by the action of metallic sodium on cyclopentadiene in tetrahydrofuran solution: THF ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 2 Na[C5H5] H2 2 Na 2 C5H6 ⎯ Sodium cyclopentadienide can then be used to react with d-metal halides to produce metallocenes. Cyclopentadiene itself is acidic enough that potassium hydroxide will deprotonate it in solution and, for example, ferrocene can be prepared with a minimum of fuss: DMSO → Fe(C5H5)2 2 H2O 2 KCl 2 KOH 2 C5H6 FeCl2 ⎯⎯⎯⎯ Because of their great stability, the 18-electron Group 8 compounds ferrocene, ruthenocene, and osmocene maintain their ligandmetal bonds under rather harsh conditions, and it is possible to carry out a variety of transformations on the cyclopentadienyl ligands. For example, they undergo reactions similar to those of simple aromatic hydrocarbons, such as FriedelCrafts acylation: O Fe MeCOCl + AlCl3 Fe Compounds 561 It also is possible to replace H on a C5H5 ring by Li: Fe BuLi Li Fe BuH As might be imagined, the lithiated product is an excellent starting material for the synthesis of a wide variety of ring-substituted products and in this respect resembles simple organolithium compounds (Section 11.17). Most Cp complexes of other metals undergo reactions similar to these two types where the five-membered ring behaves as an aromatic system. (b) Bonding in bis(cyclopentadienyl)metal complexes Key points: The MO picture of bonding in bis(Cp) metal complexes shows that the frontier orbitals are neither strongly bonding nor strongly antibonding; thus complexes that do not obey the 18-electron rule are possible. We start by looking at ferrocene, where, although details of the bonding are not settled, the molecular orbital energy level diagram shown in Fig. 22.13 accounts for a number of experimental observations. This diagram refers to the eclipsed (D5h) form of the complex which, in the gas phase, is about 4 kJ mol1 lower in energy than the staggered conformation (Section 22.19c). We shall focus our attention on the frontier orbitals. As shown in Fig. 22.13, the e1 symmetry-adapted linear combinations of ligand orbitals have the same M M e1’ a2’’ e2’ a1’ a2’’ e1’ e2’’ pz px,py e2’’ a1’ e2’ s e1’’ e1’ e1’’ a1’ e2’ e2 ’ e1’ e1’’ a2’’ a1’ a2’’ a1’ a1’ e1’’ dz 2 dx 2–y2,dxy dyz,dzx Figure 22.13 Molecular orbital energy diagram for a M(Cp)2 with D5h symmetry. The energies of the symmetry-adapted π orbitals of the C5H5 ligands are shown on the left, relevant d orbitals of the metal are on the right, and the resulting molecular orbital energies are in the centre. Eighteen electrons can be accommodated by filling the molecular orbitals up to and including the a1’ orbital in the box. The box denotes the orbitals typically regarded as frontier orbitals in these molecules. 562 22 d-Metal organometallic chemistry symmetry as the dzx and dyz orbitals of the metal atom. The lower energy frontier orbital (a1) is composed of dz2 and the corresponding SALC of ligand orbitals. However, there is little interaction between the ligands and the metal orbitals because the ligand π orbitals happen to lie, by accident, in the conical nodal surface of dz2 orbital of the metal atom. In ferrocene and the other 18-electron bis(cyclopentadienyl) complexes, the a1 frontier orbital and all lower orbitals are full but the e1 frontier orbital and all higher orbitals are empty. The frontier orbitals are neither strongly bonding nor strongly antibonding. This characteristic permits the possibility of the existence of bis(cyclopentadienyl) complexes that diverge from the 18-electron rule, such as the 17-electron complex [Fe(5-Cp)2] and the 20-electron complex [Ni(5-Cp)2]. Deviations from the 18-electron rule, however, do lead to significant changes in MC bond lengths that correlate fairly well with the molecular orbital scheme (Table 22.8). Similarly, the redox properties of the complexes can be understood in terms of the electronic structure. We have already mentioned that ferrocene is fairly readily oxidized to the ferrocinium ion, [Fe(5-Cp)2]. From an orbital viewpoint, this oxidation corresponds to the removal of an electron from the nonbonding a1 orbital. The 19-electron complex [Co(5-Cp)2] is much more readily oxidized than ferrocene because the electron is lost from the antibonding e1 orbital to give the 18-electron [Co(5-Cp)2] ion. A useful comparison can be made with octahedral complexes. The e1 frontier orbital of a metallocene is the analogue of the eg orbital in an octahedral complex, and the a1 orbital plus the e2 pair of orbitals are analogous to the t2g orbitals of an octahedral complex. This formal similarity extends to the existence of high- and low-spin bis(cyclopentadienyl) complexes. E X A M PL E 22 . 8 Identifying metallocene electronic structure and stability Refer to Fig. 22.13. Discuss the occupancy and nature of the HOMO in [Co(5-Cp)2] and the change in metalligand bonding relative to neutral cobaltocene. Answer If we consider the [Co(5-Cp)2] ion we can see that it contains 18 valence electrons (six from Co(III), 12 from the two Cp ligands). If we then assume that the molecular orbital energy level diagram for ferrocene is applicable, the 18-electron count leads to double occupancy of the orbitals up to a1. The 19-electron cobaltocene molecule has an additional electron in the e1 orbital, which is antibonding with respect to the metal and ligands. Therefore, the metalligand bonds should be stronger and shorter in [Co(5-Cp)2] than in [Co(5-Cp)2]. This conclusion is borne out by structural data. Self-test 22.8 By using the same molecular orbital diagram, comment on whether the removal of an electron from [Fe(5-Cp)2] to produce [Fe(5-Cp)2] should produce a substantial change in MC bond length relative to neutral ferrocene. (c) Fluxional behaviour of metallocenes Key point: Many metallocenes exhibit fluxionality and undergo internal rotation because the barrier to the interconversion of the various forms is low. Table 22.8 Electronic configuration and MC bond length in [M(5-Cp)2] complexes Complex Valence electrons Electron configuration MC bond length/pm [V(5-Cp)2] 15 e22 a11 228 [Cr( -Cp)2] 16 e2 a1 1 217 [Mn(5-Me-C5H4)2] 17 e23 a12 211 5 [Fe( -Cp)2] 18 e2 a1 206 [Co(5-Cp)2] 19 e24a12e11 212 20 e2 a1 e1 220 5 5 [Ni( -Cp)2] 3 4 4 2 2 2 Data are quoted for this complex because [Mn(5-Cp)2] has a high-spin configuration and hence an anomalously long MC bond (238 pm). 563 Compounds One of the most remarkable aspects of many cyclic polyene complexes is their stereochemical nonrigidity (their fluxionality). For example, at room temperature the two rings in ferrocene rotate rapidly relative to each other as there is only a low staggered-eclipsed conversion barrier. This type of fluxional process is called internal rotation, and is similar to the process by which the two CH3 groups rotate relative to each other in ethane. We have already noted how, in the gas phase, the eclipsed conformation of ferrocene is slightly more stable than the staggered one; however, the steric bulk of substituents on the metallocene rings can change the barrier and can make staggered the preferred conformation. The rings of metallocenes are often drawn in a staggered conformation simply because there is then a little more space to illustrate substitutions. Of greater interest is the stereochemical nonrigidity that is often seen when a conjugated cyclic polyene is attached to a metal atom through some, but not all, of its C atoms. In such complexes the metalligand bonding may hop around the ring; in the informal jargon of organometallic chemists this internal rotation is called ‘ring whizzing’. A simple example is found in [Ge(1-Cp)(CH3)3], in which the single site of attachment of the Ge atom to the cyclopentadiene ring hops around the ring in a series of 1,2-shifts, a motion in which a CM bond is replaced by a CM bond to the next C atom around the ring; this motion is known as a 1,2-shift, because the bond starts on atom 1 and ends up on the adjacent atom 2 (Fig. 22.14). The great majority of fluxional conjugated polyene complexes that have been investigated migrate by 1,2-shifts, but it is not known whether these shifts are controlled by a principle of least motion or by some aspect of orbital symmetry. Nuclear magnetic resonance provides the primary evidence for the existence and mechanism of these fluxional processes, as they occur on a timescale of 102 to 104 s, and can be studied by 1H- and 13C-NMR. The compound [Ru(4-C8H8)(CO)3] (71) provides a good illustration of the approach. At room temperature, its 1H-NMR spectrum consists of a single, sharp line that could be interpreted as arising from a symmetrical 8-C8H8 ligand. However, X-ray diffraction studies of single crystals show unambiguously that the ligand is tetrahapto. This conflict is resolved by 1H-NMR spectra at lower temperatures because as the sample is cooled the signal broadens and then separates into four peaks. These peaks are expected for the four pairs of protons of an 4-C8H8 ligand. The interpretation is that at room temperature the ring is ‘whizzing’ around the metal atom rapidly compared with the timescale of the NMR experiment, so an averaged signal is observed. At lower temperatures the motion of the ring is slower, and the distinct conformations exist long enough to be resolved. A detailed analysis of the line shape of the NMR spectra can be used to measure the activation energy of the migration. (d) Bent metallocene complexes u R C O CO C O 71 [Ru(η4-C8H8)(CO)3] Cp T i Cl 72 [Ti(Cp)2(Cl)2] C6H6 Cr CO 6 73 [Cr(η -C6H6)(CO)3] Ni Key point: The structures of bent sandwich compounds can be systematized in terms of a model in which three metal atom orbitals project towards the open face of the bent Cp2M fragment. In addition to the simple bis(cyclopentadienyl) and bis(arene) complexes with parallel rings, there are many related structures. In the jargon of this area, these species are referred to as ‘bent sandwich compounds’ (72), ‘half-sandwich’ or ‘piano stool’ compounds (73), and, inevitably, ‘triple deckers’ (74). Bent sandwich compounds play a major role in the . eM 3e G e G M (a ) e G M e 3 3 . Ru Ru (b ) Figure 22.14 (a) The fluxional process in [Ge(η1-Cp)(Me)3] occurs by a series of 1,2-shifts. (b) The fluxionality of [Ru(η4-C8H8)(CO)3] can be described similarly. We need to imagine that the Ru atom is out of the plane of the page with the CO ligands omitted for clarity. Ni 74 [Ni2(Cp)3]+ 564 22 d-Metal organometallic chemistry Cl M Ti Ta Cl General form 16-electron Cl Cl Cl 18-electron H Mo Mn Cl H 18-electron 18-electron Figure 22.15 Bent sandwich compounds with their electron counts. organometallic chemistry of the early and middle d-block elements, and examples include [Ti(5-Cp)2Cl2], [Re(5-Cp)2Cl], [W(5-Cp)2(H)2], and [Nb(5-Cp)2Cl3]. As shown in Fig. 22.15, bent sandwich compounds occur with a variety of electron counts and stereochemistries. Their structures can be systematized in terms of a model in which three metal atom orbitals project out of the open face of the bent M(Cp)2 fragment. According to this model, the metal atom often satisfies its electron deficiency when the electron count is less than 18 by interaction with lone pairs or agostic CH groups on the ligands. Box 22.2 describes some of the commercial uses of ferrocene. 22.20 Metalmetal bonding and metal clusters Organic chemists must go to heroic efforts to synthesize their cage-like molecules, such as cubane(75). Bycontrast, one of the distinctive characteristics of inorganic chemistry is the large number of closed polyhedral molecules, such as the tetrahedral P4 molecule (Section 15.1), the octahedral halide-bridged early d-block clusters (Section 19.11), the polyhedral carboranes (Section 13.12), and the organometallic cluster compounds that we discuss here. The structures of clusters often resemble the close-packed structures of the metal itself, and this similarity provides the main rationale for studying them—the idea that the chemical properties of the ligands of a cluster reflect the behaviour on a metal surface. In more recent years, electronic properties of clusters that depend on how large the cluster is have emerged, such as quantum dots and nanoparticles (Section 25.2). 75 Cubane, C8H8 (a) Structure of clusters Key points: A cluster includes all compounds with metalmetal bonds that form triangular or larger cyclic structures. A rigorous definition of metal clusters restricts them to molecular complexes with metal metal bonds that form triangular or larger cyclic structures. This definition excludes linear MM compounds and cage compounds, in which several metal atoms are held together exclusively by ligand bridges. However, this rigorous definition is normally relaxed, and B OX 22 . 2 Ferrocene Ferrocene and its derivatives are widely used as anti-knock agents in petrol as they are considerably less toxic than tetraethyl lead. Ferrocene is also added to diesel fuel to reduce the production of soot during combustion. Although these uses might require the large-scale production of ferrocene, they do not make much use of the more interesting properties of ferrocene. One use of ferrrocene that does rely on its electronic structure is in handheld glucose sensors for diabetics. In these sensors the redox couple of the tri-substituted ferrocene derivative (B1) is used to determine the level of glucose in a blood sample. Ferrocene compounds have also been developed as electron transfer catalysts. OH Me Fe Me B1 NH2 Compounds we shall consider any MM bonded system as a cluster. The distinction between cage and cluster compounds can seem arbitrary as the presence of bridging ligands in a cluster such as (76) raises the possibility that the atoms are held together by MLM interactions rather than MM bonds. Bond lengths are of some help in resolving this issue. If the MM distance is much greater than twice the metallic radius, then it is reasonable to conclude that the MM bond is either very weak or absent. However, if the metal atoms are within a reasonable bonding distance, the proportion of the bonding that is attributable to direct MM interaction is ambiguous. For example, there has been much debate about the extent of FeFe bonding in [Fe2(CO)9] (77). Metalmetal bond strengths in metal complexes cannot be determined with great precision, but a variety of pieces of evidence—such as the stability of compounds and MM force constants—indicate that there is an increase in MM bond strengths down a group in the d block. This trend contrasts with that in the p block, where elementelement bonds are usually weaker for the heavier members of a group. As a consequence of this trend, metalmetal bonded systems are most numerous for the 4d- and 5d-series metals. 565 CO Co 76 [Co4(CO)12] (b) Electron counting in clusters Key points: The 18-electron rule is suitable for identifying the correct number of electrons for clusters with fewer than six metal atoms; the WadeMingosLauher rules identify a correlation between the valence electron count and the structures of larger organometallic complexes. Organometallic cluster compounds are rare for the early d metals and unknown for the f metals, but a large number of metal carbonyl clusters exist for the elements of Groups 6 to 10. The bonding in the smaller clusters can be readily explained in terms of local MM and ML electron pair bonding and the 18-electron rule. If we take Mn2(CO)10 and Os3(CO)12 as examples we can arrive at a simple, yet illuminating, picture. In Mn2(CO)10 each Mn atom is considered to have 17 electrons (seven from Mn and 10 from the five CO ligands) before we take into account the MnMn bond. This MnMn bond consists of two electrons shared between the two metal atoms, and hence raises the electron count of each by 1, resulting in two 18-electron metal atoms, but with a total electron count of only 34, not 36. In Os3(CO)12, each Os(CO)4 fragment has 16 electrons before metalmetal bonding is taken into consideration and each metal shares two further electrons with adjacent metals, so increasing the number of electrons around each metal to 18, but with a total of only 48 and not 54 electrons. The bonding electrons we are dealing with are referred to as the cluster valence electrons (CVEs), and it quickly becomes apparent that a cluster of x metal atoms with y metalmetal bonds needs 18x 2y electrons. Octahedral M6 and larger clusters do not conform to this pattern, and the polyhedral skeletal electron pair rules, known as Wade’s rules (Section 13.11), have been refined by D.M.P. Mingos and J. Lauher to apply to metal clusters. These WadeMingosLauher rules are summarized in Table 22.9; they apply most reliably to metal clusters in Groups 6 to 9. In general, and as in the boron hydrides where similar considerations apply, more open structures (which have fewer metalmetal bonds) occur when there is a higher CVE count. CO Fe 77 [Fe2(CO)9] E X A MPLE 22 . 9 Correlating spectroscopic data, cluster valence electron count, and structure The reaction of chloroform (trichloromethane) with [Co2(CO)8] yields a compound of formula [Co3(CH) (CO)9]. Both NMR and IR data indicate the presence of only terminal CO ligands and the presence of a CH group. Propose a structure consistent with the spectra and the correlation of CVE with structure. Answer We assume that the CH ligand is simply C-bonded: one C electron is used for the CH bond, so three are available for bonding in the cluster. Electrons available for the cluster are then 27 for three Co atoms, 18 for nine CO ligands, and three for CH. The resulting total CVE of 48 indicates a triangular cluster (see Table 22.9). A structure consistent with this conclusion and the presence of only terminal CO ligands and a capping CH ligand is (78). Self-test 22.9 The compound [Fe4(Cp)4(CO)4] is a dark-green solid. Its IR spectrum shows a single CO stretch at 1640 cm1. The 1H NMR spectrum is a single line even at low temperatures. From this spectroscopic information and the CVE, propose a structure for [Fe4(Cp)4(CO)4]. H C Co 78 [Co3(CH)(CO)9] CO 566 22 d-Metal organometallic chemistry Table 22.9 Correlation of cluster valence electron (CVE) count and structure Number of metal atoms Structure of metal framework 1 Single atom O 2 Linear O O O O 3 Closed triangle CVE count Example 18 Ni(CO)4 (2) 34 Mn2(CO)10 48 [Co3(CH)(CO)9] (78) 60 Co4(CO)12 (76) 62 [Fe4(CO)12C]2 64 Os4(CO)16 72 Os5(CO)16 74 Fe5C(CO)15 86 Ru6C(CO)17 90 [Rh6C(CO)15]2 O O 4 Tetrahedron O O O O Butterfly O O O O O O O Square O 5 Trigonal bipyramid O O O O Square pyramid 6 O O O O O O O O O O Octahedron O O Trigonal prism O O O O O (c) Isolobal analogies Key point: Structurally analogous fragments of molecules are described as isolobal; groups of isolobal molecular fragments are used to suggest patterns of bonding between seemingly unrelated fragments, allowing the rationalization of diverse structures. We can identify analogies in the structures of apparently unrelated molecules. Thus, we may view N(CH3)3 as derived from NH3 by substitution of a CH3 fragment for each H atom. In current terminology, the structurally analogous fragments are said to be isolobal, . The origin of the name is the lobe-like and the relationship is expressed by the symbol shape of a hybrid orbital in a molecular fragment. Two fragments are isolobal if their highest energy orbitals have the same symmetry (such as the σ symmetry of the H1s and a Csp3 hybrid orbital), similar energies, and the same electron occupation (one in each case in H1s and Csp3). Table 22.10 lists some selected isolobal fragments and the first line shows isolobal fragments with a single frontier orbital. The recognition of this family permits us to anticipate by analogy with H−H that molecules such as H3C−CH3 and (OC)5Mn−CH3 can be formed. The second line of Table 22.10 lists some isolobal fragments with two frontier orbitals, and the third line lists some with three. Isolobal analogies provide a good way to picture the incorporation of hetero atoms into a metal cluster. These analogies allow us to draw a parallel between [Co3(CH)(CO)9] (78) 567 Compounds Table 22.10 Selected isolobal fragments H3C (CO)4Co (CO)5Mn – CO H H C H H N OC Pt H CO OC Fe OC H H H C B – P Co Mn OC CO OC CO CO – CO Cp – Cp 2– Co Fe Note that electrons can be added to or subtracted from each member of the isolobal group and still maintain isolobality. For example, CH3+ Mn(CO)3+ Co(CO)4 and [Co4(CO)12] (76), both of which can be regarded as triangular Co3(CO)9 fragments capped on one side either by Co(CO)3 or by CH. A minor complication in this comparison is the occurrence of Co(CO)2 groups together with bridging CO ligands in [Co4(CO)12] because bridging and terminal ligands often have similar energies. Further inspection of the isolobal fragments given in Table 22.10 shows that a P atom is isolobal with CH; accordingly, a cluster similar to [Co3(CH)(CO)9] (78) is known, but with a capping P atom. Similarly, the ligands CR2 and Fe(CO)4 are both capable of bonding to two metal atoms in a cluster; CH3 and Mn(CO)5 can bond to one metal atom. As a final example of isolobal analogies, consider the mixed manganese and platinum complex (79): the three-membered {Mn,Pt,P} rings can be considered as the metallocyclopropane form of a coordinated double bond (80). Both halves of the Mn P fragments are isolobal with CH2 if we treat them as PR2 and Mn(CO)4; the complete fragment can then be treated as analogous to an ethene molecule. This treatment means that (79) can be considered analogous to bis-(ethene)carbonylplatinum(0) (81), a known simple 16-electron organometallic compound. (d) Synthesis of clusters Key point: Three methods are commonly used to prepare metal clusters: thermal expulsion of CO from a metal carbonyl, the condensation of a carbonyl anion and a neutral organometallic complex, and the condensation of an organometallic complex with an unsaturated organometallic compound. One of the oldest methods for the synthesis of metal clusters is the thermal expulsion of CO from a metal carbonyl. The pyrolytic formation of metal cluster compounds can be viewed from the standpoint of electron count: a decrease in valence electrons around the metal resulting from loss of CO is compensated by the formation of MM bonds. One example is the synthesis of [Co4(CO)12] by heating [Co2(CO)8]: 2 [Co2(CO)8] → [Co4(CO)12] 4 CO This reaction proceeds slowly at room temperature, so samples of octacarbonyldicobalt(0) are usually contaminated with dodecacarbonyltetracobalt(0). A widely used and more controllable reaction is based on the condensation of a carbonyl anion and a neutral organometallic complex: [Ni5(CO)12]2 [Ni(CO)4] → [Ni6(CO)12]2 4 CO The Ni5 complex has a CVE of 76 whereas the Ni6 complex has a count of 86. The descriptive name redox condensation is often given to reactions of this type, which are very useful for the preparation of anionic metal carbonyl clusters. In this example, a trigonal-bipyramidal cluster containing Ni with formal oxidation number 25 and Ni(CO)4 containing Ni(0) is converted into an octahedral cluster having Ni with oxidation number 13 . The [Ni5(CO)12]2 cluster, which has four electrons in excess of the 72 expected for a trigonal R R R P P OC OC CO Pt Mn OC R Mn CO CO OC CO CO 79 R R R P OC OC OC Mn R P Pt CO Mn CO CO OC 80 t P CO 81 CO CO 568 22 d-Metal organometallic chemistry bipyramid, illustrates a fairly common tendency for the Group 10 metal clusters to have an electron count in excess of that expected from the WadeMingosLauher rules. A third method, pioneered by F.G.A. Stone, is based on the condensation of an organometallic complex containing displaceable ligands with an unsaturated organometallic compound. The unsaturated complex may be a metal alkylidene, LnM CR2, a metal alkylidyne, LnM⬅CR, or a compound with multiple metalmetal bonds: Ph OMe (OC) 5Mo Pt(cod)2 C – cod (OC) 5Mo Ph (Cp )(OC) 2WC tol + Co2(CO) 8 OMe C – 2CO Pt(cod) (CO) 3 Co tol (Cp )(O C) 2W Co(CO) O C ( C)ph R CO h R(Cp ) – 2C2H4 Pt(PR 3)2(CH 2CH 2) 2Mo ( C)ph R PR 3 Pt ( C)ph R C O (Cp )(OC) 3 PR 3 CO Mo(CO)2(Cp ) + Pt(PR 3)4 – 2 PPh3 Ph3P PPh 3 Pt (Cp )(OC) 2Mo Mo(CO)2(Cp ) Reactions The fact that most organometallic compounds can be made to react in a variety of ways is responsible for their use as catalysts. In the previous sections we looked at ligands and how to introduce them into a metal centre, and in this section we look at how ligands might react further or with each other. Implicit in the following discussion is the fact that coordinatively saturated complexes are less reactive than unsaturated ones. 22.21 Ligand substitution Key points: The substitution of ligands in organometallic complexes is very similar to the substitution of ligands in coordination complexes, with the additional constraint that the valence electron count at the metal atom does not increase above 18; steric crowding of ligands increases the rate of dissociative processes and decreases the rate of associative processes. Extensive studies of CO substitution reactions of simple carbonyl complexes have revealed systematic trends in mechanisms and rates, and much that has been established for these compounds is applicable to all organometallic complexes. The simple replacement of one ligand by another in organometallic complexes is very similar to that observed with coordination compounds, where reactions sometimes go by an associative, a dissociative, or an interchange pathway, with the reaction being either associatively or dissociatively activated (Section 21.2). The simplest examples of substitution reactions involve the replacement of CO by another electron-pair donor, such as a phosphine. Studies of the rates at which trialkylphosphines and other ligands replace CO in Ni(CO)4, Fe(CO)5, and the hexacarbonyls of the chromium group show that they are relatively insensitive to the incoming group, indicating that a dissociatively activated mechanism is in operation. In some cases a solvated intermediate such as [Cr(CO)5(THF)] has been detected. This intermediate then combines with the entering group in a bimolecular process: Cr(CO)6 sol → Cr(CO)5(sol) CO Cr(CO)5(sol) L → Cr(CO)5L sol A dissociatively activated substitution reaction would be expected with metal carbonyl complexes as associative activation would require reaction intermediates with more than 18 valence electrons, formation of which corresponds to populating high-energy antibonding MOs. 569 Reactions Whereas the loss of the first CO group from Ni(CO)4 occurs easily, and substitution is fast at room temperature, the CO ligands are much more tightly bound in the Group 6 carbonyls, and loss of CO often needs to be promoted thermally or photochemically. For example, the substitution of CO by CH3CN is carried out in refluxing acetonitrile, using a stream of nitrogen to sweep away the carbon monoxide and hence drive the reaction to completion. To achieve photolysis, mononuclear carbonyls (which do not absorb strongly in the visible region) are exposed to near-UV radiation in an apparatus like that shown in Fig. 22.16. As with the thermal process, there is strong evidence that the photoassisted substitution reaction leads to the formation of a labile intermediate complex with the solvent, which is then displaced by the entering group. Solvated intermediates in the photolysis of metal carbonyls have been detected, not only in polar solvents such as THF but also in every solvent that has been tried, even in alkanes and noble gases.3 The rates of substitution of ligands in 16-electron complexes are sensitive to the identity and concentration of the entering group, which indicates associative activation. For example, the reactions of [Ir(CO)Cl(PPh3)2] with triethylphosphine are associatively activated: [Ir(CO)Cl(PPh3)2] PEt3 → [Ir(CO)Cl(PPh3)2(PEt3)] → [Ir(CO)Cl(PPh3)(PEt3)] PPh3 Sixteen-electron organometallic compounds appear to undergo associatively activated substitution reactions because the 18-electron activated complex is energetically more favourable than the 14-electron activated complex that would occur in dissociative activation. As in the reactions of coordination complexes, we can expect steric crowding between ligands to accelerate dissociative processes and to decrease the rates of associative processes (Section 21.6). The extent to which various ligands crowd each other is approximated by the Tolman cone angle and we can see how it influences the equilibrium constant for ligand binding by examining the dissociation constants of [Ni(PR3)4] complexes (Table 22.11). These complexes are slightly dissociated in solution if the phosphine ligands are compact, such as PMe3, with a cone angle of 118º. However, a complex such as [Ni(PtBu3)4], where the cone angle is huge (182º), is highly dissociated. The ligand PtBu3 is so bulky that the 14-electron complex [Pt(PtBu3)2] can be identified. The rate of CO substitution in six-coordinate metal carbonyls often decreases as more strongly basic ligands replace CO, and two or three alkylphosphine ligands often represent the limit of substitution. With bulky phosphine ligands, further substitution may be thermodynamically unfavourable on account of ligand crowding, but increased electron density on the metal centre, which arises when a π-acceptor ligand is replaced by a net donor ligand, appears to bind the remaining CO ligands more tightly and therefore reduce the rate of CO dissociative substitution. The explanation of the influence of -donor ligands on CO bonding is that the increased electron density contributed by the phosphine leads to stronger π backbonding to the remaining CO ligands and therefore strengthens the MCO bond. This stronger MC bond decreases the tendency of CO to leave the metal atom and therefore decreases the rate of dissociative substitution. It is also observed that the second carbonyl that is replaced is normally cis to the site of the first and that replacement of a third carbonyl results in a fac complex. The reason for this regiochemistry is that CO ligands have very high trans effects (Section 21.4). E X A M PL E 22 .10 Preparing substituted metal carbonyls Starting with MoO3 as a source of Mo, and CO and PPh3 as the ligand sources, plus other reagents of your choice, give equations and conditions for the synthesis of [Mo(CO)5PPh3]. Answer Considering the materials available to us, a sensible procedure might be to synthesize Mo(CO)6 first and then carry out a ligand substitution. Reductive carbonylation of MoO3 can be performed using Al(CH2CH3)3 as a reducing agent in the presence of carbon monoxide under pressure. The temperature and pressure required for this reaction are less than those for the direct combination of molybdenum and carbon monoxide. 50 atm, 150 C, heptane MoO3 Al(CH2CH3)3 6 CO ⎯⎯⎯⎯⎯⎯⎯ → Mo(CO)6 oxidation products of Al(CH2CH3)3 3 Infrared spectroscopy has been used to demonstrate that, as quickly as can be measured (that is, in less than one picosecond), the photolysis of W(CO)6 results in the formation of W(CO)5(sol). Power Water N2 + CO N2 Mercury vapour lamp Ligand + carbonyl solution Water-cooled lamp jacket Figure 22.16 Apparatus for photochemical ligand substitution of metal carbonyls. Table 22.11 Cone angles and dissociation constants for Some Ni complexes L ␪/ Kd PMe3 118 109 PEt3 137 1.2 × 105 PMePh2 136 5.0 × 102 PPh3 145 Large PtBu3 182 Large Data are for NiL4 NiL3 L in benzene at 25°C. 570 22 d-Metal organometallic chemistry The subsequent substitution could be carried out photochemically by using the apparatus illustrated in Fig. 22.16: M THF, h ν Mo(CO)6 PPh3 ⎯⎯⎯ → Mo(CO)5PPh3 CO 82 The progress of the reaction can be followed by IR spectroscopy in the CO stretching region using small samples that are removed periodically from the reaction vessel. 18e M(CO)6 [M(CO)6] [ML(CO)5] 18e Self-test 22.10 If the highly substituted complex [Mo(CO)3L3] is desired, which of the ligands PMe3 or P(tBu)3 would be preferred? Give reasons for your choice. – 19e – [ML(CO)5] 19e 17e [M(CO)5]– L CO Figure 22.17 Schematic diagram of an electron transfer catalysed CO substitution. After addition of a small amount of a reducing initiator, the cycle continues until the limiting reagent M(CO)6 or L has been consumed. Re OC OC CO 83 [ReCp(CO)3] Although the generalizations above apply to a wide range of reactions, some exceptions are observed, especially if cyclopentadienyl or nitrosyl ligands are present. In these cases it is common to find evidence of associatively activated substitution even for 18-electron complexes. The common explanation is that NO may switch from being linear (as in 67) to being angular (as in 68), whereupon it donates two fewer electrons (Section 22.17). Similarly, the 5-Cp six-electron donor can slip relative to the metal and become an 3-Cp four-electron donor. In this case, the C5H5 ligand is regarded as having a three-carbon interaction with the metal while the remaining two electrons form a simple C C bond that is not engaged with the metal (82), and the relatively electron-depleted central metal atom becomes susceptible to substitution: [V(CO)5(NO)] PPh3 → [V(CO)4(NO)(PPh3)] CO [Re(η5-Cp)(CO)3] PPh3 → [Re(η3-Cp)(CO)2(PPh3)] CO It has been found that, for some metal carbonyls, the displacement of CO can be catalysed by electron-transfer processes that create anion or cation radicals. These radicals do not have 18 electrons, and a typical process of this type is illustrated in Fig. 22.17. As can be seen, the key feature is the lability of CO in the 19-electron anion radical compared to the metal carbonyl starting material. Similarly, the less common 19- and 17-electron metal compounds are labile with respect to substitution. The substitution of ligands at a cluster is often not a straightforward process because fragmentation is common. Fragmentation occurs because the MM bonds in a cluster are generally comparable in strength to the ML bonds, and so the breaking of the MM bonds provides a reaction pathway with a low activation energy. For example, dodecacarbonyltriiron(0) reacts with triphenylphosphine under mild conditions to yield simple mono- and disubstituted products as well as some cluster fragmentation products: 3 [Fe3(CO)12] 6 PPh3 → [Fe3(CO)11PPh3] [Fe3(CO)10(PPh3)2] [Fe(CO)5] Re OC OC [Fe(CO)4PPh3] [Fe(CO)3(PPh3)2] 3 CO CO 84 Re OC CO OC 85 However, for somewhat longer reaction times or elevated temperatures, only monoiron cleavage products are obtained. Because the strength of MM bonds increases down a group, substitution products of the heavier clusters, such as [Ru3(CO)10(PPh3)2] or [Os3(CO)10(PPh3)2], can be prepared without significant fragmentation into mononuclear complexes. E X A M PL E 22 .11 Assessing substitutional reactivity Which of the compounds (83) or (84) will undergo substitution of a CO ligand for a phosphine more readily? Answer If we consider the ligands present on the compounds, we can see that the 18-electron compound (84) contains the indenyl ligand, which can ring slip to a 16-electron 3 bound form (85) more readily than the normal Cp compound (83), as the double bond that forms becomes part of a six-membered aromatic ring. This ring slipping provides a low-energy route to coordinative unsaturation and thus the indenyl compound reacts with an incoming ligand much more rapidly than the Cp compound. Self-test 22.11 Assess the relative substitutional reactivities of indenyl and fluorenyl (86) compounds. 86 Fluorenyl– Reactions 22.22 Oxidative addition and reductive elimination Key points: Oxidative addition occurs when a molecule XY adds to a metal atom to form new MX and MY bonds with cleavage of the XY bond; oxidative addition results in increasing the coordination number of the metal atom by 2 and increasing the oxidation number by 2; reductive elimination is the reverse of oxidative addition. When we discussed the bonding of dihydrogen to a metal atom in Section 22.7, we noted that the oxidation number on the metal atom increased by 2 when the dihydrogen reacted to give a dihydride: M(Nox) H2 → [M(Nox2)(H)2] The increase in oxidation number of the metal by 2 arises because dihydrogen is treated as a neutral ligand, whereas the hydride ligands are treated as H: thus the formation of two MH bonds from a H2 molecule corresponds to a formal increase in the charge on the metal by 2. Whilst it might seem that this oxidation of the metal is just an anomaly thrown up by our method of counting electrons, two of the electrons on the metal atom have been used to backbond to the dihydrogen, and these two electrons are no longer available to the metal for further bonding. This type of reaction is quite general and is known as oxidative addition. A large number of molecules add oxidatively to a metal atom, including the alkyl and aryl halides, dihydrogen, and simple hydrocarbons. In general, the addition of any molecule XY to a metal atom, where both X and Y are more electronegative than the metal, can be classed as oxidative addition. Thus the reaction of a metal with an acid such as HCl is an oxidative addition reaction. Oxidative addition reactions are not restricted to d-block metals: the reaction of magnesium to form Grignard reagents (Section 12.13) is an oxidative addition reaction. Oxidative addition reactions result in two more ligands bound to the metal with an increase in the total electron count at the metal of 2. Thus oxidative addition reactions normally require a coordinatively unsaturated metal centre, and are particularly common for 16-electron square-planar metal complexes: Me Me PEt 3 Pt MeI PEt 3 Me Me PEt 3 Pt I Me PEt 3 16e Pt(II) 18e Pt(IV) The oxidative addition of hydrogen is a concerted reaction: dihydrogen coordinates to form a -bonded H2 ligand, and then backbonding from the metal results in cleavage of the HH bond and the formation of cis dihydrides: H Cl Rh Ph3P PPh 3 H2 PPh 3 Cl H H Rh Ph3P PPh 3 Cl PPh 3 Ph3P Five-coordinate 18e Rh(I) Four-coordinate 16e Rh(I) Rh PPh 3 H PPh 3 Six-coordinate 18e Rh(III) Other molecules, such as alkanes and aryl halides, are known to react in a concerted fashion, and in all these cases the two incoming ligands end up cis to each other. Some oxidative addition reactions are not concerted and either go through radical intermediates or are best thought of as SN2 displacement reactions. Radical oxidative addition reactions are rare and will not be discussed further here. In an SN2 oxidative addition reaction, a lone pair on the metal attacks the XY molecule displacing Y, which subsequently bonds to the metal: + Me Cl Ph3P Ir PPh 3 CO MeI Cl Ph3P Ir PPh 3 CO + I– Me Cl Ir Ph3P PPh 3 CO I There are two stereochemical consequences of this reaction. First, the two incoming ligands need not end up cis to each other and, second, unlike the concerted reaction, any 571 572 22 d-Metal organometallic chemistry chirality at the X group is inverted. An SN2-type oxidative addition is common for polar molecules such as alkyl halides. The opposite of oxidative addition, where two ligands couple and eliminate from a metal centre, is known as reductive elimination: H Et3P Et3P Me Pt I PEt 3 Pt I Me CH4 Me PEt 3 16e Pt(II) 18e Pt(IV) Reductive elimination reactions require both eliminating fragments to be cis to each other, and are best thought of as the reverse of the concerted form of oxidative addition. Oxidative addition and reductive elimination reactions are, in principle, reversible. However, in practice, one direction is normally thermodynamically favoured over the other. Oxidative addition and reductive elimination reactions play a major role in many catalytic processes (Chapter 26). E X A M PL E 22 .12 Identifying oxidative addition and reductive elimination Show that the reaction H Et3P PEt 3 Rh Et3P Et3P MeCCH CCMe Rh PEt 3 CCMe MeCC PEt 3 is an example of an oxidative addition reaction. Answer In order to identify an oxidative addition reaction, we need to establish the valence electron counts and oxidation states of both the starting material and the product. The four-coordinate squareplanar Rh starting material contains an 1-alkynyl ligand as well as three neutral phosphine ligands; it is therefore a 16-electron Rh(I) species. The six-coordinate octahedral product contains two 1-alkynyl ligands, a hydride ligand, and three neutral phosphine ligands; it is therefore an 18-electron Rh(III) species. The increase in both coordination number and oxidation number by 2 identifies it as an oxidative addition. Self-test 22.12 Show that the reaction: Cl Ph3P Cl Cl Pd Ph PPh 3 Pd Ph3P PPh 3 PhCl Cl Cl is an example of reductive elimination. 22.23 -Bond metathesis Key point: A -bond metathesis reaction is a concerted process that sometimes occurs when oxidative addition cannot take place. A reaction sequence that appears to be an oxidative addition followed by a reductive elimination may in fact be the exchange of two species by a process known as -bond metathesis. -Bond metathesis reactions are common for early d-metal complexes where there are not enough electrons on the metal atom for it to participate in oxidative addition. For instance, the 16-electron compound [ZrHMe(Cp)2] cannot react with H2 to give a trihydride as all its electrons are involved in bonding to the existing ligands. A four-membered transition state is proposed in such cases, and a concerted bond-making and bond-breaking step results in elimination of methane: H Zr H Me H2 H Zr Me H H Zr Me Zr H H C H 4 Reactions 22.24 1,1-Migratory insertion reactions Key point: 1,1-Migratory insertion reactions result from the migration of a species such as a hydride or alkyl group to an adjacent ligand such as carbonyl to give a metal complex with two fewer electrons on the metal atom. A 1,1-migratory insertion reaction is exemplified by reactions of the 1-CO ligand, where the following change can take place: X X M C O M C O The reaction is called a ‘1,1-reaction’ because the X group that was one bond away from the metal atom ends up on an atom that is one bond away from the metal atom. Typically the X group is an alkyl or aryl species and then the product contains an acyl group. In principle, the reaction could proceed by a migration of the X group, or an insertion of the CO into the MX bond. An uncertainty about the actual mechanism has led to the apparently contradictory name migratory insertion. Colloquially, however, the terms ‘migratory insertion’, ‘migration’, and ‘insertion’ are used interchangeably. The overall reaction results in a decrease in the number of electrons on the metal atom by 2, with no change in the oxidation state. It is therefore possible to induce 1,1-migratory insertion reactions by the addition of another species that can act as a ligand: [Mn(CH3)(CO)5] PPh3 → [Mn(CH3CO)(CO)4PPh3] The classic study of the migratory insertion of CO with [CH3Mn(CO)5] illustrates a number of key features of reactions of this type.4 First, in the reaction [Mn(CH3)(CO)5] 13CO → [Mn(CH3CO)(CO)4(13CO)] Me CO OC Mn CO CO Me CO CO OC O C Mn CO CO OC CO the product has only one labelled CO, and that group is cis to the newly formed acyl group. This stereochemistry demonstrates that the incoming CO group does not insert into the MnCH3 bond, and that either the methyl group migrates to an adjacent CO ligand, or a CO ligand adjacent to the methyl group inserts into the MnCH3 bond. Second, in the reverse reaction cis-[Mn(CH3CO)(CO)4(13CO)] → [MnCH3(CO)5] CO it is possible to distinguish between the migration of the methyl group and the insertion of the CO ligand: cis-[Mn(CH3CO)(CO)4(13CO)] must lose a CO ligand cis to the acyl group in order for the reaction to proceed. The scheme below summarizes the potential reaction pathways. In one quarter of the instances, this ligand will be the labelled CO and there will be no significant information gained. In half the instances, a nonlabelled CO will be lost, leaving a vacant site cis to both the labelled CO ligand and the acyl group. In this case, either (a) migration of the methyl group back to the metal atom or (b) extrusion of CO will lead to the methyl group and the 13CO ligand being cis to each other and no information is gained. However, in the remaining one quarter of the instances, the CO that is trans to the labelled CO ligand will be lost, and in this case it is possible to distinguish (c) CO ligand extrusion from (d) methyl group migration. If the methyl group migrates, it ends up trans to the labelled CO, whereas if the CO is extruded, the methyl group ends up cis to the CO. 4 The Mn species are not fluxional and do not rearrange; if they did, we would not be able to draw the conclusions made here. 573 574 22 d-Metal organometallic chemistry CO OC )a ( Me OC 0 5 % O C Mn OC CO Me )a ( O Me C b () OC CO Mn CO CO CO )b ( Me OC CO OC 5 2 % CO CO CO d () Mn )c( O Me (c) C CO Mn OC CO CO Mn CO CO CO )(d CO Me Mn CO CO OC CO Because the product with CH3 and 13CO trans to each other constitutes about 25 per cent of the product, we can conclude that the CH3 group does indeed migrate. Application of the principle of microscopic reversibility5 allows us to conclude that the forward reaction proceeds by methyl-group migration. All 1,1-migratory insertions are now thought to proceed by migration of the X group. An important consequence of this pathway is that the relative positions of the other groups on the migrating atom are left unchanged, so the stereochemistry at the X group is preserved. 22.25 1,2-Insertions and -hydride elimination Key points: 1,2-Insertion reactions are observed with ␩2 ligands such as alkenes and result in the formation of an ␩1 ligand with no change in oxidation state of the metal; ␤-hydride elimination is the reverse of 1,2-insertion. 1,2-Insertion reactions are commonly observed with 2-ligands, such as alkenes and alkynes, and are exemplified by the reaction: X CH 2 M CH 2 X CH 2 M CH2 The reaction is a 1,2-insertion because the X group that was one bond away from the metal atom ends up on an atom that is two bonds away from the metal. Typically, the X group is a hydride, alkyl, or aryl species, in which case the product contains a (substituted) alkyl group. Like 1,1-insertion reactions, the overall reaction results in a decrease in the number of electrons on the metal atom by 2, with no change in the oxidation state. If, in the above reaction with X H, another ethene molecule were to coordinate, the resultant ethyl group could migrate to give a butyl group: H CH 2 M CH 2 H CH 2 M CH2 CH 2=CH 2 H2C CH 2 M Et H2C H2 C Et M Repetition of this process gives polyethene. Catalytic reactions of this kind are of considerable industrial importance and are discussed in Chapter 26. The reverse of 1,2-insertions can occur but this is rare except when X H, when the reaction is known as -hydride elimination:6 5 The principle of microscopic reversibility states that both forward and reverse reactions proceed by the same mechanism. 6 The reaction is known as ‘-hydride elimination’ because the H atom that is eliminated is on the second carbon atom from the metal atom (the carbon atom bonded to the metal atom is the carbon, the third one the carbon, etc.). Reactions H H CH 2 M CH2 M CH 2 CH 2 The experimental evidence shows that both 1,2-insertion and -hydride elimination proceed through a syn intermediate: M H M H F 3C CF 3 M CF 3 F 3C F 3C M H A C C Y B X M H C C A Y B X A B H CF 3 M H Y X As noted in Section 22.8, a -hydride elimination reaction can provide a facile route for decomposition of alkyl-containing compounds. The 1,2-insertion reaction, coupled with the -hydride elimination, can also provide a low-energy route to alkene isomerization: M M H 1,2-insertion H β-hydride elimination M H 22.26 -, -, and -Hydride eliminations and cyclometallations Key point: Cyclometallation reactions, in which a metal inserts into a remote CH bond, are equivalent to hydride elimination reactions. -Hydride eliminations are occasionally found for complexes that have no -hydrogens and the reaction gives rise to a carbene that is often highly reactive: H M CH3 α-H elimination M CH2 -Hydride and -hydride eliminations are more commonly observed. Because the product contains a metallocycle, a cyclic structure incorporating a metal atom, these reactions are normally described as cyclometallation reactions: δ-H elimination H H M M A cyclometallation reaction is often also thought of as the oxidative addition to an adjacent CH bond. Both - and -hydride eliminations can also be considered as cyclometallation reactions. This identification is more obvious for the -hydride elimination if we consider an alkene in its metallacyclopropane form: H M H M E X A M PL E 22 .13 Predicting the outcomes of insertion and elimination reactions What product, including its stereochemistry, would you expect from the reaction between [MnMe(CO)5] and PPh3? Answer If we consider the reaction between [MnMe(CO)5] and PPh3 we can see that it is unlikely to be the simple replacement of a carbonyl ligand by the phosphine as this reaction would require a strongly bound carbonyl ligand to dissociate. A more likely reaction would be the migration of the methyl group on to an adjacent CO ligand to give an acyl group, with the phosphine filling the vacated coordination site. This reaction has a low activation barrier, and the product would therefore be expected to be cis-[Mn(MeCO) (PPh3)(CO)4]. Self-test 22.13 Explain why [Pt(Et)(Cl)(PEt3)2] readily decomposes, whereas [Pt(Me)(Cl)(PEt3)2] does not. 575 576 22 d-Metal organometallic chemistry FURTHER READING R.H. Crabtree, The organometallic chemistry of the transition metals. Wiley, New York (2005). The best single volume book on the subject. W. Scherer and G.S. McGrady, Angew. Chem., Int. Ed. Engl., 2004, 43, 1782. A review of agostic interactions that gives a good historical perspective. C. Elschenbroich, Organometallics. Wiley-VCH, Weinheim (2006). D.C. Grills and M.W. George, Adv. Inorg. Chem., 2001, 52, 113. A review of the topic of noble gas complexes. R.H. Crabtree and D.M.P. Mingos (ed.) Comprehensive organometallic chemistry III. Elsevier, Oxford, 2006. The definitive reference work that builds on the two earlier editions. See J. Organomet. Chem., 1975, 100, 273 for Wilkinson’s personal account of the development of metallocene chemistry. G.J. Kubas, Chem. Rev., 2006, 107, 4152. A historical perspective and a full account of the discovery of dihydrogen complexes. D.M.P. Mingos and D.J. Wales, Introduction to cluster chemistry. Prentice Hall, Englewood Cliffs (1990); J.W. Lauher, J. Am. Chem. Soc., 1978, 100, 5305. Descriptions of some of the concepts of cluster bonding. R. Hoffmann, Angew. Chem., Int. Ed. Engl., 1982, 21, 711. The application of isolobal analogies to metal cluster compounds (Hoffmann’s Nobel Prize lecture). EXERCISES 22.1 Name the species, draw the structures of, and give valence electron counts to the metal atoms in: (a) Fe(CO)5, (b) Mn2(CO)10, (c) V(CO)6, (d) [Fe(CO)4]2-, (e) La(5-Cp)3, (f) Fe(3-allyl)(CO)3Cl, (g) Fe(CO)4(PEt3), (h) Rh(Me)(CO)2(PPh3), (i) Pd(Me)(Cl)(PPh3)2, (j) Co(5-C5H5)(4-C4Ph4), (k) [Fe(5-C5H5)(CO)2], (l) Cr(6-C6H6) (6-C7H8), (m) Ta(5-C5H5)2Cl3, (n) Ni(5-C5H5)NO. Do any of the complexes deviate from the 18-electron rule? If so, how is this reflected in their structure or chemical properties? 22.2 (a) Sketch an 2 interaction of 1,4-butadiene with a metal atom and (b) do the same for an 4 interaction. 22.3 What hapticities are possible for the interaction of each of the following ligands with a single d-block metal atom such as cobalt? (a) C2H4, (b) cyclopentadienyl, (c) C6H6, (d) cyclooctadiene, (e) cyclooctatetraene. 22.4 Draw plausible structures and give the electron count of (a) Ni(3-C3H5)2, (b) 4-cyclobutadiene-5-cyclopentadienylcobalt, (c) Co(3-C3H5)(CO)2. If the electron count deviates from 18, is the deviation explicable in terms of periodic trends? 22.5 State the two common methods for the preparation of simple metal carbonyls and illustrate your answer with chemical equations. Is the selection of method based on thermodynamic or kinetic considerations? 22.6 Suggest a sequence of reactions for the preparation of Fe(CO)3(dppe), given iron metal, CO, dppe (Ph2PCH2CH2PPh2), and other reagents of your choice. 22.7 Suppose that you are given a series of metal tricarbonyl compounds having the respective symmetries C2v, D3h, and Cs. Without consulting reference material, which of these should display the greatest number of CO stretching bands in the IR spectrum? Check your answer and give the number of expected bands for each by consulting Table 22.7. 22.8 Provide plausible reasons for the differences in IR wavenumbers between each of the following pairs: (a) Mo(CO)3(PF3)3 2040, 1991 cm1 versus Mo(CO)3(PMe3)3 1945, 1851 cm1, (b) MnCp(CO)3 2023, 1939 cm1 versus MnCp(CO)3 2017, 1928 cm1. 22.9 The compound Ni3(C5H5)3(CO)2 has a single CO stretching absorption at 1761 cm1. The IR data indicate that all C5H5 ligands are pentahapto and probably in identical environments. (a) On the basis of these data, propose a structure. (b) Does the electron count for each metal in your structure agree with the 18-electron rule? If not, is nickel in a region of the periodic table where deviations from the 18-electron rule are common? 22.10 Decide which of the two complexes (a) W(CO)6 or (b) Ir(CO) Cl(PPh3)2 should undergo the fastest exchange with 13CO. Justify your answer. 22.11 Which metal carbonyl in each of (a) [Fe(CO)4]2 or [Co(CO)4], (b) [Mn(CO)5] or [Re(CO)5] should be the most basic towards a proton? What are the trends on which your answer is based? 22.12 Using the 18-electron rule as a guide, indicate the probable number of carbonyl ligands in (a) W(6-C6H6)(CO)n, (b) Rh(5-C5H5) (CO)n, and (c) Ru3(CO)n. 22.13 Propose two syntheses for MnMe(CO)5, both starting with Mn2(CO)10, with one using Na and one using Br2. You may use other reagents of your choice. 22.14 Give the probable structure of the product obtained when Mo(CO)6 is allowed to react first with LiPh and then with the strong carbocation reagent, CH3OSO2CF3. 22.15 Na[W(5-C5H5)(CO)3] reacts with 3-chloroprop-1-ene to give a solid, A, which has the molecular formula W(C3H5)(C5H5)(CO)3. Compound A loses carbon monoxide on exposure to light and forms compound B, which has the formula W(C3H5)(C5H5)(CO)2. Treating compound A with hydrogen chloride and then potassium hexafluorophosphate, KPF6, results in the formation of a salt, C. Compound C has the molecular formula [W(C3H6)(C5H5)(CO)3]PF6. Use this information and the 18-electron rule to identify the compounds A, B, and C. Sketch a structure for each, paying particular attention to the hapticity of the hydrocarbon. 22.16 Treatment of TiCl4 at low temperature with EtMgBr gives a compound that is unstable above 70ºC. However, treatment of TiCl4 at low temperature with MeLi or LiCH2SiMe3 gives compounds that are stable at room temperature. Rationalize these observations. 22.17 Suggest syntheses of (a) [Mo(7-C7H7)(CO)3]BF4 from Mo(CO)6 and (b) [Ir(COMe)(CO)(Cl)2(PPh3)2] from [Ir(CO)Cl(PPh3)2]. 22.18 When Fe(CO)5 is refluxed with cyclopentadiene compound A is formed which has the empirical formula C8H6O3Fe and a complicated 1 H NMR spectrum. Compound A readily loses CO to give compound B with two 1H-NMR resonances, one at negative chemical shift (relative intensity one) and one at around 5ppm (relative intensity five). Subsequent heating of B results in the loss of H2 and the formation of compound C. Compound C has a single 1H-NMR resonance and the empirical formula C7H5O2Fe. Compounds A, B, and C all have 18 valence electrons: identify them and explain the observed spectroscopic data. Problems 22.19 When Mo(CO)6 is refluxed with cyclopentadiene compound D is formed which has the empirical formula C8H5O3Mo and an absorption in the IR spectrum at 1960 cm1. Compound D can be treated with bromine to yield E or with Na/Hg to give compound F. There are absorptions in the IR spectra of E and F at 2090 and 1860 cm1, respectively. Compounds D, E, and F all have 18 valence electrons: identify them and explain the observed spectroscopic data. 22.20 Which compound would you expect to be more stable, Rh(5-C5H5)2 or Ru(5-C5H5)2? Give a plausible explanation for the difference in terms of simple bonding concepts. 22.21 Give the equation for a workable reaction that will convert Fe(5-C5H5)2 into (a) Fe(5-C5H5)(5-C5H4COCH3) and (b) Fe(5-C5H5)(5-C5H4CO2H). 22.22 Sketch the a1′ symmetry-adapted orbitals for the two eclipsed C5H5 ligands stacked together with D5h symmetry. Identify the s, p, and d orbitals of a metal atom lying between the rings that may have nonzero overlap, and state how many a1′ molecular orbitals may be formed. 22.25 The mechanism of CO insertion is thought to be RMn(CO)5 → (RCO)Mn(CO)4 (RCO)Mn(CO)4 L → (RCO)MnL(CO)4 In the first step, the acyl intermediate is formed, leaving a vacant coordination position. In the second step, a ligand enters. At high ligand concentrations, the rate is independent of [L]. At this limit, what rate constant can be extracted from rate data? 22.26 (a) What cluster valence electron (CVE) count is characteristic of octahedral and trigonal prismatic complexes? (b) Can these CVE values be derived from the 18-electron rule? (c) Determine the probable geometry (octahedral or trigonal prismatic) of [Fe6(C)(CO)16]2 and [Co6(C)(CO)16]2. (The C atom in both cases resides in the centre of the cluster and can be considered to be a four-electron donor.) 22.27 Based on isolobal analogies, choose the groups that might replace the group in boldface in (a) Co2(CO)9CH → OCH3, N(CH3)2, or SiCH3 (b) (OC)5MnMn(CO)5 → I, CH2, or CCH3 5 22.23 The compound Ni( -C5H5)2 readily adds one molecule of HF to yield [Ni(5-C5H5)(4-C5H6)] whereas Fe(5-C5H5)2 reacts with strong acid to yield [Fe(5-C5H5)2H]. In the latter compound the H atom is attached to the Fe atom. Provide a reasonable explanation for this difference. 22.24 Write a plausible mechanism, giving your reasoning, for the reactions (a) [Mn(CO)5(CF2)] H2O → [Mn(CO)6] 2 HF 577 22.28 Ligand substitution reactions on metal clusters are often found to occur by associative mechanisms, and it is postulated that these occur by initial breaking of an MM bond, thereby providing an open coordination site for the incoming ligand. If the proposed mechanism is applicable, which would you expect to undergo the fastest exchange with added 13CO, Co4(CO)12 or Ir4(CO)12? Suggest an explanation. (b) Rh(CO)(C2H5)(PR3)2 → RhH(CO)(PR3)2 C2H4 PROBLEMS 22.1 Propose the structure of the product obtained by the reaction of [Re(CO)(5-C5H5)(NO)(PPh3)] with Li[HBEt3]. The latter contains a strongly nucleophilic hydride. (For full details see: W. Tam, G. Y. Lin, W.K. Wong, W.A. Kiel, V. Wong, and J.A. Gladysz, J. Am. Chem. Soc., 1982, 104, 141.) 22.2 Treatment of TiCl4 with 4 equivalents of NaCp gives a single organometallic compound (together with NaCl byproduct). At room temperature the 1H-NMR spectrum shows a single sharp singlet; on cooling to 40ºC this singlet separates into two singlets of equal intensity; further cooling results in one of the singlets separating into three signals with intensity ratios 1:2:2. Explain these results. 22.3 When several CO ligands are present in a metal carbonyl, an indication of the individual bond strengths can be determined by means of force constants derived from the experimental IR frequencies. In Cr(CO)5(Ph3P) the cis-CO ligands have the higher force constants, whereas in Ph3SnCo(CO)4 the force constants are higher for the trans-CO. Suggest why, and explain which carbonyl C atoms should be susceptible to nucleophilic attack in these two cases. (For details see D.J. Darensbourg and M.Y. Darensbourg, Inorg. Chem., 1970, 9, 1691.) 22.4 It is often possible to assign different resonance structures to an organometallic compound, for example the structure below can be thought of as existing as either the charge separated aromatic structure on the left, or the carbene form on the right. R O N Cl O R Cl Pt- Cl R N Cl Pt- N O H+ H+ O R O R Pt Cl R O H R Cl Pt Cl H O Cl N O R Suggest ways of distinguishing the two forms. (For details see: G.W.V. Cave, A.J. Hallett, W. Errington, and J.P. Rourke, Angew. Chem., Int. Ed. Engl., 1998, 37, 3270 and C.P. Newman, G.J. Clarkson, N.W. Alcock, and J.P. Rourke, Dalton Trans., 2006, 3321.) 22.5 Describe how NMR has been used to identify alkane complexes of d metals unambiguously. See S. Geftakis and G.E. Ball, J. Am. Chem. Soc., 1998, 120, 9953 and D.J. Lawes, S. Geftakis, and G.E. Ball, J. Am. Chem. Soc., 2005, 127, 4134. 22.6 The dinitrogen complex Zr2(5-Cp)4(N2)3 has been isolated and its structure determined by single-crystal X-ray diffraction. Each Zr atom is bonded to two Cp and one terminal N2. The third N2 bridges between the Zr atoms in a nearly linear ZrNNZr array. Before consulting the reference, write a plausible structure for this compound that accounts for the 1H-NMR spectrum obtained on a sample held at 7ºC. This spectrum shows two singlets, indicating that the Cp 578 22 d-Metal organometallic chemistry rings are in two different environments. At somewhat above room temperature these rings become equivalent on the NMR timescale and 15 N-NMR indicates that N2 exchange between the terminal ligands and dissolved N2 is correlated with the process that interconverts the Cp ligand sites. Propose a way in which this equilibration could interconvert the sites of the Cp ligands. (For further details see J.M. Manriquez, D.R. McAlister, E. Rosenberg, H.M. Shiller, K.L. Willamson, S.I. Chan, and J.E. Bercaw, J. Am. Chem. Soc., 1978, 100, 3078.) What implications might this have for the oxidative addition of a hydrocarbon to a metal atom? (See, for example, R.H. Crabtree, J. Organomet. Chem., 2004, 689, 4083.) 22.9 Compare and contrast Fischer and Schrock carbenes. See E.O. Fischer, Adv. Organomet. Chem., 1976, 14, 1 and R.R. Schrock, Acc. Chem. Res., 1984, 12, 98. 22.7 How might you unambiguously identify an organometallic complex containing a noble gas atom as a ligand? See G.E. Ball, T.A. Darwish, S. Geftakis, M.W. George, D.J. Lawes, P. Portius, and J.P. Rourke, Proc. Natl Acad. Sci., 2005, 102, 1853. 22.10 Rearrangements of ligands so that differing numbers of carbons interact with the central metal atom are known as haptotropic rearrangements. Consider the (fluorenyl)(cyclo-pentadienyl)iron complex. What haptotropic isomers are possible? See E. Kirillov, S. Kahlal, T. Roisnel, T. Georgelin, J. Saillard, and J. Carpentier, Organometallics 2008, 27, 387. 22.8 What conclusions can you draw from the bonding and reactivity of dihydrogen bound to a low oxidation state d-block metal that might be applicable to the bonding of an alkane to a metal? 22.11 Suggest two plausible routes by which a carbonyl ligand in Mo(Cp)(CO)3Me might exchange for a phosphine. Neither route should invoke the initial dissociation of a CO. The f-block elements There are two series of elements in the f block, Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Rf each consisting of 14 elements, but relatively little diverse chemistry has been reported for such a large number of elements. For the 4f elements (the lanthanoids) this lack of diversity is normally ascribed to the buried nature of the 4f electrons. The chemical properties of the lanthanoids are similar to those of other electropositive metals. The applications of the lanthanoids derive mainly from the optical spectra of their ions. The fact that only two of the 5f elements (the actinoids) occur naturally, all other actinoids being synthetic and often intensely radioactive, has hindered their study, particularly for the elements with the highest atomic numbers. La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf The two series of elements in the f block derive from the filling of the seven 4f and 5f orbitals, respectively. This occupation of f orbitals from f1 to f14 corresponds to the elements cerium (Ce) to lutetium (Lu) in Period 6 and from thorium (Th) to lawrencium (Lr) in Period 7; however, given the similarity of their chemical properties, the elements lanthanum (La) and actinium (Ac) are normally included in discussion of the f block, as we do here.1 The 4f elements are collectively the lanthanoids (formerly and still commonly ‘the lanthanides’, but this terminology is potentially confusing as the suffix -ide generally refers to an anionic form of an element) and the 5f elements are the actinoids (still commonly the ‘actinides’).The lanthanoids are sometimes referred to as the ‘rare earth elements’. That name, however, is inappropriate because they are not particularly rare, except for promethium, which has no stable isotope. A general lanthanoid is represented by the symbol Ln and an actinoid by An. Because the names and symbols of the f-block elements are less familiar than those elsewhere in the periodic table, we give discreet reminders in the margins or in the text in appropriate places. The chemical properties of the lanthanoids are quite different from those of the actinoids and we discuss them separately. As we shall see, there is a striking uniformity in the properties of the lanthanoids and greater diversity in the chemistry of the actinoids. The elements We begin our account of the f-block elements by considering the properties and the extraction of the elements. 23.1 Occurrence and recovery Key points: The principal sources of the lanthanoids are phosphate minerals; the most important actinoid, uranium, is recovered from its oxide. 1 The majority of periodic tables, including the one used here, place La in Group 3 and Lu at the end of the f block; however, both these elements have ns2(n1)d1 electron configurations and never exhibit partially filled f orbitals in their compounds, so a reasonable alternative, which is adopted by some, is to place La at the start of the f block and Lu in Group 3. 23 The elements 23.1 Occurrence and recovery 23.2 Physical properties and applications Lanthanoid chemistry 23.3 General trends 23.4 Electronic, optical, and magnetic properties 23.5 Binary ionic compounds 23.6 Ternary and complex oxides 23.7 Coordination compounds 23.8 Organometallic compounds Actinoid chemistry 23.9 General trends 23.10 Electronic spectra 23.11 Thorium and uranium 23.12 Neptunium, plutonium, and americium FURTHER READING EXERCISES PROBLEMS 580 23 The f-block elements Other than promethium (Pm), the lanthanoids are reasonably common in the Earth’s crust; indeed, even the ‘rarest’ lanthanoid, thulium, has a crustal abundance that is similar to that of iodine. The principal mineral source for the early lanthanoids is monazite, (Ln,Th)PO4, which contains mixtures of lanthanoids and thorium. Another phosphate mineral, xenotime (of similar composition, LnPO4), is the principal source of the heavier lanthanoids. The common oxidation state for all the lanthanoids is Ln(III), which makes separation difficult, although cerium, which can be oxidized to Ce(IV), and europium, which can be reduced to Eu(II), are separable from the other lanthanoids by exploiting their redox chemistry. Separation of the remaining Ln3 ions is accomplished on a large scale by multistep liquidliquid extraction in which the ions are distributed between an aqueous phase and an organic phase containing complexing agents. Ion-exchange chromatography is used to separate the individual lanthanoid ions when high purity is required. Pure and mixed lanthanoid metals are prepared by the electrolysis of molten lanthanoid halides. Beyond lead (Z 82) no element has a stable isotope, but two of the actinoids, thorium (Th, Z 90) and uranium (U, Z 92), have isotopes that are sufficiently long-lived that significant quantities have persisted from their formation in the supernova that preceded the formation of the Sun (Section 1.2). Indeed the levels of Th and U in the Earth’s crust exceed those of both Sn and I. Table 23.1 gives the half-lives of the most stable actinoid isotopes. Thorium is extracted from either monazite or thorite (essentially ThSiO4); the major ores for uranium, uranite and pitchblende, have the approximate formulas UO2 and U3O8, respectively. The primary source of the other actinoids is synthesis by nuclear reactions; all of them are more radioactive than thorium and uranium. 23.2 Physical properties and applications Key points: Lanthanoids are reactive metals that find small-scale uses in specialist applications; the primary uses of actinoids are restricted by their radioactivity. The lanthanoids are soft white metals that have densities comparable to those of the 3d metals (610 g cm3). For metals, they are relatively poor conductors of both heat and electricity, with thermal and electrical conductivities 25 and 50 times less than that of copper, respectively. The light lanthanoid metals are highly reactive towards oxygen and are normally stored in sealed glass ampoules. The metals react with steam and dilute acids Table 23.1 Half-lives of the most stable actinoid isotopes Z Element Symbol Mass number t1/2 89 Actinium Ac 227 21.8 a 90 Thorium Th 232 14.1 Ga 91 Protactinium Pa 231 32.8 ka 92 Uranium U 238 4.47 Ga 93 Neptunium Np 237 2.14 Ma 94 Plutonium Pu 244 81 Ma 95 Americium Am 243 7.38 ka 96 Curium Cm 247 16 Ma 97 Berkelium Bk 247 1.38 ka 98 Californium Cf 251 900 a 99 Einsteinium Es 252 460 d 100 Fermium Fm 257 100 d 101 Mendelevium Md 258 55 d 102 Nobelium No 259 1.0 h 103 Lawrencium Lr 260 3 min a year, d day, h hour, min minute Lanthanoid chemistry but are somewhat passivated by an oxide coating. The majority of the metals adopt the hexagonal cubic close-packed structure type, although cubic close-packed forms are also known for most of the elements, particularly under high pressure. A mixture of the early lanthanoid metals, with mainly Ce and La and small amounts of Pr and Nd reflecting the composition of the monazite ore, is referred to in commerce as mischmetal. It is used in steel-making to remove impurities such as H, O, S, and As (by forming very stable lanthanoid oxysulfides), which reduce the mechanical strength and ductility of steel. Compounds of the lanthanoids find a wide range of applications, many of which depend on the optical properties associated with their ff electronic transitions (Section 23.4): europium oxide and europium orthovanadate are used as red phosphors in displays and lighting, and neodymium (as Nd3), samarium (as Sm3), and holmium (as Ho3) are used in solid-state lasers. The alloys SmCo5 and Sm2Co17 of samarium and cobalt have very high magnetic strengths, more than ten times that of iron and some magnetic iron oxides (Fe3O4). They also have excellent corrosion resistance and good stability at elevated temperatures. Neodymium iron boride, Nd2Fe14B, shows similar magnetic properties and is cheaper to produce, but because it is susceptible to corrosion the magnets are often coated with zinc, nickel, or epoxy resins. Applications of these strongly magnetic materials include headphones, microphones, magnetic switches, and components of particle beams (for guidance of the beam). Toy magnetic building sets have used them extensively. The density of the actinoids increases from 10.1 g cm3 for Ac to 20.4 g cm3 for Np, before decreasing for the remainder of the series. Plutonium has at least six phases at atmospheric pressure with density differences of more than 20 per cent. Many of the physical and chemical properties of the actinoids are unknown because only miniscule quantities have ever been isolated; in addition to being radioactive, many actinoids are known to be poisonous and all are treated as hazardous in the laboratory. Their current primary use is in nuclear reactors and nuclear weapons. Lanthanoid chemistry The lanthanoids are all electropositive metals with a remarkable uniformity of chemical properties. The significant difference between two lanthanoids is often only their atomic or ionic radius, and the ability to choose a lanthanoid of a particular size often allows the ‘tuning’ of the properties of their compounds. For example, the magnetic and electronic properties of a material often depend on the exact separation of the atoms present and degree of overlap of various atomic orbitals. By choosing a lanthanoid of an appropriate atomic radius this separation can be controlled, with consequences for its electrical conductivity or magnetic transition temperature. 23.3 General trends Key points: The lanthanoids are electropositive metals that commonly occur in their compounds as Ln(III); other oxidation states are stable only when an empty, half-filled, or full f subshell is produced. The elements La through to Lu are all highly electropositive, with standard potentials of the Ln3/Ln couple similar to that of Mg2/Mg (Table 23.2). They favour the oxidation state Ln(III) with a uniformity that is unprecedented in the periodic table. Other properties of the elements vary significantly. For example, the radii of the Ln3 ions contract steadily from 116 pm for La3 to 98 pm for Lu3. The decrease in ionic radius is attributed in part to the increase in the effective atomic number, Zeff, as electrons are added to the poorly shielding 4f subshell (Section 1.9), but detailed calculations indicate that subtle relativistic effects also make a substantial contribution. Figure 23.1 shows that the decrease in ionic radius corresponds to two gentle curves intersecting at Gd3 (f7). The effect is similar to that in the d block and can be traced to the splitting of the f-orbital energies by the crystal field generated by the ligand environment (Fig. 23.2), but the splitting is much smaller than in the d block. On the left of the series, the lower energy f orbitals (which point away from the ligands) are occupied as the number of f electrons increases (f1 to f4). For later configurations, such as f6 and f7, the higher energy 581 23 The f-block elements Table 23.2 Element names, symbols, and selected properties of the lanthanoids O Z Name Symbol Configuration (M3) E /V r(Ln3)/pm Nox† 57 Lanthanum La [Xe] 2.38 116 2(n), 3, 4 1 58 Cerium Ce [Xe]f 2.34 114 2(n), 3, 4 59 Praseodymium Pr [Xe]f2 2.35 113 2(n), 3, 4 3 60 Neodymium Nd [Xe]f 2.32 111 2(n), 3 61 Promethium Pm [Xe]f4 2.29 109 3 62 Samarium Sm [Xe]f5 2.30 108 2(n), 3 63 Europium Eu [Xe]f6 1.99 107 2, 3 64 Gadolinium Gd [Xe]f7 2.28 105 3 65 Terbium Tb [Xe]f8 2.31 104 3, 4 9 66 Dysprosium Dy [Xe]f 2.29 103 2(n), 3 67 Holmium Ho [Xe]f10 2.33 102 3 11 68 Erbium Er [Xe]f 2.32 100 3 69 Thulium Tm [Xe]f12 2.32 99 2(n), 3 70 Ytterbium Yb [Xe]f13 2.22 99 2, 3 71 Lutetium Lu [Xe]f14 2.30 98 3 Ionic radii for coordination number 8 from R.D. Shannon, Acta Cryst., 1976, A32, 751. † Oxidation numbers (Nox) in bold type indicate the most stable states; (n) indicates that the state is stable only in nonaqueous conditions. f orbitals are occupied, and as these point towards the ligands the increased electron ligand repulsion results in a smaller than expected decrease in ionic radius. The 18 per cent decrease in ionic radius from La3 to Lu3 leads to an increase in the hydration enthalpy across the series. The standard potentials of the lanthanoids are all very similar, with E O (La3/La) 2.38 V similar to E O (Lu3/Lu) 2.30 V at the other end of the series. This similarity reflects the balance of increasing hydration and atomization enthalpies as the atomic radius decreases. The common occurrence of the lanthanoids as Ln(III) is normally ascribed to the fact that once the valence s and d electrons have been removed the f electrons are held tightly by the nucleus and do not extend beyond the xenon-like core of the atom. A further 120 115 r (Ln3+)/pm 582 110 35 ∆O t1u 105 f − 15 ∆O 100 95 La Pr Ce Pm Eu Tb Ho Tm Lu Nd Sm Gd Dy Er Yb Figure 23.1 Variation of the ionic radius of Ln3. − 65 ∆O t2u a2u Figure 23.2 The splitting of f orbitals in an octahedral crystal field. Lanthanoid chemistry consequence of the burying of the f electrons is that a Ln3 ion has no frontier orbitals with directional preference, so ligand-field stabilization plays only a small part in the properties of lanthanoid complexes. For a few complexes weak stabilization energies (of a few kilojoules per mole) are observed for configurations such as f4, f10, and f11 when compared with f0, f7, and f14. This stabilization can be explained following similar arguments to those described in Section 20.1 but using f orbitals and acknowledging the much weaker effects of the ligand field. Superimposed on the common occurrence of Ln3 there are some atypical oxidation states that are most prevalent when the ion can attain the relatively more stable empty (f 0), halffilled (f7), or filled (f14) subshell (Table 23.2). Thus, Ce3 (f1) can be oxidized to Ce4 (f 0) and the latter is a strong and useful oxidizing agent. The next most common of the atypical oxidation states is Eu2 (f7), and there are a number of stable Eu2 compounds, including EuI2, EuSO4, and EuCO3, and solutions of this ion are stable. The ions Sm2 and Yb2 also have an extensive chemistry but reduce water to hydrogen in aqueous solutions. Recently Dy(II), Nd(II), and Tm(II) have been produced in molecular complexes in solution; an example is NdI2(THF)5. Other reasonably stable oxidation states include Pr(IV) and Tb(IV), and the oxides of these elements formed in air, Pr6O11 and Tb4O7, contain mixtures of Ln(III) and Ln(IV). Under very strongly oxidizing conditions Dy(IV) and Nd(IV) can be obtained. A Ln3 ion is a hard Lewis acid, as indicated by its preference for F and oxygencontaining ligands and its occurrence with PO43 in minerals. 23.4 Electronic, optical, and magnetic properties Much of the commercial and technological value of the lanthanoids stems from their optical and magnetic properties. (a) Electronic absorption spectra Key point: Lanthanoid ions typically display weak but sharp absorption spectra because the f orbitals overlap only weakly with the ligand orbitals. Most lanthanoid ions are only weakly coloured because their absorptions in the visible region of the spectrum are commonly ff transitions which are symmetry forbidden (Table 23.3). The spectra of their complexes generally show much narrower and more distinct absorption bands than those of d-metal complexes. Both the narrowness of the spectral features and their insensitivity to the nature of coordinated ligands indicate that the f orbitals have a smaller radial extension than the filled 5s and 5p orbitals. As noted in Section 1.6, the 5s and 5p orbitals are expected to shield the 4f electrons from the ligands. We shall not go into as complete an analysis of ff electronic transitions as we provided for dd electronic transitions in Chapter 20 as they can be very complex; for instance, there are 91 microstates of an f2 configuration. However, the discussion is simplified somewhat by the fact that f orbitals are relatively deep inside the atom and overlap only weakly with ligand orbitals. Hence, as a first approximation, their electronic states (and therefore electronic spectra) can be discussed in the free-ion limit and the RussellSaunders coupling scheme remains a reasonable approximation despite the elements having high atomic numbers. E X A MPL E 23 .1 Deriving the ground state term symbol of a lanthanoid ion What is the ground state term symbol of Pr3 (f2)? Answer The procedure for deriving the ground state term symbols, which have the general form 2S1{L}J, for d-block elements was summarized in Section 20.3 and we can proceed in a similar way. According to Hund’s rules, the ground state will have the two electrons in different f orbitals; noting that l 3 for an f orbital, the maximum value of ML ml1 ml2 is therefore ML (3) (2) 5, which must stem from a state with L 5, an H term. The lower spin arrangement of two electrons in different orbitals is a triplet with S 1, so the term will be 3H. According to the ClebschGordan series (Section 20.3a), the total angular momentum of a term with L 5 and S 1 will be J 6, 5, or 4. According to Hund’s rules, for a less than half-full shell the level with the lowest value of J lies lowest (in this case, J 4), so we can expect its term symbol to be 3H4. Self-test 23.1 Derive the ground state of the Tm3 ion Dy Eu Nd Pr Sm Tb Tm Yb Dysprosium Europium Neodymium Praseodymium Samarium Terbium Thulium Ytterbium 583 584 23 The f-block elements J Wavenumber, n~ /(103 cm–1) 20 10 0 2 1 0 3 2 1 4 1 4 3 2 3 6 5 4 3 P D G F The large number of microstates for each electronic configuration means a correspondingly large number of terms and hence of possible transitions between them. As electrons in f orbitals interact only weakly with the ligands there is little coupling of the electronic transitions with molecular vibrations, with the consequence that the bands are narrow. As the terms are derived almost purely from f orbitals, and there is only very little df orbital mixing or mixing with ligand orbitals, the transitions are Laporte forbidden (Section 20.6). Hence, in contrast to the d metals, which normally show one or two broad bands of moderate intensity, the visible spectra of lanthanoids usually consist of a large number of sharp, low intensity peaks that are barely affected by changing the coordination environment of the central metal ion. Figure 23.3a shows the relative energies associated with the Pr3 ion, including the ground state term 3H4; ground state terms for all the Ln3 ions are given in Table 23.3. Figure 23.3b shows the experimental absorption spectrum of Pr3(aq) from the near IR to the UV region. For this ion the absorptions occur mainly between 450 and 500 nm (blue), and at 580 nm (yellow) so the residual light that reaches the eye after reflection from a Pr3 compound is mainly green and red, giving this ion its characteristic green colour. In summary, the spectra of the lanthanoid ions are normally characterized by the following properties: r Numerous absorptions due to the large number of microstates. H r Weak absorptions due to lack of orbital mixing. Molar absorption coefficients(ε) are typically 110 dm3 mol−1 cm−1 compared with d metals (close to 100 dm3 mol−1 cm−1). r Sharp absorptions due to the weak interaction of the f orbitals with the ligand vibrations. (a) r Spectra that are to a large degree independent of the ligand type and coordination number. 3 Absorption 3 P1 Table 23.3 Electron configurations, colours, and magnetic moments (at 298 K) of the Ln3 ions Element 3 1 1 (b) P2 A Nd3 ion has a strong absorption at 580 nm due to the transition 4I9/2 ← 4F3/2. This wavelength corresponds almost exactly to the main yellow emission from excited Na atoms (Section 11.1). As a result, Nd is incorporated into goggles used by glassblowers, where it reduces the glare produced by hot sodium silicate glasses. In some cases transitions between Configuration P0 Colour in aqueous solution Ground state D2 Lanthanum G4 Cerium 10 000 20 000 30 000 [1 µm] [500 nm] [333 nm] n~ /cm–1 [l] Figure 23.3 (a) Energy level diagram for the free Pr3 (f2) ion, ground state 3H4. (b) The absorption spectrum of the Pr3(aq) ion in the visible region and excited state terms for each band. [Xe]] 1 [Xe]f 2 Magnetic moment Theory ␮ /␮B Observed§ ␮ /␮B Colourless 1 0 0 Colourless 2 2.54 2.46 S0 F5/2 Praseodymium [Xe]f Green-yellow 3 3.58 3.473.61 Neodymium [Xe]f3 Violet 4 3.68 3.443.65 4 2.83 H4 I9/2 Promethium [Xe]f 5 Samarium [Xe]f5 Yellow 6 0.84 (1.551.65)¶ 1.541.65 Europium [Xe]f6 Pink 7 0 (3.403.51)¶ 3.323.54 Colourless 8 7.94 7.98.0 Gadolinium [Xe]f 7 8 I4 H5/2 F0 S7/2 Terbium [Xe]f Pink 7 9.72 9.699.81 Dysprosium [Xe]f9 Yellow-green 6 10.63 10.010.6 10 F6 H15/2 Holmium [Xe]f Yellow 5 10.60 10.410.7 Erbium [Xe]f11 Lilac 4 9.59 9.49.5 I15/2 Thulium [Xe]f Green 3 7.57 7.07.5 Ytterbium [Xe]f13 Colourless 2 4.54 4.04.5 Lutetium [Xe]f14 Colourless 1 0 0 ¶ § 12 I8 H6 F7/2 S0 The values in parentheses include expected contribution from terms other than the ground state. From compounds such as Ln2(SO4)3.8H2O and Ln(Cp)2. Lanthanoid chemistry 585 4f and 5d orbitals can occur; as ∆l 1, these transitions are allowed and intense, but they are generally in the UV region of the spectrum. For example, the transition 4f105d1 ← 4f11 in Er3 occurs at about 150 nm. (b) Emission spectra and fluorescence Key point: Lanthanoid ions show strong emission spectra with applications in phosphors and lasers. Some of the most important applications of the lanthanoids derive from their emission spectra produced after excitation (using energetic photons or electron beams) of the f electrons. These emission spectra show many features of the absorption spectra in that they consist of sharply defined frequencies characteristic of the lanthanoid cation and mainly independent of the ligand. All the lanthanoid ions except La3 (f0) and Lu3 (f14) show luminescence, with Eu3 6 (f ) and Tb3 (f8) being particularly strong. In part, the strong luminescence is due to the large number of excited states that exist, which increases the probability of intersystem crossing (the formation of excited states of different multiplicity from the ground state, Section 20.7). The strength of the emission also stems from the excited electron interacting only weakly with its environment and so having a long nonradiative lifetime (milliseconds to nanoseconds). That is, an electronically excited lanthanoid species rarely loses energy by transfer into vibrational modes because f orbitals overlap only weakly with the ligand orbitals whereas this mechanism provides a pathway for such relaxation in many d-metal systems where there is stronger d-orbital-to-ligand-orbital overlap. These properties lead to applications of lanthanoid compounds in display systems, such as cathode-ray and plasma screens (Box 23.1). By stimulating the emission from the excited state, high intensity laser radiation can be obtained, as in Nd:YAG lasers. B OX 23 .1 Lanthanoid-based phosphors and lasers Phosphors are fluorescent materials that normally convert high energy photons, typically in the UV region of the electromagnetic spectrum, into lower energy visible wavelengths. Whereas a number of fluorescent materials are based on d-metal systems (for example, Cu and Mn doped into zinc sulfide) for some emitted energies lanthanoid-containing materials offer the best performance. This is particularly so of the red phosphors needed in displays, such as plasma screens, and fluorescent lighting where the emission spectrum of Eu3 occurs in a series of lines between 580 nm (orange) and 700 nm (red). Terbium, as Tb3, is also widely used in similar phosphors, producing emissions between 480 and 580 nm (green). The phosphor compounds, such as Eu3 doped into YVO4, are coated on the interiors of fluorescent light tubes and convert the UV radiation generated from the mercury discharge into visible light. By using a combination of phosphors fluorescing in different regions of the visible spectrum, white light is emitted by the tube (Fig. B23.1). Tb Intensity Er 400 500 600 700 Wavelength, l /nm 800 Figure B23.1 Spectrum of a fluorescent lamp showing lines due to lanthanoid cations in the phosphor. Neodymium can be doped into the garnet structure of yttrium aluminium oxide (YAG), Y3Al5O12, at levels of 1 per cent Nd for Y, to produce the material that is used in Nd:YAG lasers. The Nd3 ions are strongly absorbing at wavelengths between 730760 nm and 790820 nm, such as the high intensity light produced by krypton-containing flashlamps. The ions are excited from their ground 4I9/2 state to a number of excited states which then transfer into a relatively long-lived excited 4F3/2 state. Radiative decay of this state to 4I11/2 (which lies just above the ground state 4 I9/2) is stimulated by a photon of the same frequency and results in laser radiation. So once a large number of ions exist in an excited state, they can all be stimulated to emit light simultaneously, so producing very intense radiation. A Nd:YAG laser typically emits at 1.064 m, in the near-infrared region of the electromagnetic spectrum in pulsed and continuous modes. There are weaker transitions near 0.940, 1.120, 1.320, and 1.440 m. The high-intensity pulses may be frequency-doubled to generate laser light at 532 nm (in the visible region) or even higher harmonics at 355 and 266 nm. Applications of Nd:YAG lasers include removal of cataracts and unwanted hair, range finders, and for marking plastics and glass. In the last application, a white pigment that absorbs strongly in the infrared region, near the main Nd:YAG emission line of 1.064 m, is added to a transparent plastic. When the laser is shone on the plastic the pigment absorbs the energy and heats up sufficiently to burn the plastic at the point exposed to the laser, which becomes indelibly marked. There are several YAG gain media with other lanthanoid dopants, for example Yb:YAG emits at either 1.030 m (strongest line) or 1.050 m and is often used where a thin disk of lasing material is needed, and Er:YAG lasers emit at 2.94 μm and are used in dentistry and for skin resurfacing. Lanthanoids are widely used in other laser materials including glass fibres, where they are used as fibre lasers, and for the amplification of optical signals, In an erbium doped fibre amplifier (EDFA), a silica glass-fibre core is doped with Er3, and an optical signal passing along the glass fibre is amplified (converted to a higher intensity) when a pump laser is directed on to the fibre. 586 23 The f-block elements (c) Magnetic properties Key point: The magnetic moments of lanthanoid compounds arise from both spin and orbital contributions. The magnetic moment of many d-metal ions can be calculated by using the spin-only approximation because the strong ligand field quenches the orbital contribution (Section 20.1c). For the lanthanoids, where the spinorbital coupling is strong, the orbital angular momentum contributes to the magnetic moment, and the ions behave like almost free atoms. Therefore, the magnetic moment must be expressed in terms of the total angular momentum quantum number J: = g j { J( J + 1)} 1/2 B where the Landé g-factor is gJ = 1+ S(S + 1) − L(L + 1) + J( J + 1) 2 J( J + 1) and B is the Bohr magneton. Theoretical values of the magnetic moment of the ground states of the Ln3 ions are summarized in Table 23.3; in general these values agree well with experimental data. ■ A brief illustration. As we have seen, the ground state term symbol for Pr3 (f2) is 3H4 with L 5, S 1, and J 4. It follows that g J = 1+ 2 − 30 + 20 4 1(1+ 1) − 5 (5 + 1) + 4(4 + 1) = 1+ = 2 × 4 (4 + 1) 40 5 Therefore µ = g J {J (J + 1)}1/2 µB = 54 {4(4 + 1)}1/2 µB = 3.58 µB ■ The analysis in the brief illustration assumes that only one 2S1{L}J level is occupied at the temperature of the experiment; for most lanthanoid ions this is a good assumption. For example, the first excited state of Ce3 (5F5/2) is 1000 cm−1 above the ground state (2F3/2) and nearly unpopulated at room temperature when kT ≈ 200 cm−1. Small contributions from the higher energy terms result in the minor deviations of observed values from those based on the population of a single term. For Eu3 and to a lesser extent Sm3 the first excited state lies close to the ground state (for Eu3, 7F1 lies only 300 cm−1 above the 7F0 ground state) and is partly populated even at room temperature. Although the value of based on occupation of only the ground state is zero (because J 0), the experimentally observed value is nonzero. Long-range magnetic ordering effects, ferromagnetism, and antiferro-magnetism (Section 20.8), are observed in many lanthanoid compounds, although in general the coupling between metal atoms is much weaker than in the d-block compounds and as a result the magnetic ordering temperatures are much lower. In BaTbO3 the magnetic moments of the Tb3 ion become antiferromagnetically ordered below 36 K. 23.5 Binary ionic compounds Key points: The structures of ionic lanthanoid compounds are determined by the size of the lanthanoid ion; binary oxides, halides, hydrides, and nitrides are all known. Ce Pr Tb Cerium Praseodymium Terbium Lanthanoid(III) ions have radii that vary between 116 and 98 pm; for comparison, the ionic radius of Fe3 is 64 pm. Thus the volume occupied by a Ln3 ion is typically four to five times that occupied by a typical 3d-metal ion. Unlike the 3d metals, which rarely exceed a coordination number of 6 (with 4 being common too), compounds of lanthanoids often have high coordination numbers, typically between 6 and 12, and a wide variety of coordination environments. The binary lanthanoid(III) oxides, Ln2O3, have moderately complex structures with the coordination number of the Ln3 ions being typically 7 (or a mixture of 6 and 7). Several Lanthanoid chemistry related structure types termed A-, B-, C-Ln2O3 are known and many of the oxides are polymorphic with transitions between the structures occurring as the temperature is changed. The coordination geometries are determined by the radius of the lanthanoid ion, with the average cation coordination number in the structures decreasing with decreasing ionic radius, for example the La3 ion in La2O3 has coordination number 7, whereas the Lu3 ion in Lu2O3 has coordination number 6. In cases where Ln4 ions can be obtained (for example with Ce, Pr, and Tb), the LnO2 adopts the fluorite structure as expected from radius-ratio rules (Section 3.10). Sulfides of stoichiometry LnS may be obtained by direct reaction of the elements at 1000°C and adopt the rock-salt structure. Phases of composition Ln2S3 can also be obtained by reaction of the lanthanoid trichloride with H2S; they have been studied as replacements for the toxic CdS and CdSe as possible pigments due to their intense red-orange-yellow colours. The lanthanoid(III) trihalides have complex structural characteristics as a result of the high coordination numbers for these large ions. For example, in LaF3 the La3 ion is in an irregular 11-coordinate environment and in LaCl3 it is in a nine-coordinate, capped antisquare prismatic environment (Fig. 23.4). Towards the end of the series the trihalides of the smaller lanthanoids have different structure types with lower coordination numbers for the same halide, as expected in view of the decrease in ionic radius. In LnF3, Ln has a nine-coordinate environment which can be considered as a capped anti-square prismatic environment distorted or a tricapped trigonal prism (1) and the compounds LnCl3 have layer structures based on six-coordinate Ln in a cubic close-packed array of Cl ions. Cerium is the only lanthanoid to form a tetrahalide (CeF4); it crystallizes with a structure formed from vertex-sharing CeF8 polyhedra (Fig. 23.5). All the lanthanoids form hydrides of stoichiometry LnH2 which adopt the fluorite structure (Section 3.9a) based on cubic close-packed H ions with lanthanoid ions in all the tetrahedral holes. Cerium hydride may be further oxidized by hydrogen to form a series of nonstoichiometric phases of formula CeH2x, with additional H ions incorporated into the fluorite lattice. Some of the smaller lanthanoids (for instance Dy, Yb, and Lu) form stoichiometric trihydrides, LnH3. Complex metal hydrides containing lanthanum, such as LaNi5H6, have been studied intensively as possible hydrogen-storage materials, Box 10.4 and Sections 24.14, 24.15, as they can be prepared by heating the alloy LaNi5 in hydrogen gas under pressure; the reverse reaction occurs when LaNi5H6 is heated under ambient pressure conditions evolving H2 gas. Nitrides of composition LnN exist for all the lanthanoids and adopt the expected rocksalt structure with alternating Ln3 and N3 ions. Three different lanthanoid carbide stoichiometries are known: M3C, M2C3, and MC2. The M3C phases form for the heavier lanthanoids and contain isolated carbon atoms and are hydrolysed by water to produce methane. The M2C3 phases form for the lighter lanthanoids LaHo and contain the dicarbide anion C22 , as is also found in CaC2 (Section 3.9). The MC2 phases show metallic properties and except for the elements that form stable divalent cations, such as Yb, they can be expressed as Ln3(C22−,e−) with the electron in a conduction band formed from Ln6s Ln F 1 Ln coordination in LnF3 Dy Yb Lu Dysprosium Ytterbium Lutetium La Cl e C Figure 23.4 The structure of LaCl3 shown as vertex-linked LaCl9-capped antiprisms; a single unit is shown as inset. 587 F Figure 23.5 The structure of CeF4 contains vertex-sharing CeF8 antiprisms. 588 23 The f-block elements and Ln6p orbital overlap; they react with water to form ethyne and other hydrocarbons. The lanthanoid nickel borocarbides, LnNi2B2C, have structures containing alternating layers of the stoichiometries LnC and Ni2B2. These borocarbides are superconductors at low temperature: the transition temperature for LuNi2B2C, for instance, is 16 K. 23.6 Ternary and complex oxides Key point: Lanthanoid ions are often found in perovskites and garnets, where the ability to change the size of the ion allows the properties of the materials to be modified. The lanthanoids are a good source of stable, large, tripositive cations with a reasonable range of ionic radii. As a result, they can take one or more of the cation positions in ternary and more complex oxides. For example, perovskites of the type ABO3 can readily be prepared with La on the A cation site; an example is LaFeO3. Indeed, some distorted structure types are named after lanthanoids; an example is the structural type GdFeO3 (Fig. 23.6), which has vertex-linked FeO6 octahedra around the Gd3 ion (as in the parent perovskite structure, Fig. 3.9) but the octahedra are tilted relative to each other. This tilting allows better coordination to the central Gd3 ion. The ability to change the size of the B3 ion in a series of compounds LnBO3 allows the physical properties of the complex oxide to be modified in a controlled manner. For example, in the series of compounds LnNiO3 for Ln Pr to Eu, the insulatingmetallic transition temperature TIM (at which the properties change from that of a metal to that of an insulator, Section 3.19) increases with decreasing lanthanoid ionic radius: r(Ln3)/pm TIM/K PrNiO3 113 135 NdNiO3 111 200 EuNiO3 107 480 As the perovskite unit cell is a structural building block often found in more complex oxide structures, lanthanoids are frequently used in such materials. Famous examples are the original high-temperature superconducting cuprate (HTSC) La1.8Ba0.2CuO4 and the family of ‘123’ complex oxides, LnBa2Cu3O7, which become superconducting below 93 K. The best known of these HTSCs is the d-block (yttrium) compound YBa2Cu3O7, but they are also found for all the lanthanoids (Section 24.8). Another system where choice of lanthanoid is crucial in obtaining the required property is the complex manganites Ln1SrxMnO3 that exhibit resistance effects strongly dependent on the applied magnetic field x and temperature; the optimized properties are found when Ln Pr. Fe Figure 23.6 The GdFeO3 structure type, with the FeO6 octahedra outlined. The inset shows its relation to the ideal perovskite structure. O Gd Lanthanoid chemistry BO4 589 MO6 AO8 Figure 23.7 The garnet structure shown in the form of linked AO8, BO4, and MO6 polyhedra. The eight-coordinate A sites often occupied by yttrium can be occupied by other lanthanoids. The spinel structure (Fig. 3.44) has only small tetrahedral and octahedral holes in the close-packed O2 ion array and cannot accommodate bulky lanthanoid ions. However, the garnet structure adopted by materials of stoichiometry M3M2(XO4)3, where M and M are normally di- and tripositive cations and X includes Si, Al, Ga, and Ge, has eightcoordinate sites that can be occupied by lanthanoid ions, (Fig. 23.7). Yttrium aluminium garnet (YAG) is the host material for neodymium ions in the laser material Nd:YAG. Here we see a result of the similarities in the chemistry of yttrium from Group 3 and many lanthanoids. The contraction in the ionic radius of the lanthanoids with increasing atomic number means that many Ln3 ions have an ionic radius similar to that of Y3 and therefore partial replacement of this ion by a lanthanide ion (for instance, Nd3 in Nd:YAG) is easily achieved. Key point: High coordination numbers with ligands adopting geometries that minimize interligand repulsions are the norm for the lanthanoids. The adoption by the relatively large, hard Lewis acid Ln3 ions of structures with high coordination numbers and with a variety of coordination environments in the solid state is repeated in solution. The variation in structure adopted is consistent with the view that the spatially buried f electrons have no significant stereochemical influence, and consequently ligands adopt positions that minimize interligand repulsions. In addition, polydentate ligands must satisfy their own stereochemical constraints, much as for the s-block ions and Al3 complexes. For example, many lanthanoid complexes have been formed with crown ether and -diketonate ligands. The coordination numbers for [Ln(OH2)n]3 in aqueous solution are thought to be 9 for the early lanthanoids and 8 for the later, smaller members of the series, but these ions are highly labile and the measurements are subject to considerable uncertainty. Similarly, a striking variation is observed for the coordination numbers and structures of lanthanoid salts and complexes. For example, the small ytterbium cation, Yb3, forms the seven-coordinate complex [Yb(acac)3(OH2)], and the larger La3 is eightcoordinate in [La(acac)3(OH2)2]. The structures of these two complexes are approximately a capped trigonal prism (2) and a square antiprism (3), respectively. The partially fluorinated -diketonate ligand [C3F7COCHCOC(CH3)3]−, nicknamed fod, produces complexes with Ln3 that are volatile and soluble in organic solvents. On account of their volatility, these complexes can be used as precursors for the synthesis of lanthanoid-containing materials by vapour deposition, such as the high temperature superconductors (Section 25.5). acac 23.7 Coordination compounds b Y 2 [Yb(acac)3OH2] acac La H2O 3 [La(acac)3(OH2)2] 590 23 The f-block elements 66Dy 64Gd 69Tm 67Ho 65Tb 63Eu Concentration 71Lu 68 70 Er Yb Volume of eluent Figure 23.8 Elution of heavy lanthanoid ions from a cation exchange column using ammonium 2-hydroxyisobutyrate as the eluent. Note that the higher atomic number lanthanoids elute first because they have smaller radii and are more strongly complexed by the eluent. Yb PPh3 Pt PPh3 4 [(Cp)2Yb(C2H4)Pt(PPh3)2] Charged ligands generally have the highest affinity for the smallest Ln3 ion, and the resulting increase in formation constants from large, lighter Ln3 (on the left of the series) to small, heavier Ln3 (on the right of the series) provides a convenient method for the chromatographic separation of these ions (Fig. 23.8). In the early days of lanthanoid chemistry, before ion-exchange chromatography was developed, tedious repetitive crystallizations were used to separate the elements. Lanthanoid complexes have found applications as shift reagents in NMR spectroscopy. The chemical shift of a proton (Section 8.5) is markedly shifted when it is in the vicinity of a paramagnetic centre and the -range considerably expanded. Hence addition of a small amount of a paramagnetic lanthanoid complex to a solution of a complex organic molecule results in marked changes to the spectrum as the two molecules are in close proximity in the solvent. This technique is of considerable use when NMR spectra are collected on low- and mid-field instruments as the resonances are spread out over a greater range of values, effectively increasing the resolution of the instrument by separating otherwise overlapping resonances. Europium and Yb complexes generally induce a downfield shift whereas Pr and Dy produce upfield shifts. Commonly used shift reagents include Ln(fod)3 with Ln Eu, Pr, and Yb. Chiral lanthanide shift reagents can be used to undertake an assay of enantiomeric composition as the NMR resonances of the two enantiomers are different and their separation improved in the presence of the shift reagent, Section 27.20. 23.8 Organometallic compounds Key points: The organometallic compounds of the lanthanoids are dominated by good donor ligands, with complexes of acceptor ligands being rare; the bonding in complexes is best treated on the basis of ionic interactions; there are similarities between the organometallic compounds of the lanthanoids and those of the early d metals. In line with the picture of the lanthanoid ions as having no directional frontier orbitals, it should not be surprising that they do not exhibit a rich organometallic chemistry. In particular, the lack of any orbitals that can backbond to organic fragments (because the 5d orbitals are empty and the 4f orbitals are buried) restricts the number of bonding modes that are available. Most of the ligands discussed in Chapter 22 cannot bond in the fashion described there. In addition, the strongly electropositive nature of the lanthanoids means that they need good donor, not good acceptor, ligands. Thus alkoxide, amide, and halide ligands, which are both and π donors, are common, whereas CO and phosphine ligands, which are both donors and π acceptors, and which do such a good job in stabilizing low oxidation state d-metal complexes, are rarely seen in lanthanoid chemistry. Indeed, under normal laboratory conditions, neutral metal carbonyl compounds are unknown for the f-block elements. As an example of the contrasting behaviour of organometallic complexes of the d and f blocks it should be noted that the first 2-alkene complex of a lanthanoid was characterized only in 1987, a century and a half after the first d-metal alkene complex, Zeise’s salt, was isolated. This lanthanoid alkene complex, [(Cp)2Yb(C2H4)Pt(PPh3)2] (4), has a particularly electron-rich alkene as it is already bound to an electron-rich Pt(0) centre, and it is thought that the alkene therefore needs no backbonding from the Yb atom to stabilize it. The bonding in the organometallic lanthanoid complexes that does form is predominantly ionic and is governed by electrostatic factors and steric requirements. Consequently there is no need for the 18-electron rule to be obeyed. Although there is a chemical uniformity across the f block, there is also a gradation of change with steric factors tending to dominate: a small change in ligand size can make a significant difference to reactivity. Owing to the weakness of the bonds, the lanthanoid complexes remain strong Lewis acids and the complexes are therefore very sensitive to air and moisture. The first organometallic compounds of the lanthanoids were cyclopentadienyl compounds: G. Wilkinson made a large number of Ln(Cp)3 compounds with a wide variety of electron counts in 1954. The large lanthanoid ions can easily accommodate three cyclopentadienyl ligands, and they even tend to oligomerize, indicating that there is yet more space for additional ligands. Compounds of substituted cyclopentadienyl ligands are possible but the limit seems to have been reached with the sterically demanding pentamethylcyclopentadienyl (Cp) ligand. It was, however, nearly 40 years after the original Ln(Cp)3 Lanthanoid chemistry 591 compounds were isolated that Ln(Cp)3 compounds were obtained, and even then they exist in equilibrium: → Sm(η1-Cp)(η5-Cp) (6) Sm(η5-Cp)3 (5) ← 2 The great majority of lanthanoid organometallic compounds formally contain Ln(III) with a limited number of Ln(II) compounds: no other oxidation states are known. -Bonded alkyl groups are common, with compounds containing the cyclopentadienyl ligands tending to dominate. It is best to consider cyclopentadienyl compounds as containing Cp groups electrostatically bound to a central Ln3 (or Ln2) cation; this view is supported by the observation that the La compounds are diamagnetic. Compounds containing 8-cyclooctatetraene ligands are known, such as Ce(C8H8)2 (7), and, as noted in Section 22.11, it is best to consider these to be complexes of the electron-rich C8H82 ion as ligand. Current research into lanthanoid organometallic compounds typically involves compounds of the type [(Cp)2LnR]2, [(Cp)2LnR(sol)], [(Cp)LnRX(sol)], and [(Cp)LnR2(sol)]. A number of arene complexes are also known, such as [(C6Me6)Sm(AlCl4)3] (8), where it is thought that the bonding is largely the result of an electrostatically induced dipole between the Sm3 ion and the electron-rich ring. As well as an extensive organometallic chemistry of Ln(III) there are a number of Ln(II) molecular complexes that have been synthesized, including several for lanthanoids that rarely exist in this oxidation state. The species Er(II), Sm(II), and Yb(II) have extensive chemistries of this type, for example the compounds Ln(Cp)2, but compounds of Tm(II), Dy(II), and Tm(II) have also been obtained, including TmI2(DME)3. No lanthanoid has two stable states with oxidation numbers differing by 2, so there is no possibility of oxidative addition or reductive elimination reactions: -bond metathesis-type reactivity dominates (Section 22.23). In addition to the handling problems caused by the extreme sensitivity to air and moisture of the lanthanoid organometallic compounds, their study by NMR has been inhibited by the fact that they are all paramagnetic. Useful comparisons can be drawn through the study of compounds of the 4d-metal yttrium because Y3 has the same charge as and a size similar to a typical Ln3 ion, but is diamagnetic. In addition, yttrium is 100 per cent 89Y with I 12 , so it is possible to measure YC and YH coupling constants quite easily and thus get additional structural information. There are strong similarities between the chemical properties of the early d-block organometallic compounds (those of Groups 3 to 5) and those of the f block. These similarities are to be expected because the early d metals are also strongly electropositive, have a limited number of d electrons to backbond with ligands, and have a limited number of accessible oxidation states. An example of a lanthanoid dinitrogen complex is (μ-N2) [(C5Me4H)2La(THF)]2. Although the organometallic chemistry of the lanthanoids is less rich than that of the d metals, there are some striking examples of unusual reactions. For instance, a lanthanoid organometallic compound can be used for the activation of the CH bond in methane. This discovery was based on the observation that 13CH4 exchanges 13C with the CH3 group attached to Lu: Sm Cp 5 [Sm(η5-Cp)2(η1-Cp)] Sm Cp CH3 6 [Sm(η5-Cp)2(η1-Cp)] Dy Erbium Sm Samarium Tm Thulium Yb Ytterbium Ce 7 Ce(C8H8)2 C6M e This reaction can be carried out in deuterated cyclohexane with no evidence for activation of the cyclohexane CD bond, presumably because cyclohexane is too bulky to gain access to the Lu atom. A mechanism involving a four-centre -bond metathesis-type intermediate has been proposed (Section 22.23): Me H Lu Me Lu Dysprosium Er Lu(Cp)2CH3 13CH4 → Lu(Cp)2(13CH3) CH4 Me CH3 H Me H Me Lu Me Lanthanoid complexes are used in the ZieglerNatta polymerization of alkenes (Section 26.16). Dinitrogen complexes of lanthanoids were first reported in 1988. 6 CH3 Cl Sm l A 8 [Sm(C6Me6)(AlCl4)3] 592 23 The f-block elements E X A M PL E 23 . 2 Accounting for the organometallic reactivity of a lanthanoid Suggest a likely reaction pathway for the following transformation: Ln Bu H2 Ln H + BuH Answer The reaction with dihydrogen cannot proceed through a dihydrogen complex, oxidative addition to a dihydride, and then reductive elimination of butane because the lanthanoid is incapable of oxidative addition reactions. A possible scenario involves a -bond metathesis reaction through an intermediate such as Ln H H H Ln Bu Ln H 9 [(Cp)2Ln(µ-H)2Ln(Cp)2] Self-test 23.2 The product of the reaction above is in fact a hydride bridged dimer (9). Suggest a strategy to ensure that the hydride is monomeric. Actinoid chemistry The chemical properties of the actinoids show less uniformity across the series than those of the lanthanoids. However, the radioactivity associated with most of the actinoids has hindered their study. Because the later actinoids are available in such tiny amounts, little is known about their reactions. The early actinoids, particularly uranium and plutonium, are of great importance in the generation of power through nuclear fission and their chemical properties have been investigated thoroughly (Box 23.2). B OX 23 . 2 Nuclear fission and nuclear waste management The fission of heavy elements, such as 235U, can be induced by bombardment by neutrons. Thermal neutrons (neutrons with low velocities) bring about the fission of 235U to produce two nuclides of medium mass, and a large amount of energy is released because the binding energy per nucleon decreases steadily for atomic numbers beyond about 26 (Fe; see Fig. 1.2). That unsymmetrical fission of the uranium nucleus occurs is shown by the double-humped distribution of fission products (Fig. B23.2) with maxima close to mass numbers 95 (isotopes of Mo) and 135 (isotopes of Ba). Almost all the fission products are unstable nuclides. The most troublesome are those with half-lives in the range of years to centuries: these nuclides decay fast enough to be highly radioactive but not sufficiently fast to disappear in a convenient time. Fission yield (per cent) 10 1 10–1 10–2 10–3 10–4 60 80 100 120 140 Mass number, A 160 180 Figure B23.2 The double-humped distribution of fission products of uranium. The first nuclear power plants relied on the fission of uranium to generate heat. The heat was used to produce steam, which can drive turbines in much the same way as conventional power plants use the burning of fuels to produce heat. However, the energy produced by the fission of a heavy element is huge in comparison with the burning of conventional fuels: for instance, the complete combustion of 1 kg of octane produces approximately 50 MJ, whereas the energy liberated by the fission of 1 kg of 235U is approximately 2 TJ (1 TJ 1012 J), 40 000 times as much. Later designs of nuclear power plants used plutonium, normally mixed with uranium. Nuclear power offers the potential of enormous quantities of energy at low cost and with little contribution to carbon dioxide emissions. In some countries, such as France, nuclear energy is by far the largest contributor to their national power production. Improvements in nuclear power plant safety are being developed. A great concern, the disposal of the radioactive waste products, is a highly active area for inorganic chemistry research. At present, the best methods involve the immobilization of radioactive ions through various reactions that include vitrification (trapping the radionuclides in a glass), ion-exchange reactions with zeolites and similar porous materials, and the formation of synthetic rock materials (Synroc) which can then be buried. In the last process, high-level nuclear waste containing actinoids is mixed with several titanium-based complex oxides, such as perovskite, CaTiO3, and zirconolite, CaZrTi2O7. This mixture is then heated to produce a hard rock-like material where the actinoids have become immobilized in the complex oxides. The Synroc product can then be stored for long periods, often underground, with minimal likelihood of the radionuclides entering the environment. Actinoid chemistry 23.9 General trends Key point: The early actinoids do not exhibit the chemical uniformity of the lanthanoids; they exist in diverse oxidation states with An(III) progressively more stable across the series. The 14 elements from thorium (Th, Z 90, 5f1) to lawrencium (Lr, Z 103, 5f14) involve the progressive completion of the 5f subshell, and in this sense are analogues of the lanthanoids. However, the actinoids do not exhibit the chemical uniformity of the lanthanoids. Whereas, like the lanthanoids, a common oxidation state of the actinoids is An(III), unlike the lanthanoids the early members of the series occur in a rich variety of other oxidation states. The Frost diagrams in Fig. 23.9 show that oxidation numbers higher than 3 are easily accessible and often preferred for the early elements of the block (Th, Pa, U, Np, Pu) whereas 3 becomes predominant in Am and beyond. Linear or nearly linear AnO2 and AnO22 ions are the dominant aqua species for oxidation numbers 5 and 6 exhibited by U and Np. Unlike in the lanthanoids, the f orbitals of the early actinoids extend into the bonding region, so the spectra of their complexes are strongly affected by ligands. The 5f and 6d orbitals are less compact than the 4f and 5d orbitals and the electrons in them are more available for bonding. Thus the outermost electron configuration of U is 5f36d17s2 with all six electrons available for bonding. At around Am and Cm the 5f and 6d orbital energies are lower due to their poor screening by the other electrons, so that the later actinoids behave like the lanthanoids with only the outermost three electrons being readily ionized. The striking differences between the chemical properties of the lanthanoids and early actinoids led to controversy about the most appropriate placement of the actinoids in the periodic table. For example, before 1945, periodic tables usually showed U below W because both elements have a maximum oxidation number of 6. The emergence of oxidation state An(III) for the later actinoids was a key point in determining their current placement. The similarity of the heavy actinoids and the lanthanoids is illustrated by their similar elution behaviour in ion-exchange separation (compare Figs 23.8 and 23.10). E X A MPL E 23 . 3 Assessing the redox stability of actinoid ions Use the Frost diagram for thorium (Fig. 23.9) to describe the relative stability of Th(II) and Th(III). Answer We need to use the interpretation of Frost diagrams described in Section 5.13. The initial slope in the Frost diagrams indicates that the Th2 ion might be readily attained with a mild oxidant. However, Th2 lies above the lines connecting Th(0) with the higher oxidation states, so it is susceptible to disproportionation. Thorium(III) is readily oxidized to Th(IV) and the steep negative slope indicates that it is susceptible to oxidation by water: Th3+ (aq) + H+ (aq) → Th4+ (aq) + 21 H2 (g) We can confirm from Resource section 3 that because E Th(IV) will dominate in aqueous solution. O = +3.8 V , this reaction is highly favoured. Thus, Self-test 23.3 Use the Frost diagrams and data in Resource section 3 to determine the most stable uranium ion in acid aqueous solution in the presence of air and give its formula. Like the lanthanoids, the actinoids have large atomic and ionic radii (the radius of an An3 ion is typically about 5 pm larger than its Ln3 congener) and as a result often have high coordination numbers. For example, uranium in solid UCl4 is eight-coordinate and in solid UBr4 it is seven-coordinate in a pentagonal-bipyramidal array. Solid-state structures with coordination numbers up to 12 have been observed. Because of the small quantities of material available in most cases and their intense radioactivity, most of the chemical properties of the transamericium elements (the elements following americium, Z 95) have been established by experiments carried out on a microgram scale or even on just a few hundred atoms. For example, the transamericium ion complexes have been adsorbed on and eluted from a single bead of ion-exchange material of diameter 0.2 mm. For the heaviest and most unstable post-actinoids, such as hassium (Hs, Z 108), the lifetimes are too short for chemical separation and the identification of the element is based exclusively on the properties of the radiation it emits. Bk Berkelium Cf Californium Cm Curium Es Einsteinium 593 594 23 The f-block elements 0 0 0 c A –2 –2 –4 –4 –4 –4 –6 –6 –6 –6 –8 0 1 2 3 4 5 6 –8 0 1 2 3 4 5 6 –8 0 0 0 Concentration 96 m A m C –2 –2 –2 –4 –4 –4 –4 –6 –6 –6 –6 –8 0 1 2 3 4 5 6 –8 0 1 2 3 4 5 6 –8 0 1 2 3 4 5 6 0 Cf Bk –8 0 1 2 3 4 5 6 0 u P 0 k B 97 0 1 2 3 4 5 6 –2 0 Es U –2 –8 99 a P –2 p N 98 0 h T 0 1 2 3 4 5 6 0 f C s E m F –2 –2 –2 –2 –4 –4 –4 –4 –6 –6 –6 –6 –8 0 1 2 3 4 5 6 –8 Cm 95 Am Fm 100 Md 101 –8 0 1 2 3 4 5 6 –8 0 1 2 3 4 5 6 0 Volume of eluent Figure 23.10 Elution of heavy actinoid ions from a cation exchange column using ammonium 2-hydroxyisobutyrate as the eluent. Note the similarity in elution sequence to Fig. 23.8: the heavy (smaller) An3 ions elute first. NE°/V n A n to a id x O ,n r e b m u N 0 d M 0 1 2 3 4 5 6 0 o N r L –2 –2 –2 –4 –4 –4 –6 –6 –6 –8 0 1 2 3 4 5 6 –8 0 1 2 3 4 5 6 –8 0 1 2 3 4 5 6 Figure 23.9 The Frost diagrams for actinoids in acidic solution. (Based on J. J. Katz, G. T. Seaborg, and L. Morss, Chemistry of the actinide elements. Chapman and Hall, London (1986).) 23.10 Electronic spectra Key points: The electronic spectra of the early actinoids have contributions from ligand-to-metal charge transfer, 5f → 6d, and 5f → 5f transitions. The uranyl ion fluoresces strongly. Transitions between electronic states involving only the f orbitals, the 5f and 6d orbitals, and ligand-to-metal charge transfer (LMCT) are all possible for the actinoid ions. The ff transitions are broader and more intense than for the lanthanoids because the 5f orbitals interact more strongly with the ligands. Their molar absorption coefficients typically lie in the range 10100 dm3 mol−1 cm−1. The most intense absorptions are associated with LMCT transitions. For instance, LMCT transitions result in the intense yellow colour of the uranyl ion, UO22, in solution and its compounds. In species such as U3 (f3) transitions such as 5f26d1 ← 5f3 occur at wavenumbers between 20 000 and Actinoid chemistry 23.11 Thorium and uranium Key points: The common nuclides of thorium and uranium exhibit only low levels of radioactivity, so their chemical properties have been extensively developed; the uranyl cation is found in complexes with many different ligand donor atoms; the organometallic compounds of the elements are dominated by pentamethylcyclo-pentadienyl complexes. Because of their ready availability and low level of radioactivity, the chemical manipulation of Th and U can be carried out with ordinary laboratory techniques. As indicated in Fig. 23.9, the most stable oxidation state of Th in aqueous solution is Th(IV). This oxidation state also dominates the solid-state chemistry of the element. Eight-coordination is common in simple Th(IV) compounds. For example, ThO2 has the fluorite structure (in which a Th atom is surrounded by a cubic array of O2 ions) and in ThCl4 and ThF4 the coordination numbers are also 8 with dodecahedral and square antiprism symmetry, respectively. The coordination number of Th in [Th(NO3)4(OPPh3)2] is 10 (10), with the NO3 ions and triphenylphosphine oxide groups arranged in a capped cubic array around the Th atom. The very unusual coordination number of 11 is exhibited by Th in its hydrated nitrate Th(NO3)4.5H2O with the Th4 ion coordinated to four NO3 ions in a bidentate fashion and with three H2O molecules (11). The chemical properties of U are more varied than those of Th because the element has access to oxidation states from U(III) to U(VI), with U(IV) and U(VI) the most common. Uranium halides are known for the full range of oxidation states U(III) to U(VI), with a trend towards decreasing coordination number with increasing oxidation number. The U atom is nine-coordinate in solid UCl3, eight-coordinate in UCl4, and six-coordinate for the U(V) and U(VI) chlorides U2Cl10 (Fig 23.12) and UCl6, both of which are molecular compounds. The high volatility of UF6 (it sublimes at 57˚C) together with the occurrence of fluorine in a single isotopic form account for the use of this compound in the separation of the uranium isotopes by gaseous diffusion or centrifugation. In gas diffusion, the lighter 235UF6 molecules travel at greater average speeds and strike the walls of a container Np Neptunium Pu Plutonium Fluorescence intensity 33 000 cm−1 (500300 nm) giving solutions and compounds of this ion a deep orange red colour. For Np3 and Pu3, with increasing effective nuclear charge, the separation of the 5f and 6d levels increases and the corresponding transitions move into the UV region of the spectrum; Np3 solutions are violet and those of Pu3 are light violetblue due mainly to ff transitions. The uranyl ion, UO22, is also strongly fluorescent, with a strong emission between 500 and 550 nm on excitation with UV radiation (Fig. 23.11). This property has in the past been used for colouring glass, where the addition of 0.52 per cent of uranyl salts produces a bright golden yellow colour. The glass has a bright, greenyellow fluorescence in sunlight which adds to its attractiveness; when viewed under UV radiation it glows an intense green. However as this glass is radioactive its commercial production has been largely phased out in recent years. 595 0 4 4 0 0 5 5 5 0 0 6 6 0 0 5 7 lW e v a ,e th g n l /nm Figure 23.11 The fluorescence spectrum of the UO2 ion in aqueous solution. 2 NO3 Th O P C6H5 10 [Th(NO3)4(OPPh3)2] U Cl NO3 Th H2O Figure 23.12 The crystal structure of U2Cl10 consists of discrete molecules formed from pairs of edge-sharing UCl6 octahedra. 11 [Th(NO3)4(OH2)3] 596 23 The f-block elements M 12 M(Cp)4, M = Th, U U Cl 13 U(Cp)3Cl M 14 M(C8H8)2, M = Th, U 15 more frequently than their heavier isotopologues such as 238UF6. If the container-wall is permeable, 235UF6 will diffuse through it slightly more rapidly. A series of such ‘diffusers’, sometimes numbering thousands, enables enrichment of the gas in 235UF6. Uranium metal does not form a passivating oxide coating, so it becomes corroded on prolonged exposure to air to give a complex mixture of oxides. These oxides include UO2, U3O8, and several polymorphs of the stoichiometry UO3. The dioxide UO2 adopts the fluorite structure but also takes up interstitial O atoms to form the non-stoichiometric series UO2x, 0 x 0.25. One form of uranium trioxide, -UO3, adopts the ReO3 structure type (Section 24.8b). The most important oxide is UO3, which dissolves in acid to give the uranyl ion, UO22 [O U O]2. In water, this ion forms complexes with many anions, such as NO3 and SO42. In contrast to the angular shape of the VO2 ion and similar d0 complexes, the AnO2n unit, with An U, Np, Pu, and Am, maintains its linearity in all complexes. Both f-orbital bonding and relativistic effects have been invoked to explain this linearity. Compounds containing the UO22 ion show (2n)-coordination, n 4, 5, and 6, with the four to six additional ligands forming a planar or near planar arrangement around the uranyl unit. Thus the structure of UO2F2 has a slightly puckered ring of six F ions around the UO22 unit. The stability of the UO22 dication means that in most reactions it remains inert; recently, however, it has been shown that it is possible to reduce this species, when trapped within a rigid framework, to produce [O U(V) OR]. The separation of U from most other metals is accomplished by the extraction of the neutral uranyl nitrato complex [UO2(NO3)2(OH2)4] from the aqueous phase into a polar organic phase, such as a solution of tributylphosphate dissolved in a hydrocarbon solvent. This kind of solvent-extraction process is used to separate actinoids from other fission products in spent nuclear fuel. The organometallic chemistry of U and Th is reasonably well developed and shows many similarities to that of the lanthanoids, except that Th and U occur in a number of oxidation states and are larger than the typical Ln ion. Thus, compounds are dominated by those containing good donor ligands, such as -bonded alkyl and cyclopentadienyl groups. The increased size of Th and U compared with typical lanthanoids means that the tetrahedral species Th(Cp)4 and U(Cp)4 (12) can be isolated as monomers, and not only can U(Cp)3 be isolated but so too can U(Cp)3Cl (13). As with lanthanoid organometallic compounds, actinoid organometallic compounds do not obey the 18-electron rule (Section 22.1). Sandwich compounds are possible with the 8-cyclooctatetraene ligand, and both thorocene, Th(C8H8)2 and uranocene, U(C8H8)2 (14) are known and are sufficiently stable not to react with water. Compared with lanthanoid complexes of cyclooctatetraene, the bonding in uranocene and thorocene is complicated by the extension of the f orbitals beyond the core of the atom. In theory, therefore, not only is bonding possible (15), but so is φ bonding (16). Quite how much of a role φ bonds play in the bonding in uranocene or thorocene is the subject of considerable debate, but it is interesting to note that actinoid cyclooctatetraene compounds are the only ‘real’ compounds (as opposed to metal dimers in the gas phase) that might contain any contribution from φ bonding. 23.12 Neptunium, plutonium, and americium 16 Am Americium Np Neptunium Pu Plutonium Key points: Oxidation states higher than 3 become increasing less accessible between Np and Am, although all these actinoids can form AnO2n species in aqueous solution. The three elements Np, Pu, and Am form compounds containing similar species, although there are significant differences in the stabilities of the main oxidation states. The Frost diagrams in Fig. 23.9 summarize their behaviour. Neptunium dissolves in dilute acids to produce Np3, which is readily oxidized by air to produce Np4. Increasingly strong oxidizing agents produce NpO2 (Np(V)) and NpO22 (Np(VI)). The four oxidation states of plutonium, Pu(III), Pu(IV), Pu(V), and Pu(VI), are separated from each other by less than 1 V and solutions of Pu often contain a mixture of the species Pu3, Pu4, and PuO22 (PuO2 has a tendency to disproportionate to Pu4 and PuO22). The ion Am3 is the most stable species in solution, reflecting the tendency for the An(III) to dominate actinoid chemistry for the high atomic number elements. Under strongly Actinoid chemistry 597 oxidizing conditions AmO2 and AmO22 can be formed; Am(IV) disproportionates in acidic solutions. The An(IV) oxides NpO2, PuO2, and AmO2, which are formed by heating the elements or their salts in air, all adopt the fluorite structure. Lower oxides include Np3O8, Pu2O3, and Am2O3. The trichlorides, AnCl3, can be obtained by direct reaction of the elements at 450°C and have structures analogous to that of LnCl3 with a nine-coordinate An atom. Tetrafluorides are known for all three actinoids though only Np and Pu form tetrachlorides, further demonstrating the difficulty in raising americium to Am(IV). Both Np and Pu form hexafluorides which, like UF6, are volatile solids. All three metals have species analogous to the uranyl ion, forming NpO22, PuO22, and AmO22, which can be extracted from aqueous solution by tributylphosphate as AnO2(NO3)2(OP(OBu)3)2. The tetrahalides are Lewis acids and form adducts with electron pair donors such as DMSO, as in AnCl4(Me2SO)7. Neptunium forms a number of organometallic compounds that are analogues of those of uranium, such as Np(Cp)4. Although these elements are highly toxic due to their radioactivity, this property is put to good use in many smoke detectors, which contain a minute amount (0.2 μg) of 241Am. This isotope is an α-emitter with a half-life of 432 years and the α-particles enter a space between two electrodes where they ionize the air, so permitting a small current to flow between the electrodes. However, when smoke particles enter the space between the electrodes they absorb the α-particles (and any ions formed by the α-particles adhere to the smoke particle surfaces) and the resulting reduction in current is detected, setting off the alarm. The isotope 241Am is used in this application as it is a strong source of α-particles; however these do not escape from the smoke detector due to their short path-lengths in solids and do not present a hazard. FURTHER READING S.A. Cotton, Lanthanides and actinides. Macmillan, London (1991). N. Kaltsoyannis and P. Scott, The f elements. Oxford University Press (1999). G. Seaborg, J. Katz, and L.R. Morss, Chemistry of the actinide elements. Chapman and Hall, London (1986). D.M.P. Mingos and R.H. Crabtree (ed.). Comprehensive organometallic chemistry III. Elsevier, Oxford (2006). Volume 4 (ed. M. Bochmann) deals with Groups 3 and 4 and the lanthanoids and actinoids. W.J. Evans, Adv. Organomet. Chem., 1985, 24, 131, C.J. Schaverien, Adv. Organomet. Chem., 1994, 36, 283, and W.J. Evans and B.L. Davis, Chem. Rev., 2002, 102, 2119. Reviews of organolanthanoid chemistry. T.J. Marks and A. Streitwieser, p. 1547, and T.J. Marks, p. 1588, in G. Seaborg, J. Katz, and L.R. Morss, Chemistry of the actinide elements, Vol. 2. Chapman and Hall, London (1986). For a survey of organoactinoid chemistry. EXERCISES 23.1 (a) Give a balanced equation for the reaction of any of the lanthanoids with aqueous acid. (b) Justify your answer with reduction potentials and with a generalization on the most stable positive oxidation states for the lanthanoids. (c) Name the two lanthanoids that have the greatest tendency to deviate from the usual positive oxidation state and correlate this deviation with electronic structure. 23.2 Explain the variation in the ionic radii between La3 and Lu3. 23.3 From a knowledge of their chemical properties, speculate on why Ce and Eu were the easiest lanthanoids to isolate before the development of ion-exchange chromatography. 23.4 How would you expect the first and second ionization energies of the lanthanoids to vary across the series? Sketch the graph that you would get if you plotted the third ionization energy of the lanthanoids against atomic number. Identify elements at any peaks or troughs and suggest a reason for their occurrence. 23.5 Derive the ground state term symbol for the Tb3 ion. 23.6 Predict the magnetic moment of a compound containing the Tb3 ion. 23.7 Explain why stable and readily isolable carbonyl complexes are unknown for the lanthanoids. 23.8 Suggest a synthesis of neptunocene from NpCl4. 23.9 Account for the similar electronic spectra of Eu3 complexes with various ligands and the variation of the electronic spectra of Am3 complexes as the ligand is varied. 23.10 Predict a structure type for BkN based on the ionic radii r(Bk3) 96 pm and r(N3−) 146 pm. 23.11 Describe the general nature of the distribution of the elements formed in the thermal neutron fission of 235U, and decide which of the following highly radioactive nuclides are likely to present the greatest radiation hazard in the spent fuel from nuclear power reactors: (a) 39Ar, (b) 228Th, (c) 90Sr, (d) 144Ce. 598 23 The f-block elements PROBLEMS 23.1 Lanthanoid coordination compounds rarely exhibit isomerism in solution. Suggest two factors that might cause this phenomenon, explaining your reasoning. (See D. Parker, R.S. Dickins, H. Puschmann, C. Crossland, and J.A.K. Howard, Chem. Rev., 2002, 102, 1977.) 23.2 Neither lanthanoid nor actinoid organometallic compounds obey the 18-electron rule. Discuss the reasons, using the structures of the tris(Cp) and tris(Cp) Ln and An complexes as examples. (See W.J. Evans and B.L. Davis, Chem. Rev., 2002, 102, 2119.) 23.3 The bonding in the linear uranyl ion, OUO2, is often explained in terms of significant π bonding using 5f orbitals on the metal. Using the f orbitals illustrated in Fig. 1.16, construct a reasonable molecular orbital diagram for π bonding with the appropriate oxygen p orbitals. 23.4 The existence of a maximum oxidation number of 6 for both U (Z 92) and W (Z 74) prompted the placement of U under W in early periodic tables. When the element after uranium, Np (Z 93), was discovered in 1940 its properties did not correspond to those of Re (Z 75), and this cast doubt on the original placement of U. (See G.T. Seaborg and W.D. Loveland, The elements beyond uranium. Wiley-Interscience, New York (1990), p. 9 et seq.) Using standard potential data from Resource section 3, discuss the differences in oxidation state stability between Np and Re. 23.5 The processing of spent nuclear fuel and separation of the lighter actinoid elements, U, Np, and Pu, is an important industrial process. Discuss the chemistry involved in the various methods used to extract and separate these elements. 23.6 Summarize the arguments related to the placement of the lanthanoids and actinoids in the periodic table. See J. Chem. Ed., 2008, 85, 1482 and related papers. PART 3 Frontiers Inorganic chemistry is advancing rapidly at its frontiers, especially where research impinges on other disciplines such as the life sciences, condensed-matter physics, materials science, and environmental chemistry. These swiftly developing fields also represent many areas of inorganic chemistry where novel types of compounds are used in catalysis, electronics, and pharmaceuticals. The aim of this section of the book is to demonstrate the vigorous nature of contemporary inorganic chemistry by building on the introductory and descriptive material in Parts 1 and 2. These Frontiers chapters open with a discussion in Chapter 24 of materials chemistry, focusing on solid-state compounds, their synthesis, structure, and electronic, magnetic, and optical properties. One area that has developed enormously in the past decade has been that of nanomaterials, and in Chapter 25 we provide a comprehensive introduction to inorganic nanochemistry. Chapter 26 covers catalysis involving inorganic compounds and discusses the basic concepts relating to catalytic reactions at metal centres. Finally, we turn to the frontier where inorganic chemistry meets life. Chapter 27 discusses the function of different elements in cells and intracellular compartments and the various and extraordinarily subtle ways in which they are exploited. It also describes the structures and functions of complexes and materials that are formed in the biological environment and how inorganic elements are used in medical treatments. This page intentionally left blank Solid-state and materials chemistry The area of ‘materials chemistry’ is developing rapidly and there has been a great increase in interest in the synthesis and properties of novel inorganic solids. In this chapter we discuss a number of areas of current importance and investigation. Initially, we describe how inorganic materials are synthesized as bulk solids and on substrates. The role of defects in controlling structure and ion migration in solids is then discussed with examples from materials used in sensors and in energy generation and storage. We then extend these concepts to key classes of inorganic materials, with sections on intercalation compounds, complex electronic oxides (such as high-temperature superconductors), magnetic compounds (including materials exhibiting giant magnetoresistance), framework structures (such as zeolites and their analogues), and molecular materials. Throughout the chapter we shall see how the synthesis and properties of these solids correlate with their crystal and electronic structures. 24 Synthesis of materials 24.1 The formation of bulk material 24.2 Chemical deposition Defects and ion transport 24.3 Extended defects 24.4 Atom and ion diffusion 24.5 Solid electrolytes Metal oxides, nitrides, and fluorides 24.6 Monoxides of the 3d metals 24.7 Higher oxides and complex oxides The chemistry of the solid state is a vigorous and exciting area of inorganic research, partly on account of the technological applications of materials but also because their properties are challenging to understand. We shall draw on some of the concepts developed in Chapter 3 for discussing solids, such as lattice energetics and band structure. We also introduce some concepts that are needed to discuss the dynamics of events that occur in the interior of solids. We shall need to go beyond the views that solids have a well-defined stoichiometry and that their atoms are in fixed locations, for interesting phenomena arise from nonstoichiometry and ion mobility. The fact that in solid materials atoms and ions can interact in a cooperative manner gives rise to many of the fascinating and useful aspects of their chemical properties. Much of the current research in solid-state chemistry is motivated by the search for commercially useful materials, such as components of batteries and fuel cells, catalysts for hydrocarbon interconversion, and improved electronic and photonic devices for information processing and storage. The scope for the synthesis of new inorganic solids is enormous. For example, although it is known that 100 structural types account for 95 per cent of the known binary (AaBb, such as brass, CuZn) or ternary (AaBbCc) intermetallic compounds, there are plenty of opportunities for extending these studies to the synthesis and characterization of four-, five-, and six-component systems. Among the many other areas of materials chemistry that are currently being explored are new microporous solids for use in molecular separations and heterogeneous catalysis; these materials were discussed in Section 14.15, but recent advances in this area are developed here and in Chapter 26. Furthermore, because the properties and uses of a material are dependent on its physical size and form, there has been an explosion of work on nanomaterials, in which the focus is on inorganic solids with controlled sub-micrometre dimensions; this topic is treated in Chapter 25. Computer modelling is used to develop numerical models of the structures and properties of materials. It has been applied to a wide range of inorganic materials to understand and predict how atomic and electronic structure control their physical properties. Some of the most successful applications have been in areas related to defect structures, ionic mobility, and catalysis on surfaces. 24.8 Oxide glasses 24.9 Nitrides and fluorides Chalcogenides, intercalation compounds, and metal-rich phases 24.10 Layered MS2 compounds and intercalation 24.11 Chevrel phases and chalcogenide thermoelectrics Framework structures 24.12 Structures based on tetrahedral oxoanions 24.13 Structures based on octahedra and tetrahedra Hydrides and hydrogen-storage materials 24.14 Metal hydrides 24.15 Other inorganic hydrogen-storage materials Inorganic pigments 24.16 Coloured solids 24.17 White and black pigments Semiconductor chemistry 24.18 Group 14 semiconductors 24.19 Semiconductor systems isoelectronic with silicon Molecular materials and fullerides 24.20 Fullerides 24.21 Molecular materials chemistry FURTHER READING EXERCISES PROBLEMS 602 24 Solid-state and materials chemistry Synthesis of materials Much of synthetic inorganic chemistry, including the coordination chemistry of the metals and organometallic chemistry, makes use of the conversion of molecules by the replacement of one ligand by another in a solution-based reaction. Processes of this type generally have relatively small activation energies and can be undertaken at low temperatures, typically between 0ºC and 150ºC, and in solvents that permit the migration of reacting species. Rapid molecular migration in solvents results in fairly short reaction times. The formation of solid materials by reaction of solids, however, involves rather different reactions as the high lattice energies of their extended structures need to be overcome and ion migration in the solid state is normally slow except at very elevated temperatures. Some inorganic materials can be prepared from solutions at lower temperatures, where the building blocks are condensed together to form the extended structure. 24.1 The formation of bulk material New materials can be obtained by two main methods. One is the direct reaction of two or more solids. The other is the linking of polyhedral building units from solution and deposition of the newly formed solid. (a) Methods of direct synthesis Key point: Many complex solids can be obtained by direct reaction of the components at high temperatures. The most widely used method for the synthesis of bulk inorganic solids involves heating the solid reactants together at a high temperature, typically between 500 and 1500°C, for an extended period. Normally a complex oxide may be obtained by heating a mixture of all the oxides of the various metals present; alternatively, simple compounds that decompose to give the oxides may be used instead of the oxide itself. Thus, ternary oxides, such as BaTiO3, and quaternary oxides, such as YBa2Cu3O7, are synthesized by heating together the following mixtures for several days: 1000ºC ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → BaTiO3 (s) + CO2 (g) BaCO3 (s) + TiO2 (s) ⎯ 1 2 930ºC/air and 450ºC/O 2 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → YBa2Cu3O7 (s) + 2CO2 (g) Y2O3 (s) + 2BaCO3 (s) + 3CuO(s) + 41 O2 ⎯ High temperatures are used in these syntheses to accelerate the slow diffusion of ions in solids and to overcome the high Coulombic attractions between the ions. The direct method is applicable to many other inorganic material types, such as the syntheses of complex chlorides and dense, anhydrous metal aluminosilicates: Cs3Sc2Cl9 (s) 3CsCl(s) + 2ScCl3 (s) → NaAlO2 (s) + SiO2 (s) → NaAlSiO4 (s) Most simple binary oxides are available commercially as pure, polycrystalline powders with typical particle dimensions of a few micrometres. Alternatively, the decomposition of a simple metal salt precursor, either prior to or during reaction, leads to a finely divided oxide. Such precursors include metal carbonates, hydroxides, oxalates, and nitrates. An additional advantage of precursors is that they are normally stable in air whereas many oxides are hygroscopic and pick up carbon dioxide from the air. Thus, in the synthesis of BaTiO3, barium carbonate, BaCO3, which starts to decompose to BaO above 900ºC, would be ground together with TiO2 in the correct stoichiometric proportions using a pestle and mortar. The mixture is then transferred to a crucible, normally constructed of an inert material such as vitreous silica, recrystallized alumina, or platinum, and placed in a furnace. Even at high temperatures the reaction is slow and typically takes several days. A variety of methods can be used to improve reaction rates, including pelletizing the reaction mixture under high pressure to increase the contact between the reactant particles, regrinding the mixture periodically to introduce virgin reactant interfaces, and the use of ‘fluxes’, low-melting solids that aid the ion diffusion processes. The size of the reactant particles is a major factor in controlling the time it takes a reaction to proceed to completion. The larger the particles the lower their total surface area, and therefore the smaller Synthesis of materials the area at which the reaction can take place. Furthermore, the distances over which diffusion of ions must occur are much greater for larger particles, which are typically several microns in size for a polycrystalline material. In order to increase the rate of reaction and allow solid state reactions to occur at lower temperatures, reactants having small particle sizes, between 10 nm and 1 μm, and large surface areas are often deliberately employed. These submicron or nanoparticles of metal oxides can be prepared by using some of the methods described in Chapter 25 such as spray pyrolysis, where an aqueous solution of the metal salt is sprayed on to a hot surface to evaporate the solvent and decompose the salt to a fine-particle oxide. In the synthesis of LiNiO2, by reaction of LiOH and NiO in air 2 LiOH (s) 2 NiO(s) 1 2 O2 → 2 LiNiO2(s) H2O(g) the use of highly reactive nickel oxide particles produced by spray pyrolysis allows the reaction to occur to completion at under 700°C and in only 3 hours, which is at a lower temperature and much faster than would be needed with polycrystalline NiO. The reaction environment may need to be controlled if a particular oxidation state is required or one of the reactants is volatile. Solid-state reactions can be carried out in a controlled atmosphere, using a tube furnace in which a gas can be passed over the reaction mixture while it is being heated. An example of such a reaction is the use of an inert gas to prevent oxidation, as in the preparation of TlTaO3: N /600ºC 2 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 2 TlTaO3 (s) Tl2O(s) + Ta2O5 (s) ⎯ High gas pressures may also be used to control the composition of the reaction product. For example, Fe(III) is normally obtained in oxygen at or near normal pressures but at high pressures Fe(IV) may be formed, as in the production of Sr2FeO4 from mixtures of SrO and Fe2O3 under several hundred atmospheres of oxygen. For volatile reactants the reaction mixture is normally sealed in a glass tube, under vacuum, prior to heating. Examples of such reactions are 500ºC Ta(s) + S2 (l) ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → TaS2 (s) 860ºC Tl 2O3 (l) + 2BaO(s) + 3CaO(s) + 4CuO(s) ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → Tl 2Ba2Ca3Cu4O12 (s) The sulfur and thallium(III) oxide are volatile at the reaction temperatures and would be lost from the reaction mixture in an open vessel, leading to products of the incorrect stoichiometry. High pressures can also be used to affect the outcome of a solid-state chemical reaction. Specialised apparatus, typically based on large presses, allows reactions between solids to take place at pressures of up to about 100 GPa (1 Mbar) at temperatures close to 1500ºC. Reactions carried out under such conditions promote the formation of dense, higher coordination number structures. An example is the production of MgSiO3, with a perovskitelike structure and six-coordinate Si in an octahedral SiO6 unit, rather than the normal tetrahedral SiO4 unit. Small-scale reactions can be undertaken at very high pressures in ‘diamond anvil cells’ in which the faces of two opposed diamonds are pushed together in a vice-like apparatus to generate pressures of up to 100 GPa. (b) Solution methods Key point: Frameworks formed from polyhedral species can often be obtained by condensation reactions in solution. Many inorganic materials, especially framework structures, can be synthesized by crystallization from solution. Although the methods used are very diverse, the following are typical reactions that occur in water: ZrO2(s) 2 H3PO4(l) → Zr(HPO4)2.H2O(s) H2O(l) 12 NaAlO2(s) 12 Na2SiO3(s) (12 n) H2O 90 C ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → Na12[Si12Al12O48].nH2O (Zeolite LTA)(s) 24 NaOH(aq) Solution methods are extended by using hydrothermal techniques, in which the reacting solution is heated above its normal boiling point in a sealed vessel. Such reactions are important for the synthesis of open-structure aluminosilicates (zeolites), analogous porous 603 604 24 Solid-state and materials chemistry structures based on linked oxo-polyhedra (Section 24.13a), and related metal-organic frameworks (MOF) in which metal ions are linked by coordinating organic species, such as carboxylates (Section 24.13b). These porous structures are often thermodynamically metastable with respect as conversion to denser structure types so they cannot be made by direct high-temperature reactions. For example, the sodium aluminosilicate zeolite Na12[Si12Al12O48].nH2O formed in solution converts on heating above 800°C to the dense aluminosilicate NaSiAlO4. More recently, other solvents such as liquid ammonia, supercritical CO2, and organic amines have been used in so-called solvothermal reactions. A reaction in solution can also be used as an initial stage in the synthesis of many materials, particularly oxides, normally obtained through direct high-temperature reaction. The advantages of starting with solutions is that the reactants are mixed at the atomic level, so overcoming the problems associated with the direct reaction of two or more solid phases consisting of micrometre-sized particles. In the simplest reaction of this type, a solution of metal ions (for example, solutions of metal nitrates) is converted to a solid through a variety of methods such as evaporation of the solvent, precipitation as a simple mixed metal salt, or formation of a gel. This solid is then heated to produce the target material. Two examples are ( ) ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 2La(OH)3⋅Cu(OH)2 (s) 2La3+ (aq) + Cu2+ (aq) ⎯ 600ºC ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ La2CuO4 (s) + 4 H 2O(g) ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → OH – aq 2+ 2– n Z aq ( ) 2+eF aq 2+ ( 3C ) +O aq ( eF n Z) C→ ( 2O4 )3 (s) 2 4 2 700ºC ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → ZnFe2O4 (s) + 4CO(g) + 2CO2 (g) As well as the advantages of reduced reaction times, a result of the intimate mixing of the reactants, the final decomposition temperature is somewhat lower than that needed for the direct reaction of the oxides. The use of a lower temperature can also have the effect of reducing the size of the particles formed in the reaction. Further discussion of routes involving gel formation, or so-called ‘sol–gel processes’, are included in Sections 24.8 and 25.4. Although high-temperature direct-combination methods and solvothermal techniques are the most commonly used methods in materials chemistry, some reactions involving solids can occur at low temperatures if there is no major change in structure. These so-called ‘intercalation reactions’ are discussed in Section 24.10. 24.2 Chemical deposition Key point: Thermal decomposition of volatile inorganic compounds can be used to deposit films of materials, particularly those with uses in electronic devices, on substrates. The technological applications of the materials described in this chapter often require the inorganic material to be generated as a thin layer or film, for example on a silicon substrate, and it has become necessary to develop techniques to deposit films of many inorganic materials. The principal method is chemical vapour deposition (CVD), in which a volatile inorganic compound is decomposed above the substrate. When the compound is a metallo-organic complex (that is, a complex of a metal atom with organic ligands), this route is known as metallo-organic chemical vapour deposition (MOCVD). A large area of research has grown up around the design and synthesis of volatile inorganic compounds that can be used for CVD. For many electronic compounds, simple metal alkyls provide a route to depositing the metal or, through reaction with other gas molecules, its compounds. Thus Me2Zn, which has a vapour pressure of 0.3 bar at room temperature, can react with H2S above a substrate to generate the Group 12/16 (II/VI) semiconductor ZnS (and methane). For others, such as In, the metal alkyl is a solid with a low vapour pressure at room temperature (for Me3In, less than 2 mbar) and it cannot easily be used. In such cases other volatile compounds must be found. This search focuses on other organometallic compounds such as carbonyls, cyclopentadienyl complexes, acetoacetonates, and thiocarbamates, although it should be noted than many of these volatile metal compounds are highly toxic. For complex oxides, such as the superconducting cuprate YBa2Cu3O7, suitable precursors are required for each Defects and ion transport of the metals; the challenge is to find volatile molecules of the electropositive elements Ba and Y, which normally form ionic compounds. Metal -diketonates have been found to be useful in this respect, as compounds such as the 2,2,6,6-tetramethyl-3,5-heptanedione complex of yttrium sublime at about 150ºC. Another approach to molecules that can be used for CVD involves incorporating into a single-molecule precursor more than one of the atom types to be deposited. This procedure has the potential advantage of improving the control of the product stoichiometry. Thus, zinc sulfide can be deposited from a variety of zinc thiocomplexes, such as Zn(S2PMe2)2. A future goal is to make complex volatile molecules containing, for example, several different metal atoms that can be deposited simultaneously to make a complex oxide. Other methods of deposition of thin films, at the nanometre scale, of inorganic materials on substrates include sputtering and laser ablation; these techniques are dealt with in greater detail in Section 25.5. E X A MPL E 2 4 .1 Synthesizing complex oxides How would you synthesize a sample of the high-temperature superconductor NdBa2Cu3O7x? Answer We need to think of an analogous compound and adapt its preparation to this compound. The same method as used for preparing YBa2Cu3O7 can be used but with the appropriate lanthanoid oxide. That is, use the reaction of neodymium oxide, barium carbonate, and copper(II) oxide at 940°C followed by annealing under pure oxygen at 450°C. Self-test 24.1 How would you prepare a sample of Sr2MoO4? Defects and ion transport As discussed in Section 3.16, all solids above T 0 contain defects, imperfections of structure or composition. It is also possible to introduce defects (extrinsic defects) deliberately into a material through mechanisms such as doping. These defects, which are mainly interstitials (Frenkel-type) or vacancies (Schottky-type), are important because they influence properties such as electrical conductivity and chemical reactivity. Electrical conduction can arise from the motion of ions through the solid, and this motion is often enhanced by the presence of defects. Materials with high ionic conductivity have important applications in sensors and fuel cells of various kinds. 24.3 Extended defects Key point: Wadsley defects are shear planes that collect defects along certain crystallographic directions. The defects discussed in Chapter 3 were point defects. Such defects entail a significant local distortion of the structure and in some instances localized charge imbalances too. Therefore, it should not be surprising that defects may cluster together and sometimes form lines and planes. Tungsten oxides illustrate the formation of planes of defects. As illustrated in Fig. 24.1, the idealized structure of WO3 (which is usually referred to as the ‘ReO3 structure’; see below) consists of WO6 octahedra sharing all vertices. To picture the formation of the defect plane, we imagine the removal of shared O atoms along a diagonal. Then adjacent slabs slip past each other in a motion that results in the completion of the vacant coordination sites around each W atom. This shearing motion creates edge-shared octahedra along a diagonal. The resulting structure was named a crystallographic shear plane by A.D. Wadsley, who first devised this way of describing extended planar defects. Crystallographic shear planes randomly distributed in the solid are called Wadsley defects. Such defects lead to a continuous range of compositions, as in tungsten oxide, which ranges from WO3 to WO2.93 (made by heating and reducing WO3 with tungsten metal). If, however, the crystallographic shear planes are distributed in a nonrandom, periodic manner, so giving rise to a new unit cell, then we should regard the material as a new stoichiometric phase. Thus, when even more O2 ions are removed from tungsten oxide, a series 605 606 24 Solid-state and materials chemistry a (a) b 3.8Å (a) (b) (c) Figure 24.1 The concept of a crystallographic shear plane illustrated by the (100) plane of the ReO3 structure. (a) A plane of metal, Re, and oxygen, O, atoms. The octahedron around each metal atom is completed by a plane of oxygen atoms above and below the plane illustrated here. Some of the octahedra are shaded to clarify the processes that follow. (b) Oxygen atoms in the plane perpendicular to the page are removed, leaving two planes of metal atoms that lack their sixth oxygen ligand. (c) The octahedral coordination of the two planes of metal atoms is restored by translating the right slab as shown. This creates a plane (labelled a shear plane) vertical to the paper in which the MO6 octahedra share edges. (b) Figure 24.2 (a) High resolution electron micrograph lattice image of a crystallographic shear plane in WO3x (b) The oxygen octahedral polyhedra that surround the W atoms imaged in the electron micrograph. Note the edge-shared octahedra along the crystallographic shear plane. [Reproduced by permission from S. Iijima, J. Solid State Chem. 1975, 14, 52]. a b d c Figure 24.3 Some diffusion mechanisms for ions or atoms in a solid: (a) two atoms or ions exchange positions, (b) an ion hops from a normally occupied site in the structure to an interstitial site, which produces a vacancy which can then be filled by movement of an ion from another site, (c) an ion hops between two different interstitial sites and (d) an ion or atom moves from a normally occupied site to a vacancy, so producing a new vacant site. of discrete phases having ordered crystallographic shear planes and compositions WnO3n2 (n 20, 24, 25, and 40) are observed. Compounds with closely spaced compositions that contain shear planes are known for oxides of W, Mo, Ti, and V and some of their complex oxides, for example the tungsten bronzes M8W9O47, M Nb, Ta and the ‘Magnéli phases’, VnO2n1 (n 39). Electron microscopy (Section 25.3) provides an excellent method of observing these defects experimentally because it reveals both ordered and random arrays of shear planes (Fig. 24.2). 24.4 Atom and ion diffusion Key point: The diffusion of ions in solids is strongly dependent on the presence of defects. One reason why diffusion in solids is much less familiar than diffusion in gases and liquids is that at room temperature it is generally very much slower. This slowness is why most solid-state reactions are undertaken at high temperatures (Section 24.1). However, there are some striking exceptions to this generalization. Diffusion of atoms or ions in solids is in fact very important in many areas of solid-state technology, such as semiconductor manufacture, the synthesis of new solids, fuel cells, sensors, metallurgy, and heterogeneous catalysis. The rates at which ions move through a solid can often be understood in terms of the mechanism for their migration and the activation barriers the ions encounter as they move. The lowest energy pathway generally involves defect sites with the roles summarized in Fig. 24.3. Materials that show high rates of diffusion at moderate temperatures have the following characteristics: r Low-energy barriers: so temperatures at (or a little above) 300 K are sufficient to permit ions to jump from site to site. r Low charges and small radii: so, for example, the most mobile cation (other than the proton) and anion are Li and F. Reasonable mobilities are also found for Na and O2. More highly charged ions develop stronger electrostatic interactions and are less mobile. r High concentrations of intrinsic or extrinsic defects: defects typically provide a lowenergy pathway for diffusion through a structure that does not involve the energy penalties associated with continuously displacing ions from normal, favourable ion sites. These defects should not be ordered, as for crystallographic shear planes (Section 24.3), because such ordering removes the diffusion pathway. r Mobile ions are present as a significant proportion of the total number of ions. Figure 24.4 shows the temperature dependence of the diffusion coefficients, which are a measure of mobility, for specified ions in a selection of solids at high temperatures. The slopes of the lines are proportional to the activation energy for migration. Thus Na is 607 Defects and ion transport Any electrochemical cell, such as a battery, fuel cell, electrochromic display, or electrochemical sensor, requires an electrolyte. In many applications, an ionic solution (for example, dilute sulfuric acid in a lead–acid battery) is an acceptable electrolyte, but because it is often desirable to avoid a liquid phase due to possible spillage, there is considerable interest in the development of solid electrolytes. Two important and thoroughly studied solid electrolytes with mobile cations are silver tetraiodomercurate(II), Ag2HgI4, and sodium -alumina with the composition Na1xAl11O17x2. Other recently developed fast-cationic conductors include NASICON (a name formed from the letters in sodium, Na, Superionic Conductor) of composition Na1xZr2P3xSixO12 and a number of proton conductors that operate at or a little above room temperature, such as CsHSO4 above 160°C. Solids exhibiting high anion mobility are rarer than cationic conductors and generally show high conductivity only at elevated temperatures: anions are typically larger than cations and so the energy barrier for diffusion through the solid is high. As a consequence, fast anion conduction in solids is limited to F and O2 (with ionic radii 133 pm and 140 pm, respectively). Despite these limitations, anionic conductors play an important role in sensors and fuel cells, where a typical material is ‘yttrium-stabilized zirconia’ (YSZ), of composition YxZr1xO2x2). Table 24.1 summarizes some typical ionic conductivity values of solid electrolytes and other ionically conducting media. (a) Solid cationic electrolytes Key points: Solid inorganic electrolytes often have a low-temperature form in which the ions are ordered on a subset of sites in the structure; at higher temperatures the ions become disordered over the sites and the ionic conductivity increases. Table 24.1 Comparative values of ionic and electronic conductivity Conductivity/(S m1)† Ionic conductors Ionic crystals Example: LiI at 298°C Solid electrolytes Example: YSZ at 600°C 1016102 104 101103 1 AgI at 500°C 102 Strong (liquid) electrolytes 101103 Example: 1 M NaCl(aq) 102 Electronic conductors † Na in β-Al2O3 O in CaxZrO2–x –11 Co in CoO –13 Cr in Cr2O3 –15 Ni in NiO –17 Al in Al2O3 24.5 Solid electrolytes Material –9 log{D/(m2 s–1)} highly mobile and has a low activation energy for motion through -alumina, whereas Ca2 in CaO is much less mobile and has a high activation energy for hopping through the rock-salt structure. As with the mechanisms of most chemical reactions, the evidence for the mechanism of a particular diffusion process is circumstantial. The individual events are never directly observed but are inferred from the influence of experimental conditions on diffusion rates. For example, a detailed analysis of the thermal motion and distributions of ions in crystals based on X-ray and neutron diffraction provides strong hints about the ability of ions to move through the crystal, including their most likely paths. Computer modelling, which is often an elaboration of the ionic model (Chapter 3), also gives very useful guidance to the feasibility of these migration mechanisms. Metals 103106 Semiconductors 103102 Insulators 107 The symbol S denotes siemens; 1 S 1 Ω1, where 1 Ω (ohm) 1 V A1 Ca in CaO –19 0.4 0.6 0.8 (1000 K)/T 1.0 Figure 24.4 The diffusion coefficients (on a logarithmic scale) as a function of inverse temperature for the mobile ion in selected solids. 608 24 Solid-state and materials chemistry I Hg Ag (a) I (b) Ag, Hg, or vacancy Figure 24.5 (a) Low-temperature ordered structure of Ag2HgI4. (b) High-temperature disordered structure showing the cation disorder. Ag2HgI4 is an Ag ion conductor in the high temperature form. Below 50°C, Ag2HgI4 has an ordered crystal structure in which Ag and Hg ions are tetrahedrally coordinated by I ions and there are unoccupied tetrahedral holes (Fig. 24.5a). At this temperature its ionic conductivity is low. Above 50°C, however, the Ag and Hg2 ions are randomly distributed over the tetrahedral sites (Fig. 24.5b) and as a result there are many more sites that Ag ions can occupy within the structure than there are Ag ions present. At this temperature the material is a good ionic conductor, largely on account of the mobility of the Ag ions between the different sites available for it. The close-packed array of polarizable I ions is easily deformed and results in a low activation energy for the migration of an Ag ion from one ion site to the next. There are many related solid electrolytes having similar structures containing soft anions, such as AgI and RbAg4I5, both of which have highly mobile Ag ions so that the conductivity of RbAg4I5 at room temperature is greater than that of aqueous sodium chloride. Sodium -alumina is an example of a mechanically hard material that is a good ionic conductor. In this case, the rigid and dense Al2O3 slabs are bridged by a sparse array of O2 ions (Fig. 24.6). The plane containing these bridging ions also contains Na ions, which can move from site to site because there are no major bottlenecks to hinder their motion. Many similar rigid materials having planes or channels through which ions can move are known; they are called framework electrolytes. Another closely related material, sodium -alumina, has even less restricted motion of ions than -alumina, and it has been found possible to substitute doubly charged cations such as Mg2 or Ni2 for Na. Even the large lanthanoid cation Eu2 can be introduced into -alumina, although the diffusion of such ions is slower than that of their smaller counterparts. The material NASICON, mentioned earlier, is a nonstoichiometric, solid-solution system with a framework constructed from ZrO6 octahedra and PO4 tetrahedra, corresponding to the parent phase of composition NaZr2P3O12 (Fig. 24.7). A solid solution can be obtained by partially replacing P by Si to give Na1xZr2P3xSixO12 with an increase in the number of Na ions for charge balance. In this material, the full set of possible Na sites is only partially filled and these sites lie within a three-dimensional network of channels that allow rapid migration of the remaining Na ions. Other classes of materials currently being investigated as fast cation conductors include Li4GeO4 doped with V on the Ge sites (Li4x(Ge1xVx)O4, a lithium-ion conductor with vacancies on the Li ion sub-lattice), the perovskite La0.6Li0.2TiO3, and sodium yttrium silicate, Na5YSi4O12 (a sodium-ion conductor). E X A M PL E 24 . 2 Correlating conductivity and ion size in a framework electrolyte Conductivity data on -alumina containing monopositive ions of various radii show that Ag and Na ions, both of which have radii close to 100 pm, have activation energies for conductivity close to 17 kJ mol1 whereas that for Tl (radius 149 pm) is about 35 kJ mol1. Suggest an explanation of the difference. Answer One way of approaching this problem is to think of the constrictions to migration that are related to the sizes of the ions. In sodium -alumina and related -aluminas, a fairly rigid framework provides a two-dimensional network of passages that permit ion migration. Judging from the experimental results, the bottlenecks for ion motion appear to be large enough to allow Na or Ag (ionic radii close to 100 pm) to pass quite readily (with a low activation energy) but too small to let the larger Tl (ionic radius 149 pm) pass through as readily. Self-test 24.2 Why does increased pressure reduce the conductivity of K in -alumina more than that of Na in -alumina? Because the reactivity of a bulk material is related to the presence of crystal defects and to the processes of atom and ion diffusion, by modelling an ion diffusion process information can be obtained on both ion conduction and the reaction mechanisms of a solid. Such a study has been undertaken on Li3N with the aim of determining the energy barriers for Li ions moving through the structure. Comparison of the values for the various barriers between all the different possible sites that Li could occupy as it migrates through the solid in turn allows proposals to be made for the conduction pathway and a value for ionic conductivity to be calculated. Diffusion of Li ions in the Li2N plane (Fig. 24.8) was found to have a much lower energy barrier than for motion perpendicular to the plane. The effect of replacing the Li ions between the layers, as in Li2MN (where M is a 3d-series metal such as Ni) and thereby reducing the energy barrier can also be studied; this kind of investigation is important for understanding potential Defects and ion transport 609 Conduction channel Al2O3 spinel block Na+ 1.120 nm Conduction plane ZrO6 Na+ Bridging O2– ion (a) (P,Si)O4 O2– Figure 24.7 The Na1xZr2P3xSixO12 (NASICON) structure shown as linked (P,Si)O4 tetrahedra and ZrO6. Na+ N3– Li+ Li+ (b) Figure 24.6 (a) Schematic side view of -alumina showing the Na2O conduction planes between Al2O3 slabs. The O atoms in these planes bridge the two slabs. (b) A view of the conduction plane. Note the abundance of mobile ions and vacancies in which they can move. Figure 24.8 The Li3N structure showing possible pathways for Li ion diffusion between the Li2N planes. applications of these materials as the electrolyte in rechargeable lithium-ion batteries (Section 24.7h). (b) Solid anionic electrolytes Key point: Anion mobility can occur at high temperatures in certain structures that contain high levels of anion vacancies. Michael Faraday reported in 1834 that red-hot solid PbF2 is a good conductor of electricity. Much later it was recognized that the conductivity arises from the mobility of F ions through the solid. The property of anion conductivity is shared by other crystals having the fluorite structure. Ion transport in these solids is thought to be by an interstitial mechanism in which an F ion first migrates from its normal position into an interstitial site (a Frenkel-type defect, Section 3.16) and then moves to a vacant F site. Structures that have large numbers of vacant sites generally show the highest ionic conductivities because they provide a path for ion motion (although at very high levels of defects, clustering of the defects or the vacancies can lower the conductivity). These vacancies, which are equivalent to extrinsic defects, can be introduced in fairly high numbers into many simple oxides and fluorides by doping with appropriately chosen metal ions in different oxidation states. Zirconia, ZrO2, at high temperature has a fluorite structure, but on cooling the pure material to room temperature it distorts to a monoclinic polymorph. The cubic fluorite structure may be stabilized at room temperature by replacing some Zr4 with other ions, such as the similarly sized Ca2 and Y3 ions. Doping with these ions of lower oxidation number results in the introduction of vacancies on the anion sites to preserve the charge neutrality of the material and produces, for example, YxZr1xO2x/2, the material mentioned previously as ‘yttrium stabilized zirconia’ (YSZ). This material has completely occupied cation sites in the fluorite structure but high levels of anion vacancies, with 610 24 Solid-state and materials chemistry Potential Porous difference Pt electrodes O2 in test gas Solid ZrO2/CaO electrolyte Air (constant O2 standard) Figure 24.9 An oxygen sensor based on the solid electrolyte Zr1xCaxO2x. 0 x 0.15. These vacant sites provide a path for oxide-ion diffusion through the structure so that a typical electrical conductivity, in, for example, Ca0.15Zr0.85O1.85 is 5 S cm1 at 1000°C;1 note that this conductivity is much lower than typical solid-state cation conductivities—even at these very high temperatures—due to the large anion size. The high oxide-ion conductivity of calcium-oxide doped zirconia is exploited in a solidstate electrochemical sensor for measuring the partial pressure of oxygen in automobile exhaust systems (Fig. 24.9).2 The platinum electrodes in this cell adsorb O atoms and, if the partial pressures of oxygen are different between the sample and reference side, there is a thermodynamic tendency for oxygen to migrate through the electrolyte as the O2 ion. The thermodynamically favoured processes are: High p(O2) side: 1 2 O2 (g) + Pt(s) → O (Pt, surface) O (Pt, surface) + 2 e – → O2– (ZrO2 ) Low p(O2) side: O (Pt, surface) + 2 e – O2– (ZrO2 ) → O (Pt, surface) → 12 O2 (g) + Pt(s) The cell potential is related to the two oxygen partial pressures (p1 and p2) by the Nernst equation (Section 5.5), for the half-cell reaction O2 4 e → 2 O2, which occurs at both electrodes Ecell = RT p1 ln p2 4F so a simple measurement of the potential difference provides a measure of the oxygen partial pressure in the exhaust gases. ■ A brief illustration. According to this equation, the potential difference produced by an oxygen sensor operating at 1000 K in an exhaust system, with air on one side (p(O2) 0.2 atm) and a burnt fuel/air mixture (p(O2) 0.001 atm) on the other, is about 0.1 V. ■ Behaviour similar to that of YSZ is encountered for other compounds that adopt the fluorite structure, such as PbF2 as discovered by Faraday. As noted previously, the anionic conductivities remain low even at high temperatures, so many other complex metal oxides are currently being investigated with the aim of achieving high mobilities at low temperatures. Some compounds that show promising behaviour include La2Mo2O9, barium indate (Ba2In2O5), BIMEVOX (a d-metal doped bismuth vanadium oxide), the apatite structure of La9.33Si6O26, and strontium- and magnesium-doped lanthanum gallate (Sr,Mg-doped LaGaO3, or LSGM). As well as uses in sensor devices, materials possessing oxide- and proton-ion conductivity are important in a number of fuel cell types (Box 24.1). (c) Mixed ionic-electronic conductors Key point: Solid materials can exhibit both ionic and electronic conductivity. Most ionic conductors, such as sodium -alumina and YSZ, have low electronic conductivity (that is, conduction by electron rather than ion motion). Their application as solid electrolytes, in sensors for instance, requires this feature to avoid shorting-out the cell. In some cases a combination of electronic and ionic conductivity is desirable, and this type of behaviour can be found in some d-metal compounds where defects allow O2 conduction and the metal d orbitals provide an electronic conduction band. Many such materials are perovskite-based structures with mixed oxidation states at the B cation sites (Section 3.9). Two examples are La1xSrxCoO3y and La1xSrxFeO3y. These oxide systems are good electronic conductors with partially filled bands as a result of the nonintegral d-metal oxidation number and can The symbol S denotes siemens; 1 S 1 1, where 1 (ohm) 1 V A1. The signal from this sensor is used to adjust the air/fuel ratio and thereby the composition of the exhaust gas being fed to the catalytic converter. 1 2 Metal oxides, nitrides, and fluorides 611 B OX 2 4 .1 Solid oxide fuel cells A fuel cell consists of an electrolyte sandwiched between two electrodes; oxygen passes over one electrode and the fuel over the other, generating electricity, water, and heat. The general operation and construction of a fuel cell that converts a fuel, such as hydrogen, methane, or methanol, by reaction with oxygen into electrical energy (and combustion products H2O and CO2) was described in Box 10.1. A variety of materials can be used as the electrolyte in such cells, including phosphoric acid, proton-exchange membranes, and, in solid oxide fuel cells (SOFCs), oxide-ion conductors (Fig. B24.1). SOFCs operate at high temperatures and use an oxide-ion conductor as the electrolyte. The design of a typical SOFC is shown in the illustration. Each cell generates a limited potential difference but, as with the cells of a battery, a connected stack may be constructed in series to increase the potential difference and power supplied. Cells are connected electrically through an ‘interconnect’ that can also be used to isolate the fuel and air supplies for each cell. The attraction of SOFCs is based on several aspects, including the clean conversion of fuel to electricity, low levels of noise pollution, the ability to cope with different fuels, but most significantly a high efficiency. Their high Oxygen or air O2 + 4 e– 2 O2– (La,Sr)FeO3–x cathode YSZ electrolyte Power Ni–YSZ anode Hydrogen gas 2 H2 + 2 O2– e– e– 2 H2O + 4 e– Figure B24.1 The structure of a solid oxide fuel cell. efficiency is a result of their high operating temperatures which are typically between 500 and 1000°C. In high-temperature SOFCs the interconnect may be a ceramic such as lanthanum chromite (the perovskite LaCrO3) or, if the temperature is below 1000°C, an alloy such as Y/Cr may be used. The oxide-ion conductor used as the electrolyte in these very high-temperature SOFCs is normally yttrium-stabilized zirconia (YSZ). Intermediate-temperature SOFCs, which typically operate between 500 and 700°C, have a number of advantages over very high temperature devices in that there is reduced corrosion, a simpler design, and the time to heat the system to the operating temperature is much reduced. However, such devices require a material with excellent oxide-ion conductivities at lower temperatures to act as the electrolyte. The best developed intermediate-temperature (less than about 600°C) SOFC consists of an anode of Gd-doped CeO2 (CGO)/Ni, an electrolyte of Gd-doped cerium oxide, and a cathode of the perovskite LSCF (La,Sr)(Fe,Co)O3. The electrolyte material CGO possesses a much higher ionic conductivity than YSZ at these lower temperatures. Unfortunately, however, its electronic conductivity is also higher and the use of a CGO electrolyte can result in reduced efficiency as energy is wasted due to electrons flowing through the electrolyte. For this reason new and better oxide-ion conductors are being sought, such as those mentioned in the text. One of the main advantages of SOFCs over other fuel cell types is their ability to handle more convenient hydrocarbon fuels: other types of fuel cells have to rely on a clean supply of hydrogen for their operation. Because SOFCs operate at high temperature there is the opportunity to convert hydrocarbons catalytically to hydrogen and carbon oxides within the system (Section 26.15). Because of their size and the requirement to be heated to, and operate at, high temperatures, the applications of SOFCs focus on medium- to large-scale static systems, including small residential systems producing about 2 kW. conduct by O2 migration through the perovskite O2 ion sites. This type of material is of use in solid oxide fuel cells (SOFC, Box 24.1), one type of fuel cell mentioned in Box 5.1, in which one electrode has to allow diffusion of ions through a conducting electrode. Metal oxides, nitrides, and fluorides In this section, we explore the binary compounds of N, O, and F with metals. These compounds, particularly oxides, are central to much solid-state chemistry on account of their stability, ease of synthesis, and variety in composition and structure. These attributes lead to the vast number of compounds that have been synthesized and the ability to tune the properties of a compound for a specific application based on their electronic or magnetic characteristics. As we shall see, a discussion of the chemical properties of these compounds also provides insight into defects, nonstoichiometry, and ion diffusion, and into the influence of these characteristics on physical properties. The chemical properties of metal fluorides parallel much of that of metal oxides, but the lower charge of the F ion means that equivalent stoichiometries are produced with cations having lower charges, as in KMn(II)F3 compared with SrMn(IV)O3. Compounds containing the nitride ion, N3, in combination with one or more metal ion have only recently been developed to a significant extent; the area of metal nitride chemistry is one where rapid advances are being made in terms of new materials and structure types. 24.6 Monoxides of the 3d metals The monoxides of most of the 3d metals adopt the rock-salt structure, so it might at first appear that there is little to say about them and that their properties should be simple. 612 24 Solid-state and materials chemistry Table 24.2 Monoxides of the 3d-series metals Compound Structure Composition, x Electrical character CaOx Rock-salt 1 Insulator TiOx Rock-salt 0.651.25 Metallic VOx Rock-salt 0.791.29 Metallic MnOx Rock-salt 11.15 Semiconductor FeOx Rock-salt 1.041.17 Semiconductor CoOx Rock-salt 11.01 Semiconductor NiOx Rock-salt 11.001 Insulator CuOx PtS (linked CuO4 square-planes) 1 Semiconductor ZnOx Wurtzite Slight Zn excess Wide band gap n-type semiconductor In fact, the actual stoichiometries formed around the ideal composition MO and the structures and properties of these oxides are more interesting, and as a result they have been repeatedly investigated (Table 24.2). In particular, the compounds provide examples of how mixed oxidation states and defects lead to nonstoichiometry for solids that are nominally TiO, VO, FeO, CoO, and NiO. All these compounds are usually obtained with significant deviations from the nominal MO stoichiometry. (a) Defects and nonstoichiometry Key point: The nonstoichiometry of Fe1–xO arises from the creation of vacancies on the Fe2 octahedral sites, with each vacancy charge-compensated by the conversion of two Fe2 ions to two Fe3 ions. Fe O Defect site The origin of nonstoichiometry in FeO has been studied in more detail than that in most other MO compounds. In fact, it is found that stoichiometric FeO does not exist but rather a range of iron-deficient compounds in the range Fe1xO, 0.13 x 0.04, can be obtained by quenching (cooling very rapidly) iron(II) oxide from high temperatures. The compound Fe1xO is in fact metastable at room temperature: it is thermodynamically unstable with respect to disproportionation into iron metal and Fe3O4 but does not convert for kinetic reasons. The general consensus is that the structure of Fe1xO is derived from the rock-salt FeO structure by the presence of vacancies on the Fe2 octahedral sites, and that each vacancy is charge-compensated by the conversion of two adjacent Fe2 ions to two Fe3 ions. The relative ease of oxidizing Fe(II) to Fe(III) accounts for the fairly broad range of compositions of Fe1xO. At high temperatures, the interstitial Fe3 ions associate with the Fe2 vacancies (or defects) to form clusters distributed throughout the structure (Fig. 24.10). Similar defects and the clustering of defects appear to occur with all other 3d-metal monoxides, with the possible exception of NiO. The range of nonstoichiometry in Ni1xO is extremely narrow, but conductivity and the rate of atom diffusion vary with oxygen partial pressure in a manner that suggests the presence of isolated point defects. Both CoO and NiO occur in a metal-deficient state, although their range of compositions is again not as broad as that of FeO. As indicated by standard potentials in aqueous solution, Fe(II) is more easily oxidized than either Co(II) or Ni(II); this solution redox chemistry correlates well with the much smaller range of oxygen deficiency in NiO and CoO. Chromium(II) oxide, like FeO, spontaneously disproportionates: Cr( III ) O3 (s) + Cr(0) (s) 3 Cr( II)O(s) → 2 Figure 24.10 Defect sites proposed for Fe1xO. Note that the tetrahedral Fe3 interstitials (grey spheres) and octahedral Fe2 vacancies (circles) are clustered together. However, the material can be stabilized by crystallization in a copper(II) oxide matrix. Both CrO and TiO have structures that show high levels of defects on both the cation and anion sites forming metal-rich or metal-deficient stoichiometries (Ti1xO and TiO1x). In fact, TiO has large numbers of vacancies, in equal amounts, on both cation and anion sublattices, rather than the expected perfect, defect-free structure. Note that significant deviations from the stoichiometry MO are out of the question for Group 2 metal oxides, such as CaO, because for these elements M3 ions are chemically inaccessible. Metal oxides, nitrides, and fluorides 613 (b) Electronic properties Key points: The 3d-metal monoxides MnO, FeO, CoO, and NiO are semiconductors; TiO and VO are metallic conductors. The 3d-metal monoxides MnO, Fe1xO, CoO, and NiO have low electrical conductivities that increase with temperature (corresponding to semiconducting behaviour) or have such large band gaps that they are insulators. The electron or hole migration in these oxide semiconductors is attributed to a hopping mechanism. In this model, the electron or hole hops from one localized metal atom site to the next. When it lands on a new site it causes the surrounding ions to adjust their locations and the electron or hole is trapped temporarily in the potential well produced by this distortion. The electron resides at its new site until it is thermally activated to migrate into another nearby site. Another aspect of this chargehopping mechanism is that the electron or hole tends to associate with local defects, so the activation energy for charge transport may also include the energy of freeing the hole from its position next to a defect. Hopping contrasts with the band model for semiconductivity discussed in Section 3.20, where the conduction and valence electrons occupy orbitals that spread through the whole crystal. The difference stems from the less diffuse d orbitals in the monoxides of the mid-to-late 3d metals, which are too compact to form the broad bands necessary for metallic conduction. When NiO is doped with Li2O in an O2 atmosphere a solid solution Lix(Ni2)12x(Ni3)xO is obtained, which has greatly increased conductivity for reasons similar to the increase in conductivity of Si when doped with B. The characteristic pronounced increase in electronic conductivity with increasing temperature of metal oxide semiconductors is used in ‘thermistors’ to measure temperature. In contrast to the semiconductivity of the monoxides in the centre and right of the 3d series, TiO and VO have high electronic conductivities that decrease with increasing temperature. This metallic conductivity persists over a broad composition range from highly oxygen-rich Ti1xO to metal-rich TiO1x. In these compounds, a conduction band is formed by the overlap of the t2g orbitals of metal ions in neighbouring octahedral sites that are oriented towards each other (Fig. 24.11). The radial extension of the d orbitals of these early d-block elements is greater than for elements later in the period, and a band results from their overlap (Fig. 24.12); this band is only partly filled. The widely varying compositions of these monoxides also appear to be associated with the electronic delocalization: the conduction band serves as a rapidly accessible source and sink of electrons that can readily compensate for the formation of vacancies. O Ti Figure 24.11 Overlap of the dzx orbitals in TiO to give a t2g band. In the perpendicular directions the dyx and dzy orbitals overlap in an identical manner. 2+ O2– M p band p eg d t2g band p p band (c) Magnetic properties Key point: The 3d-metal monoxides MnO, FeO, CoO, and NiO order antiferromagnetically with Néel temperatures that increase from Mn to Ni. As well as having electronic properties that are the result of interactions between the d electrons, d-metal monoxides have magnetic properties that derive from cooperative interaction of the individual atomic magnetic moments (Section 20.8). The overall magnetic structure of MnO and the other 3d-series metal monoxides is shown in Fig. 24.13. The Néel temperatures (TN, the temperature of the paramagnetic/antiferromagnetic transition, Section 20.8) of the series of d-metal oxides are as follows: MnO FeO 122 K 198 K CoO NiO 271 K 523 K Figure 24.12 Molecular orbital energy level diagram for early d-metal monoxides. The t2g band is only partly filled and metallic conduction results. Mn O These values reflect the strength of the superexchange spin interactions along the MOM directions, which in the rock-salt type structure propagate in all three unit cell directions. As the size of the M2 ion decreases from Mn to Ni, the superexchange mechanism becomes stronger due to the increased metal–oxygen orbital overlap and TN increases. 24.7 Higher oxides and complex oxides Binary metal oxides that do not have a 1:1 metal:oxygen ratio are known as higher oxides. Compounds containing ions of more than one metal are often termed complex oxides or mixed oxides and include compounds containing three elements, ternary oxides (for instance, Figure 24.13 The arrangement of electron spins on the Mn2 ions in the antiferromagnetic state of MnO. 24 Solid-state and materials chemistry LaFeO3), and four, quaternary (for instance, YBa2Cu3O7), or more elements. This section describes the structures and properties of some of the more important complex, mixed metal oxides. (a) The M2O3 corundum structure Key point: The corundum structure is adopted by many oxides of the stoichiometry M2O3, including Cr-doped aluminium oxide (ruby). -Aluminium oxide (the mineral corundum) adopts a structure that can be modelled as a hexagonal close-packed array of O2 ions with the cations in two-thirds of the octahedral holes (Fig. 24.14). The corundum structure is also adopted by the oxides of Ti, V, Cr, Rh, Fe, and Ga in their 3 oxidation states. Two of these oxides, Ti2O3 and V2O3, exhibit metallic-to-semiconducting transitions below 410 and 150 K, respectively (Fig. 24.15). In V2O3, the transition is accompanied by antiferromagnetic ordering of the spins. The two insulators Cr2O3 and Fe2O3 also display antiferromagnetic ordering. Another interesting aspect of the M2O3 compounds is the formation of solid solutions of dark-green Cr2O3 and colourless Al2O3 to form brilliant red ruby. As remarked in Section 20.7, this shift in the ligand-field transitions of Cr3 stems from the compression of the O2 ions around Cr3 in the Al2O3 host structure. (In Al2O3, a 475 pm and c 1300 pm; in Cr2O3 the lattice constants are 493 pm and 1356 pm, respectively.) The compression shifts the absorption towards the blue as the strength of the ligand field increases, and the solid appears red in white light. The responsiveness of the absorption (and fluorescence) spectrum of Cr3 ions to compression is sometimes used to measure pressure in high-pressure experiments. In this application, a tiny crystal of ruby in one segment of the sample can be interrogated by visible light and the shift in its fluorescence spectrum provides an indication of the pressure inside the cell. (b) Rhenium trioxide Key point: The rhenium trioxide structure can be constructed from ReO6 octahedra sharing all vertices in three dimensions. The rhenium trioxide structure type is very simple, consisting of a cubic unit cell with Re atoms at the corners and O atoms at the mid-point of each edge (Fig. 24.16). Alternatively, the structure can be considered to be derived from ReO6 octahedra sharing all vertices. The structure is also closely related to a perovskite structure (Section 3.9) in which the A-type cation has been removed from the unit cell. Materials adopting the rhenium trioxide structure are relatively rare. This rarity is in part due to the requirement of the oxidation state M(VI) when M is in combination with oxygen. Rhenium(VI) oxide, ReO3, itself and one form of UO3 (-UO3) have this structure type and WO3 exists in a slightly distorted version of it. In WO3, the WO6 octahedra are slightly distorted and tilted relative to each other so that the WOW bond angle is not 180°. O –6 Al log {Conductivity/(S m–1)} 614 –2 0 2 Metallic conductor 4 6 8 Figure 24.14 The corundum structure, as adopted by Al2O3, with cations occupying two-thirds of the octahedral holes between layers of closepacked oxide ions. Semiconductor –4 0 2 4 6 8 (1000 K)/T 10 Figure 24.15 The temperature dependence of the electrical conductivity of V2O3 showing the metal-to-semiconductor transition. Metal oxides, nitrides, and fluorides Rhenium trioxide itself is a bright red lustrous solid. Its electrical conductivity at room temperature is similar to that of copper metal. The band structure for this compound contains a band derived from the Re t2g orbitals and the O2p orbitals (Fig. 24.17). This band can contain up to six electrons per Re atom but is only partially filled for the Re6 d1 configuration, so producing the observed metallic properties. 615 Re O (c) Spinels Key point: The observation that many d-metal spinels do not have the normal spinel structure is related to the effect of ligand-field stabilization energies on the site preferences of the ions. The d-block higher oxides Fe3O4, Co3O4, and Mn3O4, and many related mixed-metal compounds, such as ZnFe2O4, have very useful magnetic properties. They all adopt the structural type of the mineral spinel, MgAl2O4, and have the general formula AB2O4. Most oxide spinels are formed with a combination of A2 and B3 cations (that is, as A2B23O4 in Mg2[Al3]2O4), although there are a number of spinels that can be formulated with A4 and B2 cations (as A4B22O4 as in Ge4[Co2]2O4. The spinel structure was described briefly in Section 3.9, where we saw that it consists of an fcc array of O2 ions in which the A ions reside in one-eighth of the tetrahedral holes and the B ions inhabit half the octahedral holes (Fig. 24.18); this structure is commonly denoted A[B2]O4, where the atom type in the square bracket represents that occupying the octahedral sites. In the inverse spinel structure, the cation distribution is B[AB]O4, with the more abundant B-type cation distributed over both coordination geometries. Lattice enthalpy calculations based on a simple ionic model indicate that, for A2 and B3, the normal spinel structure, A[B2]O4, should be the more stable. The observation that many d-metal spinels do not conform to this expectation has been traced to the effect of ligand-field stabilization energies on the site preferences of the ions. The occupation factor, , of a spinel is the fraction of B atoms in the tetrahedral sites: 0 for a normal spinel and 12 for an inverse spinel, B[AB]O4; intermediate values indicate a level of disorder in the distribution, where B-type cations occupy that portion of the tetrahedral sites. The distribution of cations in (A2,B3) spinels (Table 24.3) illustrates that for d0 A and B ions the normal structure is preferred as predicted by electrostatic considerations. Table 24.3 shows that, when A2 is a d6, d7, d8, or d9 ion and B3 is Fe3, the inverse structure is generally favoured. This preference can be traced to the lack of ligand-field stabilization (Section 20.1 and Fig. 20.13) of the high-spin d5 Fe3 ion in either the octahedral or the tetrahedral site and the ligand-field stabilization of the other dn ions in the octahedral site. For other combinations of d-metal ions on the A and B sites the relative ligand-field stabilization energies of the different arrangements of the two ions on the octahedral and tetrahedral sites need to be calculated. It is also important to note that simple ligand-field stabilization appears to work over this limited range of cations. More detailed analysis is necessary when cations of different radii are present or any ions that are present do not adopt the high-spin configuration typical of most metals in spinels (for instance, Co3 in Co3O4, which is low-spin d6). Moreover, because is often found to depend on the temperature, care has to be taken in the synthesis of a spinel with a specific Figure 24.16 The ReO3 structure shown as the unit cell and the ReO6 octahedra forming the unit cell. O Re eg Re σ π 2px, 2py t2g π 2pz Re6+ σ O2– Figure 24.17 The band structure of ReO3. AO4 BO6 Table 24.3 Occupation factor, , in some spinels A Al Cr3 Mn Fe 3 3 Co3 Mn2 Fe2 Co2 Ni2 Cu2 Zn2 d10 d0 d5 d6 d7 d8 d9 0 d 0 0 0 0 0.38 0 d3 0 0 0 0 0 0 d4 0 B 3 Mg2 5 d d6 0.45 0 0 0.1 0.5 0.5 0.5 0 0 corresponds to a normal spinel; 0.5 corresponds to an inverse spinel. 0.5 0 0 Figure 24.18 A segment of the spinel (AB2O4) unit cell showing the tetrahedral environment of A ions and the octahedral environments of B ions. (Compare with Fig. 3.44.) 616 24 Solid-state and materials chemistry distribution of cations because slow cooling or quenching of a sample from a high reaction temperature can produce quite different cation distributions. E X A M PL E 24 . 3 Predicting the structures of spinel compounds Is MnCr2O4 likely to have a normal or inverse spinel structure? Answer We need to consider whether there is a ligand-field stabilization. Because Cr3 (d3) has a large ligand-field stabilization energy (1.2∆O from Table 20.2) in the octahedral site (but a much smaller one in a tetrahedral field) whereas the high spin d5 Mn2 ion does not have any LFSE, a normal spinel structure is expected. Table 24.3 shows that this prediction is verified experimentally. Self-test 24.3 Table 24.3 indicates that FeCr2O4 is a normal spinel. Rationalize this observation. B O A ) (a A O B The inverse spinels of formula AFe2O4 are sometimes classified as ferrites (the same term also applies in different circumstances to other iron oxides). When RT J, where J is the energy of interaction of the spins on different ions, ferrites are paramagnetic. However, when RT J, a ferrite may be either ferrimagnetic or antiferromagnetic. The antiparallel alignment of spins characteristic of antiferromagnetism is illustrated by ZnFe2O4, which has the cation distribution Fe[ZnFe]O4. In this compound the Fe3 ions (with S 52 ) in the tetrahedral and octahedral sites are antiferromagnetically coupled, through a superexchange mechanism (Section 20.8), below 9.5 K to give nearly zero net magnetic moment to the solid as a whole; note that Zn2 as a d10 ion makes no contribution to the magnetic moment of the material. The compound CoAl2O4 is among the normal spinels in Table 24.3 with 0 and thus has the Co2 ions at the tetrahedral sites. The colour of CoAl2O4 (an intense blue) is that expected of tetrahedral Co2. This property, coupled with the ease of synthesis and stability of the spinel structure, has led to cobalt aluminate being used as a pigment (‘cobalt blue’). Other mixed d-metal spinels that exhibit strong colours, for example CoCr2O4 (green), CuCr2O4 (black), and (Zn,Fe)Fe2O4 (orange/brown), are also used as pigments, with applications that include colouring various construction materials, such as concrete. (d) Perovskites and related phases Key points: The perovskites have the general formula ABX3, in which the 12-coordinate hole of a ReO3type BX3 structure is occupied by a large A ion; the perovskite barium titanate, BaTiO3, exhibits ferroelectric and piezoelectric properties associated with cooperative displacements of the ions. ) (b ) (c Figure 24.19 Views of the perovskite (ABO3) structure (a) emphasizing the 12-fold coordination of the larger A cation and showing the relationship with the ReO3 structure of Fig. 23.22b, (b) highlighting the octahedral coordination of the B cation. (c) A polyhedral representation accentuating the BO6 octahedra. The perovskites have the general formula ABX3, in which the 12-coordinate hole of BX3 (as in ReO3) is occupied by a large A ion (Fig. 24.19; a different view of this structure is given in Fig. 3.42) The X ion is most frequently O2 or F (as in NaFeF3) although nitrideand hydride-containing perovskites can also be synthesized, such as in LiSrH3. Perovskite itself is named after the naturally occurring oxide mineral CaTiO3 and the largest class of perovskites are those with the anion as oxide. This breadth of perovskites is widened by the observation that solid solutions and nonstoichiometry are also common features of the perovskite structure, as in Ba1xSrxTiO3 and SrFeO3y. Some metal-rich materials adopt the perovskite structure with the normal distribution of cations and anions partially inverted, for instance SnNCo3. The perovskite structure is often observed to be distorted such that the unit cell is no longer centrosymmetric and the crystal acquires an overall permanent electric polarization as a result of ion displacements. Some polar crystals are ferroelectric in the sense that they resemble ferromagnets, but instead of the electron spins being aligned over a region of the crystal, the electric dipole moments of many unit cells are aligned. As a result, the relative permittivity, which reflects the polarity of a compound, for a ferroelectric material often exceeds 1 103 and can be as high as 1.5 104; for comparison, the relative permittivity of liquid water is about 80 at room temperature. Barium titanate, BaTiO3, is the most extensively studied example of such a material. At temperatures above 120°C this compound has the perfect cubic perovskite structure. At room temperature, it adopts a lower symmetry, tetragonal unit cell in which the various ions can be considered as having been displaced from their normal high symmetry sites (Fig. 24.20). This displacement results in a spontaneous polarization of the unit cell and formation of an electric dipole; coupling between these ion displacements and therefore the induced dipoles is very weak. Application Metal oxides, nitrides, and fluorides of an external electric field aligns these dipoles throughout the material, resulting in a bulk polarization in a particular direction, which can persist after removal of the electric field. The temperature below which this spontaneous polarization can occur and the material behaves as a ferroelectric is called the Curie temperature (TC)(Section 20.8). For BaTiO3 TC 120°C. The high relative permittivity of barium titanate leads to its use in capacitors, where its presence allows up to 1000 times the charge to be stored in comparison with a capacitor with air between the plates. The introduction of dopants into the barium titanate structure, forming solid solutions, allows various properties of the compound to be tuned. For example, the replacement of Ba by Sr or of Ti by Zr causes a sharp lowering of TC. Another characteristic of many crystals, including a number of perovskites that lack a centre of symmetry, is piezoelectricity, the generation of an electrical field when the crystal is under stress or the change in dimensions of the crystal when an electrical field is applied. Piezoelectric materials are used for a variety of applications, such as pressure transducers, ultramicromanipulators (where very small movements can be controlled), sound detectors, and as the probe support in scanning tunnelling microscopy. Some important examples are BaTiO3, NaNbO3, NaTaO3, and KTaO3. Although a noncentrosymmetric structure is required for both ferroelectric and piezoelectric behaviour, the two phenomena do not necessarily occur for the same crystal. For example, quartz, which does not have the perovskite structure, is permanently polarized; although quartz is piezoelectric it is not ferroelectric because an external electric field cannot reverse the polarization. Quartz is widely used to set the clock rate of microprocessors and watches because a thin sliver oscillates at a specific frequency to produce a small oscillating electrical field. This frequency is very insensitive to temperature. Another prototypical structure, that of potassium tetrafluoridonickelate(II), K2NiF4 (Fig. 24.21), is related to perovskite. The compound can be thought of as containing individual slices from the perovskite structure that share the four F atoms from the octahedra within the layer and have terminal F atoms above and below the layer. These layers are displaced relative to each other and separated by the K ions (which are nine-coordinate, to eight F atoms of one layer and one terminal F atom from the next). Compounds with the K2NiF4 structure have come under renewed investigation because some high-temperature superconductors, such as La1.85Sr0.15CuO4, crystallize with this structure. Apart from their importance in superconductivity, compounds with the K2NiF4 structure also provide an opportunity to investigate two-dimensional magnetic domains as coupling between electron spins is much stronger within the layers of linked octahedra than between the layers. The K2NiF4 structure has been introduced as being derived from a single slice of the perovskite structure; other related structures are possible where two or more perovskite layers are displaced horizontally relative to each other. Structures with K2NiF4 at one end of the range (a single perovskite layer) and perovskite itself at the other (an infinite number of such layers) are known as Ruddlesden–Popper phases. They include Sr3Fe2O7 with double layers and Ca4Mn3O10 with triple layers (Fig. 24.22). T i 617 O Figure 24.20 The tetragonal BaTiO3 structure showing the local Ti4 ion displacement that leads to the ferroelectric behaviour of this material. (b) K NiF6 (e) High-temperature superconductors Key point: High-temperature cuprate superconductors have structures related to perovskite. The versatility of the perovskites extends to superconductivity because most of the hightemperature superconductors (which were first reported in 1986) can be viewed as variants of the perovskite structure. Superconductors have two striking characteristics. Below a critical temperature, Tc (not to be confused with the Curie temperature of a ferroelectric, TC), they enter the superconducting state and have zero electrical resistance. In this superconducting state they also exhibit the Meissner effect, the exclusion of a magnetic field. The Meissner effect is the basis of the common demonstration of superconductivity in which a pellet of superconductor levitates above a magnet. It is also the basis for a number of potential applications of superconductors that include magnetic levitation, as in MAGLEV trains. Following the discovery in 1911 that mercury is a superconductor below 4.2 K, physicists and chemists made slow but steady progress in the discovery of superconductors with higher values of Tc at a rate of about 3 K per decade. After 75 years, Tc had been edged up to 23 K in Nb3Ge. Most of these superconducting materials were metal alloys, although superconductivity had been found in many oxides and sulfides (Table 24.4); magnesium diboride is superconducting below 39 K (see Box 13.4). Then, in 1986, the (a) Figure 24.21 The K2NiF4 structure. (a) The displaced layers of NiF6 octahedra interspersed with K ions and (b) a view of one layer of composition NiF4 showing the corner sharing octahedra linked through F. 618 24 Solid-state and materials chemistry A BO6 A BO6 (a) Figure 24.22 The Ruddlesden–Popper phases of stoichiometry (a) A3B2O7 and (b) A4B3O10 formed respectively from two and three perovskite layers of linked BO6 octahedra separated by A-type cations. Table 24.4 Some materials that exhibit superconductivity below the critical temperature, Tc Element Tc /K Compound Tc /K Zn 0.88 Nb3Ge 23.2 Cd 0.56 Nb3Sn 18.0 Hg 4.15 LiTi2O4 13.7 Pb 7.19 K0.4Na0.6BiO3 29.8 Nb 9.50 YBa2Cu3O10 93 Tl2Ba3Ca3Cu4O12 134 MgB2 40 K3C60 39 PbMo6S8 15.2 NbPS 12 (b) first high-temperature superconductor (HTSC) was discovered. Several materials are now known with Tc well above 77 K, the boiling point of the relatively inexpensive refrigerant liquid nitrogen, and in a few years the maximum Tc was increased by more than a factor of five to around 134 K. Two types of superconductors are known: u Type I show abrupt loss of superconductivity when an applied magnetic field exceeds a value characteristic of the material. u Type II superconductors, which include high-temperature materials, show a gradual loss of superconductivity above a critical field denoted Hc.3 Figure 24.23 shows that there is a degree of periodicity in the elements that exhibit superconductivity. Note in particular that the ferromagnetic metals Fe, Co, and Ni do not display superconductivity; nor do the alkali metals and the coinage metals Cu, Ag, and Au. For simple metals, ferromagnetism and superconductivity never coexist, but in some of the oxide superconductors ferromagnetism and superconductivity appear to coexist on different regions of the structure of the same solid. The first HTSC reported was La1.8Ba0.2CuO4 (Tc 35 K), which is a member of the solid-solution series La2xBaxCuO4 in which Ba replaces a proportion of the La sites in La2CuO4. This material has the K2NiF4 structure type with layers of edge-sharing CuO6 octahedra separated by the La3 and Ba2 cations, although the octahedra are axially elongated by a Jahn–Teller distortion (Section 20.1g). A similar compound with Sr replacing Ba in this structure type, as in La1.8Sr0.2CuO4 (Tc 38 K), is also known. One of the most widely studied HTSC oxide materials, YBa2Cu3O7x (Tc 93 K; informally this compound is called ‘123’, from the proportions of metal atoms in the compound, or YBCO, pronounced ‘ib-co’), has a structure similar to perovskite but with missing O atoms. In terms of the structure shown in Fig. 24.11, the stoichiometric YBa2Cu3O7 unit cell consists of three simple perovskite cubes stacked vertically with Y and Ba in the A sites 3 The Chevrel phases discussed in Section 24.11 have the highest observed values of Hc. Metal oxides, nitrides, and fluorides Superconducting H He Superconducting in thin films Li Be Na Mg V C N O F Ne Al Si P S Cl Ar B Superconducting under pressure Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr K Ca Sc Ti Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te Cs Ba La Hf Ta W Re Os Ir I Xe Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Figure 24.23 Elements that show superconductivity under the specified conditions. Ba O Y Cu O Ba Cu Cu Y (a) (b) Figure 24.24 Structure of the YBa2Cu3O7 superconductor. (a) The unit cell and (b) oxygen polyhedra around the copper ions showing the layers formed from linked CuO5 square pyramids and chains formed from cornerlinked CuO4 square planes. of the original perovskite and Cu atoms in the B sites. However, unlike in a true perovskite structure, the B sites are not surrounded by an octahedron of O atoms: the 123 structure has a large number of sites that would normally be occupied by O but are in fact vacant. As a result, some Cu atoms have five O atom neighbours in a square-pyramidal arrangement and others have only four, as square-planar CuO4 units. Similarly, the Y and Ba in the A sites have less than 12-coordination. The compound YBa2Cu3O7 readily loses oxygen from some sites within the CuO4 square planes, forming YBa2Cu3O7x (0 x 1), but as x increases above 0.1 the critical temperature drops rapidly from 93 K. A sample of the 123 material made in the laboratory and heated under pure oxygen at 450°C as the final stage of its preparation is typically oxygen deficient with x ⬇ 0.1. 619 620 24 Solid-state and materials chemistry Cu O B a Cu O 2 3 Ca l 2O2 T Ca Cu O B a Cu O a B Cu O 3 2 3 Ca l 2O2 T Ca Cu O 3 B a Cu O 2 Figure 24.25 Tl2Ba3Ca2Cu3O10 structure formed from three oxygen-deficient perovskite layers produced from linked CuO4 square planes and square-based pyramids separated by Ca on the A type cation position. Double layers of stoichiometry Tl2O2 with rock-salt type arrangements of the Tl and O atoms are interleaved between the multiple perovskite layers. Me3C O – O C3F7 1 fod CMe3 Me3C CMe3 O O O Y O Me3C O O CMe3 2 Y(thd)3 CMe3 If we assign the usual oxidation numbers Nox(Y) 3, Nox(Ba) 2, and Nox(O) 2, then the average oxidation number of copper turns out to be 2.33, so it is inferred that YBa2Cu3O7x is a mixed oxidation state material that contains Cu2 and Cu3. Note that in YBa2Cu3O7x the material formally contains some Cu3 until x increases above 0.5. An alternative view is that the number of electrons in YBa2Cu3O7x is such that a partially filled band is present: this view is consistent with the high electrical conductivity and metallic behaviour of this oxide at room temperature (Section 3.19). If this band can be considered as being constructed from Cu3d orbitals, then the partial filling is a result of holes in this level (corresponding to Cu3); another possible description is that the band also involves O2p orbitals, suggesting that the material can be considered to contain Cu2 and O. The square-planar CuO4 units in YBa2Cu3O7x are arranged in chains and the CuO5 units link together to form infinite sheets. The stoichiometry of an infinite sheet of vertex-sharing CuO4 square planes is CuO2. The addition of one or two additional apical O atoms in the cases where the layers are constructed from, respectively, linked squarebased pyramids or octahedra maintains the CuO2 sheet. This structural feature is also seen in all other oxocuprate HTSCs. It is thought that it is an important component of the mechanism of superconduction in these materials. Some HTSC and other superconducting materials are listed in Table 24.4. All of them may be considered to have at least part of their structure derived from that of perovskite, as a layer of linked CuOn (n 4, 5, 6) polyhedra is a section of that structural type. Lying between these cuprate layers (which may include up to six such perovskite-derived CuO2 sheets) can be a variety of other simple structural units, containing s- and p-block metals in combination with oxygen, such as rock-salt and fluorite structures. Thus Tl2Ba2Ca2Cu3O10 can be considered as having three perovskite layers based on Cu, O, and Ca separated by double layers of a rock-salt structure built from Tl and O; the Ba lie between the rock-salt and perovskite layers (Fig. 24.25). The synthesis of high-temperature superconductors has been guided by a variety of qualitative considerations, such as the demonstrated success of the layered structures and of mixed-oxidation-state Cu in combination with heavy p-block elements. Additional considerations are the radii of ions and their preference for certain coordination environments. Many of these materials are prepared simply by heating an intimate mixture of the metal oxides to 800–900°C in an open alumina crucible. Others, such as mercury- and thallium-containing complex copper oxides, require reactions involving the volatile and toxic oxides Tl2O and HgO; in such cases the reactions are normally carried out in sealed gold or silver tubes. Thin films are needed if superconductors are to be used in electronic devices. Their preparation is an active area of research and a promising strategy is chemical vapour deposition. The general strategy is to form a thin film by decomposing a thermally unstable compound on a hot solid substrate material (Fig. 24.26). Fluorinated acetylacetonato complexes of the metals (using ligands such as fod, 1) are sometimes used because they are more volatile than simple acetylacetonato complexes. These complexes, such as Cu(acac)2, Y(thd)3 (2), and Ba(fod)2, are swept into the reaction chamber by slightly moist oxygen gas. When conditions are properly controlled, this gaseous mixture reacts on the hot substrate to produce the desired YBa2Cu3O7x film. The film may be amorphous and require subsequent heating to form a crystalline product. There is, as yet, no settled explanation of high-temperature superconductivity. It is believed that the movement of pairs of electrons, known as ‘Cooper pairs’ and responsible for conventional superconductivity, is also important in the high-temperature materials, but the mechanism for pairing is hotly debated. (f) Other superconducting oxides The observation of superconductivity in the complex cuprates is unusual in terms of the high critical temperatures reached but many other oxides and oxide phases demonstrate a transition to zero electrical resistance, albeit normally at considerably lower temperatures. Some compositionally simple examples include phases from the solid solution Li1xTi2xO4, (which adopts the spinel structure and has Tc 13.7 K for x 0) and Na0.35CoO2.H2O with Tc ⬇ 5 K (see also Table 24.4). The complex bismuth oxides of composition (K0.87Bi0.13)BiO3 (for which Tc 10.2 K) and (Ba0.6K0.4)BiO3 (Tc 30 K), which adopt perovskite structures with Bi as the Btype cation, are among a number of similar bismuthates that exhibit superconductivity. Metal oxides, nitrides, and fluorides 621 Argon, oxygen, water vapour Several complex oxides that adopt the pyrochlore structure (Fig 24.27) and have composition M2xB2O7x, where M is a Group I, Group II, or post-transition metal cation, such as Cs, Ca, or Cd and where B is a heavy d metal, show superconductivity. For example, Cd2Re2O7 is superconducting below 1.4 K and KOs2O6 is superconducting below 10 K. Very recently, interest has centred on a new family of superconductors with critical temperatures approaching the best cuprates. These new lanthanoid iron arsenic oxides of composition LnFeAs(O,F)1x were first reported for Ln La in LaFeAsO1xFx with Tc 26 K, and for compositionally similar Pr and Sm compounds critical temperatures of 52 K and 55 K, respectively, have been achieved. The structures of these compounds are based on alternating layers of composition LnO and FeAs (Fig 24.28). T1 T2 T3 Heated inlet oven (g) Colossal magnetoresistance Key point: Perovskites with Mn on the B cation sites can show very large changes in resistance on application of a magnetic field, known as colossal magnetoresistance. Manganites, which are Mn(III) and Mn(IV) complex oxides, with the generic solid solution formulation Ln1xAxMnO3 (A Ca, Sr, Pb, Ba; Ln typically La, Pr, or Nd), order ferromagnetically on cooling below room temperature, with Curie temperatures typically between 100 K and 250 K, and simultaneously transform from insulators (at the higher temperature) to poor metallic conductors. These materials also exhibit magnetoresistance, a marked decrease of their resistance on the application of a magnetic field near and just above their Curie temperatures (Fig. 24.29). Recent investigations have shown that, for these manganites, the decrease in resistance can be by as much as 11 orders of magnitude, and for this reason these compounds have been named colossal magnetoresistance manganites.4 Colossal magnetoresistance (CMR) manganites have perovskite-type structures with the A cation sites occupied by a mixture of Ln3 and A2 cations and the B site occupied by Mn. The oxidation number of Mn in these solid solutions varies between 3 and 4 as the proportion of A2 is changed. Pure LaMnO3 orders antiferromagnetically below its Néel temperature (TN 150 K), but with an increase in x in Ln1xAxMnO3, corresponding to an increase in Mn4 content, the manganites order ferromagnetically on cooling. The observation that the transition to conducting electron behaviour and ferromagnetism occurs simultaneously on cooling the Ln1xAxMnO3 manganites and the origin of the CMR effect are not completely understood, but it is known that they are based on a socalled double exchange (DE) mechanism between the Mn(III) and Mn(IV) species present in these materials. The basic process of this mechanism is the transfer of an electron from Film deposition Exhaust Infrared radiation Figure 24.26 Schematic diagram of the chemical vapour deposition of superconductor thin films. A reactive carrier gas mixture (Ar, O2, and H2O) is passed through traps of the volatile metal precursors held at temperatures T1, T2, and T3 which provide the desired vapour pressure of each reactant. Deposition occurs on the wedge-shaped block, which is heated using an IR lamp. After the deposition, the resulting film is annealed at high temperature to improve the crystallinity of the film. O La BO6 As Fe A O Figure 24.27 The pyrochlore structure adopted by many compounds of stoichiometry A2B2O7. The BO6 octahedra are shown which form channels containing the A-type cations and oxide ions. 4 Albert Fert and Peter Grünberg were awarded the 2007 Nobel Prize in Physics for their discovery of this phenomenon. Figure 24.28 The structure of the superconductor LaFeAsO delineating the two types of layer. 622 24 Solid-state and materials chemistry 0 log{Resistivity/(Ω m–1)} 165 K –1 Increasing magnetic field –2 Metallic –3 0 100 200 300 Temperature, T/K Figure 24.29 The resistivity as a function of temperature at various magnetic fields of a material displaying colossal magnetoresistance. At 165 K application of a magnetic field will cause a change in resistivity of about two orders of magnitude. CoO2 Mn3(t2g3 e1g) to Mn4(t2g3 ) through the O atom, so that the Mn(III) and Mn(IV) positions change places. In manganites at high temperatures, the electrons in these systems effectively become trapped on a specific site, leading to ordering of the Mn(III) and Mn(IV) species, that is the trapping results in charge ordering. The charge-ordered state is generally associated with insulating and paramagnetic behaviour whereas the charge-disordered state, in which the electron can move between sites, is associated with metallic behaviour and ferromagnetism. The high-temperature charge-ordered state can be transformed into a metallic, spin-ordered (ferromagnetic) state just by the application of a magnetic field, therefore application of a magnetic field to a manganite just above a critical temperature causes the transformation of charge ordering into electron delocalization and hence a massive decrease in resistance. The CMR effect is being developed for magnetic data-storage devices, such as computer hard drives. Further work using these compounds is aimed at spintronics, where, rather than use electron movements to transmit information, as is the basis of electronics and the functioning of the silicon chip, the movement of spin through materials could be used in a similar way. As spin-transfer is much faster than electron motion and does not develop heat through resistive effects, computing devices based on spintronics should have much higher processing powers and not require cooling; the latter is an increasing problem with semiconductor technology as transistors become ever more densely packed on computer processors. (h) Rechargeable battery materials Key point: The redox chemistry associated with the extraction and insertion of metal ions into oxide structures is exploited in rechargeable batteries. The existence of complex oxide phases that demonstrate good ionic conductivity associated with the ability to vary the oxidation state of a d-metal ion has led to the development of materials for use as the cathode in rechargeable batteries (see Box 11.1). Examples include LiCoO2, with a layer-type structure based on sheets of edge-linked CoO6 octahedra separated by Li ions (Fig. 24.30), and various lithium manganese spinels, such as LiMn2O4. In each of these compounds the battery is charged by removing the mobile Li ions from the complex metal oxide, as in LiCoO2 → CoO2 + Li + + e – Li Figure 24.30 The structure of LiCoO2 shown as layers of linked CoO6 octahedra separated by Li ions; lithium may be de-intercalated electrochemically from between the layers. The battery is discharged through the reverse electrochemical reaction. Lithium cobalt oxide, which is used in many commercial lithium-ion batteries, has many of the characteristics required for this type of application. The specific energy (the stored energy divided by the mass) of LiCoO2 (140 W h kg1) is maximized by using light elements such as Li and Co; the 3d metals are almost invariably used in such applications because they are the lowest density elements with variable oxidation states. The high mobility of the Li ion and good reversibility of electrochemical charging and discharging stem from the lithium ion’s small ionic radius and the layer-like structure of LiCoO2, which allows the Li to be extracted without major disruption of the structure. High capacities are obtained from the large amount of Li (one Li ion for each LiCoO2 formula unit) that may be reversibly extracted (about 500 discharge/recharge cycles) from the compound, and the current is delivered at a constant and high potential difference (of between 3.5 and 4 V). The high potential difference is partly due to the high oxidation states of cobalt (3 and 4) that are involved. Because cobalt is expensive and fairly toxic, the search continues for even better oxide materials than LiCoO2. New materials will be required to demonstrate the high levels of reversibility found for lithium cobaltate and considerable effort is being directed at doped forms of LiCoO2 and of LiMn2O4 spinels and at nanostructured complex oxides (Section 25.4), which, because of their small particle size, can offer excellent reversibility. There is also a high level of interest in LiFePO4, which shows good characteristics for a cathode material and contains cheap and nontoxic iron. The other electrode in a rechargeable lithium ion battery can simply be Li metal, which completes the overall cell reaction through the process Li(s) → Li e. Lithium ions then migrate to the cathode through an electrolyte, which is typically an anhydrous lithium salt, such as LiPF4 or LiC(SO2CF3)3 dissolved in a polymer, such as poly(propene carbonate). 623 Metal oxides, nitrides, and fluorides However, the use of Li metal has a number of problems associated with its reactivity and volume changes that occur in the cell. Therefore, an alternative anode material that is frequently used in rechargeable batteries is graphitic carbon, which can intercalate, electrochemically, large quantities of Li to form LiC6 (Section 14.5). As the cell is discharged, Li is transferred from between the carbon layers at the anode and intercalated into the metal oxide at the cathode (and vice versa on charging) with the following overall processes : charge (reaction proceeds to the right) ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ → C + Li Li y C6 + Li1–xCoO2 ← CoO2 ⎯ 6 1–x + y discharge (reaction proceeds to the leftt) The term ceramic is often applied to all inorganic nonmetallic, nonmolecular materials, including both amorphous and crystalline materials, but the term is commonly reserved for compounds or mixtures that have undergone heat treatment. The term glass is used in a variety of contexts but for our present purposes it implies an amorphous ceramic with a viscosity so high that it that can be considered to be rigid. A substance in its glassy form is said to be in its vitreous state. Although ceramics and glasses have been utilized since antiquity, their development is currently an area of rapid scientific and technological progress. This enthusiasm stems from interest in the scientific basis of their properties and the development of novel synthetic routes to new high-performance materials. We confine our attention here to glasses. The most familiar glasses are alkali-metal or alkaline-earthmetal silicates and borosilicates. (a) Glass formation Key points: Silicon dioxide readily forms a glass because the three-dimensional network of strong covalent Si–O bonds in the melt does not easily break and reform on cooling; the Zachariasen rules summarize the properties likely to lead to glass formation. A glass is prepared by cooling a melt more quickly than it can crystallize. Cooling molten silica, for instance, gives vitreous quartz. Under these conditions the solid has no longrange order as judged by the lack of X-ray diffraction peaks, but spectroscopic and other data indicate that each Si atom is surrounded by a tetrahedral array of O atoms. The lack of long-range order results from variations of the SiOSi angles. Figure 24.31a illustrates in two dimensions how a local coordination environment can be preserved but long-range order lost by variation of the bond angles around O. This loss of long-range order is readily apparent when X-rays are scattered from a glass (Fig. 24.31b); in contrast to a long-range, periodically ordered crystalline material, where diffraction gives rise to a series of diffraction maxima (Section 8.1), the X-ray diffraction pattern obtained from a glass shows only broad features as the long-range order is lost. Silicon dioxide readily forms a glass because the three-dimensional network of strong covalent SiO bonds in the melt does not readily break and reform on cooling. The lack of strong directional bonds in metals and simple ionic substances makes it much more difficult to form glasses from these materials. Recently, however, techniques have been developed for ultrafast cooling and, as a result, a wide variety of metals and simple inorganic materials can now be frozen into a vitreous state. The concept that the local coordination sphere of the glass-forming element is preserved but that bond angles around O are variable was originally proposed by W.H. Zachariasen in 1932. He reasoned that these conditions would lead to similar molar Gibbs energies and molar volumes for the glass and its crystalline counterpart. Zachariasen also proposed that the vitreous state is favoured by polyhedral corner-sharing O atoms, rather than edge- or face-shared, which would enforce greater order. These and other Zachariasen rules hold for common glass-forming oxides, but exceptions are known. An instructive comparison between vitreous and crystalline materials is seen in their change in volume with temperature (Fig. 24.32). When a molten material crystallizes, an abrupt change in volume (usually a decrease) occurs. By contrast, a glass-forming material that is cooled sufficiently rapidly persists as a metastable supercooled liquid. When cooled below the glass transition temperature, Tg, the supercooled liquid becomes rigid, and this change is accompanied by only an inflection in the cooling curve rather than an abrupt change in slope. The rates of crystallization are very slow for many complex metal silicates, phosphates, and borates, and it is these compounds that often form glasses. (a) 10 (b) 20 30 40 50 60 Figure 24.31 (a) Schematic representation of a two-dimensional crystal, left, compared with a two-dimensional glass, right. (b) The powder X-ray diffraction pattern of a glass (SiO2, orange) contrasted with that of a crystalline solid (quartz SiO2, blue). The longrange order, which gives rise to the sharp diffraction maxima in quartz, is no longer present in amorphous SiO2 and only broad features are seen. Liquid Supercooled liquid Volume 24.8 Oxide glasses Glass Crystal Tg Tf Temperature Figure 24.32 Comparison of the volume change for supercooled liquids and glasses with that for a crystalline material. The glass transition temperature is Tg. 624 24 Solid-state and materials chemistry Sol Hydrolysis Gel Dry Porous ceramic Heat Heat Dense ceramic or glass Figure 24.33 Schematic diagram of the sol–gel process. When the gel is dried at high temperatures dense ceramics or glasses are formed. Drying at low temperatures above the critical pressure of water produces porous solids known as xerogels or aerogels. Na2O O– Na+ Na+ Figure 24.34 The role of a modifier is to introduce O2 ions and cations that disrupt the lattice. Another route to glasses is the sol–gel process, which is described schematically in Fig. 24.33 and discussed more fully in Section 25.4. As described there, the sol–gel process is also used to produce crystalline ceramic materials and high-surface-area compounds such as silica gel. A typical process involves the addition of a metal alkoxide precursor to an alcohol, followed by the addition of water to hydrolyse the reactants. This hydrolysis leads to a thick gel that can be dehydrated and sintered (heated below its melting point to produce a compact solid). For example, ceramics containing TiO2 and Al2O3 can be prepared in this way at much lower temperatures than required to produce the ceramic from the simple oxides. Often a special shape can be fashioned at the gel stage. Thus, the gel may be shaped into a fibre and then heated to expel water; this process produces a glass or ceramic fibre at much lower temperatures than would be necessary if the fibre were made from the melt of the components. (b) Glass composition, production, and application Key points: Low-valence metal oxides, such as Na2O and CaO, are often added to silica to reduce its softening temperature by disrupting the silicon–oxygen framework. Other cations may be incorporated into glasses, giving applications as diverse as lasers and nuclear waste containment. Although vitreous silica is a strong glass that can withstand rapid cooling or heating without cracking, it has a high glass transition temperature and therefore must be worked at inconveniently high temperatures. Therefore, a modifier, such as Na2O or CaO, is commonly added to SiO2. A modifier disrupts some of the SiOSi linkages and replaces them with terminal SiO links that associate with the cation (Fig. 24.34). The consequent partial disruption of the SiO network leads to glasses that have lower softening points. The common glass used in bottles and windows is called ‘sodalime glass’ and contains Na2O and CaO as modifiers. When B2O3 is used as a modifier, the resulting ‘borosilicate glasses’ have lower thermal expansion coefficients than sodalime glass and are less likely to crack when heated. Borosilicate glass (such as Pyrex®) is therefore widely used for ovenware and laboratory glassware. Glass formation is a property of many oxides, and practical glasses have been made from sulfides, fluorides, and other anionic constituents. Some of the best glass formers are the oxides of elements near silicon in the periodic table (B2O3, GeO2, and P2O5), but the solubility in water of most borate and phosphate glasses and the high cost of germanium limit their usefulness. The development of transparent crystalline and vitreous materials for light transmission and processing has led to a revolution in signal transmission. For example, optical fibres are currently being produced with a composition gradient from the interior to the surface. This composition gradient modifies the refractive index and thereby decreases light loss. Fluoride glasses are also being investigated as possible substitutes for oxide glasses because oxide glasses contain small numbers of OH groups, which absorb near-infrared radiation and attenuate the signal. Doping lanthanoid ions into the glass fibres produces materials that can be used to amplify the signals by using lasing effects. Optical circuit elements are being developed that may eventually replace all the components in an electronic integrated circuit and lead to very fast optical computers. As modifier cations effectively become trapped in a glass, which is chemically inert and thermodynamically very stable with respect to transformation to a soluble, crystalline phase, such glassy materials offer a potential method for containing and storing nuclear waste. Thus vitrification of metal oxides containing a radioactive species with glass-forming oxides produces a stable glass that can be stored for long periods, allowing the radionucleide to decay (Section 23.1). Examples of such materials are borosilicate glasses and combinations of such glasses with crystalline oxide phases (‘Synroc’). The incorporation of functional inorganic compounds into glasses has led to the development of so-called ‘smart glasses’ that have switchable properties, such as electrochromism, the ability to change colour or light transmission properties in response to application of a potential difference, and reversible photochromism, the ability to change colour under certain light conditions. An electrochromic glass generally consists of a glass coated with colourless, tungsten trioxide, WO3, or a sandwich-like layer structure consisting of these two components. When a potential difference is applied to the WO3 layer, cations are inserted and the W is partially reduced to form MxW(VI,V)O3, which is dark blue. The coating maintains its dark colour until the potential difference is reversed, when Metal oxides, nitrides, and fluorides 625 the glass becomes colourless. Application of electrochromic glasses include privacy glass, auto-dimming rear view mirrors in vehicles, and aircraft windows. A similar coating is used in self-cleaning glasses where a metal oxide coating on the glass acts as a photocatalyst for the breakdown of organic dirt on their surfaces. A photochromic glass incorporates a small amount of a colourless silver halide, normally AgCl, within the glass. When exposed to UV radiation, such as that in sunlight, the AgCl dissociates to form small clusters of Ag atoms, which absorb light across the visible region of the spectrum, imparting a grey colour to the glass. When the UV radiation is removed, the AgCl reforms and the glass returns to its optically transparent state. Photochromic glasses are widely used in the lenses of spectacles and sunglasses. 24.9 Nitrides and fluorides The solid-state chemistry of the metals in combination with anions other than O2 is not as highly developed or extensive as that of the complex oxides described so far. However, complex nitrides and fluorides and mixed anion compounds are of growing importance. (a) Nitrides Key points: Complex metal nitrides and oxide nitrides are materials containing the N3 anion; many new compounds of this type have recently been synthesized. Simple metal nitrides of main-group elements, such as AlN, GaN, and Li3N, have been known for decades. Many of the recent advances in nitride chemistry have centred on d-metal compounds and complex nitrides. That nitrides are less common than oxides stems, in part, from the high enthalpy of formation of N3 compared with that of O2. Furthermore, because many nitrides are sensitive to oxygen and water, their synthesis and handling are problematic. Some simple metal nitrides can be obtained by the direct reaction of the elements, for example Li3N is obtained by heating lithium in a stream of nitrogen at 400ºC. The instability of sodium nitride allows sodium azide to be used as a nitriding agent: 750ºC, sealed Nb tube 2 NaN3 (s) + 9Sr(s) + 6Ge(s) ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 3Sr3Ge2 N 2 (s) + 2 Na(g) The ammonolysis of oxides (the dehydrogenation of NH3 by oxides with the formation of water as a byproduct) provides a convenient route to some nitrides. For instance, tantalum nitride can be obtained by heating tantalum pentoxide in a fast-flowing stream of ammonia: 700ºC 3 Ta2O5 (s) + 10 NH3 (l) ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 2 Ta3 N 5 (s) + 15 H 2O(g) In such reactions, the equilibrium is driven towards the products by removal of the steam in the gas flow. Similar reactions may be used for the preparation of complex nitrides from complex oxides, although competing reactions involving partial reduction of the metal oxide by ammonia can also occur. In all these reactions the complete elimination of O2 ions from the product can be troublesome, so the reactions give products that contain both the oxide and nitride ions: 800ºC ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 2CaTaO2 N(s) + 3H 2O(g) Ca2 Ta2O7 (s) + 2 NH3 (g) ⎯ In comparison with the O2 ion, the higher charge of the N3 ion results in a greater degree of covalence in its bonding and therefore nitrides, particularly those of less electropositive elements such as d metals, should not be described in purely ionic terms. There is also a tendency with nitrides for the formation of compounds in which the metallic element is in a lower oxidation state because nitrogen, on account of its high bond energy, is not as potent an oxidant as oxygen or fluorine. Thus, whereas heating titanium in oxygen readily produces TiO2, Ti2N and TiN are known but Ti3N4 is difficult to prepare and poorly characterized. Likewise, V3N5 is unknown whereas V2O5 is readily obtained from the decomposition of many vanadium salts in air. Many of the early d-metal nitrides are interstitial compounds and are used as hightemperature refractory ceramics. Similarly, the nitrides of Si and Al, such as Si3N4 (Fig. 24.35), are stable at very high temperatures, particularly under nonoxidizing N Si Figure 24.35 The structure of Si3N4 shown as linked SiN4 tetahedra. 626 24 Solid-state and materials chemistry Ba VN4 conditions, and are used for crucibles and furnace elements. Recently, GaN, which can exist in both wurtzite and sphalerite structural types, has been the focus of considerable research on account of its semiconducting properties. The nitride Li3N has an unusual structure based on hexagonal Li2N layers separated by Li ions (Fig. 24.8). These Li ions are highly mobile, as is expected from the existence of free space between the layers, and this compound and other structurally related materials are being studied for possible use in rechargeable batteries. Among the many complex nitrides that have been synthesized are materials of stoichiometry AMN2, such as SrZrN2 and CaTaN2, with structures based on sheets formed from MN6 octahedra sharing edges (see the discussion of LiCoO2 in Section 24.7h) and A2MN3, such as Ba2VN3, containing one-dimensional chains of cornersharing MN4 tetrahedra (Fig. 24.36). Analogues of the zeolites containing nitride bridges rather than oxide bridges have also been prepared. (b) Fluorides and other halides Key point: Because fluorine and oxygen have similar ionic radii, fluoride solid-state chemistry parallels much of oxide chemistry. Figure 24.36 Ba2VN3, containing onedimensional chains of corner-sharing VN4 tetrahedra separated by Ba2 ions. The ionic radii of F− and O2− are very similar (at between 130 and 140 pm), and as a result metal fluorides show many stoichiometric and structural analogies with the complex oxides but with lower charge on the metal ion to reflect the lower charge on the F ion. Many binary metal fluorides adopt the simple structural types expected on the basis of the radius-ratio rule (Section 3.10). For example, FeF2 and PdF2 have a rutile structure and AgF has a rock-salt structure; similarly, NbF3 adopts the ReO3 structure. For complex fluorides, analogues of typical oxide structural types are well known, including perovskites (such as KMnF3), Ruddlesden–Popper phases (for example K3Co2F7), and spinels (Li2NiF4). Synthetic routes to complex fluorides also parallel those for oxides. For instance, the direct reaction of two metal fluorides yields the complex fluoride, as in 2LiF(s) + NiF2 (s) → Li2NiF4 (s) Like some complex oxides, some complex fluorides may be precipitated from solution MnBr2 (aq) + 3KF(aq) → KMnF3 (s) + 2 KBr(aq) As with the ammonolysis of oxides to produce nitrides and oxide-nitrides, the formation of oxide-fluorides is possible by the appropriate treatment of a complex oxide, as in F / 200ºC 2 Sr2CuO3 (s) ⎯⎯⎯⎯ → Sr2CuO2F2+x (s) Sr2CuO2F2x is a superconductor with Tc 45 K. Fluoride analogues of the silicate glasses, which are based on linked SiO4 tetrahedra, exist for small cations that form tetrahedral units in combination with F; an example is LiBF4, which contains linked BF4 tetrahedra. Lithium borofluoride glasses are used to contain samples for X-ray work because they are highly transparent to X-rays on account of their low electron densities. Framework and layer structures based on linked MF4 (M Li, Be) tetrahedra have also been described. Some metal fluorides are used as fluorination agents in organic chemistry. However, few of the solid complex metal fluorides are technologically important in comparison with the wealth of applications associated with analogous complex oxides. Metal chloride structures reflect the greater covalence associated with bonding to chloride in comparison with fluoride: the chlorides are less ionic and have structures with lower coordination numbers than the corresponding fluorides. Thus, simple metal chlorides, bromides, and iodides normally adopt the cadmium-chloride or cadmium-iodide structures based on sheets formed from edge-sharing MX6 octahedra. Complex chlorides often contain the same structural unit, for example CsNiCl3 has chains of edge-sharing NiCl6 octahedra separated by Cs ions. Many analogues of oxide structures also occur among the complex chlorides, such as KMnCl3, K2MnCl4, and Li2MnCl4, which have the perovskite, K2NiF4, and spinel structures, respectively. Chalcogenides, intercalation compounds, and metal-rich phases 627 Chalcogenides, intercalation compounds, and metal-rich phases The soft chalcogens S, Se, and Te form binary compounds with metals that commonly have quite different structures from the corresponding oxides, nitrides, and fluorides. As we saw in Sections 3.9, 16.11, and 19.9, this difference is consistent with the greater covalence of the compounds of sulfur and its heavier congeners.For example,we noted there that MO compounds generally adopt the rock-salt structure whereas ZnS and CdS can crystallize with either of the sphalerite or the wurtzite structures in which the lower coordination numbers indicate the presence of directional bonding. Similarly, the d-block monosulfides generally adopt the more characteristically covalent nickel-arsenide structure rather than the rock-salt structure of alkaline-earth oxides such as MgO. Even more striking are the layered MS2 compounds formed by many d-block elements in contrast to the fluorite or rutile structures of many d-block dioxides. Before we discuss these compounds, we should note that there are many metal-rich compounds that disobey simple valence rules and do not conform to an ionic model. Some examples are Ti2S, Pd4S, V2O, and Fe3N, and even the alkali metal suboxides, such as Cs3O (Section 11.8). The occurrence of metal-rich phases is generally associated with MM interactions. Many other intermetallic compounds display stoichiometries that cannot be understood in terms of conventional valence rules. Included among them are the important permanent-magnet materials Nd2Fe17B and SmCo5. 24.10 Layered MS2 compounds and intercalation The layered metal sulfides and their intercalation compounds were introduced in Section 19.9. Here we develop a broader picture of their structures and properties. (a) Synthesis and crystal growth Key point: d-Metal disulfides are synthesized by the direct reaction of the elements in a sealed tube and purified by using chemical vapour transport with iodine. Compounds of the chalcogens with d metals are prepared by heating mixtures in a sealed tube (to prevent the loss of the volatile elements). The products obtained in this manner can have a variety of compositions. The preparation of crystalline dichalcogenides suitable for chemical and structural studies is often performed by chemical vapour transport (CVT), as described below. It is possible in some cases simply to sublime a compound, but the CVT technique can also be applied to a wide variety of nonvolatile compounds in solid-state chemistry. In a typical procedure, the crude material is loaded into one end of a borosilicate or fused quartz tube. After evacuation, a small amount of a CVT agent is introduced and the tube is sealed and placed in a furnace with a temperature gradient. The polycrystalline and possibly impure metal chalcogenide is vaporized at one end and redeposited as pure crystals at the other (Fig. 24.37). The technique is called chemical vapour transport rather than sublimation because the CVT agent, which is often a halogen, produces an intermediate volatile species, such as a metal halide. Generally, only a small amount of transport agent is needed because on crystal formation it is released and diffuses back to pick up more reactant. For example, TaS2 can be transported with I2 in a temperature gradient. The reaction with I2 to produce gaseous products Low temperature High temperature TaS2 (s) + 2 I2 (g) → TaI4 (g) + S2 (g) is endothermic, so the equilibrium lies further to the right at 850ºC than at 750ºC. Consequently, although TaI4 is formed at 850ºC, at 750ºC the mixture deposits TaS2. If, as occasionally is the case, the transport reaction is exothermic, the solid is carried from the cooler to the hotter end of the tube. (b) Structure Key points: Elements on the left of the d block form sulfides consisting of sandwich-like layers of the metal coordinated to six S ions; the bonding between the layers is very weak. TaS2 crystals Impure TaS2 powder Figure 24.37 Vapour transport crystal growth and purification of TaS2. A small quantity of I2 is present to serve as a transport agent. 628 24 Solid-state and materials chemistry S Ta (a) S Nb (b) Figure 24.38 (a) The structure of TaS2 (CdI2-type). The Ta atoms reside in octahedral sites between the AB layers of S atoms. (b) The NbS2 structure; the Nb atoms reside in trigonal-prismatic sites, between the sulfide layers. Ta S As we saw in Section 19.9, the d-block disulfides fall into two classes: layered materials are formed by metals on the left of the d block, and compounds containing formal S22 ions are formed by metals in the middle and towards the right of the block (such as pyrite, FeS2). We concentrate here on the layered materials. In TaS2 and many other layered disulfides, the d-metal ions are located in octahedral holes between close-packed AB layers (Fig. 24.38a). The Ta ions form a close-packed layer denoted X, so the metal and adjoining sulfide layers can be portrayed as an AXB sandwich. These sandwich-like slabs form a three-dimensional crystal by stacking in sequences such as AXBAXBAXB..., where the strongly bound AXB slabs are held to their neighbours by weak dispersion forces. An alternative view of these MS2 structures, which have the metal ions in octahedral holes, is as MS6 octahedra sharing edges (Fig. 24.39a), which reinforces the idea of the greater degree of covalent bonding that occurs in these materials than in, for instance, Li2S, which has the antifluorite structure. The Nb atoms in NbS2 reside in the trigonal-prismatic holes between sulfide layers that are in register with one another (AA, Fig. 24.38b and Fig 24.39b). The Nb atoms, which are strongly bonded to the adjacent sulfide layers, form a close-packed array denoted m, so we can represent each slab as AmA or CmC. These slabs form a three-dimensional crystal by stacking in a pattern such as ...AmACmCAmACmC.... Weak dispersion forces also contribute to holding these AmA and CmC slabs together. Polytypes (versions that differ only in the stacking arrangement along a direction perpendicular to the plane of the slabs) can occur. Thus, NbS2 and MoS2 form several polytypes, including one with the sequence CmCAmABmB. Molybdenum sulfide, MoS2, is used as a high performance lubricant in, for example, racing cars and machining, as it can be used at much higher temperatures and pressures than oils. Lubrication occurs with a dry coating of the material as the MoS2 layers are able to slip over each other easily due to the weak interlayer interactions. It has been convenient to describe—somewhat simplistically—the layered structures in terms of cations and anions. However, to account for the significant covalence, the electrical conductivities, and the chemical properties of these layers we need to invoke the more sophisticated band model of their electronic structure. Some approximate band structures derived primarily from molecular orbital calculations and photoelectron spectra are shown in Fig. 24.40. They show that the dichalcogenides with octahedral and trigonalprismatic metal sites have low-lying bands composed primarily of chalcogen s and p orbitals, higher-energy bands derived primarily from metal d orbitals, and still higher in energy a variety of metal and chalcogen bands. When the metal atom occupies an octahedral site, its d orbitals are split into a lower t2g set and a higher eg set, just as in localized complexes. These atomic orbitals combine to give a t2g band and an eg band. The broad t2g band may accommodate up to six electrons per metal atom. Thus TaS2, with one d electron per metal atom, has an only partly filled t2g band, and is a metallic conductor. A trigonal-prismatic ligand field leads to a lowenergy a1 level and two doubly-degenerate higher-energy orbitals designated e and e, respectively. In this case, the lowest band needs only two electrons per atom to fill it. Chalcogen bands (a) Nb S Energy d Bands EF(ZnS2) EF(MoS2) EF(NbS2) Chalcogen s and p bands (b) Figure 24.39 The metal disulfide structures of Figure 24.38 drawn as layers of MS6 polyhedra sharing edges: (a) octahedra in TaS2, (b) trigonal prisms in NbS2. Density of states Density of states Figure 24.40 Approximate band structures of dichalcogenides. (a) Octahedral MS2 compounds; for ZrS2 (d0) only the sulfur s and p bands are full and the compound is a semiconductor. (b) Trigonal prismatic MS2 compounds. NbS2 (d1) is metallic whereas MoS2 (d2) is a semiconductor. Chalcogenides, intercalation compounds, and metal-rich phases Thus MoS2, with trigonal-prismatic Mo sites and two d electrons, has a filled a1 band and is an insulator (more precisely, a large band-gap semiconductor). 629 Table 24.5 Some alkali metal intercalation compounds of chalcogenides (c) Intercalation and insertion Compound /pm Key points: Insertion compounds can be formed from the d-metal disulfides either by direct reaction or electrochemically; insertion compounds can also be formed with molecular guests. K1.0ZrS2 160 Na1.0TaS2 117 K1.0TiS2 192 Na0.6MoS2 135 K0.4MoS2 214 Rb0.3MoS2 245 Cs0.3MoS2 366 We have already introduced the idea that alkali metal ions may insert between graphite sheets (Section 14.5), metal disulfide slabs (Section 19.9), and metal oxide layers (as in LixCoO2, Section 24.7h) to form intercalation compounds. For a reaction to qualify as an intercalation, or as an insertion reaction, the basic structure of the host should not be altered when it occurs. Reactions in which the structure of one of the solid starting materials is not radically altered are called topotactic reactions. They are not limited to the type of insertion chemistry we are discussing here. For example, hydration, dehydration, and ion exchange reactions may also be topotactic. The π conduction and valence bands of graphite are contiguous in energy (we have seen in fact that graphite is formally a semimetal, Section 3.19) and the favourable Gibbs energy for intercalation arises from the transfer of an electron from the alkali metal atom to the graphite conduction band. The insertion of an alkali metal atom into a dichalcogenide involves a similar process: the electron is accepted into the d band and the charge-compensating alkali metal ion diffuses to positions between the slabs. Some representative alkali metal insertion compounds are listed in Table 24.5. The insertion of alkali metal ions into host structures can be achieved by direct combination of the alkali metal and the disulfide: The change in interlayer spacing as compared with the parent MS2 phase. e– Li 800ºC ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → Na x TaS2 (s) TaS2 (s) + x Na(g) ⎯ 2LiVS2 (s) + I2 (s) → 2LiI(s) + 2 VS2 (s) Insertion compounds also can be formed with molecular guests. Perhaps the most interesting guest is the metallocene Co(5-Cp)2, where Cp C5H5 (Section 22.14), which can be incorporated into a variety of hosts with layered structures, such as TiS2, TiSe2, and TaS2, to the extent of about 0.25 Co(5-Cp)2 per MS2 or MSe2. This limit appears to correspond to the space available for forming a complete layer of Co(5-Cp)2 ions. The organometallic compound appears to undergo oxidation upon intercalation, so the favourable Gibbs energy in these reactions arises in the same way as in alkali metal intercalation. In agreement with this interpretation, Fe(5-Cp)2, which is more difficult to oxidize than its Co analogue, does not intercalate (Section 22.19). We can imagine the insertion of ions into one-dimensional channels, between two-dimensional planes of the type we have been discussing, or into channels that intersect to form three-dimensional networks (Fig. 24.43). Aside from the availability of a site for a guest to enter, the host must provide a conduction band of suitable energy to take up R Li Li(s) MS2 + LixMS2 Li+(aq) + e– MS2 + x Li+ + x e– LixMS2 Figure 24.41 Schematic experimental arrangement for electrointercalation. A polar organic solvent (such as polypropene carbonate) containing an anhydrous lithium salt is used as an electrolyte. R is a reference electrode and CP is a coulometer (to measure the charge passed) and a potential controller. 2.6 Potential difference/V with 0.4 x 0.7. Insertion may also be achieved by using a highly reducing alkali metal compound, such as butyllithium, or the electrochemical technique of electrointercalation (Fig. 24.41). One advantage of electrointercalation is that it is possible to measure the amount of alkali metal incorporated by monitoring the current (I) passed during the synthesis (using ne It/F). It also is possible to distinguish solid-solution formation from discrete-phase formation. As illustrated in Fig. 24.42, the formation of a solid solution is characterized by a gradual change in potential as intercalation proceeds. In contrast, the formation of a new discrete phase yields a steady potential over the range in which one solid phase is being converted into the other, followed by an abrupt change in potential when that reaction is complete. Insertion compounds are examples of mixed ionic and electronic conductors. In general, the insertion process can be reversed either chemically or electrochemically. This reversibility makes it possible to recharge a lithium cell by removal of Li from the compound. In a clever synthetic application of these concepts, the previously unknown layered disulfide VS2 can be prepared by first making the known layered compound LiVS2 in a high-temperature process. The Li is then removed by reaction with I2 to produce the metastable layered VS2, which has the TiS2 structure: e– CP 2.4 2.2 2.0 1.8 1.6 0 0.2 0.4 0.6 0.8 1 Composition, x in LixTiS2 Figure 24.42 Potential versus composition diagram for the electrointercalation of lithium into titanium disulfide. The composition, x, in LixTiS2 is calculated from the charge passed in the course of electrointercalation. 630 24 Solid-state and materials chemistry electrons reversibly (or, in some cases, be able to donate electrons to the host). Table 24.6 illustrates that a wide variety of hosts are possible, including metal oxides and various ternary and quaternary compounds. We see that intercalation chemistry is by no means limited to graphite and layered disulfides. 24.11 Chevrel phases and chalcogenide thermoelectrics Key point: A Chevrel phase has a formula such as Mo6X8 or AXMo6S8, where Se or Te may take the place of S and the intercalated A atom may be a variety of metals such as Li, Mn, Fe, Cd, and Pb. (a) (b) (c) Figure 24.43 Schematic representation of host materials for intercalation reactions. (a) A three-dimensional host with intersecting channels. (b) a two-dimensional layered compound, and (c) a host containing one-dimensional channels. Table 24.6 Some three-dimensional intercalation compounds Phase Composition, x Lix[Mo6S8] 0.652.4 Nax[Mo6S8] 3.6 Nix[Mo6Se8] 1.8 HxWO3 00.6 HxReO3 01.36 Mo We close this section on sulfide materials with a brief discussion of an interesting class of ternary compounds first reported by R. Chevrel in 1971. These compounds, which illustrate three-dimensional intercalation, have formulas such as Mo6X8 and AxMo6S8; Se or Te may take the place of S and the intercalated A atom may be a variety of metals such as Li, Mn, Fe, Cd, or Pb. The parent compounds Mo6Se8 and Mo6Te8 are prepared by heating the elements at about 1000ºC. A structural unit common to this series is M6S8, which may be viewed as an octahedron of M atoms face-bridged by S atoms, or alternatively as an octahedron of M atoms in a cube of S atoms (Fig. 24.44). This type of cluster is also observed for some halides of the Periods 4 and 5 early d-block elements, such as the [M6X8]4 cluster found in Mo and W dichlorides, bromides, and iodides. Figure 24.45 shows that in the three-dimensional solid the Mo6S8 clusters are tilted relative to each other and relative to the sites occupied by intercalated ions. This tilting allows a secondary donor–acceptor interaction between vacant Mo 4dz2 orbitals (which project outward from the faces of the Mo6S8 cube) and a filled donor orbital on the S atoms of adjacent clusters. One of the physical properties that has drawn attention to the Chevrel phases is their superconductivity. Superconductivity persists up to 14 K in PbMo6S8, and it also persists to very high magnetic fields, which is of considerable practical interest because many applications involve high fields (over 25 T), for example, for the next generation of NMR instruments. In this respect the Chevrel phases appear to be significantly superior to the newer oxocuprate high-temperature superconductors. Further possible application of Chevrel phases is in the thermoelectric devices used to convert heat into electrical energy or in devices that use electrical energy directly for cooling purposes. The ideal thermoelectric materials have good electrical conductivity and with low thermal conductivities. This requirement often involves designing a material with a combination of structural elements: one that allows for rapid electron transport, a property associated with crystalline solids where there is little electron scattering by the regular placement of atoms, and a second disordered or glassy structural feature, which scatters the vibrational modes responsible for heat transport, thus producing low thermal conductivity. In Chevrel phases the ability to incorporate various cations between the M6X8 blocks that ‘rattle’ around on their sites reduces the material’s thermal conductivity but not at the expense of electronic conductivity. S Pb Figure 24.44 The Mo6S8 unit present in a Chevrel phase PbxMo6S8. Figure 24.45 The structure of a Chevrel phase showing the canted Mo6S8 units forming a slightly distorted cube around a Pb atom. An Mo atom in one cube can act as the acceptor for an electron pair donated by an S atom in a neighbouring cage. Framework structures Many other metal chalcogenide phases are being investigated for thermoelectric device applications. The compounds Bi2Te3 and Bi2Se3, in common with many metal chalcogenides, have layer-like structures similar to those of the MS2 phases described in Section 24.10b, and devices built on these materials can be designed to produce good electrical conductivity within the layers but poor thermal conductivity perpendicular to them, leading to useful thermoelectric efficiencies. Another important family of thermoelectric materials are the so-called ‘skutterudites’ (named after the mineral skutterudite, CoAs3) with the general formula Mx(Co,Fe,Ni)(P,As,Sb)3 with various inserted cations, M, such as Ln or Na, as in Na0.25FeSb3. The skutterudite structure is similar to that of ReO3 (Section 24.7) although the CoAs6 octahedra are tilted to produce some large cavities within the structure into which the cations may be inserted. These cations reduce the thermal conductivity of the structure by absorbing heat to rattle around within their cavities, while the high electronic conductivity of the (Co,Fe,Ni)(P,As,Sb)3 network is maintained. Framework structures Much of this chapter has concerned structures derived from close-packed anions accompanying d-metal ions typically with coordination number 6. Many of these structures (for example, ReO3, perovskites, and MS2) can also be described in terms of linked polyhedra in which MX6 octahedra connect through their vertices or edges to form a variety of arrays. Coordination number 6 is preferred for most d metals in their typical oxidation states but for smaller metal species (for example, the later 3d series and the lighter p-block metals and metalloids such as Al and Si), fourfold tetrahedral coordination to O is commonplace and leads to structures that are best described as based on linked MO4 tetrahedra. These tetrahedral units may be the only building block present, as in zeolites, or may link together with metal–oxygen octahedra to produce new structural types (Fig. 24.46). Many of these linked polyhedral structures are known as framework structures. 24.12 Structures based on tetrahedral oxoanions As remarked above, the elements able to form very stable tetrahedral MO4 species that can link together into framework structures are the later 3d series and the lighter p-block (a) (a ((b) b)) (c) (c) (c) (f) (e) (d) Figure 24.46 Tetrahedral (a) and octahedral (b) units are linked together through their vertices to form larger units, called secondary building units such as (c) and (d), which in turn bridge through their vertices to form framework structures such as (e) and (f). 631 632 24 Solid-state and materials chemistry metals and metalloids. These ions are so small that they coordinate strongly to four O atoms in preference to higher coordination numbers; the principal examples are SiO4, AlO4, and PO4, although GaO4, GeO4, AsO4, BO4, BeO4, LiO4, Co(II)O4, and ZnO4 are all well known in these structural types. Other tetrahedral units that have been found only rarely in framework structures include Ni(II)O4, Cu(II)O4, and InO4. We concentrate on the structures derived from the most commonly found units in the zeolites (framework aluminosilicates with large pores), aluminophosphates, and phosphates. In zeolites synthesized from solution, where all the vertices of the SiO4 and AlO4 building units are shared between two tetrahedra, Lowenstein’s rule states that no O atom is shared between two AlO4 tetrahedra. Lowenstein’s rule applies only to frameworks synthesized from solution, as directly linked AlO4 tetrahedra are readily obtained in compounds generated from reactions at high temperatures. This indicates that in aqueous solutions the bridging oxygen in a [O3AlOSiO3] unit is stabilized relative to one in a [O3AlOAlO3] unit, presumably due to the higher formal charge on Si. ■ A brief illustration. To determine the maximum Al:Si ratio in a zeolite we note that the maximum Al content is reached when alternate tetrahedra in the structure are those of AlO4 and each vertex is linked to four SiO4 units (and, similarly, each SiO4 tetrahedron is surrounded by four AlO4 units). Thus the highest ratio achievable is 1:1. ■ (a) Contemporary zeolite chemistry Key points: New zeolite framework structures are synthesized by using complex template molecules; important applications of zeolites include gas absorption and ion exchange. (a) (b) Figure 24.47 Representations as linked tetrahedra of synthesized zeolites showing the main channels in (a) UTD-1 and (b) CIT-5. The role of structure-directing agents in the synthesis of zeolitic materials was described in Section 14.15. Following the discovery of a large number of new microporous structures from the 1950s onwards, many of which were prepared in the laboratory using organic templates, more recent work on zeolites has been directed towards the systematic study of template–framework relationships. These studies can be divided into two major categories. One is to understand the interaction between the framework and template through computer modelling and experiment (Section 26.14). The other is to design templates with specific geometries to direct the formation of zeolites with particular pore sizes and connectivity. One particular area that has become the focus of much attention is the use of bulky organic and organometallic molecules as templates in the quest for new, very large pore structures. This approach has been used to make the first zeolitic materials containing 14-ring channels.5 Thus, the microporous silica UTD-1 (Fig. 24.47a) has been prepared by using a permethylated bis-cyclopentadienyl cobalt metallocene and the siliceous CIT-5 (CFI structure type) was prepared by using a polycyclic amine and lithium (Fig. 24.47b). Other more complex amines have been synthesized with the aim of using them as templates in zeolite synthesis and to obtain desirable pore geometries. The addition of fluorides into the zeolite precursor gel improves reaction rates and acts as a template for some of the smaller cage units, for example where linked TO4 (T = Si, Al, P, etc.) tetrahedra arranged at the corners of a cube surround a central F ion. As part of the overall growth in the number of known synthetic zeolites, there has been a significant increase in the proportion that can be made in (essentially) pure silica form so that over 20 structural types of zeolitic silica polymorphs are now known. The use of low H2O/SiO2 ratios is a key factor for producing these materials and the new ‘silica’ phases so produced are of unusually low density, for example a purely siliceous framework with the same topology as the naturally occurring mineral chabazite, Ca1.85(Al3.7Si8.3O24), is the least dense silica polymorph known, with—according to the normal atomic radii—only 46 per cent of the unit cell volume occupied. These pure silica zeolites have no overall framework charge and no extra-framework cations; as a result they are hydrophobic, leading to specific applications in molecular absorption of low polarity molecules and catalysis (Section 26.14). The experimental determination of the location of template molecules by X-ray and neutron diffraction has proved important for establishing template–framework relationships. Examples of such work include establishing the location of the templating ions in 5 An n-ring channel (in this case n 14) refers to the number of tetrahedral units (MO4) linked together to deﬁne the circumference of the channel: the larger n is, the greater is the diameter of the channel. Framework structures Figure 24.48 Calculated and experimental position of a pyrrolidinium cation in a zeolite cavity. The inset shows how the shape of the cavities replicates the shape of the templating amine cation. fluoride–silicalite at the channel intersections (Fig. 24.48). These approaches to the determination of structure are often used in association with computer modelling (Chapter 8). Much of the attention directed at new zeolites focuses on obtaining ever larger pore sizes or a particular pore geometry, with the eventual aim of improving their catalytic properties (Section 26.14). For example, larger pore sizes would allow larger, more complex organic molecules to undergo transformations inside the zeolite cavity. However, we should also consider the other main applications of zeolites, which are as absorbents for small molecules and as ion exchangers. Zeolites are excellent absorbents for most small molecules such as H2O, NH3, H2S, NO2, SO2, and CO2, linear and branched hydrocarbons, aromatic hydrocarbons, alcohols, and ketones in the gas or liquid phase. Zeolites with different sized pores may be used to separate mixtures of molecules based on size, and this application has led to their description as molecular sieves. Through the correct selection of zeolite pore, it is possible to control the rates of diffusion of various molecules with different effective diameters, leading to separation and purification. Figure 24.49 illustrates this application schematically. Pore diameter, d/pm 1000 (a) 800 600 400 200 NaY SF6 CCl4 CH(CH3)3 C5H12 CaA C6H6 NaA CH4 H2O C2H6 CO CO2 NH3 CH3OH N2 CH3NH2 Kr (b) Figure 24.49 The use of porous zeolite structures for the separation of molecules of various sizes. (a) Only the smaller molecule can diffuse into and then out of the zeolite pore so a membrane of this material may be used to separate this mixture. (b) The maximum molecular size that can be absorbed into various zeolite channel diameters. NaY is faujasite, a moderately large pore zeolite. 633 634 24 Solid-state and materials chemistry Industrial applications of zeolites for separation and purification include petroleum refining processes, where they are used to remove water, CO2, chlorides, and mercury, the desulfurization of natural gas (where the removal of H2S and other sulfurous compounds protects transmission pipelines and removes the undesirable smell from home supplies), the removal of H2O and CO2 from air before liquefaction and separation by cryogenic distillation, and the drying and the removal of odours from pharmaceutical products. One area of growing importance is the separation of air into its main components other than by cryogenic means. Many dehydrated zeolites adsorb N2 more strongly into their pores than O2 and this is believed to be due to a stronger interaction of the quadrupolar nitrogen (14N, I 1) compared to oxygen (16O, I 0) with the cations in the zeolite pores. By passing air over a bed of zeolite at controlled pressure it is possible to produce oxygen of over 95 per cent purity. Zeolites such as NaX (the sodium-exchanged form of an aluminium-rich zeolite of structure type X) and CaA, a calcium-exchanged form of zeolite framework structure type A, also known as LTA, were originally developed for this purpose. The selectivity of the process can be much improved by exchanging selected cations into the zeolite pores and thus changing the dimensions of the sites onto which the nitrogen and oxygen molecules adsorb. Thus lithium-, calcium-, strontium-, and magnesium-exchanged forms of the faujasite (FAU, type X) and Linde Type A (LTA) structures are now used very effectively in this process. The excellent ion-exchange properties of zeolites result from their open structures and ability to trap significant quantities of cations selectively within these pores. High capacities for ion exchange are derived from the large numbers of exchangeable cations. This is especially true of those zeolites with a high proportion of Al in the framework, which includes the zeolites with the LTA and gismondine (GIS) topologies that have the highest attainable Si:Al ratio of 1:1. The ion-exchange selectivity has led to a major application of zeolites as a ‘builder’ in laundry detergents, where they are used to remove the ‘hard’ ions Ca2 and Mg2 and replace them with ‘soft’ ions such as Na. Phosphates, which may also be used as detergent builders, have been the subject of environmental concerns linked to rapid algal growth and eutrophication of natural waters (Section 5.15). Spherical Na-form LTA zeolite particles, a few micrometres in diameter, are small enough to pass through the openings in the weave of clothing and so may be added to detergents to remove the hard cations in natural waters, and are then washed away harmlessly into the environment. A different zeolite framework, zeolite P with the GIS framework topology, has been developed as an alternative builder, with improved properties associated with its extremely high selectivity for Ca2 ions. Another important area where the ion-exchange properties of zeolites are exploited is in the trapping and removal of radionuclides from nuclear waste. Several zeolites, including the widely used clinoptilolite, have high selectivities for the larger alkali metal and alkaline earth metal cations, which in nuclear waste include 137Cs and 90Sr (Fig. 24.50). These zeolites may be vitrified by further reaction with glass-forming oxides, as discussed in Section 24.8. (b) Aluminophosphates Key point: The structures and physical properties of aluminophosphates parallel those of zeolites. Cs+ Figure 24.50 The clinoptilolite structure highlighting the relationship between the trapped Cs ions, shown as spheres with ionic radius 180 pm, and the framework. The structural and electronic equivalence of two silicate tetrahedra (SiO4)2 and the aluminophosphate unit (AlO4PO4) can be recognized in the simple compounds SiO2 and AlPO4, both of which adopt a similar range of dense polymorphs, including the quartz structure. The development of zeolites with very high Si content, which are effectively silica polymorphs, in turn led to the discovery of the aluminophosphate (ALPO) framework structures based on a 1:1 mixture of AlO4 and PO4 tetrahedra; AlPO4 itself has the same structure as quartz (SiO2) but with an arrangement of alternating Al and Si atoms at the centres of the tetrahedra. A wide range of ALPOs has been developed that parallel the zeolites in, for example, their synthesis under hydrothermal conditions (although in acid conditions rather than the basic ones used for zeolites) and their adsorption and catalytic properties. Again, organic template molecules have been designed and used to prepare many different ALPO structures, such as those with the structure codes VPI (with a large channel formed from 18 AlO4/PO4 tetrahedra), DAF, CIT, and STA (Fig. 24.51). Although aluminophosphate frameworks are neutral, substitution of either Al(III) or P(V) with metal ions of lower charge leads to the formation of ‘solid acid catalysts’ that can, for example, convert Framework structures (a) (b) Figure 24.51 The frameworks, formed from linked oxotetrahedra, of (a) DAF and (b) CIT. In each case the main channels present are emphasized by viewing the structure along them. methanol into hydrocarbons selectively, although the aluminosilicate zeolites, particularly ZSM-5, remain the best materials for this application (Section 26.14). The incorporation of Co and Mn, both of which have redox properties, into ALPO frameworks yields materials that can be used for the oxidation of alkanes. (c) Phosphates and silicates Key point: Calcium hydrogenphosphates are inorganic materials used in bone formation. Another tetrahedral oxoanion that is frequently incorporated into framework materials is the phosphate group, PO43, although, as we shall see in the next section, many other tetrahedral units also form such structures. Simple phosphate structures are described in Section 15.15 and are generally formed from linked PO4 tetrahedra in chains, cross-linked chains, and cyclic units. We consider just one metal phosphate material in more detail here, namely calcium hydrogenphosphate, and closely related materials. The principal mineral present in bone and teeth is hydroxyapatite, Ca5(OH)(PO4)3, the structure of which consists of Ca2 ions coordinated by PO43 and OH groups to produce a rigid three-dimensional structure (Fig. 24.52). The mineral apatite is the partially fluoride-substituted Ca5(OH,F)(PO4)3. Related biominerals are Ca8H2(PO4)6 and amorphous forms of calcium phosphate itself. Biominerals are discussed in more detail in Sections 25.10–12 and Chapter 27. Ca2+ PO43– OH– Figure 24.52 The structure of hydroxyapatite, Ca5(OH)(PO4)3, shown as Ca2 ions coordinated by phosphate and OH ions into a strong three-dimensional structure. The separate O atoms are actually OH ions with the H lying outside the unit cell. 635 636 24 Solid-state and materials chemistry 24.13 Structures based on octahedra and tetrahedra Many metals adopt MO6 octahedral coordination in their oxo compounds. This polyhedral structural unit can be regarded as being formed by locating metal ions in the octahedral holes of a close-packed O2 ion array in, for example, MgO with the rock-salt structure. The MO6 polyhedral building unit may also be incorporated into framework-type structures, often in combination with tetrahedral oxo species. (a) Clays, pillared clays, and layered double hydroxides Key point: Sheet-like structures, found in many metal hydroxides and clays, can be constructed from linked metal oxo tetrahedra and octahedra. Cs O Si F Mg Figure 24.53 Sheet-like structures of the clay hectorite which consist of layers of linked octahedra and tetrahedra centred on, typically, Al, Si, or Mg and separated by cations such as K or Cs. The diameters of the largest pores found in synthetic zeolites are of the order of 1.2 nm. In an attempt to increase this diameter and allow larger molecules to be absorbed into inorganic structures, chemists have turned to mesoporous materials (Section 25.9) and structures produced by ‘pillaring’ (that is, stacking and connecting together two-dimensional materials). The two-dimensional nature of many d-metal disulfides and some of their intercalation compounds are discussed in Sections 19.9 and 24.10. Similar intercalation reactions when applied to aluminosilicates from the clay family allow the synthesis of large-pore materials. Naturally occurring clays such as kaolinite, hectorite and montmorillonite have layer structures like those shown in Fig. 24.53. The layers are constructed from vertex- and edge-sharing octahedra, MO6, and tetrahedra, TO4, and exist as double layer systems (two layers, one formed from octahedra and one from tetrahedra, Fig. 24.53), as in kaolinite, and triple layer (a central layer based on octahedra sandwiched between two tetrahedrabased layers) systems, as in bentonite. The metal atoms, M and T, contained within the layers, which have an overall negative charge, are typically Si (on tetrahedral sites) and Al (occupying octahedral and tetrahedral sites). Small singly and doubly charged ions, such as Li and Mg2, occupy sites between the layers. These interlayer cations are often hydrated and can readily be replaced by ion exchange. Other materials with similar structures are the layered double hydroxides with structures similar to that of Mg(OH)2, the naturally occurring mineral brucite. In the pillaring of clays, the species exchanged into the interlayer region is selected for size. Ions such as alkylammonium ions and polynuclear hydroxometal ions may replace the alkali metal, as shown schematically in Fig. 24.54. The most widely used pillaring spen+ cies are of the polynuclear hydroxide type and include Al13O4 (OH)328+ , Zr4 (OH)16 − n , and Si8O12(OH)8, the first consists of a central AlO4 tetrahedron surrounded by octahedrally coordinated Al3 ions as Al(O,OH)6 species. The pillaring process can be followed by powder X-ray diffraction because it leads to expansion of the interlayer spacing, corresponding to an increase in the c lattice parameter. Once an ion such as Al13O4 (OH)328+ has been incorporated between the layers, heating the modified clay results in its dehydration and the linking of the ion to the layers (Fig. 24.54). The resulting product is a pillared clay with excellent thermal stability to at least 500ºC. The expanded interlayer region can now absorb large molecules in the same way as zeolites. However, because the distribution of pillaring ions between the layers is difficult to control, the pillared clay structures are less regular than zeolites. Despite this lack of uniformity, pillared clays have been widely studied for their potential as catalysts because they act in a similar way to zeolites, as acid catalysts promoting isomerization and dehydration. (b) Advances in inorganic framework chemistry Key point: Enormous structural diversity can be obtained from linked polyhedra and, with the use of templates, can lead to remarkable porous frameworks. The extensive development of aluminosilicates and zeolites has motivated synthetic inorganic chemists to seek similar structural types built from other tetrahedral and octahedral polyhedra for use in ion exchange, absorption, and catalysis. The use of different and larger polyhedra provides more flexibility in the framework topologies, and there is also the potential to incorporate d-metal ions with their associated properties of colour, Framework structures Heat, dehydrate 637 Void Figure 24.54 Schematic representation of pillaring of a clay by ion exchange of a simple monatomic interlayer cation with large polynuclear hydroxometallate followed by dehydration and cross-linking of the layers to form the cavities. TiO6 redox properties, and magnetism. Tetrahedral species that have been incorporated into such frameworks, as well as the aluminate, silicate, and phosphate groups mentioned previously, include ZnO4, AsO4, CoO4, GaO4, and GeO4. Octahedral units are mainly based on d metals and the heavier and larger metals from Groups 13 and 14. Other polyhedral units such as five-coordinate square pyramids also occur, but less commonly. Zeolite analogues are called zeotypes. Ring sizes larger than 12 tetrahedrally coordinated atoms were first seen in metallophosphate systems, and structures with 20 and 24 linked tetrahedra have been made. Aluminophosphates were the first microporous frameworks synthesized that contain polyhedra with coordination numbers greater than 4. Other socalled hypertetrahedral frameworks are now well established, such as the titanosilicate families (which have four-coordinate Si and five- and six-coordinate Ti sites) and a series of octahedral molecular sieves based on linked MnO6 units. Good examples of this structural family are the titanosilicate zeotypes built from SiO4 tetrahedra and various TiOn polyhedra with n 4–6. These compounds are made under the hydrothermal conditions similar to those used for synthesizing many zeolites but by using a source of Ti, such as TiCl4 or Ti(OC2H5)4, that hydrolyses under the basic conditions in the autoclave. Templates may also be used in such media and act as structuredirecting units giving rise to particular pore sizes and geometries. In a typical reaction the titanosilicate ETS-10 (Engelhard TitanoSilicate 10) is prepared by the reaction of TiCl4, sodium silicate, sodium hydroxide, and sometimes a template such as tetraethylammonium bromide, in a sealed polytetrafluorethylene-lined autoclave at between 150 and 230ºC. The number of titanosilicates continues to grow, but two materials, ETS-10 and Na2Ti2O3SiO4.2H2O, are worthy of further consideration here as specific examples of this type of material. ETS-10 is a microporous material built from TiO6 octahedra and SiO4 tetrahedra, with the TiO6 groups linked together in chains (Fig. 24.55a). The structure has 12-membered rings (that is, pores formed from 12-unit polyhedra) in all three directions. SiO4 (a) SiO4 TiO6 K+ (b) Figure 24.55 The structures of (a) the titanosilicate ETS-10, which is built from chains of TiO6 octahedra linked by SiO4 tetrahedra, and (b) K3H(TiO)4(SiO4)3.4H2O, a synthetic analogue of a natural mineral pharmacosiderite. 638 NiO6 24 Solid-state and materials chemistry PO4 Figure 24.56 The framework structure of VSB-1 consists of linked NiO6 octahedra and PO4 tetrahedra. The main channel is bounded by 24 such units and has a diameter such that molecules up to 0.88 nm in diameter may pass through it. (a) Na2Ti2O3SiO4.2H2O also has TiO6 octahedra, but in this structure these octahedra form clusters with four TiO6 units linked by the tetrahedral SiO4 units (as in Fig. 24.55a). This connectivity gives rise to large octagonal pores containing hydrated Na ions. This compound, in common with several other titanosilicates such as K3H(TiO)4(SiO4)3·4H2O, a synthetic analogue of a natural mineral pharmacosiderite (Fig. 24.55b), shows excellent ion-exchange properties, particularly with large cations. These large ions replace Na with very high selectivity, so that, for example, Cs and Sr2 ions may be extracted into the titanosilicate structure from their dilute solutions. This ability has led to the development of these materials for removal of radionuclides from nuclear waste, in which 137Cs and 90Sr are highly active. The aim of much of the recent work on porous framework structures has been to incorporate d-metal ions. Materials that have been synthesized include antiferromagnetic porous iron(III) fluorophosphates with Néel temperatures in the range 10–40 K. These temperatures are relatively high for iron clusters linked by phosphate groups and indicate the presence of moderately strong magnetic interactions. Zeotypic cobalt(II), vanadium(V), titanium(IV), and nickel(II) phosphates have been prepared, including the material known as VSB-1 (VersaillesSanta Barbara), which was the first microporous solid with 24-membered ring tunnels, and is simultaneously porous, magnetic, and an ion-exchange medium (Fig. 24.56). Two other rapidly growing areas of research into framework structures are materials based on sulfides and on the use of small organic molecules to link the metal ions, so giving rise to metal–organic frameworks (MOFs). Tetrahedral coordination is common in simple metal sulfide chemistry, as we saw in the discussion of ZnS in wurtzite and sphalerite (Section 3.9), and some compounds may be considered as containing linked fragments of these structures known as supertetrahedral clusters. For example, [N(CH3)4]4[Zn10S4(SPh)16] contains the isolated supertetrahedral unit Zn10S20 terminated by phenyl groups (Fig. 24.57). Metal–organic frameworks have structures that are based on bidentate or polydentate organic ligands lying between the metal atoms. Examples of simple ligands used to build these often porous frameworks are CN, nitriles, amines, and carboxylates. Many hundreds of these compounds have now been made, and include positively charged frameworks with balancing anions in the cavities, for instance Ag(4,4’-bpy)NO3, and electrically neutral frameworks, such as Zn2(1,3,5-benzenetricarboxylate)NO3.H2O. C2H5OH, from which the H2O molecules may be removed reversibly. The compounds have properties analogous to those of zeolites and of note is the framework of a chromium terephthalate (1,4-benzenedicarboxylate) which has pore diameters of about 3 nm and a specific internal surface area for N2 of over 5000 m2 g1 (Fig. 24.58). (b) Figure 24.57 (a) An isolated supertetrahedral unit of stoichiometry Zn10S20 formed from individual ZnS4 tetrahedra. (b) These supertetrahedral units may be linked together as building blocks to form large porous three-dimensional structures. Figure 24.58 A metal–organic framework (MOF) formed from chromium terephthalate units showing CrO6 octahedra (gold) linked by the organic anions, The large central sphere shows the extent of the pore in this material, which can be filled with solvent or gas molecules. Hydrides and hydrogen-storage materials 639 These materials have potential applications in gas storage, for example of H2 and CO2. The very large pores allow large, complex molecules such as many pharmaceutically active compounds to be inserted into them and a further application is the use of these MOFs as drug delivery agents. Here an active compound, for example the painkiller ibuprofen, is adsorbed within the MOF pore from which it can then be released slowly, where required, in the human body. Hydrides and hydrogen-storage materials The development of materials for use in a future hydrogen-based energy economy is a key challenge facing inorganic chemists. One of the main areas where new materials are needed is in hydrogen storage. High-pressure gas cylinders and liquification routes are unlikely to meet the requirements of some hydrogen-storage applications in, for example, transport or distributed power supplies because of their weight and safety issues. New materials for storing hydrogen will be required to have high capacities, both volumetrically and gravimetrically, while also being relatively cheap. Technical targets that have been proposed for transportation applications are materials that have 6–9 per cent of the system mass as hydrogen and cost a few dollars per kilowatt hour of stored energy. A further requirement is that the hydrogen should become available from the storage system at between 60 and 120°C. Two main approaches to these new materials are being pursued: chemically bound hydrogen in, for example, metal hydrides and new porous or high-surface-area compounds that physisorb hydrogen. In this section the inorganic chemistry behind these two approaches is discussed, together with a description of the latest advances in the materials that have been produced. 24.14 Metal hydrides Key point: Many metal hydrides and complex metal hydrides such as alanates (hydridoaluminates), amides, and borohydrides release H2 when heated and are potential materials for hydrogen storage. Hydrogen forms metal hydrides by reaction with many metals and metal alloys (Section 10.6). In many cases this process can be reversed, liberating H2 and regenerating the metal or alloy, by heating the (complex) metal hydride. These metal hydrides have an important safety advantage over pressurized or liquefied hydrogen (and many physisorbed hydrogen systems), that are also being considered for hydrogen storage, where rapid uncontrolled release of hydrogen could be a concern (Boxes 10.4, 13.3). For hydrogen storage applications compounds that contain hydrogen in combination with the light elements (such as Li, Be, Na, Mg, B, and Al) offer some of the most promising materials, particularly as H:M ratios can reach 2 or more. Simple metal hydride systems have been dealt with in the appropriate descriptive chemistry section of Chapters 10, 11 and 12, including discussion of the Group 1 and Group 2 hydrides. Here we expand on the chemistry of these compounds and the related complex metal hydrides. (a) Magnesium-based metal hydrides Magnesium hydride, MgH2 (rutile structure type, Fig. 24.59) potentially offers a high capacity of 7.7 per cent hydrogen by mass combined with the low cost of magnesium and good reversibility for hydrogen uptake and evolution at high temperatures. However, its decomposition temperature of 300°C under 1 atm of hydrogen gas is uneconomically high and considerable effort has been expended to develop new related materials that have lower hydrogen release temperatures. Doping of MgH2 with a variety of metals has been undertaken to produce the solid solutions Mg1xMxH2±y for M Al, V, Ni, Co, Ti, Ge, and La/Ni mixtures, some of which have slightly lower decomposition temperatures. More promising are changes in the microstructure of MgH2 which are caused by ball-milling (where the solid is shaken vigorously with small hard spheres inside a canister), particularly when done in combination with other materials, such as metal (Ni, Pd) and metal oxides (V2O5, Cr2O3). Ball-milling of MgH2 increases the surface area of the solid by about a factor of 10 and the number of defects in the crystallites, promoting both the adsorption and deadsorption of hydrogen. Mg H Figure 24.59 The structure of MgH2. 640 24 Solid-state and materials chemistry (b) Complex hydrides Complex hydrides include the tetrahydridoaluminates (containing AlH4−), amides, NH2−, and boranates (hydridoborates containing BH4−), which can contain very high levels of hydrogen, for example LiBH4 is 18 per cent hydrogen by mass. For these systems their decomposition to liberate hydrogen is a problem: tetrahydridoborates (borohydrides), for instance, decompose only near 500°C. For some applications that do not require reversibility, the so-called ‘one-pass hydrogen storage systems’, the hydrogen can be liberated by treatment with water. ■ A brief illustration. The molar mass of Li2BeH4 is (2 6.94) 9.01 (4 1.01) g mol−1 26.93 g mol−1 of which 4.04 g mol−1 is H, so the hydrogen content is (4.04 g mol−1/26.93 g mol−1) 100% 15.0%. ■ Both NaAlH4 and Na3AlH6 have good theoretical hydrogen storage capacities, totalling 7.4 and 5.9 per cent by mass, respectively, and low cost. Their structures are based on the tetrahedral AlH4− and octahedral AlH63− complex anions (Fig. 24.60). These systems exhibit poor reversibility in their decomposition reactions and proceed in stages, which is not ideal for applications, particularly as the final amounts of hydrogen are evolved only above 400 °C. Na AlH4 200 C 3 NaAlH4(s) Na3AlH6(s) 2 Al(s) 3 H2(g) yield: 3.6 per cent H2 by mass 260 C 2 Na3AlH6(s) 6 NaH(s) 2 Al(s) 3 H2(g) yield: 1.8 per cent H2 by mass (a) 425 C 2 NaH(s) 2 Na(s) H2(g) yield: 2.0 per cent H2 by mass AlH6 Various additives such as Ti and Zr have been used to prepare doped materials which, in some cases, have enhanced the rates of hydrogenation and dehydrogenation. Ball-milling to produce smaller particle sizes and strained materials also seems to improve rates of hydrogen evolution. The lithium tetrahydridoaluminates LiAlH4 and Li3AlH6 have higher mass percentages of hydrogen than the corresponding sodium compounds and would be attractive for hydrogen storage except for their chemical instability; LiAlH4 initially decomposes easily but the resulting products cannot be rehydrogenated back to LiAlH4. Furthermore, LiH, one of the initial products, loses hydrogen only above 680°C. Lithium nitride, Li3N, reacts with hydrogen to form LiNH2 and LiH: Na (b) Figure 24.60 The structures of (a) NaAlH4 and (b) Na3AlH6 depicting the hydride coordination around aluminium. Li NH2 BH4 Figure 24.61 The structure of Li3(NH2)2BH4. Li3N(s) 2 H2(g) → LiNH2(s) 2 LiH(s) This mixture evolves hydrogen above 230°C, also producing Li2NH, and has a theoretical hydrogen storage capacity of 6 per cent H2 by mass. One problem with nitrides is that partial decomposition to produce ammonia can occur. Lithium tetrahydridoborate, LiBH4, is theoretically a very promising material for hydrogen storage with 18 per cent H2 by mass (Box 10.4), but it does not undergo a reversible reaction involving hydrogen evolution and uptake. The compound Li3Be2H7 has excellent reversibility but only above 150°C. Other complex lithium amide-boranates, such as Li3(NH2)2BH4 (Fig 24.61) and Li4(NH2)2(BH4)2, have been proposed as hydrogen-storage materials and seem to offer reduced evolution of ammonia compared with LiNH2 but with only limited reversibility. Related metal amine complexes, for example Mg(NH3)6Cl2, have also been proposed as ‘indirect’ hydrogen storage materials (where evolved ammonia might be converted in a secondary reaction to hydrogen) as they have high hydrogen contents. (c) Intermetallic compounds Several types of intermetallic compound are being investigated as potential hydrogen-storage systems as, in general, they have excellent reversible hydrogen uptake at low pressures (1–20 bar) at just above room temperature (Table 24.7). However, the mass percentage of Hydrides and hydrogen-storage materials 641 Table 24.7 Intermetallic structure types for hydrogen storage Type Metals Typical hydride composition Mass % H2 peq, T Element Pd PdH0.6 0.56 0.020 bar, 298 K AB5 LaNi5 LaNi5H6 1.5 2 bar, 298 K AB2 (Laves) ZrV2 ZrV2H5.5 3.0 108 bar, 323 K AB FeTi FeTiH2 1.9 5 bar, 303 K A2B Mg2Ni Mg2NiH4 3.6 1 bar, 555 K BCC TiV2 TiV2H4 2.6 10 bar, 313 K The values of peq and T are the pressure and temperature conditions for phase formation/decomposition. hydrogen that can be adsorbed by these materials is relatively low due to the high molar masses of the metals. One system, the AB5 type phases, is typified by LaNi5, which absorbs hydrogen to give LaNi5H6, but this corresponds to only 1.5 per cent hydrogen by mass. A series of alloys termed Laves phases are adopted by some intermetallics of the stoichiometry AB2 where A Ti, Zr, or Ln and B is a 3d metal such as V, Cr, Mn, or Fe. These materials have high capacities and good kinetics for hydrogen adsorption, forming compounds such as ZrFe2H3.5 and ErFe2H5 with up to 2 per cent hydrogen by mass. However, the hydrides of these Laves phases are thermodynamically very stable at room temperature, restricting the reverse desorption of hydrogen. The compound Mg2NiH4, which contains 3.6 per cent hydrogen by mass, is formed by heating the alloy Mg2Ni to 300°C under 25 kbar of hydrogen Mg Mg2Ni(s) 2 H2(g) → Mg2NiH4 The structure of Mg2NiH4 consists, at low temperature, of an ordered array of Mg, Ni, and H but transforms, at high temperature or extended ball-milling, to a cubic phase with H ions randomly distributed throughout the arrangement of Mg and Ni atoms (Fig. 24.62). A third class of intermetallic compounds being studied is the so-called ‘Ti-based BCC alloys’ such as FeTi and alloys of similar composition with various quantities of other d metals, such as Ti/V/Cr/Mn. Hydrogen capacities of near 2.5 per cent by mass can be achieved for these alloys but high temperatures and pressures are needed to reach these values. 24.15 Other inorganic hydrogen-storage materials Key point: High surface area and porous inorganic compounds can adsorb high levels of hydrogen gas. Physisorption on to the surface of very high surface area materials, including any internal pores, offers high hydrogen-storage capacities. However, these high values may be reached only by cooling the system or use of very high pressures, neither of which may be applicable to eventual applications. Some inorganic systems currently being studied include metal alloys that have been templated by using zeolites, inorganic clathrates, and metal-organo frameworks (MOFs, Section 24.13b), all of which have highly porous structures formed from relatively light elements. Carbon in its various forms is also being studied as a hydrogen-storage material. Graphite itself in a nanostructure form (Section 25.7) adsorbs 7.4 mass per cent hydrogen under 1 MPa (10 bar) of hydrogen, although more typically activated carbon/graphite materials develop surface monolayers equivalent to 1.5 to 2 per cent hydrogen by mass. Carbon nanotubes (Section 25.7), with their curved surfaces, tend to show enhanced adsorption of hydrogen in comparison with graphite sheets, with reported values of up to 8 per cent by mass at 77 K. Zeolites (Section 24.12) have also been proposed as possible hydrogen-storage materials and many of the different topologies have been studied. The best framework types include faujasites (FAU, zeolites X and Y), zeolite A (LTA), and chabazite, with maximum capacities of between 2.5 and 2.0 per cent hydrogen by mass. Ni H (partially occupied) Figure 24.62 The idealized structure of cubic Mg2NiH4; hydrogen sites are only partially occupied. 642 24 Solid-state and materials chemistry Ca or Ba SiO4 CuO4 Figure 24.63 The square-planar copper/ oxygen environment formed by Si4O10 groups in Egyptian blue (CaCuSi4O10) and Chinese blue (BaCuSi4O10). Inorganic pigments Many inorganic solids are intensely coloured and are used as pigments in colouring inks, plastics, glasses, and glazes. Whereas many insoluble organic compounds (for example the C.I. Pigment Red 48, which is calcium 4-((5-chloro-4-methyl-2-sulfophenyl) azo)-3-hydroxy-2-naphthalenecarboxylic acid) are also used as pigments, inorganic materials often have advantages in terms of applications associated with their chemical, light, and thermal stability. Pigments were originally developed from naturally occurring compounds such as hydrated iron oxides, manganese oxides, lead carbonate, vermilion (HgS), orpiment (As2S3), and copper carbonates. These compounds were even used in prehistoric cave paintings. Synthetic pigments, which are often analogues of naturally occurring compounds, were developed by some of the earliest chemists and alchemists, and the first synthetic chemists were probably those involved in making pigments. Thus, the pigment Egyptian blue (CaCuSi4O10) was made from sand, calcium carbonate, and copper ores as long as 3000 years ago. This compound and a structural analogue, Chinese blue (BaCuSi4O10), which was first made about 2500 years ago, have a structure containing square-planar copper(II) ions surrounded by Si4O10 groups (Fig. 24.63). Inorganic pigments continue to be important commercial materials and this section summarizes some of the recent advances in this field. As well as producing the colours of inorganic pigments as a result of the absorption and reflection of visible light, some solids are able to absorb energy of other wavelengths (or types, for example electron beams) and emit light in the visible region. This luminescence is responsible for the properties of inorganic phosphors (Box 24.2). 24.16 Coloured solids Key point: Intense colour in inorganic solids can arise through dd transitions, charge transfer (and the analogous interband electron transfer), or intervalence charge transfer. B OX 2 4 . 2 Inorganic phosphors Luminescence is the emission of light by materials that have absorbed energy in some form. Photoluminescence occurs when photons, usually in the ultraviolet region of the electromagnetic spectrum, are the source of energy and the output usually is visible light. Cathodoluminescence uses electron beams as a source of energy and electroluminescence uses electrical energy as the energy source. Two types of photoluminescence can be distinguished: fluorescence, which has a period of less than 108 s between photon absorption and emission, and phosphorescence, for which there are much longer delay times (Section 20.7). Photoluminescent materials, frequently referred to as phosphors, generally consist of a host structure, such as ZnS (wurtzite structure; Section 3.9), CaWO4 (scheelite structure type with discrete WO42 tetrahedra separated by Ca2 ions), or Zn2SiO4, into which an activator ion is introduced by doping. These activator ions are certain d-metal or lanthanoid ions, such as Mn2, Cu2, and Eu2, that have the ability to absorb and emit light of the desired wavelengths. In some cases a second dopant is added as a sensitizer to aid the absorption of light of the desired wavelength. Well-known applications of such materials include fluorescent lamps and television screens, where there is a need for materials that fluoresce in specific regions of the visible spectrum. Many host–activator combinations have been studied as potential phosphor materials with the aim of producing a material that efficiently converts UV radiation or cathode rays (electron beams) to pure emission of a desirable colour. The modification of the host structure and the nature and environment of the activator ion allows these properties to be tuned (see table). In many fluorescent lamps, a mercury discharge produces UV radiation at 254 and 185 nm. Then a coating of ZnS that has been doped with various activators produces fluorescence at several wavelengths that combine together to give an effective white light. Phosphor host Activator Colour Zn2SiO4 Mn2 Green CaMg(SiO3)2 diopside Ti Blue CaSiO3 Mn Yellow/orange Ca5(PO4)3(F,Cl) Mn ZnS Orange 2 2 Ag , Cu , Mn Blue, green, yellow Colour television requires three primary-colour, cathodoluminescent materials to produce a picture. These phosphors are typically ZnS:Ag (blue), ZnS:Cu (green), and YVO4:Eu3 (red). In YVO4:Eu3 the vanadate group absorbs the incident electron energy and the activator is Eu3. The emission mechanism involves electron transfer between the vanadate group and Eu3 and the efficiency of this process depends on the MOM bond angle in a similar way to the superexchange process in antiferromagnets (Section 20.8). The nearer this angle is to 180°, the faster and more efficient is the electron transfer process. In YVO4:Eu this angle is 170˚ and this material is a highly efficient phosphor. Anti-Stokes phosphors convert two or more photons of lower energy to one of higher energy (such as infrared radiation to visible light). They act by absorbing two or more photons in the excitation process before emitting one photon. The best anti-Stokes phosphors have ionic host structures such as YF3, NaLa(WO4)2, and NaYF4 doped with Yb3 as a sensitizer ion (to absorb the IR radiation) and Er3 as an activator (emitting visible light). Applications include night-vision binoculars. Inorganic pigments The blue colour of CoAl2O4 and CaCuSi4O10 stems from the presence of dd transitions in the visible region of the electromagnetic spectrum. The characteristic intense colour of cobalt aluminate is a result of having a noncentrosymmetric tetrahedral site for the metal ion, which removes the constraint of the Laporte selection rule of octahedral environments (Section 20.6). The chemical and thermal stabilities are due to the location of the Co2 ion in the close-packed oxide arrangement. Other inorganic pigments with colours based on dd transitions include Ni-doped TiO2 (yellow) and Cr2O3.nH2O (green). Colour also arises in many inorganic compounds from charge transfer (Section 20.5) or what is often electronically an equivalent process in solids, the promotion of an electron from a valence band (derived mainly from anion orbitals) into a conduction band (derived mainly from metal orbitals). Charge-transfer pigments include compounds such as lead chromate (PbCrO4), containing the yellow–orange chromate(VI) anion, and BiVO4 with the yellow vanadate(V) anion. The compounds CdS (yellow) and CdSe (red) both adopt the wurtzite structure and their colour arises from transitions from the filled valence band (which is mainly derived from chalcogenide p orbitals) to orbitals based mainly on Cd. For a material with a band gap of 2.4 eV (as for CdS at 300 K) these transitions occur as a broad absorption corresponding to wavelengths shorter than 515 nm, thus CdS is bright yellow as the blue part of the visible spectrum is fully absorbed. For CdSe the band gap is smaller on account of the higher energies of the Se 4p orbitals and the absorption edge shifts to lower energies. As a result, only red light is not absorbed by the material. In some mixed-valence compounds, electron transfer between differently charged metal centres can also occur in the visible region, and as these transfers are often fully allowed they give rise to intense colour. Prussian blue, [Fe(III)]4[Fe(II)(CN)6]3 (Fig. 24.64), is one such compound and its dark blue colour has resulted in its widespread use in inks. Intensely coloured Ru compounds, such as the tris(carboxyl)-terpyridine complex [Ru(2,2,2-(COOH)3-terpy) (NCS)3], absorb efficiently right across the visible and near IR regions of the spectrum and are used as photosensitizers in Gräztel-type solar cells. Inorganic radicals often have fairly low-energy electronic transitions that can occur in the visible region. Two examples are NO2 (brown) and ClO2 (yellow). One inorganic pigment is based on an inorganic radical, but because of the high reactivity normally associated with main-group compounds containing unpaired electrons, this species is trapped inside a zeolite cage. Thus the royal-blue pigment ultramarine, a synthetic analogue of the naturally occurring semi-precious stone lapis lazuli, has the idealized formula Na8[SiAlO4]6.(S3)2 and contains the S3 polysulfide radical anion occupying a sodalite cage formed by the aluminosilicate framework (Fig. 24.65). Current developments in inorganic pigment chemistry are focused on finding replacements for some of the yellow and red materials that contain heavy metals, such as Cd and Pb. Although these materials themselves are not toxic, as the compounds are very stable and the metal is difficult to leach into the environment, their synthesis and disposal can be problematic. Compounds that have been investigated to replace cadmium chalcogenides and lead-based pigments are the lanthanoid sulfides, such as Ce2S3 (red), and early d-metal oxide-nitrides, such as Ca0.5La0.5Ta(O1.5N1.5) (orange). In both cases, the replacement of O2 by S2 or N3 ion has the effect of narrowing the band gap in these solids (compared to the colourless solids CeO2 and Ca2Ta2O7, which have large band gaps and absorb only in the UV region of the spectrum) and bringing the electron excitation energy into the visible. However, neither of these materials has the stability of the cadmium- and lead-based pigments. 24.17 White and black pigments Some of the most important compounds used to modify the visual characteristics of polymers and paints have visible-region absorption spectra that result in them appearing either white (ideally no absorption in the visible region) or black (complete absorption between 380 and 800 nm). (a) White pigments Key point: Titanium dioxide is used almost universally as a white pigment. White inorganic materials can also be classified as pigments and vast quantities of these compounds are synthesized for applications such as the production of white plastics and paints. Important commercial compounds of this class that have been used extensively CN 643 Fe Figure 24.64 Prussian blue, [Fe(III)]4[Fe(II)(CN)6]3. The iron ions are linked through cyanide, forming a cubic unit cell. – S3 (Al,Si)O4 Figure 24.65 One of the sodalite cages present in ultramarine, Na8[SiAlO4]6.(S3)2. The framework consists of linked SiO4 and AlO4 tetrahedra surrounding a cavity containing the polysulfide radical ion S3 and Na ions (the latter omitted for clarity). 644 O 24 Solid-state and materials chemistry Ti (a) O Ti historically are TiO2, ZnO, and ZnS, lead(II) carbonate, and lithopone (a mixture of ZnO and BaSO4); note that none of the metals in these materials has an incomplete d-electron shell that might otherwise induce colour through dd transitions. Titanium dioxide, TiO2, in either its rutile or anatase forms (Fig. 24.66), is produced from titanium ores, often ilmenite, FeTiO3, by the sulfate process (which involves dissolution in concentrated H2SO4 and subsequent precipitation through hydrolysis) or the chloride process (which is based on the reaction of mixed complex titanium oxides with chlorine to produce TiCl4, which is then combusted with oxygen at over 1000ºC). These routes produce very high quality TiO2 free from impurities (which is essential for a bright white pigment) of controlled particle size. The desirable qualities of TiO2 as a white pigment derive from its excellent light scattering power, which in turn is a result of its high refractive index (nr 2.70), the ability to produce very pure materials of a desired particle size, and its good light-fastness and weather resistance. Uses of titanium dioxide, which nowadays dominates the whitepigment market, include paints, coatings, and printing ink (where it is often used in combination with coloured pigments to increase their brightness and hiding power), plastics, fibres, paper, white cements, and even foodstuffs (where it can be added to icing sugar, sweets, and flour to improve their brightness). (b) Black, absorbing, and specialist pigments Key points: Special colour, light absorbing, and interference effects can be induced in inorganic materials used as pigments. (b) Figure 24.66 TiO2 exists as several polymorphs that can be described in terms of linked TiO6 octahedra, including the (a) rutile and (b) anatase forms. The most important black pigment is carbon black, which is a better defined, industrially manufactured form of soot. Carbon black is obtained by partial combustion or pyrolysis (heating in the absence of air) of hydrocarbons. The material has excellent absorption properties right across the visible region of the spectrum and applications include printing inks, paints, plastics, and rubber. Copper(II) chromite, CuCr2O4, with the spinel structure (Fig. 24.18), is used less frequently as a black pigment. These black pigments also absorb light outside the visible region, including the infrared, which means that they heat up readily on exposure to sunlight. Because this heating can have drawbacks in a number of applications, there is interest in the development of new materials that absorb in the visible region but reflect infrared wavelengths; Bi2Mn4O10 is one compound that exhibits these properties. Examples of more specialist inorganic pigments are magnetic pigments based on coloured ferromagnetic compounds such as Fe3O4 and CrO2, and anticorrosive pigments such as zinc phosphates. The deposition of inorganic pigments as thin layers on to surfaces can produce additional optical effects beyond light absorption. Thus deposition of TiO2 or Fe3O4, as thin layers a few hundred nanometres thick, on flakes of mica produces lustrous or pearlescent pigments where interference effects between light scattered from the various surfaces and layers produces shimmering and iridescent colours. Semiconductor chemistry The basic inorganic chemistry of semiconducting materials, particularly their electronic band structures, was covered in Chapter 3. The aim of this section is to discuss the inorganic semiconducting compounds themselves in more detail and to describe some of the applications that result from using chemistry to control their electronic properties. Semiconductors are classified on the basis of their composition. To produce materials with band gaps typical of a semiconductor (a few electronvolts, corresponding to 100–200 kJ mol1), compounds usually contain the p-block metals and Group 13/14 metalloids, often in combination with heavier chalcogenides and pnictides. For these combinations, the atomic orbitals form into bands with energies such that the valence and conduction band separation is in the desired range of 0.2–4 eV. Materials based on more electropositive and electronegative elements (for example, alkaline earth metal oxides) are much more ionic in character with much wider separations of the valence and conduction bands, and are insulators. A further factor that influences the band gap in semiconductors is particle size (Chapter 25). Semiconductor chemistry 24.18 Group 14 semiconductors Key point: Crystalline and amorphous silicon are cheap semiconducting materials and are widely used in electronic devices. The most important semiconducting material is Si, which in its pure crystalline form (with a diamond structure) has a band gap of 1.1 eV. As would be expected from considerations of atomic radii, orbital energies, and the extent of orbital overlap, Ge has a smaller band gap, 0.66 eV, and C as diamond has a band gap of 5.47 eV. When doped with a Group 13 or 15 element, Si—and indeed C and Ge—are extrinsic semiconductors. The conductivity of pure Si, an intrinsic semiconductor, is around 102 S cm1 at room temperature but increases by several orders of magnitude on doping with either a Group 13 element (to give a p-type semiconductor) or a Group 15 element (to give an n-type semiconductor) and thus the properties of doped Si can be tuned for a particular semiconductor application. Amorphous Si can be obtained by chemical vapour deposition or by heavy-ion bombardment of crystalline Si. The deposited material, which is often obtained by thermal decomposition of SiH4, contains a small proportion of H that is present in SiH groups in a three-dimensional glass-like structure with many SiSi links. The lack of regular structure in this material and the presence of SiH groups alters the semiconducting properties of the material considerably. One of the main applications of amorphous Si is in silicon solar cells. Thin films of p- and n-type amorphous Si forming a p–n junction generate current when illuminated (Box 24.3). The electron and hole pairs produced from the energy supplied by the incident photon separate rather than recombine because of the normal p–n junction bias, with the tendency for the electrons to travel towards the p-type Si and holes towards the n-type. If a load is connected across the junction then a current can flow and electrical energy is generated from the electromagnetic illumination. The efficiency of such devices depends on a number of factors. For example, amorphous Si absorbs solar radiation 40 times more efficiently than does single-crystal Si, so a film only about 1 μm thick can absorb 90 per cent of the usable solar energy. Also, the lifetimes and mobility of the electrons and holes are longer in amorphous Si, which results in high photoelectric efficiencies (the proportion of radiant energy converted into electrical energy) of the order of 10 per cent. Other economic advantages are that amorphous Si can be produced at a lower temperature and can be deposited on low-cost substrates. Amorphous Si solar cells are widely used in pocket calculators but, as production costs diminish, are likely to find much wider applications as renewable energy devices. 24.19 Semiconductor systems isoelectronic with silicon Key points: Semiconductors formed from equal amounts of Group 13/15 or Group 12/16 elements are isoelectronic with silicon and can have enhanced properties based on changes in the electronic structure and electron motion. Gallium arsenide, GaAs, is one of a number of so-called Group 13/15 (or, still more commonly, III/V semiconductors), which also include GaP, InP, AlAs, and GaN, formed by combination of equal amounts of a Group 13 and a Group 15 element. Ternary and quaternary Group 13/15 compounds, such as AlxGa1xAs, InAs1yPy, and InxGa1xAs1yPy, can also be formed and many of them also have valuable semiconducting properties. Note that these compositions are isoelectronic with pure Group 14 elements, but the changes in the element electronegativity and thus bonding type (for instance, pure Si can be considered as having purely covalent bonding whereas GaAs has a small degree of ionic character due to the difference in the electronegativities of Ga and As) leads to changes in the band structures and fundamental properties associated with electron motion through the structures. One of the advantageous properties of GaAs is that semiconductor devices based on it respond more rapidly to electrical signals than those based on silicon. This responsiveness makes GaAs better than silicon for a number of tasks, such as amplifying the high-frequency (1–10 GHz) signals of satellite TV. Gallium arsenide can be used with signal frequencies up to about 100 GHz. At even higher frequencies materials such as indium phosphide (InP) may be used. At present, frequencies above about 50 GHz are rarely used commercially, so most of the electronics in the world tend to be based on silicon, with some GaAs, and only a few InP devices. Gallium arsenide is also far more expensive than Si, both in terms of the cost of raw materials and the chemical processes required to produce the pure material. 645 646 24 Solid-state and materials chemistry B OX 2 4 . 3 p–n Junctions and LEDs A p–n junction can be constructed from two pieces of silicon, one of which is n-type and the other p-type (Fig B24.2), where the illustration shows the band structure of the junction. The Fermi levels in the differently doped materials are different but when they are placed in contact electrons will flow from the n-type (high potential) to the p-type (low potential) region across the junction so as to reach an equilibrium distribution in which the Fermi levels are equal. If a potential difference is applied in the right direction across the junction this process will continue, with electrons able to move from n-type to p-type and a current flows in this direction. However, current can flow only in this direction, from n-type to p–type. The p–n junction thus forms the basis of a rectifier, allowing current to pass in only one direction. A light-emitting diode (LED) is a p–n junction semiconductor diode that emits light when current is passed. The transfer of the electron from the conduction p-Type n-Type band of the n-type material to the valence band of the p-type semiconductor is accompanied by the emission of light. LEDs are highly monochromatic, emitting a pure colour in a narrow frequency range. The colour is controlled by the band gap, with small band gaps producing radiation in the infrared and red regions of the electromagnetic spectrum and larger band gaps resulting in emission in the blue and ultraviolet regions (see table). It is possible to produce white light with a single LED by using a phosphor layer (yttrium aluminium garnet) on the surface of a blue, gallium nitride, LED. These white light LEDs are highly efficient at converting electricity into light, much more so than incandescent lamps and even ‘low-energy’ fluorescent lights (Box 24.2). At present they are more expensive to produce than fluorescent light bulbs and their use is limited to small-scale battery driven devices, but they are likely to form the basis of many lighting products of the future. Another advantage of being able to produce blue LEDs is that they can be used as lasers in high-capacity optoelectronic storage devices. The wavelength of blue light produced by these LEDs (405 nm) is shorter than that used in DVD format devices (red light, 650 nm), which allows data to be written in smaller bits on an optical disc. Conduction band LED colours Dopant band – e Dopant band Valence band Figure B24.2 The structure of a p–n junction. LED colour Chip material Low brightness High brightness Red GaAsP/GaP AlInGaP Orange GaAsP/GaP AlInGaP Amber GaAsP/GaP AlInGaP Yellow GaP Green GaP GaN Turquoise GaN Blue GaN In some Group 13/15 semiconductors, such as GaN, the cubic, diamond-like, sphalerite structure is only metastable, the stable polymorph being the hexagonal, wurzite structure. Both structures can be grown by altering the synthetic routes and conditions. Because of their large and direct energy band gaps, these semiconductors allow the fabrication of luminescent devices that produce blue light at high intensity (Box 24.3), and their stabilities to high temperatures and good thermal conductivities also make them valuable for the fabrication of high-power transistors. The Group 12/16 (II/VI) semiconductors comprise the compounds containing Zn, Cd, and Hg as cations and O, S, Se, and Te as anions. These semiconductor materials can crystallize in either the cubic sphalerite phase or the hexagonal wurzite phase and the form synthesized has characteristic semiconducting properties. For example, the band gap is 3.64 eV in cubic ZnS but 3.74 eV in hexagonal ZnS. These Group 12/16 compounds are more ionic in nature than the Group 13/15 semiconductors and Group 14 elements, particularly for the lighter elements, and the band gap is around 3–4 eV for ZnO and ZnS but 1.475 eV for CdTe. Although amorphous Si is the leading thin-film photovoltaic (PV) material, cadmium telluride (CdTe) is also being studied for similar applications. Some other semiconducting oxides and sulfides were mentioned in Section 3.20, and research continues to seek other complex metal oxides and chalcogenides with improved characteristics. For example, the current world record thin-film solar cell efficiency of 17.7 per cent is held by a device based on copper indium diselenide (CuInSe2; CIS), which has a similar structure to cubic ZnS but with an ordered distribution of Cu and In atoms in the tetrahedral holes. Molecular materials and fullerides 647 Molecular materials and fullerides The majority of compounds discussed so far in this chapter have been materials with extended structures in which ionic or covalent interactions link all the atoms and ions together into a three-dimensional structure. Examples include infinite structures based on ionic, as in NaCl, or covalent interactions, as in SiO2. These materials are widely used in applications such as heterogeneous catalysis, rechargeable batteries, and electronic devices due to the chemical and thermal stability that derives from their linked structures. It is often possible to tune the properties of many solids exactly, for example by doping, introducing defects, or forming solid solutions. However, control of the arrangement of atoms into a particular structure cannot be achieved to the same degree for solids as for molecular systems. Thus a coordination or organometallic chemist can modify a molecule by introducing a wide range of ligands, often through simple substitution reactions. The desire to combine the synthetic and chemical flexibility of molecular chemistry with the properties of classical solid-state materials has led to the rapid emergence of the area of molecular materials chemistry, where functional solids are produced from linked and interacting molecules or molecular ions. 24.20 Fullerides Figure 24.67 The arrangement of C60 molecules in a face-centred cubic lattice in the crystalline material. Key points: Solid C60 can be considered as a close-packed array of fullerene molecules interacting only weakly through van der Waals forces; holes in arrays of C60 molecules may be filled by simple and solvated cations and small inorganic molecules. The chemical properties of C60 span many of the conventional borders of chemistry and include the chemistry of C60 as a ligand (Section 14.6). In this section we describe the solidstate chemistry of solid fullerene, C60(s), and the MnC60 fulleride derivatives that contain n molecular anions. The synthesis and chemistry of the more complex carbon discrete C60 nanotubes are discussed in Section 25.7. Crystals of C60 grown from solution may contain included solvent molecules, but with the correct crystallization and purification methods—for example, using sublimation to eliminate the solvent molecules—pure C60 crystals may be grown. The solid structure has a face-centred cubic array of the C60 molecules as shown in Fig. 24.67, as would be expected on the basis of efficient packing of these almost spherical molecules. At room temperature the molecules can rotate freely on their lattice positions and powder X-ray diffraction data collected from crystalline C60 are typical of an fcc lattice with a lattice parameter of 1417 pm. The molecules are separated by a distance of 296 pm, which is similar to the value found for the interlayer separation in graphite (335 pm). On cooling the solid, the rotation halts and adjacent molecules align relative to each other such that an electron-rich region of one C60 molecule is close to an electron-poor region in its neighbour. Exposure of solid C60 to alkali metal vapour results in the formation of a series of compounds of formula MxC60, the precise stoichiometry of the product depending on the composition of the reactant mixture. With excess alkali metal, compounds of composition M6C60, M K, Rb, Cs, and sometimes Na and Li, are formed. The structure of K6C60 is body-centred cubic; C606 molecular ions occupy sites at the cell corners and body centre and the K ions fill a portion of the sites, with approximately tetrahedral coordination to four C60 molecular ions, near the centre of each of the faces (Fig. 24.68). Of most interest are the compounds with the stoichiometries M3C60, which become superconducting in the temperature range 10–40 K depending on the type of metal. The stoichiometry K3C60 is obtained by filling all the tetrahedral and all the octahedral holes in the cubic close3 arrangement (Fig. 24.69). K3C60 becomes superconducting on cooling to 18 K, packed C60 although gradual replacement of K by the larger alkali metal ions raises Tc, so that for Rb3C60 Tc 29 K and for CsRb2C60 Tc 33 K. Note that Cs3C60 does not form the same fcc structure as the other M3C60 phases (in fact it has a structure based on a body-centred 3 anions) and is not superconducting at normal pressures; however, it arrangement of C60 can be made superconducting, with a critical temperature of 40 K at 12 kbar. Other more complex species can be incorporated into a matrix of C60 units. Molecular species such as iodine (I2) or phosphorus molecules (P4 tetrahedra) can fill spaces between the C60 molecules in close-packed arrays. Solvated cations can also occupy the tetrahedral and octahedral holes in a similar way to the simple alkali metal cations. K+ Figure 24.68 The structure of K6C60 with 6 a body-centred cubic unit cell with C60 molecular ions at the cell corners and body centre, and K ions occupying half of the sites in the faces of the cell which have approximate tetrahedral coordination to four C60 molecular ions. K+ Figure 24.69 The structure of K3C60 is obtained by filling all the tetrahedral and all 3 the octahedral holes in the close-packed C60 lattice with K ions. 648 24 Solid-state and materials chemistry Thus, Na(NH3)4CsNaC60, which is obtained by the ammoniation of Na2CsC60, contains Na ions solvated with ammonia molecules on the octahedral site and uncoordinated Cs 6 and Na ions on the (twice as abundant) tetrahedral sites in an fcc arrangement of C60 molecular ions. 24.21 Molecular materials chemistry The ability to modify the shapes, and thus the packing and arrangements of inorganic molecules in the solid state, is one valuable aspect of molecular materials chemistry. This capability, when associated with some of the specific properties of inorganic compounds such as the unpaired d electrons of d metals, can allow control of magnetic and electronic properties. This section considers a number of such inorganic molecular materials being developed at the frontiers of the subject. (a) One-dimensional metals Key points: A stack of molecules that interact with each other along one dimension, as occurs in a number of crystalline platinum complexes, can show conductivity in that direction; a Peierls distortion ensures that no one-dimensional solid is a metallic conductor below a critical temperature. Pt CN Figure 24.70 A representation of the infinite chain structure of KCP (K2Pt(CN)4Br0.3.3H2O) and a schematic illustration of its d band. (a) (b) Figure 24.71 The formation of a Peierls distortion; the energy of the line of atoms with alternating bond lengths (b) is lower than that of the uniformly spaced atoms (a). A one-dimensional metal is a material that exhibits metallic properties along one direction in the crystal and nonmetallic properties orthogonal to that direction. Such properties arise when the orbital overlap occurs along a single direction in the crystal (as in VO2). Several classes of one-dimensional metals are known and include (SN)x and organic polymers, such as doped polyacetylenes [(CH)I0.25]n, but this section is concerned specifically with chains of interacting d metals, particularly Pt. In such materials, the structural requirements for a one-dimensional metal are satisfied by the presence of square-planar complexes that stack one above another (Fig. 24.70). The ligands surrounding the metal atom ensure large interchain separations, of at least 900 pm, while the average intrachain metal–metal distance is less than 300 pm. Square-planar complexes are commonly found for metal ions with d8 configurations, and the overlap of orbitals between d8 species is greatest for the heavy d metals of Period 6 (which use 5d orbitals). Hence the compounds of interest are mainly associated with Pt(II) and Ir(I), where a band is formed from overlapping dz2 and pz orbitals. The dz2 band is full for Pt(II), and a partially full level is achieved by oxidation of the platinum. Many d8 integral oxidation number tetracyanoplatinate(II) complexes are semiconductors with dPtPt 310 pm, and the partial oxidation of these salts results in PtPt distances of less than 290 pm and metallic behaviour. Typical means of oxidizing the chain are the incorporation of extra anions into the structure or the removal of cations. The first one-dimensional metal Pt complex was made in 1846 by oxidation of a solution of K2Pt(CN)4.3H2O with bromine, which on evaporation gave crystals of K2Pt(CN)4Br0.3.3H2O, known as KCP (as in Fig. 24.70). The electronic properties of one-dimensional solids are not quite as simple as has been implied by the discussion so far, as a theorem due to Rudolph Peierls states that, at T 0, no one-dimensional solid is a metal! The origin of Peierls’ theorem can be traced to a hidden assumption in the discussion so far: we have supposed that the atoms lie in a line with a regular separation. However, the actual spacing in a one-dimensional solid (and any solid) is determined by the distribution of the electrons, not vice versa, and there is no guarantee that the state of lowest energy is a solid with a regular lattice spacing. In fact, in a one-dimensional solid at T 0, there always exists a distortion, a Peierls distortion, which leads to a lower energy than in the perfectly regular solid. An idea of the origin and effect of a Peierls distortion can be obtained by considering a one-dimensional solid of N atoms and N valence electrons (Fig. 24.71). Such a line of atoms distorts to one that has long and short alternating bonds. Although the longer bond is energetically unfavourable, the strength of the short bond more than compensates for the weakness of the long bond and the net effect is a lowering of energy below that of the regular solid. Now, instead of the electrons near the Fermi surface being free to move through the solid, they are trapped between the longer-bonded atoms (these electrons have antibonding character, and so are found outside the internuclear region between strongly bonded atoms). The Peierls distortion introduces a band gap in the centre of the original conduction band, and the filled orbitals are separated from the empty orbitals. Hence, the distortion results in a semiconductor or insulator, not a metallic conductor. 649 Molecular materials and fullerides The conduction band in KCP is a d band formed principally by overlap of Pt5dz2 orbitals. The small proportion of Br in the compound, which is present as Br, removes a small number of electrons from this otherwise full d band, so turning it into a conduction band. Indeed, at room temperature, doped KCP is a lustrous bronze colour with its highest conductivity along the axis of the Pt chain. However, below 150 K the conductivity drops sharply on account of the onset of a Peierls distortion. At higher temperatures, the motion of the atoms averages the distortion to zero, the separation is regular (on average), the gap is absent, and the solid is metallic. Mixed-valence platinum chain complexes, such as [Pt(en)2]PtCl2(en)24, may be isolated as insulated, one-dimensional wires by surrounding the Pt chain with a sheath of electronically inert molecules such as anionic lipids. Nanowires of this type are discussed more fully in Section 25.7. The observation of metallic behaviour in one-dimensional solids formed from interacting molecules has in turn led to a search for superconductivity in this class of materials. One type of material in which superconductivity has been found is a series of metal complexes derived from the organic metal and superconductors based on stacked molecules with interacting π systems. Thus salts of TCNQ (7,7,8,8-tetracyano-p-quinodimethane (3) with TTF (tetrathiafulvalene, 4) show metallic properties, and salts of tetramethyltetraselenafulvalene (TMTSF, 5) such as (TMTSF)2ClO4 show superconductivity, albeit at less than 10 K and often only under pressure. Molecular metal complexes involving types of sulfur-containing ligand, such as dmit (dmit2 1,3-dithiol-2-thione4,5-dithiolato), can also show superconductivity. The compound [TTF][Ni(dmit)2]2, which consists of stacks of both TTF and Ni(dmit)2, shows superconductivity at 10 kbar and below 2 K. C N N C C N N C 3 TCNQ S S S S 4 TTF Se Se Se Se 5 TMTSF (b) Molecular magnets Key point: Molecular solids containing individual molecules, clusters, or linked chains of molecules can show bulk magnetic effects such as ferromagnetism. Molecular inorganic magnetic materials, in which individual molecules, or units constructed from such molecules, contain d-metal atoms with unpaired electrons, is a class of compounds of growing interest. Generally the phenomena associated with long-range interaction of electron spins, such as ferromagnetism and antiferromagnetism, are much weaker as the short, superexchange-type pathways that are found in metal oxides do not exist. However, as with all molecular systems the opportunity exists to tune interactions between metal centres by tailoring the ligand properties. Examples of ferromagnetic molecular inorganic compounds are decamethylferrocene tetracyanoethenide (TCNE), [Fe(5-Cp)2(C2(CN)4)] (6, Cp C5Me5) and the analogous manganese compound. These materials, which have structures based on chains of alternating [M(5-Cp)2] and TCNE ions, show ferromagnetism along the chain direction below TC 4.8 K (for M Fe) and 6.2 K (for M Mn). An alternative approach to molecularbased compounds exhibiting magnetic ordering consists of assembling chains of magnetically interacting centres. For example, MnCu(2-hydroxy-1,3-propenebisoxamato).3H2O (7) consists of chains of alternating Mn(II) and Cu(II) ions bridged by the ligand. The magnetic moments on the metal ions in the chains order ferromagnetically below 115 K. This ferromagnetic ordering occurs initially only along the chains, that is in one dimension, due to the stronger interactions through the ligands and shorter MnCu distances, but this material orders fully in three dimensions at 4.6 K once the individual chains interact with each other magnetically. The incorporation of several d-metal ions into a single complex provides an opportunity to produce a molecule that acts as a tiny magnet. Such compounds have been termed single-molecule magnets (SMMs). One example is the complex manganese acetate Mn12O12 (O2CMe)16(H2O)4.2MeCO2H.4H2O, which contains a cluster of 12 Mn(III) and Mn(IV) ions linked through O atoms with the metal oxide unit terminated by the acetate groups (Fig. 24.72). Another is [Mn84O72(O2CMe)78 (OMe)24(MeOH)12(H2O)42(OH)6].xH2O. yCHCl3, which contains 84 Mn(III) ions in a large doughnut-shaped molecule 4 nm in diameter. The ability to magnetize such individual SMMs potentially provides a route to storing information at extremely high densities because their dimensions, a few nanometres, are much less than that of the typical domain of magnetic material used in a conventional magnetic data storage medium. NC CN Fe NC CN 6 [(η5-Cp)2Fe(TCNE)] H2O n M CH3 Cu N 7 O 650 24 Solid-state and materials chemistry CH3CO2– Mn (a) Figure 24.72 The core of the single molecule magnetic compound Mn12O12(O2CMe)16(H2O)4.2MeCO2H .4H2O that contains 12 Mn(III) and Mn(IV) ions. (b) Figure 24.73 Schematic diagram of liquid crystalline materials based on (a) calamitic (rod-like) or (b) discotic (disc-like) molecules. (c) Inorganic liquid crystals Key point: Inorganic metal complexes with disc- or rod-like geometries can show liquid crystalline properties. R Me O O Cu O O Me R 8 Liquid crystalline, or mesogenic, compounds possess properties that lie between those of solids and liquids and include both. For instance, they are fluid, but with positional order in at least one dimension. Such materials have become widely used in displays. The molecules that form liquid crystalline materials are generally calamitic (rod-like) or discotic (disc-like), and these shapes lead to the ordered liquid-type structures in which the molecules align in a particular direction (Fig. 24.73). Although most liquid crystalline materials are totally organic there is a growing number of inorganic liquid crystals based on the coordination compounds of metals and on organometallic compounds. These metalcontaining liquid crystals show similar properties to the purely organic systems but offer additional properties associated with a d-metal centre, such as redox and magnetic effects. As the requirement for liquid crystalline behaviour is a rod- or disc-shaped molecule, many of the metal-containing systems are based around the low coordination geometries of the later d metals, particularly the square-planar complexes found for Groups 10 and 11. Thus the -diketone complex (8) has a square-planar Cu2 ion coordinated to four O atoms from two -diketones with long pendant alkyl groups. This copper(II) material is paramagnetic but also forms a nematic phase in which the rod-shaped molecules align predominantly in one direction (as in Fig. 24.73). FURTHER READING A.R. West, Basic solid state chemistry. Wiley, New York (1999). A good comprehensive guide to the fundamentals of the solid state from an inorganic chemist’s perspective. D.W. Bruce and D. O’Hare, Inorganic materials. Wiley, New York (1997). Collection of reviews on various topics of materials chemistry, highlighting molecular systems. A.K. Cheetham and P. Day (ed.), Solid state chemistry: compounds. Oxford University Press (1992). A useful collection of chapters covering the key compound types in materials chemistry. M.T. Weller, Inorganic materials chemistry. Oxford Chemistry Primers 23. Oxford University Press (1994). Introductory text covering some aspects of solid-state chemistry and materials characterization. R.M. Hazen, The breakthrough: the race for the superconductor. Summit Books, New York (1988). A readable narrative of the discovery of high-temperature superconductors. R.C. Mehrotra, Present status and future potential of the sol–gel process. Struct. Bonding, 1992, 77, 1. A good review of sol–gel chemistry. A.K. Cheetham, G. Férey, and T. Loiseau, Open-framework inorganic materials. Angew. Chem., Int. Ed. Engl., 1999, 38, 3268. An excellent review of progress in, and the structural chemistry of, framework solids. C.N.R. Rao and J. Golalakrishnan, New directions in solid state chemistry. Cambridge University Press (1997). Review of current hot topics and research in solid-state chemistry. L.V. Interrante, L.A. Casper, and A.B. Ellis (ed.), Materials chemistry: an emerging discipline. Advances in Chemistry Series no. 245. American Chemical Society, Washington, DC (1995). A series of chapters on a broad range of inorganic and organic solids. Problems S.E. Dann, Reactions and characterization of solids. Royal Society of Chemistry, Cambridge (2000). A good introductory text on solidstate chemistry. A.F. Wells, Structural inorganic chemistry. Oxford University Press (1985). A comprehensive and systematic volume on structural solidstate chemistry. U. Müller, Inorganic structural chemistry. Wiley, New York (1993). A useful text on structural solid-state chemistry with numerous illustrations. B.D. Fahlman Materials chemistry. Springer, Dordrecht (2007). Good coverage of semiconductors, metals and alloys and characterization methods. P. Day. Molecules into materials: case studies in materials chemistrymixed valency, magnetism and superconductivity. World Scientific Publishing, Singapore (2007). A collection of papers demonstrating how this important area of materials chemistry developed. 651 P. Ball Made to measure: new materials for the 21st century. Princeton University Press, Princeton, (1997). A very readable overview of materials from the perspective of applications, covering fuel cells, ultra-hard materials, and smart materials. J.N. Lalena and D.A. Cleary. Principles of inorganic materials design. John Wiley and Sons, New Jersey (2005) Good overview of inorganic materials with a strong theoretical perspective. A Züttel, A Borgschulte, and L. Schlapbach. Hydrogen as a future energy carrier. Wiley VCH, Weinheim (2008). The latest developments and thinking on the hydrogen energy economy. M.D. Hampton, D.V. Schur, S.Yu Zaginaichenko, and V.I. Trefilov. Hydrogen materials science and chemistry of metal hydrides. Kluwer Academic Publishers, Dordrecht (2002). A comprehensive review of hydrogen storage materials. EXERCISES 24.1 When NiO is doped with small quantities of Li2O the electronic conductivity of the solid increases. Provide a plausible chemical explanation for this observation. (Hint: Li occurs on Ni2 sites.) 24.2 Is a crystallographic shear plane a defect, a way of describing a new structure, or both? Explain your answer. 24.3 How might you distinguish experimentally the existence of a solid solution from a series of crystallographic shear plane structures for a material that appears to have variable composition? 24.4 Sketch the wurtzite crystal structure and locate on the sketch the interstitial sites to which a cation might migrate. What is the nature of the bottleneck between the normal and interstitial sites? 24.5 Outline how you could prepare samples of (a) MgCr2O4, (b) SrFeO3Cl, (c) Ta3N5. 24.6 Identify the likely products of the reactions 800 C, O 2 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → (a) Li2CO3 CoO ⎯ 900 C, O2 ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → (b) 2 Sr(OH)2 WO3 MnO ⎯ 24.11 Which of the materials (a) YBa2Cu4O8, (b) Ca1.8Na0.2CuO2Cl2, (c) Gd2Ba2Ti2Cu2O11, (d) SrCuO2.12, which all contain layers of the stoichiometry CuO2, might be expected to be high-temperature superconductors? 24.12 Classify the oxides (a) BeO, (b) TiO2, (c) La2O3, (d) B2O3, (e) GeO2 into glass forming and nonglass forming. 24.13 Which metal sulfides might be glass forming? 24.14 Write possible formulas for a sulfide and a fluoride that might adopt the spinel structure. 24.15 Describe two methods that could be used to prepare the intercalation compound LiTiS2. 24.16 Which of the elements Be, Ca, Ga, Zn, P, and Cl might form structures in which the element is incorporated into a framework as an oxotetrahedral species? 24.17 Propose formulas for structures that would be isomorphous with SiO2 and zeolites of the same stoichiometry involving Al, P, B, and Zn, or mixtures thereof, replacing Si. 24.7 Draw one unit cell in the ReO3 structure showing the M and O atoms. Does this structure appear to be sufficiently open to undergo Na ion intercalation? If so, where might the Na ions reside? 24.18 Calculate the mass percentage of hydrogen in NaBH4 and state whether or not this material might be suitable for hydrogen storage. 24.8 Explain the variation of the magnetic susceptibility of a material that orders antiferromagnetically with TN 240 K, with temperature between 4 K and 600 K. 24.19 Substitution of Mg by small amounts of Li and Al into MgH2 improves its hydrogen-storage properties. Write a formula for this lithium aluminium magnesium dihydride and explain how Li and Al would be incorporated into the structures. 24.9 Magnetic measurements on the ferrite CoFe2O4 indicate 3.4 spins per formula unit. Suggest a distribution of cations between octahedral and tetrahedral sites that would satisfy this observation. 24.10 Given ligand-field ∆O and ∆T values of 1400 and 620 cm1, respectively, for Fe3 and 860 and 360 cm1, respectively, for Ni2, determine the site preference for A Ni2 and B Fe3 in normal compared with inverse spinel, assuming that ligand-field stabilization is dominant. 24.20 Egyptian blue, CaCuSi4O10, is pale blue and the spinel CuAl2O4 is an intense blue–green. Explain the difference. 24.21 Place the semiconductors AlP, BN, InSb, and C(diamond) in order of the expected band gap. 24.22 Describe the structures of Na2C60 and Na3C60 in terms of hole filling in a close-packed array of fulleride molecular ions. PROBLEMS 24.1 The compound FexO generally has x 1. Describe the probable metal ion defect that leads to x being less than 1. 24.2 The compound Ag2HgI4 is a good electrical conductor above 50ºC. Speculate on why Ag rather than Hg2 is the mobile species. 24.3 Solid PbF2 and ZrO2 at elevated temperatures are good anion conductors. Describe their crystal structure and why it may be conducive to ion transport. 24.4 Describe the inorganic chemistry involved in the operation of a lambda sensor (an exhaust gas oxygen sensor) in a vehicle engine. 652 24 Solid-state and materials chemistry 24.5 To obtain high oxidation states for d metals in complex oxides, compounds are normally prepared at as low a temperature as possible commensurate with the reaction. Discuss the thermodynamic reasons for the use of these conditions and explain why the optimum temperature for producing YBCO involves a final annealing stage at about 450ºC under pure oxygen. 24.13 State Zachariasen’s two generalizations that favour glass formation and apply them to the observation that cooling molten CaF2 leads to a crystalline solid whereas cooling molten SiO2 at a similar rate produces a glass. 24.6 Heating a complex oxide under ammonia can lead to nitridation or reduction. Describe possible products that might be formed when Sr2WO4 is heated in ammonia. 24.15 Reaction of ZrS2 (c lattice parameter 583 pm) with Co(5-C5H5)2 gives a compound with a c lattice parameter of 1164 pm and reaction with Co(5-C5Me5)2 gives a product with a corresponding lattice parameter of 1161 pm: ZrS2 does not react with Fe(5-C5H5)2. Explain these observations. 24.7 What are the advantages of sol–gel routes, in comparison with direct high-temperature reaction methods, in the synthesis of complex metal oxides? 24.8 Describe the issue of occupation factors in Fe3O4, Cr3O4, and Mn3O4 and how the occupation factor is related to ligand-field stablization factors. 24.9 The perovskites, such as ABO3, often have the dipositive ion in the A site and an ion of higher oxidation state in the B site. What are the factors that lead to this site preference? 24.10 Superconductors are often classified as type I or II. Describe the physical characteristic that determines the classification of a superconductor into one or other of these two types. 24.11 The superconducting compound YBa2Cu3O7 is described as having a perovskite-like structure. Ignoring problems of charge on the ions, describe the difference between this structure and that of perovskite. 24.12 Discuss how various chemical substitutions led to the discovery, and subsequent optimization of their properties, of the superconducting materials with the LnFeOAs structure type. 24.14 Starting with TiCl4 and AlCl3, give a plausible procedure for the formation of a ceramic containing polycrystalline Al2O3 and TiO2. 24.16 Describe the interactions between Mo6S8 units in a Chevrel phase. 24.17 Discuss differences in the properties of zeolites, zeotypes (frameworks built from oxotetrahedral species other than AlO4 and SiO4), and metal organic frameworks. 24.18 Which class of material is the most promising for hydrogen storage in transport applications? What are the different requirements of a static system that might be used to store hydrogen produced from renewable energy? 24.19 What are the properties of an ideal inorganic pigment? 24.20 Compare and contrast the chemistries of graphite and C60 with respect to their compounds in association with the alkali metals. 24.21 ‘The increased reactivity that allows the design and synthesis of materials based on molecular units also means that these compounds are unsuitable for many applications that currently use inorganic materials.’ Discuss this remark. Nanomaterials, nanoscience, and nanotechnology Inorganic chemistry often focuses on entities with atomic and molecular dimensions ranging from 0.1 to 10 nm, whereas solid-state chemistry and materials chemistry have traditionally been concerned with solid materials with dimensions greater than 100 nm. There is currently a great deal of interest in materials that exist between these two scales because they can exhibit unique properties. This chapter focuses on materials that have a critical dimension between 1 and 100 nm. It introduces the fundamental physical and chemical principles that illustrate why these nanomaterials have generated such intense and broad interest, and describes the technology used to generate and exploit them. In the first part of this chapter we introduce nanomaterials, nanoscience, and nanotechnology from a historical perspective, together with definitions and examples. In the second part we discuss the advances in characterization and fabrication methods that allow us to make high-quality nanomaterials. The bulk of the chapter is the third part, where we use examples of nanomaterials to introduce how chemical principles and materials properties are different or superior in the nanoscale regime. All materials exhibit a hierarchy of structural levels. Chemists have long been familiar with the atomic, molecular, unit-cell, and electronic structural levels because atomic-scale interactions play a primary role in many of the bulk properties exhibited by real materials. Moreover, deviations on an atomic level from the basic structural motif are of great interest because they can affect the bulk properties of materials, as we have seen in connection with the electrical conductivities of semiconductors and solids that contain point defects (Chapter 3). Materials scientists have long been interested in the structure of materials from the micro (106 m, 1 µm, ‘1 micron’) to macro (bulk) scales, as well as in the properties of materials that are controlled by structures that exist on and interact over these scales. They have studied how the physical properties at these scales deviate from those predicted by extrapolation from longer scales and deviate from the basic structural motif, such as the presence of line, plane, and volume defects. For example, crystals often contain an enormous number of regions where one part of the crystal is displaced relative to the other part. These dislocations affect strength, crystal growth, electrical conductivity, and other physical properties. This chapter is concerned with nanomaterials, which are materials that have some critical dimension between 1 and 100 nm. Over this intermediate scale, effects from the submicrometre- and micrometre-length scales can play equal roles and quantum effects can intervene to give rise to fascinating properties. Fundamentals It will prove helpful to understand nanomaterials from historical, empirical, and chemical perspectives. 25.1 Terminology and history Key points: A nanomaterial is any material that has a critical dimension on the scale of 1 to 100 nm; a more exclusive definition is that a nanomaterial is a substance that exhibits properties absent in both the molecular and bulk solid state on account of it having a critical dimension in this range. 25 Fundamentals 25.1 Terminology and history 25.2 Novel optical properties of nanomaterials Characterization and fabrication 25.3 Characterization methods 25.4 Top-down and bottom-up fabrication 25.5 Templated synthesis using frameworks, supports, and substrates Self-assembled nanostructures 25.6 Control of nanoarchitecture 25.7 One-dimensonal control: carbon nanotubes and inorganic nanowires 25.8 Two-dimensional control: quantum wells and solid-state superlattices 25.9 Three-dimensional control Bioinorganic nanomaterials 25.10 DNA and nanomaterials 25.11 Natural and artificial nanomaterials: biomimetics 25.12 Bionanocomposites FURTHER READING EXERCISES PROBLEMS 654 25 Nanomaterials, nanoscience, and nanotechnology The prefix ‘nano’ will occur in many combinations throughout this chapter. Nanoscience is the study of the properties of matter that have length scales between 1 and 100 nm. Nanotechnology is the collection of procedures for manipulating matter on this scale in order to build nanosized entities for useful purposes. Some definitions, however, are more restrictive. Thus, a ‘nanomaterial’ is taken to be a solid material that exists over the scale of 1 to 100 nm and exhibits novel properties that are related to its scale. Likewise, ‘nanoscience’ is also sometimes restricted to the study of new effects that arise only in materials that exist on the nanoscale, and ‘nanotechnology’ is similarly restricted to the procedures for creating new functionalities that are possible only by manipulating matter on the nanometre-scale. The original version of nanotechnology occurred in nature, where organisms developed an ability to manipulate light and matter on an atomic scale to build devices that perform specific functions, such as storing information, reproducing themselves, and moving about. In this sense, DNA is the ultimate nanomaterial, as it stores information as the sequence of base pairs that are spaced about 0.3 nm apart. Folded DNA molecules have an information density of more than about 1 Tb cm2 (1 Tb 1012 bits). Photosynthesis is another example of biological nanotechnology in which nanostructures are exploited to absorb light, separate electric charge, shuttle protons around, and ultimately convert solar energy into biologically useful chemical energy. Our quest to harness solar energy has also turned to the nanoscale, where nanocatalysts are improving our ability to convert light to electrical energy by using photovoltaic materials. Humans have practised nanotechnology for centuries, although it was more of an art than a science or engineering discipline. For example, gold and silver salts have long been used to colour glass: gold has been used to produce red stained glass and silver used to produce yellow. In stained glass, the metal atoms form nanoparticles (known previously as ‘colloidal particles’) with optical properties that depend strongly on their size. Metallic nanopigments are now becoming a focal point of biomedical nanotechnology because they can be used to tag DNA and other nanoparticles. Other traditional examples of nanotechnology include the photosensitive nanosized particles in silver halide emulsions used in photography and the nanosized carbon granules in the ‘carbon black’ used for reinforcing tyres and in printer’s ink. The science and engineering of nanotechnology began to take shape in the latter half of the twentieth century. One significant leap forward in nanotechnology occurred when Gerd Binnig and Heinrich Rohrer developed the scanning tunnelling microscope, which was the forerunner to all other forms of scanning probe microscopy (Section 25.3). Later, a scanning probe tip was used to rearrange atoms on a surface to spell out words, so demonstrating an ability to manipulate and characterize nanoscale structures. Work in the areas of nanoscience and nanotechnology has thus been multidisciplinary as well as diverse in scope. This chapter aims to introduce this broad field, highlighting the importance of inorganic chemistry in the multidisciplinary research directed towards understanding nanomaterials. 25.2 Novel optical properties of nanomaterials Confinement effects lead to some of the most fundamental manifestations of nanoscale phenomena in materials and are frequently used as a point of departure for the study of nanoscience. Novel optical properties appear in nanoparticles as a result of such effects and are being exploited for information, biological, sensing, and energy technologies. (a) Semiconducting nanoparticles Key points: The colour of quantum dots is dictated by quantum confinement phenomena and particle localization; quantization of the HOMO–LUMO bands leads to novel optical effects involving interband and intraband transitions. Semiconducting nanoparticles have been investigated intensively for their optical properties. These particles are often called quantum dots (QD) because quantum effects become important in these three-dimensionally confined particles (dots). Two important effects occur in semiconductors when electrons are confined to tiny regions. First, the HOMO– LUMO energy gap increases from the value observed in bulk crystals. Second, the energy levels of electrons in the LUMOs (and holes—the absence of electrons—in the HOMOs) are quantized, like those of a particle in a box. Both effects play an important role in determining the optical properties of QDs. Fundamentals Quantum confinement, the trapping of electrons and holes in tiny regions, provides a method of tailoring or engineering the band gap of materials. The crucial feature is that as the critical dimension of a material decreases, the band gap increases. Transitions of electrons between states in the valence band (the so-called HOMO states) and the conduction band (the LUMO states) are called interband transitions, and the minimum energy for these transitions is increased in QDs relative to those in bulk semiconductors. The wavelengths of interband transitions depend on the size of the dots and it is possible to tailor their luminescence simply by changing their size. A key example of these QD materials is CdSe. By varying the size of the CdSe nanoparticle it is possible to tune emission over the entire visible spectrum, making them ideal for LED and fluorescent display technologies (Box 25.1). Another exciting use of QDs is as chromophores for ‘biotags’, where dots of different sizes are functionalized to detect different biological analytes. It turns out that, although they emit at specific tuneable wavelengths, QDs exhibit broadband absorption for energies above the band gap. The intriguing point for bioapplications is that it is possible to excite an array of distinct QD chromophores with a single broadband excitation and to simultaneously detect multiple analytes by their distinct optical emissions. These materials have been used to image breast cancer cells and live nerve cells to track small molecule transport to specific organelles. The second manifestation of quantum confinement is that the available quantized energy levels inside a QD have no net linear momentum and therefore transitions between them do not require any momentum transfer. As a result, the transition probabilities between any two states are high. This lack of momentum dependence also explains the broadband absorption nature of QDs because probabilities are high for most transitions from the occupied valence band states to unoccupied conduction band states. Probabilities are also high for intraband transitions, transitions of electrons between states in the LUMO band or of holes in the HOMO band. These relatively intense intraband transitions are typically in the infrared region of the spectrum and are currently being exploited to make devices such as infrared photodetectors, sensors, and lasers. (b) Metallic nanoparticles Key points: The colours of metallic nanoparticles dispersed in a dielectric medium are dominated by localized surface plasmon absorption, the collective oscillation of electrons at the metal–dielectric interface. The optical properties of metallic nanoparticles arise from a complex electrodynamic effect that is strongly influenced by the surrounding dielectric medium. Light impinging on metallic particles causes optical excitations of their electrons. The principal type of optical excitation that occurs is the collective oscillation of electrons in the valence band of the metal. Such coherent oscillations occur at the interface of a metal with a dielectric medium and are called surface plasmons. In bulk particles, the surface plasmons are travelling waves and are characterized by a linear momentum. To excite plasmons using photons in bulk metals, the momenta of the plasmon and the photon must match. This matching is possible only for very specific geometries of the interaction between light and matter, and is a weak contributor to the optical properties of the metal. In nanoparticles, however, the surface plasmons are localized and have no characteristic momentum. As a result, the momenta of the plasmon and the photon do not need to match and plasmon excitation occurs with a greater intensity. The peak intensity of the surface plasmon absorption for gold and silver occurs in the optical region of the spectrum, and so these metallic nanoparticles are useful as pigments. The characteristics of plasmon absorption depend strongly on the metal and the dielectric surroundings as well as on the size and shape of the nanoparticle. To control the dielectric surroundings, so-called core–shell composite nanoparticles have been designed in which metallic shells of nanometre-scale thickness encapsulate a dielectric nanoparticle. Metallic nanoparticles and metallic nanoshells are used as dielectric sensors because their optical properties change when they come into contact with different dielectric materials. In particular, biological sensing is of interest because biological analytes can bind to the surface of the nanoparticle, causing a detectable shift in the plasmon absorption band. Gold nanoparticles are common examples of metallic nanoparticles and have found practical applications as biological and chemical sensors, ‘smart bombs’ for cancer therapy, and optical switching and fluorescent display materials. Many of these applications are possible owing to the development of techniques to bind photoresponsive chromophores 655 656 25 Nanomaterials, nanoscience, and nanotechnology B OX 25 .1 CdSe Nanocrystals for LEDs Cadmium selenide (CdSe) nanocrystals have found application in a wide variety of commercial venues including LEDs (Fig. B25.1), solar cells, fluorescent displays, and in vivo cellular imaging of cancer cells. Their high photoluminescence efficiencies and emission colour tunability based on nanocrystal size make them attractive for full-colour displays. Close-packed QD monolayers have been self-assembled using soft nanolithography contact printing to achieve light-emitting devices. In soft nanolithography, photolithography or electron beam lithography is used to produce a pattern in a layer of photoresist on the surface of a silicon wafer. This process generates a bas-relief master that contains islands of photoresist that stand out from the Si surface. A chemical precursor to polydimethylsiloxane (PDMS), a free-flowing liquid, is then poured over the bas-relief master and cured into the rubbery solid. After curing, a PDMS stamp that matches the original pattern reproduces features from the master as small as a few nanometres. Whereas making a master is expensive, copying the pattern on PDMS stamps is inexpensive and easy. One stamp can be used in various inexpensive ways, including microcontact printing and micromoulding to make nanostructures. These techniques can be employed to produce subwavelength optical devices, waveguides, and optical polarizers used in optical fibre networks and eventually perhaps in all-optical computers. Ag Ag/Mg Electron transport Hole blocking Hole transport PEDOT:PSS ITO anode 1 mm (a) (b) (c) 100 nm 0.9 Red QD-LED y Blue QD-LED Electroluminescence Green QD-LED HDTV colour triangle 0 x 0 400 0.8 500 600 Wavelength, l /nm 700 . (d) (e) Figure B25.1 (a) Electroluminescent (EL) red, green, and blue QD-LED pixels with the device structure shown in (b). (b) Schematic diagram shows the cross section of a typical QD-LED. (c) A high-resolution AFM micrograph shows a close-packed monolayer of QDs deposited on top of the holetransporting polymer layer prior to deposition of hole-blocking and electron-transporting layers. (d) Chromaticity diagram shows the positions of red, green, and blue QD-LED colours, an HDTV colour triangle is shown for comparison. (e) Normalized EL spectra of fabricated QD-LEDs corresponding to the colour coordinates in (d). QD-LED images and EL spectra are taken at video brightness (100 cd m–2), which corresponds to the applied current density of 10 mA cm–2 for red QD-LEDs, 20 mA cm–2 for green QD-LEDs, and 100 mA cm–2 for blue QD-LEDs. (Based on L. Kim, et al., Nano Lett., 2008, 8, 4513.) to the surface of the nanoparticles. Gold nanoparticles can be used tagged with biomolecules that afford delivery to specific cells and have found applications as immunoprobes for early detection of disease. The nanomaterials have also been used as key electron-transfer agents in a new generation of solar materials that couples photoresponsive π-conjugated molecules such as pyrene to gold nanoparticle surfaces. Chromophore-functionalized gold nanoparticles afford unique device architectures and flexibility in design as these hybrid Characterization and fabrication 657 materials can be coupled covalently to conductive glass substrates that serve as electrodes, so leading to improved charge transport and photoefficiencies. Silver nanoplates with sizes in the range 40–300 nm have been synthesized by a simple room-temperature solution-phase chemical reduction method in the presence of diluted cetyltrimethylammonium bromide (CTAB, (C16H33)(CH3)3NBr). These plates are single crystals having as their basal plane the (111) plane of face-centred cubic silver. The stronger adsorption of CTAB on the (111) basal plane than on the (100) side plane of these plates may account for the anisotropic growth of nanoplates. As discussed earlier, metal nanoparticles have interesting optical properties related to their surface plasmon excitations. The optical (in-plane dipole) plasmon resonance peaks can be shifted to wavelengths of 1000 nm in the near-IR when the aspect ratio (the ratio of long-axis to short-axis, or width to thickness) of the nanoplates reaches 9. Such control over the optical properties of simple metals opens up new possibilities for various near-IR applications, including remote sensing (Fig. 25.1). Characterization and fabrication The burgeoning activities during the late twentieth century in the areas of nanomaterials, nanoscience, and nanotechnology were intimately tied to advances in characterization and fabrication. This section introduces the basic concepts of nanoscale characterization and fabrication methods that underly the design of nanomaterials and the development of nanotechnologies. 25.3 Characterization methods (a) (b) (c) (d) (e) (f) The great advances made in nanosciences and nanotechnology would not have occurred without the ability to characterize the nanoscale structural, chemical, and physical properties of materials. Moreover, the direct observation of nanostructure allows meaningful relationships between processing and properties to be made. (a) Scanning probe microscopy Key points: Scanning tunnelling microscopy uses tunnelling current from a sharp conductive tip to image and characterize a conductive surface; atomic force microscopy responds to intermolecular forces to image the surface. a b c d e f Relative absorbance Scanning tunnelling microscopy (STM) and atomic force microscopy (AFM) were the first of a series of scanning probe microscopies (SPM), which are techniques that allow threedimensional imaging of the surface of materials by using a sharply pointed probe brought into close proximity (or in contact) with the specimen and constructing an image by scanning the probe over the surface of the specimen and monitoring the spatial variation in the value of a physical parameter, such as potential difference, electric current, magnetic field, or mechanical force. In STM, an atomically sharp conductive tip is scanned at about 0.3–10 nm above the surface of the sample. Electrons in the conductive tip, which is held either at a constant potential or at a constant height above the sample, can tunnel through the gap with a probability that is related exponentially to the distance from the surface. As a result, the electron tunnelling current reflects the distance between the sample and the tip. If the tip is scanned over the surface at a constant height, then the current represents the change in the topography of the surface. The movement of the tip at a constant height with great precision is accomplished by using piezoelectric ceramics for the displacement of the tip. Typically, STM probes are sharpened to have only a few atoms at the end. The simplest method of creating such sharp tips is through electrochemical etching. In one technique, a thin tungsten wire is placed in a solution of potassium hydroxide and an electric current is passed through it. STM probes have been used to manipulate individual atoms and molecules on cooled surfaces to achieve patterning on the atomic scale. Figure 25.2 shows an STM study of trimesic acid (TMA) molecules adsorbed on a graphite substrate. The TMA molecules act as a host for the incorporation of C60 as a molecular guest. The STM tip was used to nudge these adsorbed C60 molecules from one cavity of the host structure to the next, giving rise to what is jocularly referred to as ‘nanosoccer’. 200 603 605 777 895 997 f e d c b a 933 400 600 800 1000 1200 Wavelength, ␭/nm Figure 25.1 (Top) TEM images of the silver nanoplates obtained with different seed amounts: (a) 0.1 cm3, (b) 0.25 cm3, (c) 0.5 cm3, (d) 1.0 cm3, (e) 2.0 cm3, and (f) 3.0 cm3. (Bottom) Absorption spectra of the corresponding silver nanoplate solutions. (S. Chen and D.L. Carroll, J. Phys. Chem. B, 2004, 108, 5500.) 658 25 Nanomaterials, nanoscience, and nanotechnology In AFM, the atoms at the tip of the probe interact with the surface atoms of the sample through intermolecular forces (such as van der Waals interactions). The cantilever holding the probe bends up and down in response to the forces and the extent of deflection is monitored with a reflected laser beam. Variations on AFM include: frictional force microscopy, which measures variations in the lateral forces on the tip that are based on chemical variations on the surface magnetic force microscopy, which uses a magnetic tip to image magnetic structure electrostatic force microscopy, which uses the tip to sense electric fields scanning capacitance microscopy, in which the tip is used as an electrode in a capacitor. (a) Molecular recognition AFM is also now being carried out in which the tip is functionalized with specific ligands and the interaction between the tip and surface is measured. Such microscopes can provide resolution of the chemical properties on the surface. AFM can also act as a patterning tool. In dip-pen nanolithography (DPN, Fig. 25.3) the tip of an AFM is used as an ink pen. By using an ink containing molecular entities, self-assembled monolayers (SAMs) can be formed through molecular transport of the nanoink to the solid substrate surface. These monolayers often involve specific covalent interactions between the S atom of organothiols and Au surfaces. Scanning near-field optical microscopy (SNFOM) combines the local interaction of a scanning probe and a specimen with well-established methods of optical spectroscopy. Conventional optical spectroscopy is limited in resolution by the wavelength of the light source, but when the light source is brought to within a few atomic diameters of the surface and a nanometre-scale aperture is used, the local confinement of the electromagnetic field allows for much improved spatial resolution (of about 20 nm) to investigate local chemical properties of nanoscale structures. (b) Figure 25.2 Playing nanosoccer: (a) STM topograph images of the starting situation with a single C60 guest molecule inside the trimesic acid host network. (b) Final result of the lateral manipulation with the whole molecule imaged in the adjacent target cavity. (S.J.H. Griessl, et al., J. Phys. Chem. B 2004, 108, 1155.) AFM tip Water Substrate Figure 25.3 The process of dip-pen nanolithography. Organothiols move from an AFM tip through a water meniscus forming a self-assembled monolayer on a gold substrate. (Adapted from C. Mirkin, Nanoscience Boot Camp at Northwestern University 2001.) (b) Scanning electron microscopy techniques Key point: Transmission and scanning electron microscopes use electrons to image the sample in a similar fashion to optical microscopes. An imaging electron microscope operates like a conventional optical microscope but instead of imaging photons, as in the visible microscope, electron microscopes use electrons. In these instruments, electron beams are accelerated through 1–200 kV and electric and magnetic fields are used to focus the electrons. In transmission electron microscopy (TEM) the electron beam passes through the thin sample being examined and is imaged on a phosphorescent screen. In scanning electron microscopy (SEM) the beam is scanned over the object and the reflected (scattered) beam is then imaged by the detector. The ultimate resolution of SEM depends on how sharply the incident beam is focused on the sample, how it is moved over the sample, and how much the beam spreads out into the sample before reflecting. In both microscopes, the electron probes cause the production of X-rays with energies characteristic of the elemental composition of the material. As such, energy-dispersive spectroscopy of these characteristic X-rays is used to quantify the chemical make-up of materials by using electron microscopes. The primary advantage of SEM over TEM is that it can form images of electron-opaque samples without the need for difficult specimen preparations. It is therefore the electron microscopy method of choice for the straightforward characterization of materials. However, SEM samples need to be conductive because otherwise electrons can collect on the sample and interact with the electron beam itself, resulting in blurring. Nonconductive samples must therefore be coated with a thin layer of metal, usually gold or aluminium. The use of focused ion beams (FIBs; high energy streams of Ga3 ions) to cut slices of samples has revolutionized TEM sample preparation, shortening times and allowing for in situ studies of materials’ architectures. The FIB technique can be used in concert with SEM to examine internal morphologies. Damaged or dysfunctional integrated circuits can be sliced open to investigate the origin of failure and examine links between materials’ defects and their conductive behaviour. 25.4 Top-down and bottom-up fabrication Key points: Top-down fabrication methods carve out or add on nanoscale features to a bulk material by using physical methods; bottom-up fabrication methods assemble atoms or molecules in a controlled manner to build nanomaterials piece by piece. Characterization and fabrication There are two basic techniques for the fabrication of nanoscale entities. The first is to take a macroscale (or microscale) object and carve out nanoscale patterns. Methods of this sort are called top-down approaches. In top-down approaches, patterns are first designed on a large scale, and their lateral dimensions are reduced and then used to transfer the nanoscaled features into or on to the bulk material. Physical interactions are used in top-down fabrication approaches, such as photolithography, e-beam lithography, and soft lithography. The most common and well-known approach is photolithography, the technique used to fabricate very large-scale integrated circuits having feature dimensions on the 100 nm scale (Fig. 25.4). The second technique is to build larger objects by controlling the arrangement of their component smaller-scale objects. Methods of this sort are called bottom-up approaches and start with control over the arrangements of atoms and molecules. This chapter emphasizes the bottom-up approach to nanoscale fabrication because of its focus on the interactions of atoms and molecules and their arrangement into larger functional structures. The two basic approaches most widely used to prepare nanomaterials, solution methods and vapour-phase methods, are bottom-up methods because control over the arrangement of individual atoms is exerted to achieve larger-scale structures. Top down 1m As discussed in Chapter 24, the two basic techniques used to generate inorganic solids are direct-combination methods and solution-based methods. The former does not lend itself well to the synthesis of nanoparticles because the reactants tend to be larger than nanoparticles and therefore require long times to reach equilibrium. Moreover, the use of elevated temperatures leads to particle growth and coarsening during the reaction period, resulting in large crystallite sizes. There are, however, some examples of using mechanical ball-milling at low temperature to break macroscopic powders into nanoparticles. However, solution-based methods permit excellent control over the crystallization of inorganic materials and are widely used. The techniques used to make nanoparticles are similar to those described in Chapter 24 to make solid-state compounds. Special care must be taken to control the size and shape of the particles during nanoparticle synthesis, as well as their uniformity in size and shape. By fine-tuning the crystallization process from solution, highly monodisperse, uniformly shaped nanoparticles of a wide range of compositions can be prepared from combinations of elements from throughout the periodic table. Because the reactants in solution-based methods are mixed on an atomic scale and solvated in a liquid medium, diffusion is fast and diffusion distances are typically small. Therefore, reactions can be carried out at low temperature, which minimizes the thermally driven particle growth that is problematic in direct combination methods. Although the specifics of each reaction differ greatly, the basic stages in solution chemistry are: 1. Solvate the reactant species and additives. 2. Form stable solid nuclei from solution. 3. Grow the solid particles by addition of material until the reactant species are consumed. The basic aim in solution synthesis is to generate in a controlled manner the simultaneous formation of large numbers of stable nuclei that undergo little further growth. If growth is to occur, it should occur independently of the nucleation step because then all particles have a chance to grow to similar sizes. If performed successfully, the particles will be monodisperse and in the nanometre range. The drawback to the solution method is that the particles can undergo Ostwald ripening, in which smaller particles in the distribution redissolve and their solvated species reprecipitate on to larger particles, so increasing the size distribution and decreasing the total particle count. To prevent this unwanted ripening, stabilizers, surfactant molecules that help to stabilize the particles against growth and dissolution, are added. Bulk m materia material Thin films ilms Heterostructures ostructu o Lithographic wires ithographic wire ith ires Quantum dots um m do d 1 nm Nanocrystals Molecular wires lecular w lec wi Proteins (a) Solution-based synthesis of nanoparticles Key points: Solution-based synthetic methods are the main techniques for nanoparticle synthesis because they have atomically mixed and highly mobile reagents, allow for the incorporation of stabilizing molecules, and have been widely successful in practice; the two stages of crystallization from solution are nucleation and growth. 659 Molecules ecules e 100 pm Atoms ms m Bottom up Figure 25.4 The two techniques for making nanoscale structures. The top-down technique starts with larger objects that are whittled down into nanoscale objects; the bottom-up technique starts with smaller objects that are combined into nanoscale objects. 660 25 Nanomaterials, nanoscience, and nanotechnology E X A M PL E 25 .1 Controlling nucleation during nanoparticle synthesis Use thermodynamic arguments related to the crystallization of a phase to describe how to generate a short burst of homogeneous nucleation in a solution. Answer We need to be aware that the formation of stable solid nuclei is controlled through the thermodynamic driving force of the reaction, which can be controlled by adjusting the composition of the different solution species and the temperature (as well as the pressure for hydrothermal techniques). As with most chemical reactions, we need to note that there is an activation barrier to the crystallization reaction, which prevents the instantaneous formation of many stable nuclei. To overcome this barrier in a short single burst, we convert the solution quickly to a highly nonequilibrium state so that the tendency to nucleate becomes very strong. Widespread homogeneous nucleation of many particles then occurs. This nucleation quickly moves the system back towards equilibrium and suppresses further nucleation. Self-test 25.1 Use a similar argument to describe the synthesis of core-shell nanoparticles. As there are many methods to synthesize nanoparticles, we limit this discussion to a few well-known examples. The first example is metallic nanoparticles, such as gold. In 1857, Michael Faraday found that reduction of an aqueous solution of [AuCl4] with phosphorus in CS2 produced a deep-red suspension that contained nanoparticles of gold. Because sulfur forms chemical bonds to gold, sulfur-containing species are good stabilizing agents and the most widely used stabilizer for gold nanoparticles contain a thiol group (SH). An approach has been developed in the same spirit as Faraday’s to control the size and dispersity of gold nanoparticles by using [AuCl4] and thiol stabilizers. The reaction is relatively simple and the procedure produces air-stable gold nanoparticles with diameters between 1.5 and 5.2 nm. In the so-called Brust-Schiffrin method, [AuCl4] is first transferred from water to methylbenzene (toluene) by using tetraoctylammonium bromide as a phase-transfer agent. The methylbenzene contains dodecanethiol as a stabilizer and, after transfer, NaBH4 is used as a reducing agent to precipitate Au nanoparticles with dodecanethiol surface groups: Transfer: AuCl4 N(C8H17)4(sol) → N(C8H17)4(sol) AuCl4(sol) Precipitation (of the [Aum(C12H25SH)n] nanoparticle): m AuCl4 n C12H25SH(sol) 3m e → 4m Cl(sol) Aum(Cl12 H25 SH)n where sol is methylbenzene. The ratio of stabilizer (C12H25SH) to metal (Au) controls the particle size in the sense that higher stabilizer:metal ratios lead to the smaller metal core sizes. By adding the NaBH4 reductant quickly and cooling the system as soon as possible after the reaction terminates, smaller and more monodisperse nanoparticles are formed. The rapid addition of reductant improves the probability of simultaneous formation of all nuclei. By cooling the solution quickly, both post-nucleation growth and dissolution of particles are minimized. Similar approaches can be used for other metal nanoparticles. Quantum dots of materials such as GaN, GaP, GaAs, InP, InAs, ZnO, ZnS, ZnSe, CdS, and CdSe have been investigated for their optical properties (Section 25.2) because their interband absorption and fluorescence occur in the visible spectrum. As an early example of their preparation, dimethylcadmium is dissolved in mixture of trioctylphosphine (TOP) and trioctylphosphine oxide (TOPO), and the selenium source, which is often Se dissolved in TOP or TOPO, is added at room temperature. The solution is injected into a reactant vessel containing vigorously stirred, hot TOPO, which permits widespread nucleation of TOPO-stabilized CdSe QDs. The addition of the room-temperature liquid lowers the temperature of the solution and prevents further nucleation or growth (because the activation barriers are so high). The solution is then reheated to a temperature that allows for slow growth but no further nucleation. This step leads to nanoparticles with narrow size distributions and sizes in the range 2–12 nm. Alternative, less hazardous synthetic methods have been developed and are being pursued. Cadmium sulfide can be grown in pH-controlled aqueous solutions of Cd(II) salts with polyphosphate stabilizers by the addition of a sulfur source. For example, at pH 10.3 the addition of Na2S causes the precipitation of CdS nanoparticles from aqueous solutions Characterization and fabrication 661 containing Cd(NO3)2 and sodium polyphosphate. These QDs range in size from 1 to 10 nm, and the size is controllable through the reactant concentrations and the rate of addition of the reactant. Oxide nanoparticles can also be grown by solution methods. Many applications use colloidal particles of oxides such as SiO2 and TiO2 for food, ink, paints, coatings, and so on. Many of the efforts to achieve controlled oxide nanoparticle growth stem from earlier work in traditional ceramic and colloidal applications, where particle sizes from 1 nm to 1 µm are used. Silica, SiO2, and titania, TiO2, are probably the best-known oxides grown from solution. Typical schemes involve the controlled hydrolysis of metal alkoxides. All successful hydrolysis reactions of metal alkoxides aim to follow the same basic rules as described above: the controlled nucleation stage and slow growth stage are performed independently. In all cases, strict monitoring of the pH, precursor chemistry, reactant concentration, rate of addition of reactant, and temperature is required to control the final size and shape of the particles. An important example of nanoparticle oxide use is in the photoelectrochemical solar cell known as the Grätzel cell. This cell uses nanocrystalline TiO2 as a medium to transfer electrons from an organoruthenium dye to a conductor. Nucleation occurs in the hydrolysis of titanium isopropoxide that is added dropwise into vigorously stirred 0.1 m HNO3(aq). The filtered nanoparticles are then placed in an autoclave and allowed to grow by the hydrothermal addition of material. The size, shape, and state of agglomeration are controlled by adjusting the conditions of either the nucleation or the growth stage. (b) Vapour-phase synthesis of nanoparticles Key points: Vapour-phase synthetic methods are alternative techniques for nanoparticle synthesis because they have atomically mixed and highly mobile reagents, can be controlled by varying the conditions, and have been widely successful in practice. The same fundamentals concerning nucleation and growth that are relevant to solution synthesis apply to vapour-phase synthesis. The vapour needs to be supersaturated to the point at which a high density of homogeneous nucleation events produces solid particles in one short burst, and growth must be limited and controlled in a subsequent step, if it is to occur at all. Commercially, vapour-phase synthesis is carried out to produce nanoscale carbon black and fumed silica in large quantities. Metals, oxides, nitrides, carbides, and chalcogenides can also easily be formed by using vapour-phase techniques. There are, however, several differences between vapour-phase and solution-based techniques. In the latter, stabilizers can be added in a straightforward and controllable fashion, and particles remain dispersed and independent of one another. In vapour-phase techniques, however, surfactants or stabilizers are not easily added (although solid-state stabilizers are available). Without surface stabilizers, nanoparticles tend to agglomerate into larger particles. If temperatures during the process are high enough, the particles sinter and coalesce into one larger particle. Although the ‘soft agglomerates’ formed under the influence of weak intermolecular forces can usually be redispersed, the ‘hard agglomerates’ (partially sintered-formed at intermediate temperature) and coalesced particles (formed at high temperatures) cannot be. Therefore, techniques that lead to soft agglomerates are generally best suited to nanoparticle production. Because of these differences, the size dispersion of nanoparticles tends to be better for solution-based techniques than for vapourphase techniques. Vapour-phase techniques are attractive synthesis methods for particles when continuous operation is required or when solution methods do not produce good quality nanoparticles. They can also produce complex materials in a straightforward manner, such as doped binary and ternary compounds or composite particles. Vapour-phase techniques are classified by the physical state of the precursor used as a reagent and by the reaction method, such as ‘plasma synthesis’ or ‘flame pyrolysis’. In each case, the reagent is converted to a supersaturated or superheated vapour that is allowed to react or to cool to force nucleation. Solid reagents are vaporized and then recondensed in gas condensation methods (thermal evaporation to produce vapour), laser ablation, sputtering, and spark discharge methods. Liquid or vapour precursors are used in spray pyrolysis, flame synthesis, laser pyrolysis, plasma synthesis, and chemical vapour deposition. In chemical vapour deposition, the vapour is transported to a substrate where reaction and solid nucleation occurs (Fig.25.5). In a variation of this procedure, gaseous precursors are delivered into a hot-walled Chemical vapour in carrier gas Substrate Heater Heterogeneous nucleation (a) Exhaust vapour in carrier gas Heater (b) Homogeneous nucleation Figure 25.5 Chemical vapour methods to achieve (a) thin film growth and (b) nanoparticle production. 662 25 Nanomaterials, nanoscience, and nanotechnology reactor and allowed to react homogeneously in the vapour phase to nucleate solids, and particles are collected downstream. Particle sizes are controlled by the flow rates, precursor chemistry, concentrations, and residence times in the reactor. Examples of materials synthesized by this technique include metals, oxides, nitrides, and carbides such as SiC, SiO2, Si3N4, SiCxOyNz, TiO2, TiN, ZrO2, and ZrN. Both solution and vapour-phase methods can be used to make composite nanoparticles, such as core–shell nanocomposites. In both techniques the approach is to grow a second phase on an initial nucleus or nanoparticle. Reaction design to produce core– shell nanoparticles by solution is straightforward provided the solution characteristics of both materials are similar. In practice, however, it is difficult to find materials where the synthesis conditions overlap. Vapour-phase techniques offer another approach to the design of core–shell particles, by injecting a second vapour into a reactor at the growth stage. Another straightforward approach to vapour-phase nanoparticle production is plasma synthesis, which can be used to synthesize elemental solids, alloys, and oxides, as well as core–shell nanoparticles. In this method, gas or solid particles are fed into a plasma where they vaporize and ionize to highly energetic charged species. Inside a plasma, the temperatures can exceed 10 kK. On leaving the plasma, the temperature falls rapidly and crystallization occurs under conditions far from equilibrium. The nanoparticles are collected downstream from the plasma zone. Depending on the carrier gas (that is, the oxidation conditions), elemental solids, core–shell, or compound particles can be formed. Figure 25.6 shows two examples of nickel/zinc ferrite particles. 10 nm (a) 25.5 Templated synthesis using frameworks, supports, and substrates 20 nm (b) Figure 25.6 TEM images of nanoparticles of Ni0.5Zn0.5Fe2O4 synthesized by plasma torch synthesis. The particle shapes are superimposed: (a) an octahedron and (b) a truncated octahedron. (Based on images provided by M. McHenry and R. Swaminatham.) Heterogeneous nucleation, nucleation that occurs on an existing surface, can be used to generate important kinds of nanostructures, including zero-, one-, and two-dimensional materials. The methods are similar to those already described: they are either physical or chemical and involve crystallization from either liquids or vapours. The principal difference is that an outside agent is also involved and permits the direct control of the nanoparticle formation. The outside agent can be a framework or a support structure that limits the size of the reaction volume, as described in Section 25.5a for nanoparticle synthesis using inverse micelle frameworks and embossed supports. During thin film growth, the outside agent is a substrate. Because of its importance in nanomaterials synthesis, thin film growth on substrates is described in Sections 25.5b and c. (a) Nanosized reaction vessels Key points: By carrying out reactions in nanoscale reaction vessels, the ultimate dimensions of solid products are confined to the vessel size; a reverse micelle has an aqueous core in which reactions can occur. Water droplet (reaction volume) Figure 25.7 An inverse micelle. The hydrophilic heads of molecules surround the water droplet and the hydrophobic tails contact the nonpolar solvent, which acts as the dispersed medium. Reactants are solvated in the water droplet and then caused to react in these spatially confined vessels. When the synthesis of a particle is carried out in a nano-vessel, the ultimate particle size is limited by the vessel size. One popular route is the inverse micelle synthesis approach. An inverse micelle consists of a two-phase dispersion of immiscible liquids, such as water and a nonpolar oil. By including amphipathic surfactant molecules (molecules having a polar and a nonpolar end), the aqueous phase can be stabilized as dispersed spheres with a size dictated by the water:surfactant ratio (Fig. 25.7). The size of the crystalline particles is limited by the micelle volume, which can be controlled on the nanoscale. Examples of nanoparticles formed by this technique are Cu, Fe, Au, Co, CdS, CdSe, ZrO2, ferrites, and core–shell particles such as Fe/Au. An alternative approach to nanopatterning is laser-assisted embossing, which has been used to generate close-packed arrays of hemispherical nanowells having volumes of the order of zeptolitres (1 zL 1021 L 1025 m3 103 nm3). The volumes of the nanowells are manipulated by changing the intensity of the laser, by altering the pressure between the mould and the substrate, and by growing an amorphous oxide on the patterned silicon prior to nanocrystallization. Nanowells with diameters as small as 50 nm have been used as reaction chambers for the preparation of semiconducting nanocrystals. Under most growth conditions, independent nanocrystals are formed in individual nanowells (Fig. 25.8). Characterization and fabrication 400 nm 100 nm Figure 25.8 An SEM image of an array of nanoscale wells that contain one CdS nanocrystal per well. The lower inset magnifies a small area of the array; the white arrows indicate crystals formed at the edges of the wells. The upper inset is a TEM image of an individual CdS nanocrystal removed from the array of nanowells by sonication. (J.E. Barton and T.W. Odom, Nano Lett., 2004, 4, 1525.) E X A MPL E 25 . 2 Assessing the role of the volume of nano-reaction vessels The use of nano-reaction vessels combines solution-based synthesis techniques directly with size-limited growth. Assume that a surface is embossed with 100 nm-wide hemispherical imprints that are then used as reaction vessels. (a) What is the volume of the reaction vessel? (b) What will be the size of an NaCl particle that is crystallized in this vessel to produce a single spherical particle? Assume a solution of 1.00 M NaCl(aq) and that the crystallization goes to completion. The density of NaCl is 2170 kg m3 and its molar mass is 58.44 g mol1. Answer (a) The volume of each hemispherical imprint is half the volume of a sphere of radius 50 nm: v solution = 21 ( 34 π r 3 ) = 32 π (5.0 × 10 − 8 m)3 = 2.62 × 10 − 22 m3 (b) Because the concentration of the solution is 1.00 mol dm3, the amount of NaCl in the hemispherical imprint is n = (1.00 × 103 mol m–3 ) × (2.62 × 10 − 22 m3 ) = 2.62 × 10 −19 mol The mass of this amount of NaCl is m = (2.62 × 10 −19 mol) × (58.44g mol−1 ) = 1.53 × 10 −17 g or 1.53 × 1020 kg of NaCl in the solution. The volume of this NaCl is therefore VNaCl = 1.53 × 10 −20 kg = 7.06 × 10 −24 m3 2170 kg m− 3 To determine the size of the particle, we use the definition of volume and solve for the diameter (twice the radius): d = 2r = 2( 43πv ) 1/ 3 = 2( 43π × 7.06 × 10 − 24 m3 ) 1/ 3 = 2.38 × 10 − 8 m Hence, the diameter of the particle is 23.8 nm. Self-test 25.2 For the same solution, how small would the hemispherical imprint need to be to make a particle 2 nm in diameter? As discussed earlier, high-temperature (150–250ºC) solution synthesis is one of the most successful methods for achieving size and shape control of nanocrystalline materials through nucleation and growth in the presence of selective surfactant molecules, including TOPO. However, inhomogeneity in the injection of precursors, mixing of the reactants, 663 664 25 Nanomaterials, nanoscience, and nanotechnology and temperature gradients in the reaction flask can lead to problems with control over the size distribution of nanoparticles. Small reaction vessels, such as reverse micelles and laserembossed surfaces, provide an appealing alternative for controlling chemical and thermal homogeneity. The use of nanofabricated structures made from these materials enables the growth and assembly of nanocrystals that are potentially more interesting than inorganic salts and organic molecular crystals. This approach to the growth of nanocrystals has several advantages, including facile syntheses based on bulk reactions, flexible size control achieved by using different concentrations of precursor materials, realization of isolated nanocrystals in ordered arrays, and reusable templates. (b) Physical vapour deposition Key points: In the physical vapour deposition methods, a vapour of atoms, ions, or clusters physically adsorb to the surface and combine with other species to create a solid; molecular beam epitaxy is a technique where evaporated species from elemental charges are directed as a beam at a substrate where growth occurs. (a) Photoresponse Unpolarized y-Polarized x x-Polarized y 6 8 10 12 14 Wavelength, l /µm (b) Figure 25.9 (a) Photoresponse (in arbitrary units) of (In0.2Ga0.8)As quantum dots formed using MBE. The IR absorption band arises from intraband absorption. The inset is an AFM image of the quantum dots. The quantum dots vary in lateral dimensions between 15 and 20 nm and in vertical height in the range 3–7 nm. (b) A crosssectional TEM micrograph of an ordered superlattice of (In0.2Ga0.8) As quantum dots in a GaAs matrix. (Courtesy of E. Towe.) In physical vapour deposition (PVD) methods, vapours are delivered from their source to a solid substrate on which they crystallize. The arriving gaseous species are typically atoms, ions, or clusters of elements. There are several general forms of PVD that are widely used: molecular beam epitaxy (MBE), sputtering, and pulsed laser deposition (PLD). The gas-phase species can have either relatively low kinetic energies on arrival (as in MBE) or relatively high kinetic energies (as in sputtering and PLD). The most important feature that all PVD methods have in common is the ability to achieve complex film stoichiometries, either through the transfer of the different species between the source and the film or by using multisource systems to compensate for deviations from the ideal film composition. Because vapour deposition methods allow single atomic layers to be deposited in a controlled fashion on a support or substrate, nanoscale architectures can be built from the bottom up. In this section we describe some fundamental principles for these processes in systems that allow for real-time, in situ control over the nanostructure of materials. Molecular beam epitaxy is an ultrahigh vacuum technique for growing thin epitaxial films, films that have a definite crystallographic relationship with the underlying substrate. In MBE, molecular beams are formed by heating elemental sources until atoms evaporate and are transported ballistically to the substrate surface with relatively low kinetic energies (of the order of 1 eV). The overall pressure is kept very low to minimize collisions within the beam. To prevent species from bouncing off other surfaces of the system and back towards the substrate, much of the system is cooled with liquid nitrogen. In that way, stray elemental species adsorb onto surfaces and remain there. This cooling also helps to maintain the vacuum. Film stoichiometry and film growth rates are highly dependent on the beam fluxes, which can be controlled by adjusting the temperatures of the elemental sources. Homoepitaxy is the epitaxial growth of a thin film of a material on a substrate of the same material. Heteroepitaxy is the epitaxial growth of a thin film of a material on a substrate of a different material. Heteroepitaxy introduces a strain between the growing material and the substrate caused by their crystallographic mismatch. This strain can lead to a self-assembly process where nano-islands form in ordered arrays on the surface to minimize strain energies. By alternating the deposition of two materials by MBE, a multilayer structure of ordered QDs can be created (Fig. 25.9). By tailoring the structure of the substrate appropriately, nanowires and nanowells can also be created, as well as the superlattices and artificially layered crystals we describe later. Pulsed laser deposition is a versatile PVD technique that can be used to synthesize a wide variety of high-quality thin films (Fig. 25.10). In PLD, a pulsed laser is used to ablate a target, which releases a plume of atomized and ionized particles from its surface that condenses onto a nearby target. The PLD process usually produces films of composition identical to that of the substrate, which is a major simplification compared with techniques that require fine-tuning or expensive control equipment to achieve a specific stoichiometry. This simplicity makes PLD attractive for exploratory materials chemistry because the cation stoichiometry is controlled externally and independently of the deposition parameters. The technique also allows control over a number of deposition conditions that alter the thermodynamics and kinetics of growth, including substrate temperature, laser energy, and atmospheric conditions (partial pressures of O2, N2, or Ar); it can be carried out at pressures at or below 500 mTorr (about 100 Pa). Pulsed laser deposition can achieve average growth Characterization and fabrication rates similar to those of MBE (for example, 100 pm s1) but the instantaneous growth is much higher because growth occurs only in the short intervals (of about 1 µs) immediately after the 1020 ns laser pulse ablates the target. Therefore, for a very short period there is a very high arrival rate and a very considerable supersaturation at the substrate surface, leading to very high nucleation rates. An additional difference from MBE is that in PLD the energies of the particles arriving at the substrate are usually 10–100 eV, which is large enough that there is the potential that arriving particles could damage the growing film. Nevertheless, PLD has been used to grow a variety of high-quality superlattices, such as the SrMnO3 and PrMnO3 superlattice film shown in Fig. 25.11 and described later. (c) Chemical vapour deposition Key point: In chemical vapour deposition methods, a vapour of molecules chemically interact or decompose at or near the substrate, where they adsorb on the surface and combine with other species to create a solid and residual gaseous products. The control over complex stoichiometries, the ability to achieve monolayer-by-monolayer growth, and the attainment of high-quality films is not limited to physical vapour techniques. Chemical techniques, such as metal/organic chemical vapour deposition (MOCVD) and atomic layer deposition (ALD), also provide these levels of control. In contrast to physical methods, in which species condense directly onto a substrate and react with one another, chemical techniques require that a precursor decomposes chemically on or near the substrate in order to deliver the reactant species to the growing film. Therefore, in chemical vapour methods, the decomposition thermodynamics of the selected precursors must be considered because the vapour often contains elements that must not be incorporated in the growing films. Typically, the kinetic energy of the arriving species is relatively low in chemical techniques, and the temperature of the substrate can be controlled externally. The layout of a typical chemical vapour deposition (CVD) system was shown Fig. 25.5. Such systems normally operate at moderate vacuum or even at atmospheric pressure (in the region of 0.1–100 kPa). Their growth rates can be quite high, more than 10 times that of MBE or PLD. In CVD techniques, chemical decomposition of the feed molecules proceeds upstream of a substrate surface on which the desired product will be grown. The decomposition of the gaseous reactants is activated by high temperatures, lasers, or plasmas. Large numbers of different materials have been grown using CVD methods. For Group 13/15 (III/V) semiconductors (such as GaAs), the typical sources are organometallic precursors for the Group 13 element (such as Ga(CH3)3), and hydrides or chlorides for the Group 15 element (such as AsH3). A typical reaction is 550650ºC ⎯ → GaAs(s) + 3CH 4 (g) Ga(CH3 )3 (g) + AsH3 (g) ⎯⎯⎯⎯ carried out in a hydrogen environment. The CVD technique can be fine-tuned to produce very high-quality films. The drawbacks of CVD include the use of toxic chemicals, turbulent flow in the reaction chamber, the incorporation of unwanted chemical species owing to incomplete decomposition or exhaust, and the use of high pressure (which prevents the use of in situ diagnostics). This same technique, however, can be used without a substrate to create nanoparticles by pyrolysis of the vapour species. Furthermore, MOCVD and MBE have been combined to allow for in situ monitoring of growth. This technique has been called chemical beam epitaxy (CBE). These low pressures decrease the interactions in the reactant stream and lead to more ballistic transport of the chemical vapour species to the substrate, where they decompose and react to form a film. A final chemical approach aims to control the precise chemical interactions that occur at a surface. In the process called atomic layer deposition (ALD), chemical species are delivered sequentially to a substrate on which a single monolayer deposits. The excess reactant is removed. Repetition of this monolayer coverage, subsequent reaction, and removal of excess reactants allows for precise control over the growth of complex materials. In this process, it is necessary to control the chemical species and their interactions to ensure monolayer coverage only and facile subsequent reaction. To do so means that the vapour of each reagent must interact in the proper manner with the film layer deposited previously. The technique produces flat, homogeneously coated layers. Some examples of ALD-grown nanomaterials are Al2O3, ZrO2, HfO2, CuS, and BaTiO3. Target Deposition chamber Substrate 665 Laser beam Plume Substrate heater Figure 25.10 A pulsed laser deposition chamber used to produce nanostructured super lattices and artificially layered thin films. Figure 25.11 Cross-sectional TEM image of a 2 2 superlattice of SrMnO3 and PrMnO3 prepared using laser-MBE. The inset shows a calculated image confirming the (SrMnO3)2(PrMnO3)2 structure. (Reprinted with permission from B. Mercey, et al., In situ monitoring of the growth and characterization of (PrMnO3)n (SrMnO3)n superlattices. J. Appl. Phys., 2003, 94, 2716, American Institute of Physics.) 666 25 Nanomaterials, nanoscience, and nanotechnology Self-assembled nanostructures One of the frontiers of nanoscience lies at methods that bridge the bottom-up and topdown approaches to achieve nanoscale structures. This frontier is the focus of the section that follows. The specific examples that are discussed reflect the key advantages of, and challenges facing, bottom-up methods. 25.6 Control of nanoarchitecture In order to achieve specific commercial applications, it is important to be able to control the architecture of nanomaterials. Self-assembly, supramolecular chemistry, and morphosynthesis all offer routes to the control of the dimensionality of self-assembled nanostructures. (a) Self-assembly Key point: Components that can self-assemble fall between the sizes that can be controlled chemically and those that can be manipulated by conventional manufacturing; self-assembly offers the crucial technique to bridge top-down and bottom-up methods. Various definitions of self-assembly have been proposed. They include the noncovalent interaction of two or more molecular subunits to form an aggregate with novel structure and properties that are determined by the nature and positioning of the components, the spontaneous assembly of molecules into structured, stable, noncovalently joined aggregates, the spontaneous formation of higher-ordered structures, and the process by which specific components spontaneously assemble in a highly selective fashion into a welldefined, discrete supramolecular architecture. Because the components that assemble fall between the sizes that can be controlled chemically and those that can be manipulated by conventional manufacturing, self-assembly may offer the crucial technique to bridge top-down and bottom-up methods. An example of the application of methods that bridge top-down and bottom-up assembly has been the development of microdevices capable of heterogeneous catalysis, acting as lasers, and as the basis of gas sensing. Potential self-assembly components must be mobile, therefore self-assembly usually takes place in fluid phases or on smooth surfaces. Static self-assembly occurs when the system is at a global or local equilibrium, such as in liquid crystals. Dynamic selfassembly occurs when the system is dissipating energy, such as in an oscillating chemical reaction. Templated self-assembly occurs when systems are organized based on interactions bet-ween the components and regular features in the environment. The QD self-assembly that occurs during strained heteroepitaxy in MBE and the use of laserembossed surfaces for small reaction vessel synthesis in ordered arrays (Section 25.5) are examples of templated self-assembly. Finally, biological self-assembly occurs in systems involving life, such as cells and tissues: entire organisms are elaborate examples of biological self-assembly. (b) Morphosynthesis Key point: Morphosynthesis is the control of architecture and morphology and the patterning of inorganic materials with nanoscale to macroscopic-scale dimensions through changes in synthesis parameters. Figure 25.12 SEM illustrating the multiple levels of order in BaSO4 nanostructures controlled through morphosynthesis: (a) funnel-like structure with well-aligned cones; (b) magnified section showing the alignment in the superstructure; (c) magnified section showing the alignment of the nanostructured bundles; (d) magnified section showing the surface of the bundles. (Reproduced with permission from S.-H. Yu, et al., Nano Lett., 2003, 3, 379.) Morphosynthesis is the control of architecture and morphology and the patterning of inorganic materials with nanoscale to macroscale dimensions through changes in synthesis parameters. As an example, sodium polyacrylate has been used as a structuredirecting or templating agent that, when coupled with controls over temperature, pH, and the concentration of reagents, can be used to tune the nanoarchitectures of BaSO4 (Fig. 25.12). Another intriguing example of morphosynthesis is the use of small molecules, such as ethylenediamine, to control crystal face growth to achieve ZnS nanowires. Zinc sulfide is a large band-gap semiconductor that has been widely used commercially as a phosphor in luminescent devices because of its emission in the visible range. Ethylenediamine molecules can act as templates in a solvothermal route to direct the formation of nanorods. The structures Self-assembled nanostructures 667 B OX 25 . 2 Cu2S/CNT: Novel solar cell and glucose sensor materials Nanocrystals (NCs) and carbon nanotubes (CNTs) have found practical applications as solar cells and sensing devices. Semiconducting nanocrystalline oxides have been used to generate efficient photocurrent by an interaction between the excited NCs and the conductive CNTs, thus demonstrating their importance as foundations to build light-harvesting assemblies. These third-generation solar cells could push the efficiency of solar energy conversion beyond what traditional silicon-based solar cells can offer. Silicon-based solar cells range in 15–20 per cent conversion of solar power to usable electricity, whereas new third-generation solar cells can reach an efficiency of 35–40 per cent. The size and morphology of the Cu2S NCs depend strongly on the concentration of the precursors, Cu(acac)2 and elemental S (Fig. B25.2). When the concentration of Cu(acac)2 is near 0.05 M, the Cu2S NCs are all spherical with diameters in the range 3–5 nm, with an average of 4 nm. On increasing the concentration of Cu(acac)2 to 0.10 M, the spherical NCs increase in size, averaging about 8 ± 1 nm in diameter. As the concentration of Cu(acac)2 is increased to 0.15 M, the NCs become triangular in shape, with an edge size of about 12 nm. The density of the NCs on the MWCNTs (a) (b) (c) decreases with this change of shape. The lattice structure of the MWCNTs acts as a substrate and template for the growth of the Cu2S nanocrystals. This triangular shape of the Cu2S NCs has been observed only in the presence of the MWCNTs. It is thought that the formation of triangular NCs at higher precursor concentration can be explained by kinetic arguments, assisted by the lattice matching with the graphite layers. These NC–CNT structures can also serve as biosensor platforms owing to their ability to promote electron-transfer reactions with enzymes and other biomolecules. NCs deposited on CNTs have been shown to be excellent amperometric sensors for glucose over a range of concentrations. The key to these devices is to have the active Cu2S NCs in electrical contact with the backing electrodes with the MWCNTs acting as a bridge for electrical conductivity. The redox reaction provides mobile electrons and the electrode coated with these NC–CNT hybrid materials acts as an electrical conductor. Amperometric measurements can then be made to monitor the extent of the redox reaction to detect glucose levels. Glucose detection using these nanostructures was very good, with a 105 M selectivity and a 5 108 M detection limit. (d) Figure B25.2 (a, b) TEM and HRTEM images of Cu2S-MWCNT nanostructures. (c) TEM image of higher-density spherical Cu2S NCs using 0.10 M Cu(acac)2(aq). (d) HRTEM image of the 0.10 M Cu(acac)2 Cu2S NCs with average size of 8 ± 1nm. (H. Lee, et al. Nano Lett., 2007, 7, 778.) of the nanoparticles for solar cell and glucose sensor applications can be modified by changing the concentration of the copper(II) acac complexes that are used as templates for the growth of copper(I) sulfide nanoparticles on carbon nanotubes (Box 25.2). (c) Supramolecular chemistry and dimensonality Key point: Supramolecular chemistry is regarded as a most promising method for designing and controlling the bottom-up assembly of nanometre-scale objects with rationally designed dimensionality. A general concept for generating ordered structures is based on the recognition-derived spontaneous assembly of complementary subunits. In the field of supramolecular chemistry, one of the key challenges is the characterization of the supramolecules themselves, in particular the characterization of discrete, highly symmetrical, large, self-assembled entities. A great number of biological systems are formed as a result of the formation of many weak hydrogen bonds and van der Waals interactions. Synthetic systems can take advantage not only of these bonds but also of metal–ligand bonding interactions, the latter being much stronger and generally highly directional. There are some important examples of biological systems using coordination chemistry for self-assembly, such as the metal-ion activated proteins, including Ca2 signalling proteins and Zn fingers (Sections 27.4 and 27.5).As a result of various d-metal coordination geometries, metal complexes provide a pool of different acceptor subunits that can be linked together by donor building blocks to form rigid 668 25 Nanomaterials, nanoscience, and nanotechnology frameworks. The final shape of the self-assembled entity is defined by both the metal atom coordination geometry and the orientation of the interaction sites in a given ligand. This methodology has been successfully used to synthesize the cuboctahedron shown in Fig. 25.13. Many self-assembled nanostructures use similar approaches to take advantage of encoded information within the components of the nanostructure building block. Molecular simulations have been performed to study the self-assembly of nanoparticles functionalized with oligomeric tethers attached to specific locations on the nanoparticle surface (Fig. 25.14). These results suggest that tethered nanobuilding blocks constitute a unique class of macromolecule that can be used to assemble novel nanomaterials. 12 OTf Ph3P Pt 24+ PPh3 O O N N 8 PPh3 Ph3P Pt Pt TfO Ph3P PPh3 OTf 24 TfO– Figure 25.13 A cuboctahedron synthesized through self-assembly of two distinct building blocks. (Adapted from S. Leininger, et al., Chem. Rev., 2000, 100, 853.) (a) (c) (b) (e) (d) (g) (f) (h) (i) (j) (m) (l) (k) (o) (p) (n) (q) (r) (s) (t) (w) (u) (v) Figure 25.14 Representative tethered nanobuilding blocks: (a–d) tethered nanospheres; (e–g) tethered nanorods; (h–p) tethered nanoplates, including circular, triangular, and rectangular plates; (q, r) tethered nanowheels; (s, t) tethered nanocubes; (u–w) triblock nanobuilding blocks created by joining two nanoobjects with a polymer tether. (Adapted from Z. Zhang, et al., Nano Lett., 2003, 3, 1341.) Self-assembled nanostructures (a) 0D Density of states (b) 1D (c) 2D 669 (d) 3D Scale Figure 25.15 Effects of confinement of charge carriers on the density of states for nanomaterials with specific dimensionality. (Adapted from A. Mrumjadar, Nonoscience Boot Camp at Northwestern University 2001.) Specific tethering groups influence the nature of the self-assembly of the nanobuilding blocks into larger ordered nanostructures. Control over dimensionality in materials can yield unique control over their physical properties. For example, dimensionality has a noted influence on the density of electronic states (Fig. 25.15). In the next few sections we examine specific examples of how that control has been achieved and the novel properties that have been reported. 25.7 One-dimensonal control: carbon nanotubes and inorganic nanowires Key point: Dimensionality plays a crucial role in determining the properties of materials. The elongated one-dimensional morphology of nanorods, nanowires, nanofibres, nanowhiskers, nanobelts, and nanotubes has been studied extensively because one-dimensional systems are the lowest dimensional structures that can be used for efficient transport of electrons and optical excitation. They are therefore expected to be crucial to the function and integration of nanoscale devices. Not much is known, however, about the nature of localization that could preclude transport through one-dimensional systems. Such systems should possess discrete molecular-like states extending over large distances and exhibit some exotic phenomena, such as the effective separation of the spin and charge of an electron. There are also many applications where one-dimensional nanostructures could be exploited, including nanoelectronics, very strong and tough composites, functional nanostructured materials, and novel probe microscopy tips. To address these fascinating fundamental scientific issues and potential applications, two important questions pose key challenges to the fields of condensed matter chemistry and physics research. First, how can atoms or other building blocks be assembled rationally into structures with nanometre-sized diameters but much greater lengths? Second, what are the intrinsic properties of these quantum wires and how do these properties depend, for example, on their diameter and their structure? A key class of nanomaterials that offers potential answers to these questions is carbon nanotubes (CNTs). Carbon nanotubes are perhaps the best example of novel nanostructures fabricated through bottom-up chemical synthesis approaches. They have very simple chemical composition and atomic bonding configuration but exhibit remarkably diverse structures and unparalleled physical properties. These novel materials have found application as chemical sensors, fuel cells, fieldeffect transistors, electrical interconnects, and mechanical reinforcers. The structure and properties of buckminsterfullerene, C60, and related species have produced much activity in the chemistry and materials sciences communities. The now familiar soccer-ball structure of C60 itself was discussed in Chapter 14. Carbon nanotubes were discovered in the early 1990s by electron microscopy. The bonding, local coordination, and general structure of CNTs are similar to those of buckminsterfullerene, but CNTs can have a greatly extended length, leading to a tube rather than a ball structure (Fig. 25.16). Figure 25.16 STM image of a single-walled carbon nanotube. (J. Hu, et al., Acc. Chem. Res., 1999, 32, 435.) (Reproduced with permission from the American Chemical Society.) 670 25 Nanomaterials, nanoscience, and nanotechnology (a) (8,8) (10,–2) (8,0) a2 (b) (c) a1 (d) Figure 25.17 (a) The honeycomb structure of a graphene sheet. Single-walled carbon nanotubes can be formed by folding the sheet along lattice vectors, two of which are shown as a1 and a2. Folding along the (8,8), (8,0), and (10,–2) vectors leads to armchair (b), zigzag (c), and chiral (d) tubes, respectively. (Based on H. Dai, Acc. Chem. Res., 2002, 35, 1035.) (Reproduced with permission from the American Chemical Society.) Carbon nanotubes are cylindrical shells formed conceptually by rolling graphene (graphitelike) sheets into closed tubular nanostructures with diameters matching that of C60 (0.5 nm) but lengths up to micrometres. A single-walled nanotube (SWNT) is formed by rolling a sheet of graphene into a cylinder along an (m,n) lattice vector in the graphene plane (Fig. 25.17). The (m,n) indices determine the diameter and chirality of the CNT, which in turn control its physical properties. Most CNTs have closed ends where hemispherical units cap the hollow tubes. Carbon nanotubes self-assemble into two distinct classes, SWNTs and multiwalled carbon nanotubes (MWNTs). In MWNTs, the tube wall is composed of multiple graphene sheets. Morphosynthetic control provides routes to tune the details of these self-assembled nanostructures. This self-assembly occurs during synthesis using several of the fabrication techniques we have already described, including pulsed laser ablation, laser-assisted catalytic growth, chemical vapour deposition (CVD) based on hydrocarbon gases, and carbon arc discharge. All these strategies rely on vaporizing carbon and condensing some fraction into extended nanostructures. Laser vaporization methods typically make relatively small amounts of nanocarbons; specialized CVD techniques have been developed to synthesize CNTs in quantities in excess of a few milligrams. In the CVD approach, a hydrocarbon gas such as methane is decomposed at elevated temperatures and C atoms are condensed on to a cooled substrate that may contain various catalysts, such as Fe. This CVD method is attractive because it produces open-ended tubes (which are not produced in the other methods), allows continuous fabrication, and can easily be scaled up to large-scale production. Because the tubes are open, the method also allows for the use of the nanotube as a templating agent. In the arc discharge method, extremely high temperatures are obtained by shorting two carbon rods together, which causes a plasma discharge. Such plasmas easily achieve temperatures in excess of where carbon vaporizes (at about 4500 K). Low potential differences and moderately high currents are needed to produce this arc. The typical CNT formed by either the arc method or CVD is multiwalled. To encourage SWNT formation, it is necessary to add a metal catalyst such as Co, Fe, or Ni to the carbon source. These metal catalyst particles block the end-cap of each carbon hemisphere and thus promote SWNT growth. In addition, the growth directions of the nanotubes can be controlled by van der Waals forces, applied electric fields, and patterning of the metal catalyst onto different substrates. The patterned growth approach is feasible with discrete catalytic nanoparticles and scalable on large wafers to achieve large arrays of nanowires (Fig. 25.18). Figure 25.18 Ordered carbon nanotube structures obtained by direct chemical vapour deposition synthesis. (a) An SEM image of self-oriented MWNT arrays. Each tower-like structure is formed by many closely packed multiwalled nanotubes. Nanotubes in each tower are oriented perpendicular to the substrate. (b) SEM top view of a hexagonal network of SWNTs (the line-like structures) suspended on top of silicon posts (the bright dots). (c) SEM top view of a square network of suspended SWNTs. (d) Side view of a suspended SWNT power line on silicon posts (the bright lines). (e) SWNTs suspended by silicon structures (the bright regions). The nanotubes are aligned along the electric field direction. (H. Dai, Acc. Chem. Res., 2002, 35, 1035.) (Reproduced with permission from the American Chemical Society.) Self-assembled nanostructures The repeating axial hexagonal patterns of CNTs are graphitic structures; however, the electrical properties of nanotubes depend on the relative orientation of the repeating hexagons. The nanotubes can be either semiconductors or metallic conductors. When oriented in the chair configuration, CNTs exhibit remarkably high electrical conductivity. Electrons can travel through the micrometre-length nanowire with zero scattering and zero heat dissipation. CNTs also have very high thermal conductivity, comparable to the best thermal conductors known (such as their counterparts, diamond and graphite). Therefore, they are being heralded as the ideal nanomaterial for the development of interconnects for integrated circuits. They may solve two key challenges in the computer industry: heat dissipation and increased processing speeds. Open-ended CNTs can also be used for nanorobotic spot welding to deposit metal at specific sites for connections in integrated circuits (Fig. 25.19). Copper is a good choice for these applications because it has high thermal and electrical conductivity. It also has a very low bonding energy to C (0.10–0.14 eV/atom), so can be deposited easily on surfaces and structures from the encapsulating CNT. The deposition is accomplished by making physical contact between a STM probe and a copper-filled CNT tip, and then applying a potential difference to create a circuit with the probe as the anode. This procedure generates sufficient heat for the Cu atoms to be able to migrate along the tube and be deposited where they are required. The nanowelding process is of interest because its advantages include efficiency (a very low current can induce melting and flow), three-dimensional manipulation of welding site by using piezoelectric distortion, rapid melting, precise control of small mass delivery, and the compatibility of Cu with conventional semiconductors. In addition to CNTs, parallel methods have been discovered for making nanotubes out of materials that share bonding characteristics with C, including semiconductors and metal oxides. More specifically, BN, ZnO, ZnSe, ZnS, InP, GaAs, InAs, and GaN have all been made into nanotubes. The novel electronic properties and small sizes of these nanotubes make them appealing as inorganic nanowires. Core–sheath wires similar to macroscopic coaxial wires are of great interest. One method to prepare them uses a colloid-templating strategy that involves a layer-by-layer assembly of polyelectrolytes and inorganic nanoparticles on submicrometre- and micrometre-sized polystyrene (PS) latex particles. A number of methods, including laser ablation, carbothermal reduction, and several solution-based methods, have been developed to generate one-dimensional nanostructures with coaxial structures. For instance, by using a novel nanowire-templating technique based on the layer-by-layer approach and calcining (heating in air to drive off volatile templating agents), ordered Au/TiO2 core–sheath nanowire arrays have been fabricated. A template-grown gold nanowire array is used as a positive template, and then a cationic polyelectrolyte and an inorganic precursor are assembled on gold nanowires by the layer-by-layer technique. Calcination then converts the inorganic precursor to titanium dioxide (Fig. 25.20). Figure 25.19 FESEM image of Cu-filled CNTs; all have sharp tips filled with metal nanoneedles. These CNTs are up to 5 µm long, with outer diameters in a range of 40–80 nm. (L. X. Dong, et al., Nano Lett. 2007, 7, 58.) 671 672 25 Nanomaterials, nanoscience, and nanotechnology (a) (b) Figure 25.20 (a) Low-magnification and (b) high-magnification SEM images of Au/TiO2 core–sheath nanowire arrays. (Y.-G. Guo, et al., J. Phys. Chem. B, 2003, 107, 5441.) An advantage of the layer-by-layer nanowire-templating approach over other methods is that nanoscale control can be exerted over the sheath thickness by varying the number of deposition cycles. Titanium dioxide has been one of the most investigated oxide materials because of its technological importance. It has been widely used for photocatalysis and environmental clean-up applications, on account of its strong oxidizing power, chemical inertness, and nontoxicity. These core–sheath nanowires have well-defined diameters and lengths, largely determined by the dimensions of the templates, and porous sheaths with thicknesses controlled by the number of layers deposited. This method opens a door to the use of a wide variety of nanowires as the positive templates for the fabrication of core–sheath nanostructured materials for chemical sensors, photocatalysis, light energy conversion devices, and nanoscale electronic and optoelectronic devices. More specifically, Au and Ag nanowires have been targeted as scaffolds for optical sensing of environmentally and biologically important analytes with very high selectivity and sensitivity. 25.8 Two-dimensional control: quantum wells and solid-state superlattices In this section, we describe the design of so-called two-dimensional materials, which are materials that have macroscopic length scales in two dimensions and a nanoscale length scale along the third dimension. Thin-film processing methods permit the deposition of films only one atomic (or one unit cell) layer thick. By varying sequentially the types of atomic (or unit cell) layers being deposited, it is possible to control the material architecture along the growth direction at a sub-nanometre scale, thereby allowing the bottom-up development of artificially layered nanostructures. A quantum well (QW) is a thin layer of one material sandwiched between two thick layers of another material and is the two-dimensional equivalent of a zero-dimensional QD. In a superlattice, two (or more) materials alternate with an artificially induced periodicity along the growth direction. Superlattices often have repeat periods of about 1.5–20 nm or greater, and sublayer thicknesses that range from two unit cells to many tens of unit cells. Artificial crystal structures usually have repeat distances similar to bulk crystals (about 0.3–2.0 nm) and have sublayer thicknesses that range from an atomic layer to two unit cells (about 1 nm). These structures have found broad commercial application as key device elements in computer chip manufacturing, including hard disk read heads. To control a physical property such as absorption and emission, a quantum well can be used in a similar fashion to the QDs described earlier. The optical properties of quantum wells can be enhanced in quantum well superlattices (often called multiple quantum well structures), which are used in lasers. To control a physical property such as hardness, which is tied to dislocation formation and motion, it is known that many interfaces between dissimilar materials are required. Hardness can be increased substantially in such superlattices, as compared to monolithic films, ranking them Self-assembled nanostructures among the hardest known materials and making them successful coatings in the tooling industry. To control a property such as ferroelectricity or superconductivity, which are intimately tied to the crystal structures, it is important to be able to engineer artificial crystal structures. (a) Quantum wells AlxGa1–xAs GaAs AlxGa1–xAs Key points: Quantum wells consist of a thin small band-gap material sandwiched between thick layers of large-gap materials; multiple quantum wells can enhance the effects of quantum wells when the wells do not interact. A quantum well is typically composed of two semiconductor materials with different band gaps, such as Al1xGaxAs and GaAs. The smaller band-gap material (GaAs) is sandwiched between layers of the larger band-gap material (Al1xGaxAs), and the thickness of the layer of the small band-gap material is confined to the nanometre scale (Fig. 25.21). Electrons are confined in the layer of the small band-gap material along the growth direction; the barrier height is the difference in the energy levels of the band edges for the two materials. Therefore, the energy states in the smaller gap material are quantized along the growth direction, similar to the quantization of energy in a QD. The electronic wave functions are spatially extended in the direction parallel to the layer and therefore the energy levels are not quantized. As for a QD, the optical properties of quantum wells can be tailored and both interband (between the valence and conduction band) and intraband (between the quantized sub-bands) absorption and emission can be controlled. Both In1xGaxAs/GaAs and Al1xGaxAs/GaAs quantum wells have been widely studied, and the optical transition has been observed to move to higher energies compared to the bulk when the thickness drops down to about 20 nm. The main use of quantum wells is in semiconductor lasers, where the small-gap quantum well is the active layer in the device. Typical QW lasers use the interband transitions, but intraband lasers are also being developed and are discussed below. Many of the effects that occur in quantum wells can be enhanced by using superlattice structures, the periodic repetition of quantum wells along the special direction (the z-axis). In the context of semiconductors these superlattices are called multiple quantum well (MQW) structures. If the active layers (the small band-gap layers) do not interact, then the electrons are confined to a given layer and are unable to tunnel between them. The use of MQW structures in this case increases the absorption or emission from a given device, as there are multiple levels. For example, an MQW laser has much higher power output than the corresponding single quantum-well laser. If the wide band-gap layers are thin enough, one QW interacts with the adjacent QW and electrons can tunnel between them. This phenomenon is used in quantum cascade (QC) lasers, which operate at high powers in the IR region. The laser characteristics of these materials are fundamentally different from those of semiconductor diodes and MQW lasers in a variety of ways. In a QC laser, only one type of carrier (electrons) is necessary to achieve laser action; in the other two, both electrons and holes are required. In addition, in the QC laser, the transitions are intraband transitions that arise from the quantization of the valence band. The QC lasers are constructed from 13/15 (III/V) semiconducting materials such as GaAs, InAs, and AlAs. All these superlattice systems have been grown by solid-source molecular beam epitaxy on single-crystal substrates. The heterostructures that comprise an individual layer in such a superlattice are far more complex than the simple unit-cell periodicities found in most artificially layered structures or superlattices discussed below. For example, a single functional unit of an AlAs/GaAs superlattice QC laser has a layer sequence (in nanometres) of 2.4 4.3 1.8 4.3 1.4 4.4 1.3 4.5 1.2 4.5 1.1 4.8 0.8 5.2 0.6 5.8 0.6 6.2 0.5 6.6 0.5 7.2 0.5 z 673 8.0 with GaAs layers in red, AlAs in blue, and Si-doped layers in green. This sequence leads to a functional layer thickness of 72.8 nm, which is much larger than most superlattice periods. The superlattice itself consists of 40 such layers. (b) Solid-state superlattices Key points: Artificially layered materials have a periodic repeat along the growth direction of a thin film; the periodic repeat is controlled by the number and type of sublayers deposited in sequence. The periodic repeat along the growth direction of a thin film in a superlattice is controlled by the number and type of sublayers deposited in sequence, whereas the lateral periodicity is determined by the coherency—the matching of lattice characteristics—between the Figure 25.21 An (AlxGa1xAs)(GaAs) (AlxGa1xAs) quantum well; the thickness of the GaAs layer is on the nanoscale. 674 25 Nanomaterials, nanoscience, and nanotechnology Al N AlN c TiN 20 AlN/TiN TiN 40 TiC2 60 Aluminium Tool steel AlN Hardness, h/Pa 0 (b) (a) Figure 25.22 (a) The structure of an ultrahard AlN/TiN superlattice and (b) the hardness of a nitride superlattice and commonly used hard materials. (Adapted from S.A. Barnett and A. Madan, Physics World, 1998, 11, 45.) Relative hardness 6 5 4 3 2 TiN VN 1 0 10 20 30 40 Superlattice period, l /nm Figure 25.24 The dependence of hardness on the superlattice period for (TiN)m(VN)m superlattices. (Adapted from U. Helmersson, et al., J. Appl. Phys., 1987, 62, 481.) c l Superlattice 80 Artificial crystal TiN 100 NbN/TiN c-BN Diamond Ti ccr T iO 2 SrO Substrate (a) (b) Figure 25.23 (a) The structure of an AB artificial crystal and an AB superlattice; c and represent the repeat period during growth of the artificial structure and the superlattice period, respectively. (b) The structure of Sr2TiO4 as an artificially layered oxide. The polyhedra represent Ti-centred, corner-sharing TiO6 octahedra and the spheres represent Sr2 cations. The lateral coherency between the SrO and TiO2 sublayers is excellent in this layered structure and the crystallographic repeat parameter, ccr, is twice the growth repeat period, c. sublayers (Figures 25.22 and 25.23). It should be kept in mind that the superlattice is built bottom-up with periods in the nanometre range and total thicknesses in the micrometre range. The goal of materials chemists and engineers is to use these synthesis skills to control the properties and functionality of materials of this kind. Superlattice nitrides are ranked among the hardest known materials. The superlattice period and the chemical composition play important roles in determining the mechanical properties of such compounds, as the interfaces between the two nitride layers are responsible for the enhanced hardness. Figure 25.24 shows, for instance, that the maximum hardness of a typical superlattice occurs for periodicities in the range 5–10 nm. These superlattices have been successfully deposited using sputtering, pulsed laser deposition, and molecular beam epitaxy. Such superlattices are used on cutting tools and are made economically by using sputtering to form them. Perovskite oxides (Section 24.7) are used in numerous applications on account of their ferroelectric, acoustic, microwave, electronic, magnetic, and optical properties. The most widely used perovskite is BaTiO3, which is a very important dielectric material. It has been discovered that layering ferroelectric BaTiO3 with the isostructural perovskite SrTiO3 can lead to enhanced dielectric properties arising from lattice strain. The techniques of PLD, MBE, and CVD have all been used to create SrTiO3/BaTiO3 superlattice thin films (Fig. 25.25). Although the misfit between the two crystal structures is small enough to allow for epitaxial growth and the formation of coherent interfaces between each bilayer, the stress introduced at the interface is sufficient to improve the dielectric response, specifically the remanent polarization (the polarization in the absence of an applied field) of the superlattices, compared to undoped BaTiO3. It is important to control layer thickness precisely to ensure the formation of coherently strained interfaces that lead to improved dielectric response. The technique of PLD has been used to deposit alternating blocks of SrTiO3 and BaTiO3 having unequal layer thicknesses of SrTiO3/BaTiO3 15/3, 10/3, 3/10, and 3/15, and the net dipole moment density of superlattices has been reported to be in the order of 46 µC cm2, whereas that of bulk films of the parent phases is about 14 µC cm2. Perovskite-based structures can also exhibit interesting magnetic effects. In particular, manganese-based perovskite films such as (La,Sr)MnO3 possess useful ferromagnetic and magnetoresistive properties. In general, these characteristics arise from electron exchange Self-assembled nanostructures interactions occurring between Mn and O that are mediated by the various cations in the structure, and can be altered significantly when these cations are spatially ordered. The PLD procedure has been used to deposit superlattices of LaMnO3/SrMnO3; the Mn3 cations are found in the LaMnO3 layers and the Mn4 are found in the SrMnO3 layers. Thus, the thin-film superlattice technique allows for precise ordering of A-site cations (La, Sr), which in turn causes ordering of the Mn in its different oxidation states. In superlattices of (LaMnO3)m(SrMnO3)m, samples with m ≤ 4 possess magnetic properties just like solid solutions of La0.5Sr0.5MnO3, whereas superlattices with larger periods have significantly higher resistivities and lower Curie temperatures. Similar effects are observed in superlattices with unequal layer thicknesses, such as (LaMnO3)m(SrMnO3)n and (PrMnO3)m(SrMnO3)m systems (Fig. 25.11). The magnetic properties of thin-film superlattices in the family La(Sr)MnO3/LaMO3 (M Fe, Cr, Co, and Ni) exhibit a wide range of properties depending on both superlattice repeat period and the identity of M. For instance, when M Co or Ni the Curie temperature is increased but when M Fe or Cr it is decreased. Interesting electronic effects are observed in superlattice thin films consisting of spinel Fe3O4 and rock-salt NiO layers. Superlattices of these two structures are well matched because both have the same O sublattice and differ only in the locations of the cations. Superlattices of composition Fe3O4/NiO have been grown by MBE for a variety of layer thicknesses. The electronic conduction mechanism in many d-metal oxides such as these is electron hopping, a process that has a short mean free path. If this distance is approximately the same as the crystal lattice spacings it can lead to novel electrical properties for artificially layered materials with short lattice spacings. Thus, Fe3O4/NiO superlattices with bilayer repeat periods of 6.8 nm have electrical conductivities that differ by factors in the range 106–108 in different directions, which is among the highest known electrical anisotropy for any material. This result effectively demonstrates both the high degree of chemical modulation available by superlattice deposition and the extent of electrical conduction localization that can be achieved as a consequence. 675 Figure 25.25 TEM image of a SrTiO3 / BaTiO3 superlattice. (Reprinted from D. G. Schlom, et al., Oxide nano-engineering using MBE. Mater. Sci. Eng. B, 2001, 87, 282, with permission from Elsevier.) 25.9 Three-dimensional control The design and synthesis of three-dimensional (3D) supramolecular architectures with tunable, nanoporous, open channel structures have attracted considerable attention because of their potential applications as molecular sieves, sensors, size-selective separators, and catalysts. 1.5–10 nm (a) Mesoporous materials Key points: Mesoporous materials and metal–organic frameworks have ordered pore structures that are defined over the nanoscale; the synthesis of these materials relies upon control of self-assembly; guests can be incorporated into the inorganic host framework for novel applications. An important class of three-dimensionally ordered nanomaterials is mesostructured nanomaterials. Mesoporous materials are well known in heterogeneous catalysis (Chapter 26) and are of great interest because of our ability to tune their pore sizes from 1.5 to 10 nm (Fig. 25.26). Mesoporous inorganic nanomaterials are synthesized in a multistep process based on the initial self-assembly of surfactant molecules and block copolymers that self-organize into supramolecular structures (which are liquid crystalline assemblies of cylindrical, spherical, or lamellar micelles; Fig. 25.27). These supramolecular frameworks serve as structuredirecting templates for the growth of mesostructured inorganic materials (often silica or titania). During a solvothermal reaction step, oxide particles (silica, Fig. 25.27) form at the surfaces of the hexagonal rods, assembling around the supramolecular structures. The templating agent can be removed through an acid wash or by calcination to yield inorganic materials having hexagonal pores with uniform and controllable dimensions. This porosity offers unique capabilities for both catalysis and inclusion chemistry. A wide variety of mesostructured and mesoporous inorganic materials have been obtained by varying the choice of templating agent and reaction conditions (Fig. 25.28). For instance, the family known as M41S has silica or alumina-silica inorganic phases and various cationic surfactants leading to three distinct types of structures: hexagonal lamellar (MCM-50), cubic (MCM-48), and hexagonal (MCM-41) (Fig. 25.28). (a) (b) Figure 25.26 A hexagonal mesoporous structure with (a) controlled nanoporosity and (b) functionalized pores. 676 25 Nanomaterials, nanoscience, and nanotechnology Remove copolymer (RO)3SiCH2CH2CH2R 4–5nm Si(OR´)4 + copolymer ) (a OH –S R i–O O –S OH –S i–O O i– O 14 nm ) (b Figure 25.27 Block-copolymer structure-directing agents self-assemble into micellar rods that form hexagonal arrays that can be removed to yield nanoporous silica matrices for inclusion chemistry and catalysis. (Adapted from M.E. Davis, Chem. Rev., 2002, 102, 3601.) 7nm (c) Figure 25.28 Representations of three types of ordered mesoporous solids. (a) Lamellar (layered materials), (b) cubic (complex arrangements), and (c) hexagonal (honeycomb). (Adapted from A. Mueller and D.F. O’Brien, Chem. Rev., 2002, 102, 729.) The shape of the surfactant, and the water content used during synthesis controls the resulting nanoarchitecture, as can be seen from the illustration. Surfactants can also be used to tailor the structure. For example, surfactant cetyltrimethylammonium cations (C16TMAC) have been used to fabricate silica nanofibres with hexagonal pores. Mesoporous nanomaterials also offer routes to functionalize pores to increase catalytic activity and selectivity. They have received much attention as host materials for the inclusion of numerous guests such as organometallic complexes, polymers, d-metal complexes, macromolecules, and optical laser dyes. The silica nanofibres shown in Fig. 25.29 can even be used as hosts to grow nanowires of various other oxide materials. E X A M PL E 25 . 3 Controlling nanoporosity in molecular sieves (a) Compare the nanoarchitecture of ZSM-5 zeolite and MCM-41; include in your comparison descriptions of the dimensionality of tunnels and relative pore sizes. (b) Cetyltrimethylammonium [C16H33N(CH3)3] cation and tetrapropylammonium [N(CH2CH2CH3)4] cation are surfactants used in the syntheses of these materials. Which surfactant is a better choice for the synthesis of MCM-41? Answer We need to consider the relative size of the pores in these two catalytic materials and the chain length and size of the surfactant used for templating their porosity. (a) ZSM-5 is a microporous zeolite with pore sizes of approximately 0.5 nm. It has three-dimensional intersecting pores similar to the cubic mesoporous phases but with smaller dimensions. MCM-41 is a mesoporous solid with hexagonal, onedimensional pores with tunable dimensions of 2 nm up to 10 nm. (b) The choice of surfactant must match the pore size of the solid. The longer chain surfactant (cetyltrimethylammonium cation) is a better choice for the self-assembly of the MCM-41 material owing to its longer hydrocarbon tails and larger spontaneous curvature. This increased tail size encourages larger pore dimensions that are a direct consequence of the packing of the surfactants within the micellar rods that lead to the hexagonal mesophase. Self-test 25.3 Which of these materials, MCM-41 or ZSM-5, is more likely to be used as a host material for the entrapment of QDs? Figure 25.29 (a) SEM image of mesoporous silica nanofibres. (b) Lowmagnification TEM image of the nanofibres. (c) High magnification TEM image of one nanofibre. The inset is a selected-area electron diffraction pattern of the nanofibre. (d) High-resolution TEM image recorded at the edge of one nanofibre. (J. Wang, et al., Chem. Mater., 2004, 16, 5169.) (b) Metal–organic polyhedra and frameworks Key points: Metal–organic polyhedra and frameworks offer tunable catalytic surfaces and high surface areas; they are assembled using supramolecular chemistry. A second class of three-dimensionally controlled nanostructures is metal–organic frameworks (MOFs). These frameworks are self-assembled from a careful choice of metal ions, bridging organic ligands, and/or metal–organic polyhedra (MOP). They offer an alternative approach to the synthesis of open porous materials using supramolecular assembly and are being called the ‘new zeolites’. These macroscopic materials have some of the Self-assembled nanostructures highest reported surface areas and, as such, have found practical application for methane and hydrogen storage, catalysis, and drug delivery. As discussed earlier, supramolecular chemistry relies on bridging together nanobuilding blocks (NBBs) to assemble macroscopic materials. One new strategy in design is the incorporation of nanoporosity (pores with dimensions of 1 to 100 nm) within the NBB or MOP. The synthesis of a nanoporous cuboctahedron NBB (Fig. 25.30a) was achieved by linking rigid square molecules of Cu2(CO2)4 with m-BDC (1,3-benzenedicarboxylate). The overall NBB formula is Cu25(m-BDC)24(DMF)14(H2O)10(H2O)50(DMF)6 (C2H5OH)6 and is termed more conveniently MOP-1. This compound is constructed from 12 paddle-wheel units (themselves NBBs) bridged by m-BDC to yield a large metal-carboxylate polyhedron. The simplest way to view the overall structure is by considering its relationship to the cuboctahedron (shown in Fig. 25.13), where each square and link has been replaced by the paddle-wheel NBB and the m-BDC (two-connector) units, respectively, to give an expanded-augmented cuboctahedron (or a truncated cuboctahedron). The ability to synthsize the cuboctahedron illustrates the feasibility of obtaining crystals of large porous metal–organic polyhedra in which rigid NBBs are an integral part of a well-defined structure. More recently, a zeolitic imidazolate framework compound, ZIF-69, has been reported that can store 83 dm3 of carbon dioxide per litre (1 dm3) of material at 0°C and 1 atm. MOPs have also been designed for selective catalysis that mimic active sites in enzymes. Enzymatic catalysis often relies on the highly constrained environment of substrates within the active site. Recent examples in supramolecular organometallic chemistry mimic this approach by using host–guest interactions. A supramolecular M4L6 tetrahedral assembly has been developed to provide a host that mediates a variety of organic and organometallic transformations, including the isomerization of allylic alcohols. The M4L612 host is formed from the self-assembly of four octahedral metal centres (M Fe(III) or Ga(III)) at the vertices and six bis-catecholamide naphthalene ligands (L4) that span the edges of a tetrahedron (Fig. 25.30b). The host has been found to encapsulate monocationic guests Figure 25.30a The crystal structure of MOP-1 (drawn using coordinates obtained for c-MOP-1) showing (a) 12 paddle-wheel units (Cu, red; O, blue; C, grey) linked by m-BDC to form (b) a large truncated cuboctahedron of 1.5 nm diameter void (yellow sphere); the grey spheres (carbon atoms) represent the polyhedron constructed by linking together only the carboxylate C atoms in (a) to form linked square NBBs, an arrangement that provides for (c) a large porous polyhedron with triangular and square windows. Hydrogen atoms in light grey; otherwise same colouring scheme as in (a). (M. Eddaoudi, et al., J. Am. Chem. Soc., 2001, 123, 4368.) (Reproduced with permission from the American Chemical Society.) (a) (b) Figure 25.30b (a) Schematic of MOP structure of an M4L6 assembly with one of the edge-spanning ligands displayed containing an organometallic guest. (b) Crystal structure of Fe4L6 assembly with an organometallic guest. (D. H. Leung, et al., J. Am. Chem. Soc., 2007, 129, 2746.) (Reproduced with permission from the American Chemical Society.) 677 678 25 Nanomaterials, nanoscience, and nanotechnology ranging from NH4 cations to cationic organometallic species containing Rh metal centres. The catalytic activity is controlled by the size and shape of the host cavity. Some of the key challenges in the area of mesostructured materials are achieving control over pore size and shape, the presence of counterions and solvents within the channels, the interpenetration of different networks, the framework structural instability in the absence of guest molecules, and the often low thermal stability of the host framework. In some cases, the incorporation of solvent molecules can be used to achieve desired structures. For example, a robust three-dimensional porous structure of formula [Ln2(PDC)3(DMF)2]∞ has been constructed from lanthanoid (Ln) cations (Er3 or Y3) and the non-linear anionic bridging ligand, pyridine-3,5-dicarboxylate (PDC2–) in dimethylformamide (DMF). The solvated framework polymers undergo a solid-state, crystal-to-crystal reaction on heating. Through loss of both sorbed and coordinated solvent and rearrangement of the framework core a porous MOF structure is formed with retention of structural integrity (Fig. 25.31). These mesoporous materials have proven to be effective absorbants for H2, N2, and benzene. (c) Inorganic–organic nanocomposites Key points: Class I inorganic–organic materials have noncovalent interactions and class II have some covalent interactions; sol–gel and self-assembly methods are key chemical routes to hybrid nanocomposite design and synthesis; the properties of polymer nanocomposites are controlled through the nature of the polymeric and inorganic phases, as well as through their dispersions and interactions. Inorganic–organic nanocomposites are a third class of three-dimensional ordered materials that possess chemical and physical properties that can be tuned by using the synergistic association of organic and inorganic components at nanoscales. These hybrid materials originated in the paint and polymer industries where inorganic fillers and pigments were dispersed in organic materials (including solvents, surfactants, and polymers) to fabricate commercial products with improved materials performance. Hybrid nanomaterials are also of interest because their mechanical properties occupy a niche between glasses and polymers, so achieving enhanced strength and robustness. Hybrid nanomaterials have been reported with excellent laser efficiencies, photostability, and ultrafast photochromic response. They also can have very large second-order and third-order nonlinear optical response, which is important in frequency conversion and optical switching for telecommunications. Nanocomposites have already entered the market-place in sunscreens, fire-retardant fabrics, stain-resistant clothing, thermoplastics, water filters, and automobile parts. Examples include the use of nylon-6/montmorillonite clay nanocomposites for timing-belt covers, television screens that are coated with indigo dyes embedded in a silica/zirconia matrix, organically doped sol–gel glassware, and sol–gel-entrapped enzymes. Hybrid nanomaterials offer the materials chemist novel routes that use rational materials design to optimize structure–property relationships. Their nanoarchitectures and resulting properties depend on the chemical nature of the components and the synergy between them. A key part of the design of these hybrids is the selective tuning of the nature, extent, and accessibility of the interfaces between the inorganic and the organic building blocks. x y x y z z z (a) (b) x (c) (d) Fig 25.31 Views of the MOF structures of [Er2(PDC)3(DMF)2]∞: (a) zigzag connection between adjacent erbium centres; (b) channels with a view along the orthorhombic x axis showing coordinated DMF molecules protruding into the channel space. Views of the structure after DMF solvent removal and rearrangement: (c) the shorter linear connection between adjacent Er3 centres; (d) view of the open channels along the hexagonal z axis. (J. Jia, et al., Inorg. Chem., 2006, 45, 8838.) (Reproduced with permission from the American Chemical Society.) Self-assembled nanostructures The interfaces fall into two main classes. Class I corresponds to hybrid materials where no covalent or ionic bonds are present between the organic and inorganic phases. In Class I materials, the various components interface through weak interactions such as hydrogen bonding, van der Waals contacts, and electrostatic forces. In Class II materials, at least some of the organic and inorganic components are linked through strong chemical bonds (covalent, ionic, or Lewis acid–base bonds). An important feature in the tailoring of hybrid networks is the chemical pathway used to design a given material. The main chemical routes are summarized in Fig. 25.32. It is possible to control the nature of the inorganic precursors (by using the methods described in Section 25.5), the nature of the organic material, and the method of assembling the composite material. The templated-growth fabrication processes often rely on organic molecules and macromolecules as structure-directing agents to achieve the construction of complex hierarchical architectures. It turns out that molecular and supramolecular interactions between template molecules (surfactants, amphiphilic block copolymers, organogelators, etc.) and hybrid or metal–oxo-based networks and NBBs allow a rich variety of nanocomposite materials to be prepared. Figure 25.33 illustrates two approaches to the assembly of NBBs into larger, ordered structures. R–M R) –(O + M(OR)n x OrganoMetal functional alkoxide metal alkoxide Controlled H2O Pre- or post-functionalized metal–oxo clusters or nanoparticles (c) Self-assembly (a) Conventional sol–gel route (b) (d) Organic connectors Oxo-polymers Organic component R Periodic RR functional porosity Mesostructured NBB-based hybrids Figure 25.32 Different routes to obtain hybrid nanomaterials. Route (a) is a conventional sol–gel route that leads to ‘ordinary’ hybrids. Routes (b) and (d) involve the use of templates capable of self-assembly, giving rise to mesoorganized phases. Routes (c) and (d) involve the assembly of nanobuilding blocks (NBBs). (Adapted from C. Sanchez, et al., Chem. Mater., 2001, 13, 3061.) Figure 25.33 Different approaches to the assembly of nanobuilding blocks (NBBs) involving clusters and different connectors. (a) Ordered cluster dispersions based on the covalent linkage of complementary organic and inorganic NBBs. (b) Creation of a bicontinuous mesostructured hybrid arrangement. Interactions between the template and the inorganic NBB include covalent bonding and van der Waals forces (hydrophobic contacts). (Adapted from C. Sanchez, et al., Chem. Mater., 2001, 13, 3061.) 679 680 25 Nanomaterials, nanoscience, and nanotechnology Polymer solution Syringe High voltage source Figure 25.34 The electrospinning apparatus. (Adapted from R. Sen, et al., Nano Lett., 2004, 4, 459.) O C12H25 O Figure 25.35 Ester-functionalized SWNTs (SWNT-COO(CH2)11CH3, EST-SWNTs). (Based on R. Sen et al., Nano Lett., 2004, 4, 459.) Important examples of Class I are polymer nanocomposites (PNCs). These materials are composed of inorganic nanoparticles dispersed in a polymeric matrix. Early commercial PNCs used two-dimensional ordered or lamellar clays, such as sodium montmorillonite (Na-MMT), dispersed in the polymer matrix. Such dispersants (or fillers) have sandwich-type structures (with channels between layers), a total thickness of 0.3–1 nm, and a length of 50–100 nm for each layer; these sandwich structures typically agglomerate into micrometre-sized agglomerates. The dispersion of the inorganic phase within the organic polymeric matrix in a PNC is obtained by intercalation (inserting the polymer between the layered sheets of the clay). Intercalation involves the expansion of the interlamellar spacing as a consequence of ion exchange by using organic amines or quaternary ammonium salts. Exfoliation is achieved through reactive chemical compounding or by intensive melt mixing of the clay and polymer phases. Without proper dispersion, the nanosized filler particles aggregate into larger clusters, causing degradation in the properties of the composite. Many techniques can be used to control the state of dispersion as well as the nature of the bond between nanoparticle and matrix, including the use of silanes, grafting, and CVD. Silanes are being produced that perform both tasks simultaneously: they both tailor the surface properties of the filler, which promotes the coupling of the nanoparticle to the polymer matrix, and act as a surfactant, reducing the surface energy of the filler to prevent agglomeration and promote dispersion. Radiation copolymer grafting using -rays, electron beams, or X-rays is very effective in modifying polymer surfaces. In most cases the polymer chains that are created strengthen the polymer–filler bonds. This strengthening is also evident in CVD as the gaseous molecules that form a film on the substrate typically increase the interactions between filler and polymer. The nature of the filler (dispersant) and any cavities (voids) present in a PNC greatly influences the mechanical properties of the composites because they can control the distribution of stress through the composite matrix. Both the yield strength (a measure of the material’s ability to resist permanent deformation) and toughness (a measure of the energy absorbed prior to fracture) of nanocomposites are controlled by the nanoparticle size and dispersion and nanoparticle-to-polymer contact interactions. Therefore, control of the nanoparticle dispersion and alignment of these fillers offers a way to tailor the mechanical properties of the nanocomposites. One important class of PNCs uses nanocarbons as the dispersants. Single-walled nanotubes (SWNTs) have exceptional mechanical properties; their very high Young’s modulus (their resistance to mechanical stress), low densities, and high aspect ratios make them very attractive as fillers to enhance polymer strength. SWNTs have high tensile strength, almost 100 times that of steel, whereas their density is only about one-sixth that of steel. SWNTreinforced composites have been developed by the physical mixing of SWNTs in solutions of preformed polymers, in situ polymerization in the presence of SWNTs, surfactant-assisted processing of SWNT–polymer composites, and chemical modification of the incorporated SWNTs. Melt processing can be carried out easily by compression moulding at high temperatures and pressures followed by rapid quenching. The so-called electrospinning methods use electrostatic forces to distort a droplet of polymer solution into a fine filament, which is then deposited on a substrate (Fig. 25.34). The nanofibres created from the electrospinning can be configured into a variety of forms including membranes, coatings, and films, and can be deposited on targets of different shapes. Fibres can be prepared with diameters smaller than 3 nm, with high surface-to-volume and length-to-diameter ratios, and with controlled pore sizes. Electrospun polymer nanocomposites have been made with homogeneously dispersed SWNTs and have exhibited significantly improved mechanical strength. Different organo-functional groups covalently attached to the nanotubes through oxidation reactions have been used to improve their chemical compatibility with specific polymers (Fig. 25.35). The use of SWNTs with different functionalities allows the study of the importance of the interfacial interactions between the filler and the polymer matrix. This chemical functionalization is an effective approach towards improving the processability of the nanocomposite and the chemical compatibility of the components. As described before, specific functionalization can be used to separate the SWNT bundles and prevent their agglomeration. Metal oxide fillers have also been used to optimize polymer strengths and thermal properties. A particularly interesting example involves the control over the mechanical properties of alumina/polymethylmethacrylate (PMMA) nanocomposites by engineering the distribution of weak particle-to-polymer interactions. The filler nanoparticles Bioinorganic nanomaterials Bioinorganic nanomaterials Biological phenomena, such as DNA condensation, intercellular transport, tissue assembly, respiration, photosynthesis, and reproduction, originate and operate at the nanoscale. Because biological processes use and manipulate materials at this length scale, it is not surprising that scientists and engineers are looking to natural systems for inspiration and to reach a better understanding of the design of robust and useful nanomaterials. This interest has spawned an extensive research effort termed biomimetics, the mimicking of biological systems. Of particular interest to us are materials that bridge solid-state inorganic materials and living cells, the so-called bioinorganic materials (Chapter 27). Their nano counterparts, the bioinorganic nanomaterials, promise to have a profound impact on the way we live; key applications include drug delivery, medical diagnostics, cancer therapy, and environmental pollution control, as well as chemical and biological sensing. Many biological structures (people, for instance) are produced from self-assembly of functional building blocks, which have complex morphological architectures themselves. Despite success in understanding the basic principles of the biological assembly process, as well as in making inorganic materials through biological templating, it remains a key challenge to mimic natural pathways as efficient routes for fabricating artificial bionanomaterials. In this section we discuss some recent advances and offer a taste of the complexity, diversity, and architectures of exquisite bionanomaterials by using key examples of these types of materials. 25.10 DNA and nanomaterials Key point: Interactions between gold nanoparticles and DNA can cause the self-assembly of DNA into condensates and the ordering of nanoparticles into regular arrays that can be used for biosensors. DNA condensates are self-assembled nanostructures. The architecture of these condensates is driven by the components that become woven into the nanoassembly. Electrostatic interactions drive the formation of DNA condensates. DNA is a negatively charged polyelectrolyte that interacts with positively charged ions, molecules, or modified nanoparticles to form ordered structures and convert from their random coils to a more compact form. This compact form, called a condensate, is essential for the organization of DNA into chromatin, a component of genes. Mimicking or controlling the DNA condensation process is the basis of the design of nonviral gene delivery vectors that have practical application in the treatment of cancer, drug delivery, and biosensing. Studies of DNA condensation in vitro have focused on the use of complex cations, including hexaamminecobalt(III) or polyamines as the positive charge centre in the condensate. Complex formation in these systems can be expected to mimic DNA wrapping around charged particles in living cells, such as positively charged histone proteins. Because the cationic particles can be fine-tuned with respect to size and charge density, the effect of those parameters on the complex formation patterns with DNA have begun to be understood. Gold nanoparticles have been suggested as an effective transfection agent (a device for introducing exogeneous DNA into a cell) and, as such, it is important to understand the interactions of nanoparticles with DNA. Figure 25.37 shows AFM images of complexes of lysine-modified gold nanoparticles and DNA at different nanoparticle/DNA ratios. These results indicate the possibility of developing 80 (a) (b) Stress/MPa (alumina) are dispersed in the composite matrix in the form of ‘net-like’ structures as opposed to ‘isolated island’ structures formed when microparticles are dispersed in the polymer matrix. In contrast to microparticle–polymer composites, these net-like structures inhibit crazing, the propagation of microscopic cracks during tensional loading. Alumina/ PMMA composites also display higher yield strengths and a shift from brittle to ductile behaviour when nanoparticulate alumina is used (Fig. 25.36). Ductility is advantageous as the composite can then be drawn into thin wires and can withstand sudden impact, thus making the composite more resilient mechanically. The addition of nano-alumina with the anti-agglomerant methacrylic acid lowers the glass transition temperature enough to change the stress–strain curve of the composite and shift from brittle to ductile under tensile loading. 681 60 (c) 40 20 0 0 0.1 0.2 Strain 0.3 Figure 25.36 Typical stress–strain curves for (a) neat PMMA, (b) 2 per cent by mass as-received micrometre-sized alumina-filled PMMA composite, and (c) 2.2 per cent by mass 38 nm (MAA) alumina/PMMA nanocomposite. Although the strength decreases slightly for (c) (related to the decreased stress at the curve maximum), the overall ductility (related to the total strain) and toughness (related to the area under the curve) are greatly improved. (Adapted from B.J. Ash, et al., Macromolecules, 2004, 37, 1358.) (Reproduced with permission from the American Chemical Society.) 682 25 Nanomaterials, nanoscience, and nanotechnology 2.5 µm 2.5 µm (a) (b) (c) (d) Figure 25.37 AFM images of complexes of lysine-modified gold nanoparticles and DNA at different nanoparticle/DNA ratios (a) 10:1, (b) 20:1, (c) 50:1, and (d) 100:1. Several wrapped molecules are seen along with free nanoparticles in (a) and (b). Network formation is seen in (c) and (d). The inset in (a) shows a single DNA molecule wrapping around six nanoparticles. (Reproduced by permission from M. Ganguli, et al., Langmuir, 2004, 20, 5165.) functionalized gold nanoparticle-DNA into a model system to study DNA condensation in vivo. These smart nanomaterials have far-reaching consequence for the design of nanoscale self-assembled materials that can be used as building blocks for active nanodevices. Another exciting accomplishment in the synthesis of bionanomaterials is the use of DNA to drive the assembly of inorganic nanoparticles (Fig. 25.38). First, nanoparticles are functionalized with single strands of DNA. Then, the complementary strand of DNA is introduced from an analyte or bioagent, leading to dimerization with the DNA on the nanoparticles and the self-assembly of the nanoparticles into ordered arrays. Detectable differences in optical properties occur between the disordered and ordered states and result in colour changes. This work has helped to make sophisticated optical-based biosensor arrays with remarkable sensitivity and selectivity for different analytes and bioagents. T < Tm T > Tm 25.11 Natural and artificial nanomaterials: biomimetics Key point: Biological materials can be used as templates in the design of nanoinorganics having specific architectures that mimic the structure of the natural material. Figure 25.38 DNA templating of inorganic semiconducting nanoparticles. (N.L. Rosi and C.A. Mirkin, Chem. Rev., 2005, 105, 1547.) The formation mechanisms in fossilization, including those for siliceous woods, offer efficient methods to reproduce morphological hierarchies of original plant matter through the replacement of the organic components by silica. Artificial fossilization processes can be realized by carefully lining the morphologically complex surfaces of the biological structure with inorganic layers followed by removal of the organic template. Natural materials such as wood and eggshell membrane have been used as templates for the preparation of macroporous silica, zeolites, and titanium dioxide from both precursor sol–gel solutions and suspensions of nanocrystals. Unfortunately, morphological replication has been achieved only over the micrometre scale and the nanoscale details of the biological templates have not yet been reproduced. An artificial fossilization process has been developed by taking advantage of a surface sol–gel process that can replicate nanoscale features of biological templates (Box 25.3). The surface sol–gel process consists of two steps. First, metal alkoxides are adsorbed from the solution on to hydroxylated substrate surfaces. Then the adsorbed species are hydrolysed to yield nanometre-thick oxide films. Natural cellulose fibres possess surface –OH groups and provide a template for using the surface sol–gel process. The outer diameter of the TiO2 nanotube varies from 30 to 100 nm, and the thickness of the tube is uniform along its length, with a wall thickness of about 10 nm. The nanotube assembly exhibits the original morphology of interwoven cellulose fibres. The ‘titania paper’ produced in this way records the morphological information of the original paper at the nanoscale and offers a remarkable example of successful biotemplating of metal oxide nanomaterials. 25.12 Bionanocomposites Key points: Bionanomaterials synthesis offers new routes to realize smart materials with advanced mechanical strengths and improved performance over natural materials; protein engineering offers an opportunity to insert species that attach selectively to inorganic materials. Bioinorganic nanomaterials 683 B OX 25 . 3 Artificial fossils An artificial ‘titania fossil’ of paper has been created. A titania gel film was deposited on the morphologically complex surface of paper, and the resultant paper/titania composite was calcined to remove the original filter paper. In a typical procedure, a piece of commercial filter paper was placed in a suction filtering unit, washed by suction filtration with ethanol, and then dried by air flow. The 10 cm3 of titanium butoxide solution was passed through the filter paper for 2 minutes. Two 20 cm3 portions of ethanol were filtered immediately to remove the unreacted metal alkoxide and 20 cm3 of water was passed through to promote hydrolysis and condensation. Finally, the filter paper was dried in flowing air. By repeating this filtration/ deposition cycle, thin titania gel layers covered the surface of the cellulose (a) fibres. The resultant paper/titania composite was calcined in air at 723 K for 6 hours to remove the filter paper. The resulting titania fossils possessed, except for some shrinkage in size owing to the calcination step, the morphological characteristics of the original filter paper, as shown in Fig. B25.3. The titania sheet is self-supporting and 0.22 mm thick. In this case, deposition of titania thin films was repeated 20 times. Besides paper, other morphologies have been produced using a similar approach (Fig. B25.2d,e). The sheet size and thickness depend on the original filter paper used. The original morphology of the filter paper was found to be replicated by titania films, and the cellulose fibres were precise copies of irregular titania nanotubes, which are clearly seen in the illustrations. (b) (c) (d) (e) Figure B25.3 (a) A field emission scanning electron micrograph (FE-SEM) of titania paper. The inset shows the photograph of a sheet of titania paper. (b, c) Transmission electron micrographs of individual titania nanotubes, isolated from the assembly. The inset in (c) gives a schematic illustration of the boxed area, showing that the titania nanotube wall is composed of fine anatase particles. (d) An SEM image of titania cloth and (e) an SEM image of titania thread. (J. Huang and T. Kunitake, J. Am. Chem. Soc., 2003, 125, 11834.) Millions of bone fractures resulting in hospitalization occur worldwide every year. Modern treatments for severe bone injuries replace a nonuniform defect with a permanent biomaterial that may corrode, wear, and ultimately cause severe infection. In addition, these biomaterials typically exhibit mechanical strength far greater than that of bone, resulting in stress shielding and eventual bone resorption around the implant. There is a social need for a bone tissue engineering scaffold that has mechanical properties similar to bone and that will facilitate bone growth into the defect, degrade slowly and naturally, and eventually be replaced by natural bone tissue without threat of infection. The primary obstacle towards this goal is matching the unique mechanical properties of natural bone tissue with a degradable, synthetic nanomaterial. The versatile and important structural properties of bone are derived from the nanoscale interactions between its inorganic and organic components. There are two types of bone, trabecular bone, the weaker and porous central part, and cortical bone, the stronger and denser outer shell of bone. Hydroxyapatite nanocrystals provide compressive strength as they precipitate onto bundles of collagen fibres that impart tensile strength. Degradable biomaterials composed of inorganic and organic components can have mechanical properties closely matching natural bone tissue, and they may be suitable for load-bearing bone tissue engineering applications. Here we describe one illustrative example of these approaches that combines polymeric components with inorganic components. 684 25 Nanomaterials, nanoscience, and nanotechnology Poly(propene fumarate) (PPF) cross-linked with poly(propene fumarate)-diacrylate (PPF-DA) is an injectable, biodegradable material. Although PPF shows great potential as a material for bone tissue engineering where mechanical properties such as tensile strength are not crucial, such as the trabecular bone, its mechanical properties are inferior to those required for the more demanding human bone applications, such as cortical bone. A promising strategy to improve the mechanical properties of a polymer is to mimic the architecture of bone or to incorporate inorganic particles into the polymer matrix. For example, significant increases have been observed in the compressive mechanical properties of PPF by the incorporation of calcium phosphate. The extent to which inorganic particles modify the polymer properties is associated with their size, shape, and uniformity of dispersion, as well as with the degree of interaction between the inorganic and organic components (similar to the red abalone shell, Box 25.4). Chemically modified inorganic materials that allow for optimal dispersion and interaction with the polymer matrix therefore seem to be ideal candidates for use in bionanocomposites. A hybrid alumoxane nanoparticle, or a partially hydrolysed aluminium oxyhydroxide with surrounding organic material (Fig. 25.39), with a long carbon chain and a reactive double bond dispersed in PPF/PPF-DA, shows over a threefold increase in strength over polymer resin alone. Several approaches have been successful for attaching nucleic acids to nanoparticles, which is then followed by DNA hybridization like that mentioned in Section 25.10. Various functional inorganic nanomaterials, such as semiconductors, fullerenes, metals, and oxides, have been considered for use in bottom-up approaches to the design of systems that bridge inorganic and biological materials. One approach is to functionalize the surface of nanoparticles for selective molecular attachment; such materials are used in the development of biosensors, nanoprobes, drug transporters, and other smart devices (Box 25.5). B OX 25 . 4 A natural nanocomposite: the shell of red abalone Nacre (mother-of-pearl) is an outstanding example of a natural nanocomposite that achieves enhanced structural integrity by combining (on the nanoscale) individual components that are either brittle or structurally weak. Red abalone shell is composed of about 95 per cent by mass inorganic aragonite (a polymorph of CaCO3), with only a small percentage of an organic biopolymer. This ceramic/polymer composite material exhibits natural beauty in conjunction with a nanostructure that affords high-quality mechanical properties. Compared to its constituent materials, the laminated structure of red abalone achieves approximately a twofold increase in strength and a thousandfold increase in toughness, owing to the interfaces between the ceramic matrix and the biopolymer assemblies. These remarkable increases in material strength and toughness have inspired chemists and materials scientists to develop synthetic, biomimetic nanocomposites that attempt to reproduce Nature’s composite systems. The abalone shell is a highly ordered nanocomposite material (Fig. B25.4) that is extraordinarily tough, hard, and strong. Nanoscale asperities (spikes) exist on the ceramic platelet surfaces that serve to lock together neighbouring platelets, thus resisting fracture and enhancing material strength. The formation mechanisms of these nanoscale asperities are still unknown but they play an essential role in stabilizing the material against fracture. In the nacreous layer, cobble-like polygonal nanograins are the basic building blocks that construct individual aragonite platelets. The nanograin-structured aragonite platelets are somewhat ductile and the organic biopolymer serves as an adhesive to hold the aragonite platelets together. The nacreous region of the abalone shell has such a great increase in toughness over its constituent materials because this nanocomposite has excellent crack-tip deflection properties (or the ability to prevent catastrophic failure by crack propagation), the ability to deform by nanograin or platelet slip, and its possession of a well-dispersed organic adhesive holding everything together. By contrast, the outer prismatic layer does not show these crack diversion mechanisms and serves mainly as a brittle outer shield for the abalone. (a) (b) (c) Figure B25.4 SEM images of the nacreous fracture shell section taken at increasing magnification. The brick-and-mortar architecture is shown in (a) and (c). Nanoscale asperities on the aragonite platelet surface are readily observed in (b). (X. Li, et al., Nano Lett., 2004, 4, 613.) Bioinorganic nanomaterials O (a) (b) (C H 2)71 C H 3 O O H N O O HO H N OH Al Al O O O O O O HO O Al O Al OH O OH Figure 25.39 Chemical structures of modified alumoxanes: (a) diacryloyl lysine– alumoxane (activated), (b) stearic acid– alumoxane (surfactant), and (c) acryloyl undecanoic amino acid–alumoxane (hybrid). (R.A. Horch, et al., Biomacromolecules, 2004, 5, 1990.) (Reproduced with permission from the American Chemical Society.) OH O O Al Al O OH OH O Al O Al O Al Al O O 685 O OH H N (c) OH O (C H 2)11 O O HO O OH O Al O Al O O O Al O Al O OH Al O OH OH B OX 25 . 5 Clay/DNA bioinorganic nanostructures Biomolecules have been used frequently in the design and assembly of functional nanostructured materials and as templates for the directed growth of nanoparticles and nanowires. The advantage of using biomolecules to develop the self-assembly of extended nanostructures into a functional circuit is that molecular recognition already exists for these materials. Recent studies have led to the push for the development of DNA-based nanostructures as nonviral vectors for cellular delivery and transfection and as active elements in nanocircuitry and biosensors. Clay minerals are attractive for a large number of applications owing to their versatility, ease of synthetic control, wide range of compositions, low toxicity and low cost. Clays may be classified into two main types: cationic clays with negatively charged alumino silicate layers and anionic clays with positively charged hydroxide layers. Bottom-up self-assembly of DNA/organoclay nanocomposites occurs by two methods which incorporate aminopropyl (AMP) groups, with approximate unit cell composition [H2N(CH)3)2]8[Si8Mg6O16(OH)4], as the structural organoclay building block. The AMP clay exercises electrostatic induction and capture of DNA to form both intercalative mesolamellar assemblies and wrapped single molecules (Fig. B25.5). This method for developing nanocomposites is facile and results in ordered mesolamellar structures with biofunctionality. The overall positive charge of AMP clay layers shield negatively charged DNA, reducing the electrostatic repulsion interaction between cell membrane and anionic DNA to facilitate the potential for cell uptake by endocytosis. The development of intercalated mesolamellar structures has widespread applications, ranging from pharmaceuticals and cosmetics to agricultural and environmental uses. Protonation of aminopropyl side chains in AMP leads to exfoliation (delamination or separation of the clay nanoparticles into dispersed nanosheets). Charge transport within the clay is enhanced by this exfoliation process and leads to reassembly with negatively charged drug molecules and biomolecules. The denaturation of biomolecules occurs easily under a number of conditions, including sensitivity to pH, temperature, solvent/buffer, and ionic effects. Organoclay wrapping has been utilized as an effective method for increasing protein stability. Similarly, organoclay wrapping has been employed as a method for increasing DNA stability. The increased thermal stability for intercalated DNA nanocomposites affords new applications in drug delivery and transfection studies for gene therapy. DNA (a) (b)) Figure B25.5 Preparation of DNA/organoclay nanostructures. (a) Protonation of aminopropyl side chains of as-synthesized AMP clay in water results in exfoliation and formation of dispersed nanosheets. Addition of stoichiometric quantities of DNA gives rise to electrostatically induced reassembly of the organoclay layers in association with biomolecule intercalation to produce an ordered mesolamellar nanocomposite. (b) Exfoliation and fractionation by gel chromatography results in organoclay polycationic clusters that bind and condense to produce an ultrathin organoclay sheath on individual DNA molecules, leading to molecular-scale isolation of the double-helical strands. (A.J. Patil, et al., Nano Lett. 2007, 7, 2660.) 686 25 Nanomaterials, nanoscience, and nanotechnology O (a) N O O N (b) Figure 25.40 (a) Structure of N-(3maleimidopropionyl)-3,4-fulleropyrrolidine. (b) The protein conjugated to N-(3maleimidopropionyl)-3,4-fulleropyrrolidine (shown to scale). (P. Nednoor, et al., Biocon. Chem., 2004, 15, 12.) Proteins have been bound to nanoparticles by using a single site on the protein to yield hybrid nanomaterials. An important example of this type of material consists of proteins attached to fullerenes, which take advantage of the latter’s rich chemical and electronic properties, small dimensions, and potential applications. Alterations to the amino acid sequence of a protein are carried out by site-directed mutagenesis (SDM). By using SDM, a single cysteine residue, which selectively attaches to the fullerene, has been introduced in the protein sequence away from the active site to avoid deactivation on binding to the nanoparticles. Figure 25.40 illustrates one approach to create a covalent linkage between nanosized fullerenes and an active protein by using a functional tether (a bridging molecule). Many of these functionalized fullerenes have found biological applications as inhibitors of HIV protease, as light-sensitive biochemical probes, and in photodynamic tumour therapies. Engineering covalent linkages between nanomaterials and proteins has outstanding potential for future applications in drug delivery and enhanced drug efficacy. There is a new world of inorganic chemistry, as we have sought to show, waiting in the wings of our subject: this intermediate domain between the atomic and the bulk is waiting for discovery and exploitation. With exploitation comes the responsibility of careful consideration of the effects that nanomaterials will have on our environment both as novel materials for water filtration and remediation of biohazards and as sources of toxic pollutants (Box 25.6). It is essential to have a reliable and extensive grounding in the conventions of inorganic chemistry, to note what Nature has already achieved, and—above all—to build on it with imagination. B OX 25 .6 Environmental ceria nanocomposites for water treatment and CO removal Clean water and air are crucial for human survival. Toxic substances like As, Cr, and CO can contaminate air and drinking water, causing injury and death for those who consume them. Arsenic has been used for years as a poison and is a potential carcinogen. Ingested Cr(VI) can cause liver and kidney damage and is a potential carcinogen; inhaled Cr(VI) is known to cause lung cancer. Several researchers have explored the use of ceria (CeO2) for the removal of these and other pollutants from water and air. Nanoscale ceria has a higher surface area and catalytic activity, making it even more suitable for catalysis and adsorption. Ceria nanocrystals with included Au nanoparticles can oxidize CO at lower temperatures compared to bulk ceria. Ceria nanoparticles supported on carbon nanotubes are able to adsorb high amounts of As and Cr. However, nanoparticles are too small to be removed by conventional microfiltration, making the nanoparticles difficult to recover for re-use and potentially creating new health risks as these nanomaterials may easily enter drinking water streams and have shown high toxicity when inhaled. One possible solution to this dilemma is to create ceria microparticles with nanoscale features. This approach preserves enhanced nanoscale properties such as surface area while making the composite easier to handle and filter. Ceria micro/nanocomposites for use in water treatment and CO removal have been prepared by dissolving cerium(III) chloride hydrate, urea, and tetrabutylammonium bromide (TBAB) in ethylene glycol and stirring the solution for 30 minutes at 180°C. The resulting ceria precursor is then calcined in air at 450°C for 2 hours to remove organic components of the precursor, leaving only ceria (CeO2). In order to fabricate the ceria-supported gold catalyst, HAuCl4 and NaOH are combined in water and added to a suspension of the ceria nanocomposite in water. Ceria nanocomposites with a self-assembled flowerlike structure have been fabricated by using TBAB as a templating agent. The self-assembly process of the ceria nanocomposite was studied by collecting samples of the ceria precursor at various times after the appearance of precipitate in the ethylene glycol solution (Fig B25.6). After the precipitate appears the ceria precursor consists of nanoparticles about 100 nm in diameter; 11 minutes later there are some microscale petals among the nanoparticles, after 19 minutes, flowerlike microstructures are visible, and at 30 minutes the precursor consists entirely of flowerlike microparticles. SEM imaging reveals microparticles 4–6 μm in diameter consisting of multiple microscale petals both before and after calcination. The urea was necessary to (a) (b) (c) (d) Figure B25.6 SEM images of the as-prepared ceria precursors collected at different time intervals after precipitation of ceria: (a) 3 min, (b) 11 min, (c) 19 min, and (d) 30 min. (L.-S. Zhong, et al., Chem. Mater. 2007, 19, 1648.) Exercises maintain the proper pH for precipitate formation and the TBAB is necessary for the assembly of the petals into the flowerlike microstructure. This is an interesting example of morphosynthetic control of structure by careful control of pH, temperature, and templating agent. These ceria nanocomposites can be used to remove As(V) and Cr(VI) from water. The maximum adsorption capacity of the nanocomposite is 14.4 mg g1 for As(V) and 5.9 mg g1 for Cr(VI). The microcrystalline ceria nanocomposite can be removed from the water after adsorption by centrifugation, so demonstrating the benefit of an overall microscale size. Commercial ceria powder was used to remove the pollutants under 687 the same conditions and found to be much less effective. This is probably due to the higher surface area of the ceria nanocomposite (34.1 m2 g1) compared to that of the commercial ceria (2.0 m2 g1). Ceria-supported gold catalysts have also been used to remove CO by oxidation. The ceria-supported gold catalyst was capable of converting 100 per cent of the CO at a temperature of only 125°C. Commerical ceria powders currently in use require 500°C to convert all of the CO. It is likely that the high catalytic ability of the ceria-supported gold catalyst is due to the CO adsorbing on the Au nanoparticles and reacting with the oxygen stored in the ceria. FURTHER READING G.A. Ozin and A.C. Arsenault, Nanochemistry: A chemical approach to nanomaterials. Springer-Verlag, New York (2005). An invaluable reference book for undergraduate and graduate students looking for an easy way to educate themselves with the up-to-date advances made in chemical patterning, self-assembly, and nanomaterial synthesis. C.P. Poole and F.J. Owens, Introduction to nanotechnology. WileyInterscience, Hoboken (2003). This book provides key chapters on a variety of nanomaterials systems, including quantum structures, magnetic nanomaterials, nanoelectro-mechanical systems (NEMS), carbon nanotubes, and nanocomposites, with emphasis on characterization and synthesis strategies for each system. The rise of nanotech. Reprinted by Scientific American, 2007. This collection of easy-to-understand articles that originally appeared in Scientific American in 2001 offers an outstanding overview of the key issues of nanoscience, including coverage of top-down versus bottom-up methods of synthesis and applications of nanotechnology in medicine, computers, and telecommunications. M. Ratner and D. Ratner, Nanotechnology: the next big idea. Prentice Hall, Upper Saddle River (2003). A wonderful general approach to nanoscience with broad coverage of key issues in the field, including molecular electronics and smart materials. M. Wilson, K. Kannangara, G. Smith, M. Simmons, and B. Raguse (ed.), Nanotechnology: basic science and emerging technologies. CRC Press, Boca Raton (2002). This is another good introductory book on many areas of nanotechnology. B. Bhushan (ed.), Handbook of nanotechnology. Springer, Berlin (2004). An overarching technical introduction to nanotechnology. P.N. Prasad, Nanophotonics. Wiley-Interscience, Hoboken (2003). A good introduction to optical properties and processes in nanostructured materials. Supramolecular chemistry and self-assembly. Special issue of Science (March 2002). This collection of articles describes key research on the overlap between supramolecular chemistry and self-assembled nanostructures; it includes surveys of chemical approaches to link nanobuilding blocks to generate extended framework structures. Inorganic–organic nanocomposites. Special issue of Chemistry of Materials (October 2001). This collection of articles describes nanocomposites with broad applications in photonics, electronics, and chemical sensing. J. Hu, T.W. Odom, and C.M. Leiber, Chemistry and physics in one dimension: synthesis and properties of nanowires and nanotubes. Acc. Chem. Res., 1999, 32, 435. An excellent review of issues of controlled dimensionality in nanomaterials, including important pioneering work on the transport properties of carbon nanotubes. T. Masciangioli and W.-X. Zhang, Environmental technologies at the nanoscale. Environ. Sci. Technol., A-Pages, 2003, 37, 102A. A short survey of the future impacts of nanoscience on our environment. C.J. Murphy, Optical sensing with quantum dots. Anal. Chem., 2002, 74, 520A. A well-constructed survey of the key advantages and limitations of using QDs for optical sensing. K.G. Thomas and P.V. Kamat, Chromophore-functionalized gold nanoparticles. Acc. Chem. Res., 2003, 36, 888. A thorough review of Au nanoparticles and the chemistry of surface functionalization with various chromophores to achieve metal hybrid nanomaterials for optoelectronic, light harvesting, and sensing applications. EXERCISES 25.1 (a) Compare the surface area for two spherical objects: one has a diameter of 10 nm and the other has a diameter of 1000 nm. (b) Describe whether these two objects are considered nanoparticles by using the size-based definition of a nanomaterial. (c) Using a surfacearea-related property, describe what must hold true for either of these objects to be considered nanoparticles using the size/properties-based definition of a nanomaterial. 25.2 Quantum confinement bestows unique properties on semiconductor nanocrystals compared to the bulk semiconductor. For this to happen, some characteristic length that describes an electron in a crystal and the size of the crystal must be similar. Describe a characteristic length for an electron and how it becomes comparable to the particle size in quantum confinement. 25.3 Explain why quantum dots might be superior to organic fluorophores for bio-imaging applications. 25.4 Compare and contrast the band energies for a quantum dot nanocrystal and a bulk semiconductor. 688 25 Nanomaterials, nanoscience, and nanotechnology 25.5 (a) Explain the difference between the top-down and bottomup methods of fabrication of materials. Be specific and provide one example of each. (b) Give one advantage and one disadvantage for each synthesis method. 25.14 (a) Give two examples of applications of quantum wells. (b) Describe why quantum wells are used and if either molecular materials or traditional solid-state materials can exhibit similar properties. (c) How are quantum wells made? 25.6 (a) Give a definition for scanning probe microscopy that makes clear both what it is and why it is called scanning probe microscopy. (b) Using a material of interest to you, pick any scanning probe microscopy method and describe how you might use it to characterize an important aspect of your material. 25.15 Use any of the solid-state superlattices described in the chapter to explain why the use of nanostructured materials led to enhanced properties. 25.7 Distinguish between SEM and TEM. What is the principal difference in sample preparation and detection? 25.17 Discuss briefly the features common to self-assembly processes. 25.8 (a) Describe the three basic steps in nanoparticle formation from solution. (b) Explain why two steps should occur independently to achieve a uniform size distribution. (c) What are stabilizer molecules used for in nanoparticle synthesis? 25.18 Distinguish between static and dynamic self-assembly. Give an example of each type. 25.9 Explain whether vapour-phase or solution-based techniques typically lead to (a) larger size distributions in nanoparticle synthesis, (b) agglomerated particles that are strongly bonded to one another in so-called hard agglomerates. 25.10 (a) Draw a schematic diagram of a core–shell nanoparticle. (b) Briefly describe how core–shell nanoparticles could be made using either vapour-phase or solution-based techniques. (c) For what purpose would core–shell nanoparticles be used? 25.11 (a) Discuss the difference between homogeneous and heterogeneous nucleation from the vapour phase. (b) Which type of nucleation is preferred for the growth of a thin film in this process? (c) Which type of nucleation is preferred for the growth of nanoparticles in this process? 25.12 Describe the difference between a physical vapour and a chemical vapour with respect to the type and stability of the vapour species. 25.13 (a) Superlattices of ordered quantum dots (QDs) and an overlaying semiconductor can be considered core–shell QD arrays. What is the purpose of building multiple layers on the QD? (b) What are the limitations on the order and types of semiconductors that can be put together? 25.16 (a) What is the relevance of self-assembly to the fabrication of nanomaterials? (b) What role will it play in nanotechnology? 25.19 How is a self-assembled monolayer (SAM) comprising goldorganothiol linkages related to the cell membrane within our bodies? Use a sketch to make your comparison, define the term surfactant, and distinguish between the hydrophilic and hydrophobic regions. 25.20 Define morphosynthesis. Give an example of how this approach can be used to control nanoarchitecture. 25.21 (a) Describe the two classes of inorganic–organic nanocomposites based on their bonding types. (b) Give one example of a nanocomposite in each class. 25.22 (a) Explain why the state of dispersion of inorganic nanoparticles is important in inorganic–organic nanocomposites. (b) Use the concept of oil and water dispersions to explain why highly dispersed inorganic nanoparticles might be difficult to achieve in these nanocomposites. 25.23 Give an example of a bionanomaterial and its application in nanotechnology. 25.24 (a) Define biomimetics. (b) Describe biomimetics with respect to how artificial fossilization is used to create titania paper. 25.25 Describe why mechanical properties are a key metric of the quality of artificial bone materials. Use the example described in the text to explain how chemistry plays a major role in enhancing the mechanical properties of bionanocomposite bone material. PROBLEMS 25.1 The synthesis method described in the chapter to generate CdSe quantum dots involves the use of rather toxic compounds. Explore the chemical literature to find a more recent example that uses less toxic substances. Describe the solvation step, the nucleation step, and the growth step in each case. Comment on the size dispersions in each case. (See the following articles as a start: G.C. Lisensky and E.M. Boatman, J. Chem. Educ., 2005, 82, 1360; W. William Yu and X.-G. Peng, Angew. Chem., Int. Ed. Engl., 2002, 41, 2368.) 25.2 The Grätzel cell has been described as a useful photoelectrochemical cell. Describe how the photoelectrochemical cell differs from the photovoltaic cell. Describe why the nanostructure TiO2 is important for the improved performance. What other inorganic species play an important role in functionalizing the Grätzel cell? (See M. Grätzel, Nature, 2001, 414, 338.) 25.3 In Fig. 25.11, a TEM micrograph is shown of a superlattice of (PrMnO3)2(SrMnO3)2 deposited on SrTiO3 epitaxially. (a) What is meant by ‘epitaxy’? (b) Describe whether the diagram illustrates that there is atomic order between all of the layers (the contrast represents columns of atoms). (c) The material was grown layer-by-layer by using a physical vapour method. Of the following combinations of growth, which refer to homoepitaxial situations and which refer to heteroepitaxial combinations: PrMnO3 on SrTiO3; PrMnO3 on PrMnO3; SrMnO3 on PrMnO3? 25.4 Compare and contrast the atomic-layer deposition (based on chemical interactions) and the MBE deposition (based on physical interactions of atomic beams) with respect to system cost, speed of process, ambient atmospheric conditions, in situ monitoring abilities, and precision ability to design artificial structures and superlattices. Describe which has been used to produce more materials and describe the general quality of the materials. 25.5 One of the key challenges of nanotechnology is achieving mechanical devices that function on the nanoscale. One such device is a Brownian ratchet. Describe how the Brownian ratchet mechanism has been used to sort DNA. (See J.S. Bader et al., Proc. Natl. Acad. Sci. USA, 1999, 96, 13165.) 25.6 Carbon nanotubes have been suggested for their use as wires in molecular electronics. Describe the challenges of using Problems 689 nanotubes as wires in terms of connecting two functional electronic devices. Describe a possible technique to overcome some of the problems. its neighbours) of a square mark need to be to attain an information density of 1 Tb cm2? Describe the method proposed to make such marks and to read them. 25.7 Show that folded DNA has an information density (density of base-pairs) equal to 1 Tb cm2. IBM has proposed a scanning probe data storage device called the Millipede, in which an array of scanning probe tips reads and writes nanoscale marks on a substrate. What would the footprint (the mark plus the space between it and one of 25.8 Nanotubes are widely known for carbon. Find an example of an inorganic nanotube not based on carbon and describe its synthesis and properties as compared to the corresponding bulk material. Contrast its structure compared to the carbon nanotubes discussed in this chapter. 26 General principles 26.1 The language of catalysis 26.2 Homogeneous and heterogeneous catalysts Homogeneous catalysis 26.3 Alkene metathesis 26.4 Hydrogenation of alkenes 26.5 Hydroformylation 26.6 Wacker oxidation of alkenes 26.7 Asymmetric oxidations 26.8 Palladium-catalysed CC bondforming reactions 26.9 Methanol carbonylation: ethanoic acid synthesis Heterogeneous catalysis 26.10 The nature of heterogeneous catalysts 26.11 Hydrogenation catalysts 26.12 Ammonia synthesis 26.13 Sulfur dioxide oxidation 26.14 Catalytic cracking and the interconversion of aromatics by zeolites 26.15 Fischer–Tropsch synthesis 26.16 Alkene polymerization 26.17 Electrocatalysis 26.18 New directions in heterogeneous catalysis Hybrid catalysis 26.19 Tethered catalysts 26.20 Biphasic systems FURTHER READING EXERCISES PROBLEMS Catalysis In this chapter we apply the concepts of organometallic chemistry, coordination chemistry, and materials chemistry to catalysis. We emphasize general principles, such as the nature of catalytic cycles, in which a catalytic species or surface is regenerated in a reaction, and the delicate balance of reactions required for a successful cycle. We see that there are numerous requirements for a successful catalytic process: the reaction being catalysed must be thermodynamically favourable and fast enough when catalysed; the catalyst must have an appropriate selectivity towards the desired product and a lifetime long enough to be economical. We then survey homogeneously catalysed reactions and show how proposals about mechanisms are invoked. The final part of the chapter develops a similar theme in heterogeneous catalysis, and we shall see that many parallels exist between homogeneous and heterogeneous catalysis. In neither type of catalysis are mechanisms necessarily finally settled and there is still considerable scope for making new discoveries. A catalyst is a substance that increases the rate of a reaction but is not itself consumed. Catalysts are widely used in nature, in industry, and in the laboratory, and it is estimated that they contribute to one-sixth of the value of all manufactured goods in industrialized countries. As shown in Table 26.1, 16 of the top 20 synthetic chemicals in the USA are produced directly or indirectly by catalysis. For example, a key step in the production of a dominant industrial chemical, sulfuric acid, is the catalytic oxidation of SO2 to SO3. Ammonia, another chemical essential for industry and agriculture, is produced by the catalytic reduction of N2 by H2. Inorganic catalysts are also used for the production of the major organic chemicals and petroleum products, such as fuels, petrochemicals, and polyalkene plastics. Catalysts play a steadily increasing role in achieving a cleaner environment, through, for example, the destruction of pollutants (as with the catalytic converters found on the exhaust systems of vehicles), the development of better industrial processes that are more efficient with higher product yields and fewer unwanted byproducts and in clean energy generation in fuel cells. Industrially important catalysts are almost invariably inorganic (which justifies their discussion in this book). Enzymes, a class of biochemical catalysts, often with a metal ion at the centre of a complex molecule, are discussed in Chapter 27. In addition to their economic importance and contribution to the quality of life, catalysts are interesting in their own right: the subtle influence a catalyst has on reagents can completely change the outcome of a reaction. The understanding of the mechanisms of catalytic reactions has improved considerably in recent years with the greater availability of isotopically labelled molecules, improved methods for determining reaction rates, improved spectroscopic and diffraction techniques, and much more reliable molecular orbital calculations. General principles A catalysed reaction is faster than an uncatalysed version of the same reaction because the catalyst provides a different reaction pathway with a lower activation energy. The term negative catalyst is sometimes applied to substances that retard reactions. Substances that block one or more elementary steps in a catalytic reaction are called catalyst poisons. General principles 691 Table 26.1 The top 20 synthetic chemicals in the USA in 2008 (based on mass) Rank Chemical Catalytic process Rank Chemical Catalytic process 1 Sulfuric acid SO2 oxidation, heterogeneous 12 Ammonium nitrate Precursors catalytic 2 Ethene Hydrocarbon cracking, heterogeneous 13 Urea NH3 precursor catalytic 3 Propene Hydrocarbon cracking, heterogeneous 14 Ethylbenzene Alkylation of benzene, homogeneous 3 Polyethene Polymerization, heterogeneous 15 Styrene Dehydrogenation of ethylbenzene, 5 Chlorine Electrolysis, not catalytic 16 HCl heterogeneous 17 Cumene Precursors catalytic 6 Ammonia N2 H2, heterogeneous 7 Phosphoric acid Not catalytic Alkylation of benzene, 8 1,2-Dichloroethane Ethene Cl2, heterogeneous 18 Ethylene oxide heterogeneous 9 Polypropene Polymerization, heterogeneous 19 Ammonium sulfate Ethene O2, heterogeneous 20 Sodium carbonate Precursors catalytic 10 Nitric acid NH3 O2, heterogeneous 11 Sodium hydroxide Electrolysis, not catalytic Not catalytic Source: Facts & Figures for the Chemical Industry, Chem. Eng. News, 2009, 87, 33. 26.1 The language of catalysis Before we discuss the mechanism of catalytic reactions, we need to introduce some of the terminology used to describe the rate of a catalytic reaction and its mechanism. (a) Energetics Key point: A catalyst increases the rates of processes by introducing new pathways with lower Gibbs energies of activation; the reaction profile contains no high peaks and no deep troughs. (b) Catalytic cycles Key point: A catalytic cycle is a sequence of reactions that consumes the reactants and forms products, with the catalytic species being regenerated after the cycle. The essence of catalysis is a cycle of reactions in which the reactants are consumed, the products are formed, and the catalytic species is regenerated. A simple example of a catalytic 1 That is, G depends only on the current state of the system and not on the path that led to the state. (a) ∆‡G Gibbs energy A catalyst increases the rates of processes by introducing new pathways with lower Gibbs energies of activation, ∆‡G. We need to focus on the Gibbs energy profile of a catalytic reaction, not just the enthalpy or energy profile, because the new elementary steps that occur in the catalysed process are likely to have quite different entropies of activation. A catalyst does not affect the Gibbs energy of the overall reaction, ∆rG O , because G is a state function.1 The difference is illustrated in Fig. 26.1, where the overall reaction Gibbs energy is the same in both energy profiles. Reactions that are thermodynamically unfavourable cannot be made favourable by a catalyst. Figure 26.1 also shows that the Gibbs energy profile of a catalysed reaction contains no high peaks and no deep troughs. The new pathway introduced by the catalyst changes the mechanism of the reaction to one with a very different shape and with lower maxima. However, an equally important point is that stable or nonlabile catalytic intermediates do not occur in the cycle. Similarly, the product must be released in a thermodynamically favourable step. If, as shown by the blue line in Fig. 26.1, a stable complex were formed with the catalyst, it would turn out to be the product of the reaction and the cycle would terminate. Similarly, impurities may suppress catalysis by coordinating strongly to catalytically active sites and act as catalyst poisons. (b) ∆rG ° (c) Extent of reaction Figure 26.1 Schematic representation of the energetics of a catalytic cycle. The uncatalysed reaction (a) has a higher ∆‡G than a step in the catalysed reaction (b). The Gibbs energy of the overall reaction, ∆rG O , is the same for routes (a) and (b). The curve (c) shows the profile for a reaction mechanism with an intermediate that is more stable than the product. 692 26 Catalysis cycle involving a homogeneous catalyst is the isomerization of prop-2-en-1-ol (allyl alcohol (CH2 CHCH2OH)) to prop-1-en-1-ol (CH3CH CHOH) with the catalyst [Co(CO)3H]. The first step is the coordination of the reactant to the catalyst. That complex isomerizes in the coordination sphere of the catalyst and goes on to release the product and reform the catalyst (Fig. 26.2). Once released the prop-1-en-1-ol tautomerises to propanal (CH3CH2CHO). As with all mechanisms, this cycle has been proposed on the basis of a range of information like that summarized in Fig. 26.3. Many of the components shown in the diagram were encountered in Chapter 21 in connection with the determination of mechanisms of substitution reactions. However, the elucidation of catalytic mechanisms is complicated by the occurrence of several delicately balanced reactions, which often cannot be studied in isolation. Two stringent tests of any proposed mechanism are the determination of rate laws and the elucidation of stereochemistry. If intermediates are postulated, their detection by magnetic resonance and IR spectroscopy also provides support. If specific atom-transfer steps H OH OC OC Co OH CO H OH OH H OC OC OC OC Co CO Co CO OH H OC OC Co CO Figure 26.2 The catalytic cycle for the isomerization of prop-2-en-l-ol to prop-1-en-l-ol. Heterogeneous Determine adsorption isotherms Identify surface species; note analogies with organometallic compounds Note support effects Note differential poisoning General Homogeneous Determine overall rate law and selectivity as a function of concentration Determine formation of metal complex Postulate mechanism Identify reaction intermediates; note analogies with known reactions Determine rate laws of individual steps Note isotope effects Infer best mechanism Figure 26.3 The determination of catalytic mechanisms. Note stereochemistry Note solvent and ligand effects General principles 693 are proposed, then isotopic tracer studies may serve as a test. The influences of different ligands and different substrates are also sometimes informative. Although rate data and the corresponding laws have been determined for many overall catalytic cycles, it is also necessary to determine rate laws for the individual steps in order to have reasonable confidence in the mechanism. However, because of experimental complications, it is rare that catalytic cycles are studied in this detail. (c) Catalytic efficiency and lifetime Key points: A highly active catalyst, one that results in a fast reaction even in low concentrations, has a large turnover frequency. A catalyst must be able to survive a large number of catalytic cycles if it is to be of use. The turnover frequency, f, is often used to express the efficiency of a catalyst. For the conversion of A to B catalysed by Q and with a rate v, Q A ⎯⎯ → B v = d[B] dt (26.1) then provided the rate of the uncatalysed reaction is negligible, the turnover frequency is f = v [Q] (26.2) A highly active catalyst, one that results in a fast reaction even in low concentrations, has a high turnover frequency. In heterogeneous catalysis, the reaction rate is expressed in terms of the rate of change in the amount of product (in place of concentration) and the concentration of catalyst is replaced by the amount present. The determination of the number of active sites in a heterogeneous catalyst is particularly challenging, and often the denominator [Q] in eqn 26.2 is replaced by the surface area of the catalyst. The turnover number is the number of cycles for which a catalyst survives. If it is to be economically viable, a catalyst must have a large turnover number. However, it may be destroyed by side reactions to the main catalytic cycle or by the presence of small amounts of impurities in the starting materials (the feedstock). For example, many alkene polymerization catalysts are destroyed by O2, so in the synthesis of polyethene (polyethylene) and polypropene (polypropylene) the concentration of O2 in the ethene or propene feedstock should be no more than a few parts per billion. Some catalysts can be regenerated quite readily. For example, the supported metal catalysts used in the reforming reactions that convert hydrocarbons to high-octane gasoline become covered with carbon because the catalytic reaction is accompanied by a small amount of dehydrogenation. These supported metal particles can be cleaned by interrupting the catalytic process periodically and burning off the accumulated carbon. (d) Selectivity Key point: A selective catalyst yields a high proportion of the desired product with minimum amounts of side products. A selective catalyst yields a high proportion of the desired product with minimum amounts of side products. In industry, there is considerable economic incentive to develop selective catalysts. For example, when metallic silver is used to catalyse the oxidation of ethene with oxygen to produce oxirane (ethylene oxide, 1), the reaction is accompanied by the more thermodynamically favoured but undesirable formation of CO2 and H2O. This lack of selectivity increases the consumption of ethene, so chemists are constantly trying to devise a more selective catalyst for oxirane synthesis. Selectivity can be ignored in only a very few simple inorganic reactions, where there is essentially only one thermodynamically favourable product, as in the formation of NH3 from H2 and N2. One area where selectivity is of considerable and growing importance is asymmetric synthesis, where only one enantiomer of a particular compound is required and catalysts may be designed to produce one chiral form in preference to any others. 1 Oxirane (ethylene oxide) 694 26 Catalysis 26.2 Homogeneous and heterogeneous catalysts Key points: Homogeneous catalysts are present in the same phase as the reagents, and are often well defined; heterogeneous catalysts are present in a different phase from the reagents. CO, hydrocarbons, NOx, N2, O2, SOx CO2, H2O, N2, O2, H2S Catalyst Support Figure 26.4 A heterogeneous catalyst in action. The automobile catalytic converter oxidizes CO and hydrocarbons, and reduces nitrogen and sulfur oxides. The particles of a metal catalyst are supported on a robust, ceramic honeycomb. Catalysts are classified as homogeneous if they are present in the same phase as the reagents; this normally means that they are present as solutes in liquid reaction mixtures. Catalysts are heterogeneous if they are present in a different phase from that of the reactants; this normally means that they are present as solids with the reactants present either as gases or in solution. Both types of catalysis are discussed in this chapter and will be seen to be fundamentally similar. From a practical standpoint, homogeneous catalysis is attractive because it is often highly selective towards the formation of a desired product. In large-scale industrial processes, homogeneous catalysts are preferred for exothermic reactions because it is easier to dissipate heat from a solution than from the solid bed of a heterogeneous catalyst. In principle, every homogeneous catalyst molecule in solution is accessible to reagents, potentially leading to very high activities. It should also be borne in mind that the mechanism of homogeneous catalysis is more accessible to detailed investigation than that of heterogeneous catalysis as species in solution are often easier to characterize than those on a surface and because the interpretation of rate data is frequently easier. The major disadvantage of homogeneous catalysts is that a separation step is required. Heterogeneous catalysts are used very extensively in industry and have a much greater economic impact than homogeneous catalysts. One attractive feature is that many of these solid catalysts are robust at high temperatures and therefore tolerate a wide range of operating conditions. Reactions are faster at high temperatures, so at high temperatures solid catalysts generally produce higher outputs for a given amount of catalyst and reaction time than homogeneous catalysts operating at lower temperatures in solutions. Another reason for their widespread use is that extra steps are not needed to separate the product from the catalyst, resulting in efficient and more environmentally friendly processes. Typically, gaseous or liquid reactants enter a tubular reactor at one end, pass over a bed of the catalyst, and products are collected at the other end. This same simplicity of design applies to the catalytic converter used to oxidize CO and hydrocarbons and reduce nitrogen oxides in automobile exhausts (Fig. 26.4), see also Box 26.1. Homogeneous catalysis Here we concentrate on some important homogeneous catalytic reactions based on organometallic compounds and coordination complexes. We describe their currently favoured mechanisms, but it should be noted that, as with nearly all mechanistic proposals, B OX 26 .1 Catalytic converters Catalytic convertors are used to reduce toxic emissions from internal combustion engines, which include nitrogen oxides (NOx), carbon monoxide, and unburnt hydrocarbons (HC). By 2009, when the Euro V emission standards came into use in Europe, the levels of these compounds in exhaust gases were restricted to 0.50 (CO), 0.23 (NOx HC) and 0.18 (HC) g km1, respectively; these values are similar to those demanded in California. The catalytic converter consists of a honeycomb stainless steel or ceramic structure on to which first silica and alumina are deposited followed by a mixture of platinum, rhodium, and palladium as nanoparticles, with diameters typically between 10 and 50 nm. A three-way catalytic converter, used with petrol (gasoline) engines undertakes the following three reactions 2 NOx(g) → x O2(g) N2(g) 2 CO(g) O2(g) → 2 CO2(g) CxH2x2(g) 2x O2(g) → x CO2(g) 2x H2O(g) The first stage of the catalytic converter involves reduction of the NOx on a reduction catalyst, which consists of a mixture of platinum and rhodium; rhodium is highly reactive towards NO. The second stage involves catalytic oxidation, which removes the unburnt hydrocarbons and carbon monoxide by oxidizing them on the mixed metal platinum/palladium catalyst. A two-way catalytic converter used in conjunction with most diesel engines undertakes only the oxidation reactions (the second and third of those above). In order to facilitate the near complete conversion of the gases emerging from the engine the correct initial fuel/air mixture is used. The ideal stoichiometric air:fuel mixture entering the engine is 14.7:1 so that after combustion the gases entering the catalytic converter are about 0.5 per cent oxygen. If richer or leaner air:fuel mixtures are used (that is, with lower and higher air contents, respectively), then the level of oxygen entering the exhaust stream may be too high or low for effective operation of the catalytic converter. For these reasons, various metal oxides, particularly Ce2O3 and CeO2, are incorporated into the catalytic coating to store and release oxygen as the oxygen content of the gases emerging from the engine changes. Homogeneous catalysis catalytic mechanisms are subject to refinement or change as more detailed experimental information becomes available. Unlike simple reactions, a catalytic process frequently contains many steps over which the experimentalist has little control. Moreover, highly reactive intermediates are often present in concentrations too low to be detected spectroscopically. The best attitude to adopt towards these catalytic mechanisms is to learn the pattern of transformations and appreciate their implications but be prepared to accept new mechanisms that might be indicated by future work. The scope of homogeneous catalysis ranges across hydrogenation, oxidation, and a host of other processes. Often the complexes of all metal atoms in a group will exhibit catalytic activity in a particular reaction, but the 4d-metal complexes are often superior as catalysts to the complexes of their lighter and heavier congeners. In some cases the difference may be associated with the greater substitutional lability of 4d organometallic compounds in comparison with their 3d and 5d analogues. It is often the case that the complexes of costly metals must be used on account of their superior performance compared with the complexes of cheaper metals. Chapters 21 and 22 described reactions that take place at metal centres: these reactions lie at the heart of the catalytic processes we now consider. In general, we need to invoke a variety of the processes described in those chapters, and it will be useful to review the following sections: ligand substitution reactions: Sections 21.0–9 and Section 22.21 redox reactions: Sections 21.10–12 oxidative addition and reductive elimination reactions: Section 22.22 migratory insertion reactions: Section 22.24 1,2-insertions and β-hydride eliminations: Section 22.25 Together with the direct attack on coordinated ligands, these reaction types (and in some cases their reverse), often in combination, account for the mechanisms of most of the homogeneous catalytic cycles that have been proposed for organic transformations. Other reactions yet to be fully investigated will undoubtedly use other reaction steps. 26.3 Alkene metathesis Key points: Alkene metathesis reactions are catalysed by homogeneous organometallic complexes that allow considerable control over product distribution; a key step in the reaction mechanism is the dissociation of a ligand from a metal centre to allow an alkene to coordinate. In an alkene metathesis reaction carbon–carbon double bonds are redistributed, as in the cross metathesis reaction: R1 R1 R2 R2 PCy 3 Alkene metathesis was first reported in the 1950s with poorly defined mixtures of reagents, such as WCl6/Bu4Sn and MoO3/SiO2, being used to bring about a number of different reactions (Table 26.2). In recent years, a number of newer catalysts have been introduced, and the development of the well-defined ruthenium alkylidene compound (2) by Grubbs in 1992, now known as Grubbs’ catalyst, was of seminal importance.2 Alkene metathesis reactions proceed through a metallacyclobutane intermediate: R1 R1 [M] R2 R1 [M] R3 R2 [M] R3 R2 R3 2 The 2005 Nobel prize was awarded to Robert Grubbs, Yves Chauvin, and Richard Schrock for their work on developing metathesis catalysts. Cl Ru Cl PCy 3 2 Ph 695 696 26 Catalysis Table 26.2 The scope of the alkene metathesis reaction X X n Ring-opening metathesis polymerization (ROMP) n X X −n n Acyclic diene metathesis polymerization (ADMET) n X X − Diene ring-closing metathesis (RCM) X Enyne ring-closing metathesis (RCM) X X X R Ring-opening metathesis (ROM) R R1 − R1 R2 Cross-metathesis (CM or XMET) R2 N Cl Cl N Ru PCy3 3 Ph In the case of Grubbs’ catalyst, it is known that the dissociation of a PCy3 ligand from the Ru metal centre is crucial in allowing the alkene molecule to coordinate prior to metallacyclobutane formation. The identification of this mechanism led Grubbs to replace one of the PCy3 ligands with a bis(mesityl) N-heterocyclic carbene (NHC) ligand, reasoning that the stronger σ-donor and poorer π-acceptor ability of the NHC ligand would both encourage PCy3 dissociation and stabilize the alkene complex. In a triumph of rational design, the so-called second-generation Grubbs’ catalyst (3) proved to be more active than the original bisphosphine complex. The second-generation Grubbs’ catalyst is active in the presence of a large number of different functional groups on substrates and can be used in many solvent systems. It is commercially available and has been widely used, including in the total synthesis of a number of natural products. The driving force for alkene metathesis reactions varies. For ROM and ROMP (see Table 26.2) it is the release of ring strain from a strained starting material that provides the energy to drive the reaction. For metathesis reactions that result in the generation of ethene (such as RCM or CM) it is the removal of the liberated ethene that can be used to encourage the formation of the desired products. Where there is no clearly identifiable thermodynamically favourable product possible, mixtures of alkenes result, with their relative proportions being determined by the statistical likelihood of their formation. 26.4 Hydrogenation of alkenes Key points: Wilkinson’s catalyst, [RhCl(PPh3)3], and related complexes are used for the hydrogenation of a wide variety of alkenes at pressures of hydrogen close to 1 atm or less; suitable chiral ligands can lead to enantioselective hydrogenations. The addition of hydrogen to an alkene to form an alkane is favoured thermodynamically (∆rG O 101 kJ mol1 for the conversion of ethene to ethane). However, the reaction rate is negligible at ordinary conditions in the absence of a catalyst. Efficient homogeneous and heterogeneous catalysts are known for the hydrogenation of alkenes and are used in such diverse areas as the manufacture of margarine, pharmaceuticals, and petrochemicals. One of the most studied catalytic systems is the Rh(I) complex [RhCl(PPh3)3], which is often referred to as Wilkinson’s catalyst. This useful catalyst hydrogenates a wide variety of alkenes and alkynes at pressures of hydrogen close to 1 atm or less at room temperature. The dominant cycle for the hydrogenation of terminal alkenes by Wilkinson’s catalyst is shown in Fig. 26.5. It involves the oxidative addition of H2 to the 16-electron complex [RhCl(PPh3)3] (A), to form the 18-electron dihydrido complex (B). The dissociation of a Homogeneous catalysis H Ph3P Cl H2 Rh H PPh3 PPh 3 PPh3 Ph3P Cl PPh3 PPh3 Rh H B Ph3P Cl A Rh H PPh 3 C R R F PPh3 Ph3P Cl R H Rh H H PPh3 e M D E Ph3P Cl PPh 3 R H H Rh Ph3P Cl Rh H PPh3 R PPh 3 e M Figure 26.5 The catalytic cycle for the hydrogenation of terminal alkenes by Wilkinson’s catalyst. phosphine ligand from (B) results in the formation of the coordinatively unsaturated complex (C), which then forms the alkene complex (D). Hydrogen transfer from the Rh atom in (D) to the coordinated alkene yields a transient 16-electron alkyl complex (E). This complex takes on a phosphine ligand to produce (F), and hydrogen migration to carbon results in the reductive elimination of the alkane and the reformation of (A), which is set to repeat the cycle. A parallel but slower cycle (which is not shown) is known in which the order of H2 and alkene addition is reversed. Another cycle is known, based around the 14-electron intermediate [RhCl(PPh3)2]. Even though there is very little of this species present, it reacts much faster with hydrogen than [RhCl(PPh3)3] and makes a significant contribution to the catalytic cycle. In this cycle, (E) would eliminate alkane directly, regenerating [RhCl(PPh3)2], which rapidly adds H2 to give (C). Wilkinson’s catalyst is highly sensitive to the nature of the phosphine ligand and the alkene substrate. Analogous complexes with alkylphosphine ligands are inactive, presumably because they are more strongly bound to the metal atom and do not readily dissociate. Similarly, the alkene must be just the right size: highly hindered alkenes or the sterically unencumbered ethene are not hydrogenated by the catalyst, presumably because the sterically crowded alkenes do not coordinate and ethene forms a strong complex that does not react further. These observations emphasize the point made earlier that a catalytic cycle is usually a delicately poised sequence of reactions, and anything that upsets its flow may block catalysis or alter the mechanism. Wilkinson’s catalyst is used in laboratory-scale organic synthesis and in the production of fine chemicals. Related Rh(I) phosphine catalysts that contain a chiral phosphine ligand have been developed to synthesize optically active products in enantioselective reactions (reactions that produce a particular chiral product). The alkene to be hydrogenated must be prochiral, which means that it must have a structure that leads to R or S chirality when complexed to the metal. The resulting complex will have two diastereomeric forms depending on which face of the alkene coordinates to the metal atom. In general, diastereomers have different stabilities and labilities, and in favourable cases one or the other of these effects leads to product enantioselectivity. Enantioselectivities are normally measured in terms of the enantiomeric excess (ee), which is defined as the percentage yield of the major enantiomeric product minus the percentage yield of the minor enantiomeric product. ■ A brief illustration. A reaction that gives 51 per cent of one enantiomer and 49 per cent of another would be described as having an enantiomeric excess of 2 per cent; a reaction that gave 99 per cent of one enantiomer and 1 per cent of another would have ee 98 per cent. ■ 697 698 26 Catalysis P OMe P OMe An enantioselective hydrogenation catalyst containing a chiral phosphine ligand referred to as DiPAMP (4) is used to synthesize l-dopa (5), a chiral amino acid used to treat Parkinson’s disease. An interesting detail of the process is that the minor diastereomer in solution leads to the major product. The explanation of the greater turnover frequency of the minor isomer lies in the difference in activation Gibbs energies (Fig. 26.6). Spurred on by clever ligand design and using a variety of metals, this field has grown rapidly and provides many clinically useful compounds; of particular note are systems derived from ruthenium(II) BINAP (6).3 4 DiPAMP 26.5 Hydroformylation HO COOH NH2 HO 5 L-Dopa In a hydroformylation reaction, an alkene, CO, and H2 react to form an aldehyde containing one more C atom than in the original alkene: RCH CH2 CO H2 → RCH2CH2CHO 1 X X RuBr2 ∆‡G(S) ∆‡G(R) 6 [Ru(BINAP)Br2], X = PPh2 Gibbs energy Key point: The mechanism of hydrocarbonylation is thought to involve a pre-equilibrium in which octacarbonyldicobalt combines with hydrogen at high pressure to give a monometallic species that brings about the actual hydrocarbonylation reaction. Minor isomer (S) Major isomer (R) The term ‘hydroformylation’ derived from the idea that the product resulted from the addition of methanal (formaldehyde, HCHO) to the alkene, and the name has stuck even though experimental data indicate a different mechanism. A less common but more appropriate name is hydrocarbonylation. Both cobalt and rhodium complexes are used as catalysts. Aldehydes produced by hydroformylation are normally reduced to alcohols that are used as solvents and plasticizers, and in the synthesis of detergents. The scale of production is enormous, amounting to millions of tonnes annually. The general mechanism of cobalt-carbonyl-catalysed hydroformylation was proposed in 1961 by Heck and Breslow by analogy with reactions familiar from organometallic chemistry (Fig. 26.7). Their general mechanism is still invoked, but has proved difficult to verify in detail. In the proposed mechanism, a pre-equilibrium is established in which octacarbonyldicobalt combines with hydrogen at high pressure to yield the known tetracarbonylhydridocobalt complex (A): [Co2(CO)8] H2 → 2 [Co(CO)4H] H Extent of reaction CO Figure 26.6 Kinetically controlled stereoselectivity. Note that ∆‡GS < ∆‡GR, so the minor isomer reacts faster than the major isomer. OC OC R OC OC CO H B H Co R Co OC OC CO CO Co C A R CO CHO CO D E H2 OC OC CO Co CO CO R OC OC CO Co CO R CO Figure 26.7 The catalytic cycle for the hydroformylation of alkenes by a cobalt carbonyl catalyst. 3 Ryoji Noyori and William Knowles were jointly awarded the 2001 Nobel prize for their work on asymmetric hydrogenation. The prize was shared with Barry Sharpless for his work on asymmetric oxidations (Section 26.7). Homogeneous catalysis This complex, it is proposed, loses CO to produce the coordinatively unsaturated complex [Co(CO)3H] (B): [Co(CO)4H] → [Co(CO)3H] CO It is thought that [Co(CO)3H] then coordinates an alkene, producing (C) in Fig. 26.7, whereupon the coordinated hydrido ligand migrates onto the alkene, and CO recoordinates. The product at this stage is a normal alkyl complex (D). In the presence of CO at high pressure, (D) undergoes migratory insertion and coordinates another CO, yielding the acyl complex (E), which has been observed by IR spectroscopy under catalytic reaction conditions. The formation of the aldehyde product is thought to occur by attack of either H2 (as depicted in Fig. 26.7) or the strongly acidic complex [Co(CO)4H] to yield an aldehyde and generate [Co(CO)4H] or [Co2(CO)8], respectively. Either of these complexes will regenerate the coordinatively unsaturated [Co(CO)3H]. A significant portion of branched aldehyde is also formed in the cobalt-catalysed hydroformylation. This product may result from a 2-alkylcobalt intermediate formed when reaction of (C) leads to an isomer of (D), with hydrogenation then yielding a branched aldehyde as set out in Fig. 26.8. When the linear aldehyde is required, such as for the synthesis of biodegradable detergents, the isomerization can be suppressed by the addition of an alkylphosphine to the reaction mixture. One plausible explanation is that the replacement of CO by a bulky ligand disfavours the formation of complexes of sterically crowded 2-alkenes: OC OC Co K 90 per cent) and their high surface areas lead to very high catalytic activities. One further application of these catalysts is to the conversion of naturally occurring polyunsaturated fats, which are liquids, to solid polyhydrogenated fats, as in margarine. 26.12 Ammonia synthesis Key point: Catalysts based on iron metal are used for the synthesis of ammonia from nitrogen and hydrogen. The synthesis of ammonia has already been discussed from several different viewpoints (Section 15.6). Here we concentrate on details of the catalytic steps. The formation of ammonia is exergonic and exothermic at 25°C, the relevant thermodynamic data being ∆fG O 116.5 kJ mol1, ∆fH O 146.1 kJ mol1, and ∆fS O 199.4 J K1 mol1. The negative entropy of formation reflects the fact that two NH3 molecules form in place of four reactant molecules. The great inertness of N2 (and to a lesser extent H2) requires that a catalyst be used for the reaction. Iron metal, together with small quantities of alumina and potassium salts and other promoters, is used as the catalyst. Extensive studies on the mechanism of ammonia synthesis indicate that the rate-determining step under normal operating conditions is the dissociation of N2 coordinated to the catalyst surface. The other reactant, H2, undergoes much more facile dissociation on the metal surface and a series of insertion reactions between adsorbed species leads to the production of NH3: N2(g) N2(g) N N H2(g) H2(g) H H N + H NH NH2 H NH3 H NH3(g) Figure 26.20 Schematic diagram of the stages involved in the hydrogenation of ethene by deuterium on a metal surface. 710 26 Catalysis Because of the slowness of the N2 dissociation, it is necessary to run the ammonia synthesis at high temperatures, typically 400ºC. However, because the reaction is exothermic, high temperature reduces the equilibrium constant of the reaction. To recover some of this reduced yield, pressures in the order of 100 atm are used to favour the formation products. A catalyst operating at room temperature that could give good equilibrium yields of NH3, such as the enzyme nitrogenase (Section 27.13), has long been sought. In the course of developing the original ammonia synthesis process, Haber, Bosch, and their co-workers investigated the catalytic activity of most of the metals in the periodic table and found that the best are Fe, Ru, and U promoted by small amounts of alumina and potassium salts. Cost and toxicity considerations led to the choice of iron as the basis of the commercial catalyst. The role of the various promoters, particularly K, in the Fe metal catalyst had been the subject of much scientific research. G. Ertl4 found that, in the presence of potassium, N2 molecules adsorb more readily on the metal surface and the adsorption enthalpy is made more exothermic by about 12 kJ mol−1, probably as a result of the increased electron-donating abilities of the Fe/K surface. The more strongly adsorbed N2 molecule is then cleaved more easily in the rate-determining step in the process. 26.13 Sulfur dioxide oxidation Key point: The most widely used catalyst for the oxidation of SO2 to SO3 is molten potassium vanadate supported on a high-surface-area silica. The oxidation of SO2 to SO3 is a key step in the production of sulfuric acid (Section 16.13). The reaction of sulfur with oxygen to produce SO3 gas is exergonic (∆rG O 371 kJ mol1) but very slow, and the principal product of combustion is SO2: S(s) + O2 (g) → SO2 (g) The combustion is followed by the catalytic oxidation of SO2: SO2 (g) + 12 O2 (g) → SO3 (g) SO3 O2 O2– + 2 V(V) SO2 2 V(IV) Figure 26.21 Cycle showing the key elements involved in the oxidation of SO2 by V(V) compounds. This step is also exothermic and thus, as with ammonia synthesis, has a less favourable equilibrium constant at elevated temperatures. The process is therefore generally run in stages. In the first stage, the combustion of sulfur raises the temperature to about 600ºC, but by cooling and pressurizing before the catalytic stage the equilibrium is driven to the right and high conversion of SO2 to SO3 is achieved. Several quite different catalytic systems have been used to catalyse the combination of SO2 with O2. The most widely used catalyst at present is potassium or caesium vanadate molten salt covering a high-surface-area silica. The current view of the mechanism of the reaction is that the rate-determining step is the oxidation of V(IV) to V(V) by O2 (Fig. 26.21). In the melt, the vanadium and oxide ions are part of a polyvanadate complex (Section 19.8), but little is known about the evolution of the oxo species. 26.14 Catalytic cracking and the interconversion of aromatics by zeolites Key points: Zeolite catalysts have strongly acidic sites that promote reactions such as isomerization via carbonium ions; shape selectivity may arise at various stages of the reaction due to the relative dimensions of the zeolite channels and reactant, intermediate, and product molecules. The zeolite-based (Sections 14.15 and 24.12) heterogeneous catalysts play an important role in the interconversion of hydrocarbons and the alkylation of aromatics as well as in oxidation and reduction. Two important zeolites used for such reactions are faujasite (Fig 26.22), also known as zeolite X or zeolite Y (the X or Y terminology is defined by the Si:Al ratio of the material, X has a higher Al content) and zeolite ZSM-5, an aluminosilicate 4 Gerhard Ertl was awarded the 2007 Nobel Prize for chemistry. Heterogeneous catalysis Figure 26.22 The zeolite faujasite (also known as zeolite X or Y) framework structure showing the large pore in which catalytic cracking occurs. Tetrahedra are SiO4 or AlO4. Figure 26.23 The zeolite ZSM-5 structure highlighting the channels along which small molecules may diffuse. Tetrahedra are SiO4 or AlO4. H O O O O Al Si O O O Si O O BH+ O B H O O O O O O O Al Si O O Al Si O O O O Si O O O O Si O O O Figure 26.24 The Brønsted acid site in H-ZSM5 and its interaction with a base, typically an organic molecule. (Based on W.O. Haag, R.M. Lago, and P.B. Weisz, Nature, 1984, 309, 589.) zeolite with a high Si content.5 The channels of ZSM-5 consist of a three-dimensional maze of intersecting tunnels (Fig. 26.23). As with other aluminosilicate catalysts, the Al sites are strongly acidic. The charge imbalance of Al3 in place of tetrahedrally coordinated Si(IV) requires the presence of an added positive ion. When this ion is H (Fig. 26.24) the Brønsted acidity of the aluminosilicate can be higher than that of concentrated H2SO4 and is termed a superacid (Section 4.15); the turnover frequency for hydrocarbon reactions at these sites can be very high. Natural petroleum consists of only about 20 per cent of alkanes suitable for use in petrol (gasoline) and diesel with chain lengths ranging from C5H12, pentane, to C12H26. Conversion of the higher molar mass hydrocarbons to the valuable lighter ones involves breaking the CC bonds but also structural rearrangement of the hydrocarbons through dehydrogenation, isomerization, and aromatization reactions. All these processes are catalysed by a solid acid catalyst based on alumina, silica, and zeolites. Acidic clays and mixed Al2O3/SiO2 were originally used for this process in the 1940s but since the 1960s these catalysts have been largely superseded by zeolites. The principal zeolite used for catalytic cracking is zeolite Y, in which the extra-framework cations have been replaced with lanthanoid ions, typically a mixture of La, Ce, and Nd. The mechanism of the catalytic cracking initially involves protonation of the alkane or alkene chain by the Brønsted acid sites in the zeolite pores followed by cleavage of the CC bond in the -position to the C atom carrying the positive charge. For example RCH2CHCH2(CH2)2R → RCH2CH CH2 CH2CH2R 5 The catalyst was developed in the research laboratories of Mobil Oil; the initials stand for Zeolite Socony-Mobil. 711 712 26 Catalysis Acidic zeolite catalysts also promote rearrangement reactions via carbonium ions. For example, the isomerization of 1,3-dimethylbenzene to 1,4-dimethylbenzene probably occurs by the following steps: CH 3 CH 3 H+ H - + CH 3 H CH 3 H CH 3 CH 3 -H + H+ H H CH 3 CH 3 Reactions such as dimethylbenzene (xylene) isomerization and methylbenzene (toluene) disproportionation illustrate the selectivity that can be achieved with acidic zeolite catalysis. The shape selectivity of these zeolite catalysts has been attributed to a variety of processes. In reactant selectivity the molecular sieving abilities of zeolites are important as only molecules of appropriate size and shape can enter the zeolite pores and undergo a reaction. In product selectivity, a reaction product molecule that has dimensions compatible with the channels will diffuse faster, allowing it to escape; molecules that do not fit the channels diffuse slowly and, on account of their long residence in the zeolite, have ample opportunity to be converted to the more mobile isomers that can escape rapidly. A currently more favoured view of zeolite selectivity is based on transition state selectivity, where the orientation of reactive intermediates within the zeolite channels favours specific products. In the case of dimethylbenzene (xylene) isomerization the narrower intermediates formed during the generation of 1,4-dialkylbenzene molecules fit better within the pores. Another common reaction in zeolites is the alkylation of aromatics with alkenes. E X A M PL E 26 . 3 Proposing a mechanism for the alkylation of benzene In its protonated form, ZSM-5 catalyses the reaction of ethene with benzene to produce ethylbenzene. Write a plausible mechanism for the reaction. Answer We should recall that protonated forms of zeolite catalysts are very strong acids. Therefore, a mechanism involving protonation of the organic species present in the system is the likely pathway. The acidic form of ZSM-5 is strong enough to generate carbocations from aliphatic hydrocarbons so the initial stage would be: CH2CH2 + H+ → CH2CH3+ As we saw in Section 4.10, the carbocation can attack benzene as a strong electrophile. Subsequent deprotonation of the intermediate yields ethylbenzene: CH2CH3+ + C 6H6 → C 6H5CH2CH3 + H+ Self-test 26.3 A pure silica analogue of ZSM-5 can be prepared. Would you expect this compound to be an active catalyst for benzene alkylation? Explain your reasoning. Mesoporous silicates discovered in the 1990s (Sections 24.13 and 25.9) have large ordered arrays of pores in the range 120 nm and generate very high specific surface areas (of over 1000 m2 g1). The large pores allow larger molecules to undergo catalytic processes, although their acidity is weak compared with the zeolites. More importantly other catalytic centres, such as nanoparticles of metals, alloys, and metal oxides such as platinum or Pt/Sn, may be deposited within the mesostructured channels. As one example, Co deposited on the mesoporous silica support MCM-416 promotes the cycloaddition of alkynes with alkenes and carbon monoxide to produce cyclopentenones in the Pauson–Khand reaction: EtO2C EtO2C 20 atm CO 130°C EtO2C O EtO2C Other porous inorganic framework materials (Section 24.14) are also being investigated as potential catalysts, either in their own right or as hosts for active species. For example, 6 MCM-41 stands for Mobile Crystalline Material. Heterogeneous catalysis the d-metal constituents of metal-organic frameworks can take part in redox reactions and also have very large pores that can incorporate nanoparticles of metals, such as Pt. 26.15 Fischer–Tropsch synthesis Key point: Hydrogen and carbon monoxide can be converted to hydrocarbons and water by reaction over iron or cobalt catalysts. The conversion of syngas, a mixture of H2 and CO, to hydrocarbons over metal catalysts was first discovered by Franz Fischer and Hans Tropsch at the Kaiser Wilhelm Institute for Coal Research in Müllheim in 1923. In the Fischer–Tropsch reaction, CO reacts with H2 to produce hydrocarbons, which can be written symbolically as the formation of the chain extender (CH2), and water: CO + 2 H 2 → –CH 2 – + H 2O The process is exothermic, with ∆rH O 165 kJ mol1. The product range consists of aliphatic straight-chain hydrocarbons that include methane (CH4) and ethane, LPG (C3 to C4), gasoline (C5 to C12), diesel (C13 to C22), and light and heavy waxes (C23 to C32 and >C33, respectively). Side reactions include the formation of alcohols and other oxygenated products. The distribution of the products depends on the catalyst and the temperature, pressure, and residence time. Typical conditions for the Fischer–Tropsch synthesis are a temperature range of 200–350ºC and pressures of 15–40 atm. There is general agreement that the first stages of the hydrocarbon synthesis involve the adsorption of CO on the metal, followed by its cleavage to give a surface carbide (and water), and the successive hydrogenation of such species to surface methyne (CH), methylene (CH2), and methyl (CH3) species, but there is still debate on what happens next and how chain growth occurs. One proposal suggests a polymerization of bridging surface CH2 groups initiated by a surface CH3 group. However, the fact that many such species have been isolated and are stable as metal complexes suggests that the mechanism is probably not so simple. Other mechanistic investigations of these processes have suggested an alternative possibility for chain growth in the hydrocarbon synthesis, namely that it proceeds by combination of surface bridging CH2 groups and alkenyl chains (MCH CHR), rather than by the combination of alkyl chains (MCH2CH2R) with methylene groups in the surface. Several catalysts have been used for the Fischer–Tropsch synthesis; the most important are based on Fe and Co. Cobalt catalysts have the advantage of a higher conversion rate and a longer life (of over 5 years). The cobalt catalysts are in general more reactive for hydrogenation and produce fewer unsaturated hydrocarbons and alcohols than iron catalysts. Iron catalysts have a higher tolerance for sulfur, are cheaper, and produce more alkene products and alcohols. The lifetime of the iron catalysts is short, however, and in commercial installations generally limited to about 8 weeks. 26.16 Alkene polymerization Key points: Heterogeneous Ziegler–Natta catalysts are used in alkene polymerization; the Cossee–Arlman mechanism describes their function; low molar mass homogeneous catalysts also catalyse the alkene polymerization reaction; considerable control over polymer tacticity is possible with judicious ligand design. Polyalkenes, which are among the most common and useful class of synthetic polymers, are most often prepared by use of organometallic catalysts, either in solution or supported on a solid surface. The development of alkene polymerization catalysts in the second half of the twentieth century, producing polymers such as polypropene and polystyrene, ushered in a revolution in packaging materials, fabrics, and constructional materials. In the 1950s J.P. Hogan and R.L. Banks discovered that chromium oxides supported on silica, a so-called Philips catalyst, polymerized alkenes to polyenes. Also in the 1950s K. Ziegler, working in Germany, developed a catalyst for ethene polymerization based on a catalyst formed from TiCl4 and Al(C2H5)3, and soon thereafter G. Natta in Italy used this type of catalyst for the stereospecific polymerization of propene (see below). Both the Ziegler–Natta catalysts and the chromium-based polymerization catalysts are widely used today. 713 714 26 Catalysis lC iT .cte tE lA 4 3 tE lC lC lC lC lC tE i T lC i T tE lC lC lC lC lC i T lC tE i T Figure 26.25 The Cossee–Arlman mechanism for the catalytic polymerization of ethene. Note that the Ti atoms are not discrete but are part of an extended structure containing bridging chlorides. Zr Me L 12 [Zr(η5-Cp)2(CH3)L]+ 13 The full details of the mechanism of Ziegler–Natta catalysts are still uncertain, but the Cossee–Arlman mechanism is regarded as highly plausible (Fig. 26.25). The catalyst is prepared from TiCl4 and Al(C2H5)3, which react to give polymeric TiCl3 mixed with AlCl3 in the form of a fine powder. The alkylaluminium alkylates a Ti atom on the surface of the solid and an ethene molecule coordinates to the neighbouring vacant site. In the propagation steps for the polymerization, the coordinated alkene undergoes a migratory insertion reaction. This migration opens up another neighbouring vacancy, and so the reaction can continue and the polymer chain can grow. The release of the polymer from the metal atom occurs by -hydrogen elimination, and the chain is terminated. Some catalyst remains in the polymer, but the process is so efficient that the amount is negligible. The proposed mechanism of alkene polymerization on a Philips catalyst involves the initial coordination of one or more alkene molecules to a surface Cr(II) site followed by rearrangement to metallocycloalkanes on a formally Cr(IV) site. Unlike Ziegler–Natta catalysts, the solid-phase catalyst does not need an alkylating agent to initiate the polymerization reaction; instead, this species is thought to be generated by the metallocycloalkane directly or by formation of an ethenylhydride by cleavage of a CH bond at the chromium site. Homogeneous catalysts related to the Philips and Ziegler–Natta catalysts provide additional insight into the course of the reaction and are of considerable industrial significance in their own right, being used commercially for the synthesis of specialized polymers. Most examples are from Group 4 (Ti, Zr, Hf) and are based on a bis(cyclopentadienyl) metal system: the tilted ring complex [Zr(5-Cp)2(CH3)L] (12) is a good example. These Group 4 metallocene complexes catalyse alkene polymerization by successive insertion steps that involve prior coordination of the alkene to the electrophilic metal centre. Catalysts of this type are used in the presence of the so-called methyl aluminoxane (MAO), a poorly defined compound of approximate formula (MeAlO)n, which, among other functions, serves to methylate a starting chloride complex. Additional complications arise with alkenes other than ethene. We shall discuss only terminal alkenes such as propene and styrene, as these are relatively simple. The first complication to consider arises because the two ends of the alkene molecule are different. In principle, it is possible for the polymer to form with the different ends head-to-head (13), head-to-tail (14), or randomly. Studies on catalysts such as (12) show that the growing chain migrates preferentially to the more highly substituted C atom of the alkene, thus giving a polymer chain that contains only head-to-tail orientations: + Zr R R R Zr Zr R 14 Zr Heterogeneous catalysis 715 If we consider propene, we can see that the coordinated alkene induces less steric strain if its smaller CH2 end is pointing into the cleft of the (Cp)2Zr catalyst (15), rather than the larger methyl substituted end, (16). The migrating polymer chain is thus adjacent to the methyl-substituted end of the propene molecule, and it is to this methyl-substituted C atom that the chain attaches, giving a head-to-tail sequence to the whole polymer chain. The second structural modification of polypropene is its tacticity, the relative orientations of neighbouring groups in the polymer. In a regular isotactic polypropene, all the methyl groups are on the same side of the polymer backbone (17). In regular syndiotactic polypropene, the orientation of the methyl groups alternates along the polymer chain (18). In an atactic polypropene, the orientation of neighbouring methyl groups is random (19). Control of the tacticity of a polymer is equivalent to controlling the stereospecificity of the reaction steps. The orientation of neighbouring groups is not simply of academic interest because the orientation has a significant effect on the properties of the bulk polymer. For example, the melting points of isotactic, syndiotactic, and atactic polypropene are 165ºC, 130ºC, and below 0ºC, respectively. It is not possible to control the tacticity of polypropene with a Zr catalyst such as (15), and an atactic polymer results. However, with other catalysts it is possible to control the tacticity. The type of catalyst normally used to control the tacticity has a metal atom bonded to two indenyl groups that are linked by a CH2 CH2 bridge. Reaction of the bis(indenyl) fragment with a metal salt gives rise to three compounds: two enantiomers (20) and (21), which have C2 symmetry, and a nonchiral compound (22). These compounds are called ansa-metallocenes (the name is derived from the Latin for handle and used to indicate a bridge). It is possible to separate the two enantiomers from the nonchiral compounds and both enantiomers of these ansa-metallocenes can catalyse the stereoregular polymerization of propene. If we now consider the coordination of propene to one of the enantiomeric compounds (20) or (21) there is a second constraint in addition to the steric factor mentioned above (the CH2 group pointing towards the cleft). Of the two potential arrangements of the methyl group shown in (23) and (24), the latter is disfavoured by a steric interaction with the phenyl ring of the indenyl group. During the polymerization reaction, the R group migrates preferentially to one side of the propene molecule; coordination of another alkene is then followed by migration, and so on. Figure 26.26 shows how an isotactic polypropene then results. Me Zr 15 Zr Me 16 17 18 E X A MPL E 26 . 4 Controlling the tacticity of polypropene Show that polymerization of propene with a catalyst containing CH2 CH2-linked fluorenyl and cyclopentadienyl groups (25) should result in syndiotactic polypropene. Answer We need to consider how the propene reactant will coordinate to the catalyst: in complex (25) a coordinated propene will always coordinate with the methyl group pointing away from the fluorenyl and towards the cyclopentadienyl ring. A series of sequential alkene insertions, as outlined in Fig. 26.27, will therefore lead to a product that should be syndiotactic. 19 Self-test 26.4 Demonstrate that polymerization of propene with a simple Zr(Cp)2Cl2 catalyst would give rise to atactic polypropene. Zr Cl Cl Cl Cl Cl Zr Cl 20 Zr Zr Cl Zr Cl 21 Cl 22 Cl Zr R Zr R Cl Zr R Zr Cl 23 Zr R 24 Zr R Cl Cl Zr R 25 716 26 Catalysis Zr Zr R R H Zr R R Zr H Zr R Zr R H R Zr R Zr Figure 26.26 When propene is polymerized with an indenyl metallocene catalyst, isotactic polypropene results. The zirconium species all have a single positive charge; this has been omitted for clarity. Zr Zr R H R Zr R Zr R H Zr R Zr R H H R R Zr Zr Figure 26.27 When a propene is polymerized with a fluorenyl metallocene catalyst, syndiotactic polypropene results. The zirconium species all have a single positive charge; this charge has been omitted for clarity. 717 Heterogeneous catalysis 26.17 Electrocatalysis Key points: Overpotentials represent the kinetic barrier for electrochemical reactions and electrocatalysts may be used to increase current densities in such processes. Kinetic barriers are quite common for electrochemical reactions at the interface between a solution and an electrode and, as mentioned in Section 5.18, it is common to express these barriers as overpotentials (eta), the potential in addition to the zero-current cell potential (the emf) that must be applied to bring about an otherwise slow reaction within the cell. The overpotential is related to the current density, j (the current divided by the area of the electrode), that passes through the cell by7 (26.3) where j0 and a are best regarded for our purposes as empirical constants. The constant j0, the exchange current density, is a measure of the rates of the forward and reverse electrode reactions at dynamic equilibrium. For systems obeying these relations, the reaction rate (as measured by the current density) increases rapidly with increasing applied potential difference when a > 1. If the exchange current density is high, an appreciable reaction rate may be achieved with only a small overpotential. If the exchange current density is low, a high overpotential is necessary. There is therefore considerable interest in increasing the exchange current density. In an industrial process an overpotential in a synthetic step is very costly because it represents wasted energy. A catalytic electrode surface can increase the exchange current density and hence dramatically decrease the overpotential required for sluggish electrochemical reactions, such as H2, O2, or Cl2 evolution and consumption. For example, ‘platinum black’, a finely divided form of platinum, is very effective at increasing the exchange current density and hence decreasing the overpotential of reactions involving the consumption or evolution of H2. The role of platinum is to dissociate the strong HH bond and thereby reduce the large barrier that this bond strength imposes on reactions involving H2. Palladium also has a high exchange current density (and hence requires only a low overpotential) for H2 evolution or consumption. The effectiveness of metals can be judged from Fig. 26.28, which also gives insight into the process. The volcano-like plot of exchange current density against MH bond enthalpy suggests that MH bond formation and cleavage are both important in the catalytic process. It appears that an intermediate MH bond energy leads to the proper balance for the existence of a catalytic cycle and the most effective metals for electrocatalysis are clustered around Group 10. Ruthenium dioxide is an effective catalyst for both O2 and Cl2 evolution and it also is a good electrical conductor. It turns out that at high current densities RuO2 is more effective for the catalysis of Cl2 evolution than for O2 evolution. RuO2 is therefore extensively used as an electrode material in the commercial production of chorine. The electrode processes that contribute to this subtle catalytic effect do not appear to be well understood. There is great interest in devising new catalytic electrodes, particularly ones that decrease the O2 overpotential on surfaces such as graphite. Thus, tetrakis(4-N-methylpyridyl) porphyriniron(II), [Fe(TMPyP)]4+ (26), has been deposited on the exposed edges of graphite electrodes (on which O2 reduction requires a high overpotential) and the resulting electrode surface was found to catalyse the electrochemical reduction of O2. A plausible explanation for this catalysis is that [FeIII(TMPyP)] attached to the electrode (indicated below by an asterisk) is first reduced electrochemically: 2 Pt Re Rh Ir 3 Au 5 Ni Co –log j0 j = j0 ea Fe W 7 9 10 0 Sn Bi Zn Ag Pb Ga Cd Tl In Mo Ti Nb Ta 100 200 300 400 B(M–H)/(kJ mol–1) Figure 26.28 The rate of H2 evolution expressed as the logarithm of the exchange current density plotted against MH bond energy. N N N [ FeIII (TMPyP)] + e − → [ FeII (TMPyP)] N Fe The resulting [FeII(TMPyP)] forms an O2 complex: N N [ FeII (TMPyP)] + O2 → [ FeII (O2 )(TMPyP)] N 7 The exponential relation between the current and the overpotential is accounted for by the Butler– Volmer equation, which is derived by applying transition-state theory to dynamical processes at electrodes. See P. Atkins and J. de Paula, Physical chemistry. Oxford University Press and W.H. Freeman & Co (2010). 4(ClO 4− ) 26 N 718 26 Catalysis This iron(II) porphyrin oxygen complex then undergoes reduction to water and hydrogen peroxide: [ FeII (O2 )(TMPyP)] + n e − + n H + → [ FeII (TMPyP)] + (H 2O2 , H 2O)n Although details of the mechanism are still elusive, this general set of reactions is in harmony with electrochemical measurements and the known properties of iron porphyrins. The investigation of porphyrin iron complexes as catalysts was motivated by Nature’s use of metalloporphyrins for oxygen activation (Section 27.10). One application where cheap and efficient electrocatalysts are much needed is in PEM fuel cells (PEM stands for ‘proton exchange membrane’ and sometimes ‘polymer electrolyte membrane’; Box 5.1). As these systems operate at quite low temperatures, 50–100°C, they are suitable for applications that involve transport and mobile power, such as mobile telephones (cell phones). The electrolyte, a polymer that conducts protons but not electrons, separates an anode in contact with hydrogen gas and a cathode over which flows oxygen gas. As noted above hydrogen gas is readily dissociated at the anode using a platinum metal catalyst but the oxygen reduction reaction at the cathode presents more of a problem. There is a substantial overpotential associated with this reduction, which considerably reduces the efficiency of the cell, giving operating voltages near 0.7 V as compared with the theoretical value of 1.23 V. Platinum may again be used to catalyse the reaction but even so the reaction is inefficient and use of large quantities of platinum costly. Considerable research effort is being applied to finding better electrocatalysts and recently the alloy Pt3Ni (111 surface) has proved to have greatly enhanced properties. 26.18 New directions in heterogeneous catalysis Key point: Continuing developments in heterogeneous catalysis include research on new compositions for the controlled partial oxidation of hydrocarbons. The development of solid-phase catalysts is very much a frontier subject for inorganic chemistry with the continuing discovery of compositions for promoting reactions, particularly for petrochemicals. One very active area is the investigation of selective heterogeneous oxidation catalysts, which allow partial oxidation of hydrocarbons to useful intermediates in, for example, the polymer and pharmaceutical industries. Examples of such reactions include alkene epoxidation, aromatic hydroxylation, and ammoxidation (an oxidation in the presence of ammonia that generates nitriles) of alkanes, alkenes, and alkyl aromatics. In all these cases it is desirable to produce the products without complete oxidation of the hydrocarbon to carbon dioxide. One example where new catalysts are being investigated is in the partial oxidation of benzene to phenol. At present, the three-step cumene process produces about 95 per cent of the phenol used in the world and gives propanone (acetone) as a byproduct, although the market for acetone is oversupplied from other industrial processes. The cumene process involves three stages, namely alkylation of benzene with propene to form cumene (a process catalysed by phosphoric acid or aluminium chloride), direct oxidation of cumene to cumene hydroperoxide using molecular oxygen, and finally cleavage of cumene hydroperoxide to phenol and acetone, which is catalysed by sulfuric acid. Far better would be a single-stage process that accomplishes the reaction C6 H6 + 12 O2 → C 6 H 5OH Examples of catalysts investigated for this and similar processes include iron-containing zeolites with the silicalite framework type (Section 24.13), mixtures of FeCl3/SiO2, photocatalysts based on Pt/H2SO4/TiO2, and various vanadium salts. Hybrid catalysis Chemists have begun to look at catalytic systems that cannot simply be defined as either homogeneous or heterogeneous. Research has centred on trying to achieve the best of both systems: the high selectivity of homogeneous catalysts coupled with the ease of separation of heterogeneous catalysts. This approach is sometimes referred to as ‘heterogenizing’ homogeneous catalysts. Hybrid catalysis 26.19 Tethered catalysts Key point: The tethering of a catalyst to a solid support allows easy separation of the catalyst with little loss in catalyst activity. One popular technique has been the tethering of a homogeneous catalyst to a solid support. Thus, a hydrogenation catalyst such as Wilkinson’s catalyst can be attached to a silica surface by means of a long hydrocarbon chain: (PPh 3)3RhCl Si PPh 2 Si PPh 2 Ph3P Rh Cl PPh 3 When the silica monolith is immersed in a solvent, the rhodium-based catalytic site behaves as though it is in solution, and reactivity is largely unaffected. Separation of the products from the catalyst simply requires the decanting of the solvent. Commercially available functionalized silica precursors include amino-, acrylate-, allyl-, benzyl-, bromo-, chloro-, cyano-, hydroxy-, iodo-, phenyl-, styryl-, and vinyl-substituted reagents. Relatively simple reactions can result in the synthesis of a whole host of further reagents (such as the phosphine compound in the scheme above). In addition to silica, polystyrene, polyethene, polypropene, and various clays have been used as the solid support and have led to reports of the successful heterogenization of most reactions that rely on soluble metal complexes. In some cases the activity of a supported catalyst is greater than that of its unsupported analogue. This improvement normally takes the form of enhanced selectivity brought about by the steric demands of approaching a catalyst constrained to a surface or an increase in catalyst turnover frequencies brought about by protection from the support. Often, however, supported catalysts suffer from catalyst leaching and reduced activity. 26.20 Biphasic systems Key point: Biphasic systems offer another way of combining the selectivity of homogeneous catalysts with the ease of separation of heterogeneous catalysts. Another popular method of attempting to combine the best of both homogeneous and heterogeneous catalysts has been the use of two liquid phases that are not miscible at room temperature but are miscible at higher temperatures. The existence of immiscible aqueous/ organic phases together with a compound that facilitates the transfer of reagents between the two phases will be familiar; two other systems worthy of note are ionic liquids and the ‘fluorous biphase’ system. Ionic liquids are typically derived from 1,3-dialkylimidazolium cations (27) with counterions such as PF6, BF4, and CF3SO3. These systems have melting points of less than (and often much less than) 100ºC, a very low viscosity, and an effectively zero vapour pressure. If it can be arranged that the catalysts are preferentially soluble in the ionic phase (for instance, by making them ionic), immiscible organic solvents can be used to extract organic products. As an example, consider the hydroformylation of alkenes by a Rh catalyst with phosphine ligands. When the ligand used is triphenylphosphine, the catalyst is extracted from the ionic liquid together with the products. However, when an ionic sulfonated triphenylphosphine is used as the ligand, the catalyst remains in the ionic liquid and product separation from the catalyst is complete. It should be borne in mind that the ionic liquid phase is not always unreactive, and may induce alternative reactions. The fluorous biphase system, which typically consists of a fluorinated hydrocarbon and a ‘normal’ organic solvent such as methylbenzene, offers two principal advantages over aqueous/organic phases: the fluorous phase is unreactive, so sensitive groups (such as hydrolytically unstable groups) are stable in it, and polyfluorinated and hydrocarbon solvents, which are not miscible at room temperatures, become so on heating, giving rise to a genuinely homogeneous system. A catalyst that contains polyfluorinated groups is preferentially R N 27 N R' 719 720 26 Catalysis Hydrocarbon phase (contains reactants) Hydrocarbon phase (contains products) C 6F13 C 2H 4 O C 6 F13 x x PPh 3-x 9 2 F3 C Cool C 6 F13 Fluorous phase (contains catalyst) PPh 3-x 28 Homogeneous mixture Heat O O P O CF F 2C C 6F 13 2 2 Fluorous phase (contains catalyst) Figure 26.29 The sequence of processes that occurs during the use of a fluorous biphase catalyst. P F2 C CF C F2 CF 3 CF2 30 31 retained in the fluorous solvent, with the reactants (and products) preferentially soluble in the hydrocarbon phase. Figure 26.29 indicates the type of sequence that is used. Separation of the products from the catalyst becomes a trivial matter of decanting one liquid from another. A number of ligand systems that confer fluorous solubility on a catalyst have been developed; typically they are phosphine based such as (28), (29), and (30). Rhodium- and Pd-based catalysts of these ligands have then been prepared. With a fluorous solvent such as perfluoro-1,3-dimethylcyclohexane (31), miscibility with an organic phase can be achieved at 70ºC, and catalysts have been used in hydrogenation (Section 26.4), hydroformylation (Section 26.5), and hydroboration reactions. FURTHER READING R. Whyman, Applied organometallic chemistry and catalysis. Oxford University Press (2001). G.W. Parshall and S.D. Ittle, Homogeneous catalysis. Wiley, New York (1992). See H.H. Brintzinger, D. Fischer, R. Mülhaupt, B. Rieger, and R.M. Waymouth, Angew. Chem., Int. Ed. Engl., 1995, 34, 1143 for a review of the area of control of polymer tacticity. For a good review of the Stille reaction that touches on the mechanism of all palladium-catalysed coupling reactions, see P. Espinet and A.M. Echavarren, Angew. Chem., Int. Ed. Engl., 2004, 43, 4704. A whole issue of a major journal has been devoted to the subject of supported homogeneous catalysts. See Chem. Rev., 2002, 102, 3215. Fluorous biphase systems are discussed in greater detail by E.G. Hope and A.M Stuart, J. Fluorine Chem., 1999, 100, 75. For a review of the development of alkene metathesis catalysts, see T.M. Trnka and R.H. Grubbs, Acc. Chem. Res., 2001, 34, 18. V. Ponec and G.C. Bond, Catalysis by metals and alloys. Elsevier, Amsterdam (1995). Comprehensive discussion of the basis of chemisorption and catalysis by metals. R.D. Srivtava, Heterogeneous catalytic science. CRC Press, Boca Raton (1988). A survey of experimental methods and several major heterogeneous catalytic processes. M. Bowker, The basis and applications of heterogeneous catalysis. Oxford Chemistry Primers 53. Oxford University Press (1998). Concise coverage of heterogeneous catalysis. J.M. Thomas and W.J. Thomas, Principles and practice of heterogeneous catalysis. VCH, Weinheim (1997). Readable introduction to the fundamental principles of heterogeneous catalysis written by worldrenowned experts. K.M. Neyman and F. Illas, Theoretical aspects of heterogeneous catalysis: applications of density functional methods. Catalysis Today, 2005, 105, 15. Modelling methods applied to heterogeneous catalysis. M.A. Keane, Ceramics for catalysis. J. Mater. Sci., 2003, 38, 4661. Overview of heterogeneous catalysis illustrated with three established methods: (i) catalysis using zeolites; (ii) catalytic converters; (iii) solid oxide fuel cells. F.S. Stone, Research perspectives during 40 years of the Journal of Catalysis. J. Catal., 2003, 216, 2. A historical perspective on developments in catalysis. G. Rothenberg, Catalysis: concepts and green applications. Wiley VCH (2008). Catalysis and sustainability. D.K. Chakrabarty and B. Viswanathan, Heterogeneous catalysis. New Age Science Ltd (2008). EXERCISES 26.1 Which of the following constitute genuine examples of catalysis and which do not? Present your reasoning. (a) The addition of H2 to C2H4 when the mixture is brought into contact with finely divided platinum. (b) The reaction of an H2/O2 gas mixture when an electrical arc is struck. (c) The combination of N2 gas with lithium metal to produce Li3N, which then reacts with H2O to produce NH3 and LiOH. 26.2 Define the terms (a) turnover frequency, (b) selectivity, (c) catalyst, (d) catalytic cycle, (e) catalyst support. Problems 26.3 Classify the following as homogeneous or heterogeneous catalysis and present your reasoning. (a) The increased rate in the presence of NO(g) of SO2(g) oxidation by O2(g) to SO3(g). (b) The hydrogenation of liquid vegetable oil using a finely divided nickel catalyst. (c) The conversion of an aqueous solution of d-glucose to a d,l mixture catalysed by HCl(aq). 26.4 You are approached by an industrialist with the proposition that you develop catalysts for the following processes at 80ºC with no input of electrical energy or electromagnetic radiation: (a) The splitting of water into H2 and O2. (b) The decomposition of CO2 into C and O2. (c) The combination of N2 with H2 to produce NH3. (d) The hydrogenation of the double bonds in vegetable oil. The industrialist’s company will build the plant to carry out the process and the two of you will share equally in the profits. Which of these would be easy to do, which are plausible candidates for investigation, and which are unreasonable? Describe the chemical basis for the decision in each case. 26.5 Addition of PPh3 to a solution of Wilkinson’s catalyst, [RhCl(PPh3)3], reduces the turnover frequency for the hydrogenation of propene. Give a plausible mechanistic explanation for this observation. 26.6 The rates of H2 gas absorption (in dm3 mol1 s1) by alkenes catalysed by [RhCl(PPh3)3] in benzene at 25ºC are: hexene, 2910; cis4-methyl-2-pentene, 990; cyclohexene, 3160; 1-methylcyclo-hexene, 60. Suggest the origin of the trends and identify the affected reaction step in the proposed mechanism (Fig. 26.5). 26.7 Infrared spectroscopic investigation of a mixture of CO, H2, and 1-butene under conditions that bring about hydroformylation indicate the presence of compound (E) in Fig. 26.7 in the reaction mixture. The same reacting mixture in the presence of added tributylphosphine was studied by infrared spectroscopy and neither (E) nor an analogous phosphine-substituted complex was observed. What does the first observation suggest as the rate-limiting reaction in the absence of 721 phosphine? Assuming the sequence of reactions remains unchanged, what are the possible rate-limiting reactions in the presence of tributylphosphine? 26.8 Show how reaction of MeCOOMe with CO under conditions of the Monsanto ethanoic acid process can lead to ethanoic anhydride. 26.9 Suggest reasons why (a) ring-opening alkene metathesis polymerization and (b) ring-closing metathesis reactions proceed. 26.10 (a) Starting with the alkene complex shown in Fig. 26.9 with trans-DHC CHD in place of C2H4, assume dissolved OH attacks from the side opposite the metal. Give a stereochemical drawing of the resulting compound. (b) Assume attack on the coordinated transDHC CHD by an OH ligand coordinated to Pd, and draw the stereochemistry of the resulting compound. (c) Does the stereochemistry differentiate these proposed steps in the Wacker process? 26.11 Aluminosilicate surfaces in zeolites act as strong Brønsted acids, whereas silica gel is a very weak acid. (a) Give an explanation for the enhancement of acidity by the presence of Al3 in a silica lattice. (b) Name three other ions that might enhance the acidity of silica. 26.12 Why is the platinum/rhodium catalyst in automobile catalytic converters dispersed on the surface of a ceramic rather than used in the form of a thin metal foil? 26.13 Alkanes are observed to exchange hydrogen atoms with deuterium gas over some platinum metal catalysts. When 3,3-dimethylpentane in the presence of D2 is exposed to a platinum catalyst and the gases are observed before the reaction has proceeded very far, the main product is CH3CH2C(CH3)2CD2CD3 plus unreacted 3,3-dimethylpentane. Devise a plausible mechanism to explain this observation. 26.14 The effectiveness of platinum in catalysing the reaction 2 H(aq) 2 e → H2(g) is greatly decreased in the presence of CO. Suggest an explanation. 26.15 Describe the role of electrocatalysts in reducing the overpotential in the oxygen reduction reaction in fuel cells. PROBLEMS 26.1 Consider the validity of each of the following statements and provide corrections where required. (a) A catalyst introduces a new reaction pathway with lower enthalpy of activation. (b) As the Gibbs energy is more favourable for a catalytic reaction, yields of the product are increased by catalysis. (c) An example of a homogeneous catalyst is the ZieglerNatta catalyst made from TiCl4(l) and Al(C2H5)3(l). (d) Highly favourable Gibbs energies for the attachment of reactants and products to a homogeneous or heterogeneous catalyst are the key to high catalytic activity. 26.2 A catalyst might not just lower the enthalpy of activation, but might make a significant change to the entropy of activation. Discuss this phenomenon. (See A. Haim, J. Chem., Educ., 1989, 66, 935.) 26.3 The addition of promoters can further enhance the rate of a catalysed reaction. Describe how the promoters allowed the iridium-based Cativa process to compete with the rhodium-based process in the carbonylation of methanol. (See A. Haynes, P.M. Maitlis, G.E. Morris, G.J. Sunley, H. Adams, and P.W. Badger, J. Am. Chem. Soc., 2004, 126, 2847.) 26.4 When direct evidence for a mechanism is not available, chemists frequently invoke analogies with similar systems. Describe how J.E. Bäckvall, B. Åkermark, and S.O. Ljunggren (J. Am. Chem. Soc., 1979, 101, 2411) inferred the attack of uncoordinated water on 2-C2H4 in the Wacker process. 26.5 Whereas many enantioselective catalysts require the precoordination of a substrate, this is not always the case. Use the example of asymmetric epoxidation to demonstrate the validity of this statement, and indicate the advantages that such catalysts might have over catalysts that require precoordination of a substrate. (See M. Palucki, N.S. Finney, P.J. Pospisil, M.L. Güler, T. Ishida, and E.N. Jacobsen, J. Am. Chem. Soc., 1998, 120, 948.) 26.6 Discuss shape selectivity with respect to catalytic processes involving zeolites. 26.7 Summarize the potential impact of heterogeneous oxidation catalysts in chemistry. (See J.M. Thomas and R. Raja, Innovations in oxidation catalysis leading to a sustainable society. Catalysis Today, 2006, 117, 22. 26.8 Discuss the applications and mechanisms of oxidation and ammoxidation catalysts such as bismuth molybdate. (See, for example, R.K. Grasselli, J. Chem. Educ., 1986, 63, 216.) 26.9 Discuss the advantages of a solid support in catalysis by reference to the use of (Ni(POEt)3)4 in alkene isomerization. (See A.J. Seen, J. Chem. Educ., 2004, 81, 383 and K.R. Birdwhistell and J. Lanza, J. Chem. Educ., 1997, 74, 579.) 27 The organization of cells 27.1 The physical structure of cells 27.2 The inorganic composition of cells Transport, transfer, and transcription Biological inorganic chemistry Organisms have exploited the chemical properties of the elements in remarkable ways, providing examples of coordination specificities that are far higher than observed in simple compounds. This chapter describes how different elements are taken up selectively by different cells and intracellular compartments and the various ways they are exploited. We discuss the structures and functions of complexes and materials that are formed in the biological environment in the context of the chemistry covered earlier in the text. 27.3 Sodium and potassium transport 27.4 Calcium signalling proteins 27.5 Zinc in transcription 27.6 Selective transport and storage of iron 27.7 Oxygen transport and storage Biological inorganic chemistry (‘bioinorganic chemistry’) is the study of the ‘inorganic’ elements as they are utilized in biology. The main focus is on metal ions, where we are interested in their interaction with biological ligands and the important chemical properties they are able to exhibit and impart to an organism. These properties include ligand binding, catalysis, signalling, regulation, sensing, defence, and structural support. 27.8 Electron transfer Catalytic processes 27.9 Acid–base catalysis 27.10 Enzymes dealing with H2O2 and O2 27.11 The reactions of cobalt-containing enzymes 27.12 Oxygen atom transfer by molybdenum and tungsten enzymes Biological cycles The organization of cells To appreciate the role of the elements (other than C, H, O, and N) in the structure and function of organisms we need to know a little about the organization of the ‘atom’ of biology, the cell, and its ‘fundamental particles’, the cell’s constituent organelles. 27.1 The physical structure of cells 27.13 The nitrogen cycle 27.14 The hydrogen cycle Sensors 27.15 Iron proteins as sensors 27.16 Proteins that sense Cu and Zn levels Biomineralization The chemistry of elements in medicine 27.17 Chelation therapy 27.18 Cancer treatment 27.19 Anti-arthritis drugs 27.20 Imaging agents Perspectives 27.21 The contributions of individual elements 27.22 Future directions FURTHER READING EXERCISES PROBLEMS Key points: Living cells and organelles are enclosed by membranes; the concentrations of specific elements may vary greatly between different compartments due to the actions of ion pumps and gated channels. Cells, the basic unit of any living organism, range in complexity from the simplest types found in prokaryotes (bacteria and bacteria-like organisms now classified as archaea) and the much larger and more complex examples found in eukaryotes (which include animals and plants). The main features of these cells are illustrated in the generic model shown in Fig. 27.1. Crucial to all cells are membranes, which act as barriers to water and ions and make possible the management of all mobile species and of electrical currents. Membranes are lipid bilayers, approximately 4 nm thick, in which are embedded protein molecules and other components. Bilayer membranes have great lateral strength but they are easy to bend. The long hydrocarbon chains of lipids make the membrane interior very hydrophobic and impermeable to ions, which must instead travel through specific channels, pumps, and other receptors provided by special membrane proteins. The structure of a cell also depends on osmotic pressure, which is maintained by high concentrations of solutes, including ions, imported during active transport by pumps. Prokaryotic cells consist of an enclosed aqueous phase, the cytoplasm, which contains the DNA and most of the materials used and transformed in the biochemical reactions. Bacteria are classified according to whether they are enclosed by a single membrane or have an additional intermediate aqueous space, the periplasm, between the outer membrane and the cytoplasmic membrane, and are known as ‘Gram-positive’ or ‘Gram-negative’, respectively, depending on their response to a staining test with the dye crystal violet. The much The organization of cells more extensive cytoplasm of eukaryotic cells contains subcompartments (also enclosed within lipid bilayers) known as organelles, which have highly specialized functions. Organelles include the nucleus (which houses DNA), mitochondria (the ‘fuel cells’ that carry out respiration), chloroplasts (the ‘photocells’ that harness light energy), the endoplasmic reticulum (for protein synthesis), Golgi (vesicles containing proteins for export), lysosomes (which contain degradative enzymes and help rid the cell of waste), peroxisomes (which remove harmful hydrogen peroxide), and other specialized processing zones. Nucleus Pump 723 Endoplasmic reticulum Peroxisome 27.2 The inorganic composition of cells Key points: The major biological elements are oxygen, hydrogen, carbon, nitrogen, phosphorus, sulfur, sodium, magnesium, calcium, and potassium. The trace elements include many d metals, as well as selenium, iodine, silicon, and boron. Table 27.1 lists many of the elements known to be used in living systems, although not necessarily by higher life forms. All the second- and third-period elements except Be, Al, and the noble gases are used, as are most of the 3d elements, whereas Cd, Br, I, Mo, and W are the only heavier elements so far confirmed to have a biological function. Several others, such as Li, Ga, Tc, Ru, Gd, Pt, and Au, have important and increasingly well-understood applications in medicine. The biologically essential elements can be classified as either ‘major’ or ‘trace’. Although a good idea of the biological abundances of different elements is given in Table 27.1, the levels vary considerably among organisms and different components of organisms. For example, Ca has little role in microorganisms but is abundant in higher life forms, whereas the use of Co by higher organisms depends on it being incorporated into a special cofactor (cobalamin) by microorganisms. There is probably a universal requirement for K, Mg, Fe, and Mo. Vanadium is used by lower animals and plants as well as some bacteria. Nickel is essential for most microorganisms, and is used by plants, but there is no evidence for any direct role in animals. Nature’s use of different elements is largely based on their availability. For example, Zn has widespread use (and, together with Fe, ranks among the Table 27.1 The approximate concentrations, log ([J]/mol dm−3), where known, of elements (apart from C, H, O, N, P, S, and Se) in different biological zones Element External fluids (sea water) Free ions in external fluids (blood plasma) Cytoplasm (free ions) Comments on status in cell Na 101 101 102 Not bound K 102 4 103 3 101 Not bound Mg 10 10 c. 10 Weakly bound as ATP complex Ca 103 103 c. 107 Concentrated in some vesicles Cl 101 101 102 Not bound 2 Too much unbound Fe is toxic (Fenton chemistry) in and out of cells 109 1011 Totally bound, but may be exchangeable 1012 1015 (Cu(I)) Totally bound, not mobile. Mostly outside cytoplasm 109 c. 106 Higher in chloroplasts and vesicles 1011 (Fe(III)) 10 Zn 108 Cu 1010 (Cu(II)) Mn Co Mo 3 10 (Fe(II)) 17 Fe Ni 3 16 10 (Fe(III)) 7 109 Totally bound (cobalamin) 9 1010 Totally bound 7 10 Mostly bound 10 10 7 Channel Chloroplast Golgi Cytoplasm apparatus Mitochondrion Figure 27.1 The layout of a generic eukaryotic cell showing the cell membrane, various kinds of compartments (organelles), and the membrane-bound pumps and channels that control the flow of ions between compartments. 724 27 Biological inorganic chemistry most abundant biological trace elements) whereas Co (a comparatively rare element) is essentially restricted to cobalamin. The early atmosphere (over 2.3 Ga ago1), being highly reducing, enabled Fe to be freely available as soluble Fe(II) salts, whereas Cu was trapped as insoluble sulfides (as was Zn). Indeed, Cu is not found in the archaea (which are believed to have evolved in pre-oxygenic times), including the hyperthermophiles, organisms that are able to survive at temperatures in excess of 100ºC. These organisms are found in deep sea hydrothermal vents and terrestrial hot springs and are good sources of enzymes that contain W, the heaviest element known to be essential to life. The finding that W, Co, and for the most part Ni are used only by more primitive life forms probably reflects their special role in the early stages of evolution. (a) Compartmentalization Key point: Different elements are strongly segregated inside and outside a cell and among different internal compartments. Compartmentalization is the distribution of elements inside and outside a cell and between different internal compartments. The maintenance of constant ion levels in different biological zones is an example of ‘homeostasis’ and it is achieved as a result of membranes being barriers to passive ion flow. An example is the large difference in concentration of K and Na ions across cell membranes. In the cytoplasm, the K concentration may be as high as 0.3 m whereas outside it is usually less than 5 × 103 m. By contrast, Na is abundant outside a cell but scarce inside; indeed, the low intracellular concentration of Na, which has characteristically weak binding to ligands, means that it has few specific roles in biochemistry. Another important example is Ca2, which is almost absent from the cytoplasm (its free concentration is below 1 × 107 m) yet is a common cation in the extracellular environment and is concentrated in certain organelles, such as mitochondria. That pH may also vary greatly between different compartments has particularly important implications because sustaining a transmembrane proton gradient is a key feature in photosynthesis and respiration. The distributions of Cu and Fe provide another example: Cu enzymes are often extracellular, that is they are synthesized in the cell and then secreted outside the cell, where they catalyse reactions involving O2. By contrast, Fe enzymes are contained inside the cell. This difference can be rationalized on the basis that the inactive trapped states of these elements are Fe(III) and Cu(I) (or even metallic Cu) and organisms have stumbled on the expediency of keeping Fe in a relatively reducing environment and Cu in a relatively oxidizing environment. The selective uptake of metal ions has potential industrial applications, for many organisms and organs are known to concentrate particular elements. Thus, liver cells are a good source of cobalamin2 (Co) and milk is rich in Ca. Certain bacteria accumulate Au and thus provide an unusual way for procuring this precious metal. Compartmentalization is an important factor in the design of metal complexes that are used in medicine (Sections 27.17–20). The very small size of bacteria and organelles raises an interesting point about scale, as species present at very low concentrations in very small regions may be represented by only a few individual atoms or molecules. For example, the cytoplasm in a bacterial cell of volume 1015 dm3 at pH 6 will contain less than 1000 ‘free’ H ions. Indeed, any element nominally present at less than 1 nmol dm3 may be completely absent in individual cases. The word ‘free’ is significant, particularly for metal ions such as Zn2 that are high in the Irving–Williams series; even a eukaryotic cell with a total Zn concentration of 0.1 mmol dm3 may contain very few uncomplexed Zn2 ions. Two important issues arise in the context of compartmentalization. First, the process requires energy because ions must be pumped against an adverse gradient of chemical potential. However, once a concentration difference has been established, there is a difference in electrical potential across the membrane dividing the two regions. For instance, if the concentrations of K ions on either side of a membrane are [K]in and [K]out, then the 1 Current geological and geochemical evidence date the advent of atmospheric O2 at between 2.2 and 2.4 Ga ago (1 Ga 109 a). It is likely that this gas arose by the earliest catalytic actions of the photosynthetic Mn cluster described in Section 27.10. 2 In nutrition, the common complexes of cobalamin that are ingested are known as vitamin B12. The organization of cells contribution to the potential difference ∆ across the membrane is = RT [ K + ]in ln + F [ K ]out (27.1) This difference in electrical potential is a way of storing energy, which is released when the ions flood back to their natural concentrations. Second, the selective transport of ions must occur through ion channels built from membrane-spanning proteins, some of which release ions on receipt of an electrical or chemical signal whereas others, the transporters and pumps, transfer ions against the concentration gradient by using energy provided by adenosine triphosphate (ATP) hydrolysis. The selectivity of these channels is exemplified by the highly discriminatory transport of K as distinct from Na (Section 27.3). Proteins, the most important sites for metal ion coordination, are not permanent species but are ceaselessly degraded by enzymes (proteases), releasing both amino acids and metal ions to provide materials for new molecules. E X A M PL E 27.1 Assessing the role of phosphate ions Phosphate is the most abundant small anion in the cytoplasm. What implications does this abundance have for the biochemistry of Ca2? Answer We can approach this problem by considering how Ca2 is compartmentalized. In a eukaryotic cell Ca2 is pumped out of the cytoplasm (to the exterior or into organelles such as mitochondria) using energy derived from ATP hydrolysis. Spontaneous influx of Ca2 occurs under the action of special channels or if the cell boundary is damaged. The solubility product of Ca3(PO4)2 is very low and it could precipitate inside the cell if the Ca2 concentration rises above a critical value. Self-test 27.1 Is Fe(II) expected to be present in the cell as uncomplexed ions? (b) Biological metal-coordination sites Key points: The major binding sites for metal ions are provided by the amino acids that make up protein molecules; the ligation sites range from backbone peptide carbonyls to the side chains that provide more specific complexation; nucleic acids and lipid head groups are usually coordinated to major metal ions. Metal ions coordinate to proteins, nucleic acids, lipids, and a variety of other molecules. For instance, ATP is a tetraprotic acid and is always found as its Mg2 complex (1); DNA is stabilized by weak coordination of K and Mg2 to its phosphate groups but destabilized by binding of soft metal ions such as Cu(I) to the bases. Macromolecules known as ribozymes may represent an important stage in the early evolution of life forms and are catalytic molecules composed of RNA and Mg2. The binding of Mg2 to phospholipid head groups is important for stabilizing membranes. There are a number of important small ligands, apart from water and free amino acids, which include sulfide, sulfate, carbonate, cyanide, carbon monoxide, and nitrogen monoxide, as well as organic acids such as citrate that form reasonably strong polydentate complexes with Fe(III). As will be familiar from introductory chemistry, a protein is a polymer with a specific sequence of amino acids linked by peptide bonds (2). A ‘small’ protein is generally regarded as one with molar mass below 20 kg mol1, whereas a ‘large’ protein is one having a molar mass above 100 kg mol1. The principal amino acids are listed in Table 27.2. Proteins are NH2 Mg O –O – O N P O P O P O O O N O O O H N H H H OH OH 1 Mg–ATP complex N H N H C O N C H H 2 Peptide bond O 725 726 27 Biological inorganic chemistry synthesized, a process called translation (of the genetic code carried by DNA), on a special assembly called a ribosome. A protein may be processed further by post-translational modification, a change made to the protein structure, which includes the binding of cofactors such as metal ions. Metalloproteins, proteins containing one or more metal ions, perform a wide range of specific functions. These functions include oxidation and reduction (for which the most Table 27.2 The amino acids and their codes Amino acid Alanine Structure in peptide chain (side chain shown in blue) CH Ala A Arg R Asn N Asp D Cys C Glu E Gln Q Gly G His H Ile I O H N H2N NH CH NH2 Asparagine One-letter abbreviation NH H3C C Arginine Three-letter abbreviation C O NH H2N CH O C O NH CH HO Aspartic acid O C O NH Cysteine HS CH C O O Glutamic acid NH CH HO C O O Glutamine NH H2N CH C H Glycine O NH CH C O NH Histidine CH HN N C O CH3 Isoleucine H3C NH CH C O The organization of cells Leucine Lysine NH CH H3C CH3 C NH CH +H N 3 H3C C C Proline Serine NH CH C Met M Phe F Pro P Ser S Thr T Trp W Tyr Y Val V O N CH C O HO K O NH CH Phenylalanine Lys O NH CH S L O C Methionine Leu O CH3 Threonine NH CH HO C O NH CH Tryptophan C HN O NH CH Tyrosine C HO O CH3 Valine H3C NH CH C O important elements are Fe, Mn, Cu, and Mo), radical-based rearrangement reactions and methyl-group transfer (Co), hydrolysis (Zn, Fe, Mg, Mn, and Ni), and DNA processing (Zn). Special proteins are required for transporting and storing different metal atoms. The action of Ca2 is to alter the conformation of a protein (its shape) as a step in cell signalling (a term used to describe the transfer of information between and within cells). Such proteins are often known as metal ion-activated proteins. Hydrogen bonding between main-chain −NH and CO groups of different amino acids results in secondary structure (Fig. 27.2). The -helix regions of a polypeptide provide flexible mobility 727 728 27 Biological inorganic chemistry O – O (a) O– Ca Figure 27.2 The most important regions of secondary structure, (a) helix, (b) sheet, showing hydrogen bonding between main-chain amide and carbonyl groups and their corresponding representations. O 3 Ca2+ coordination H N N H N N C u 4 Cu–imidazole coordination Zn S– 5 Zn–cysteine coordination Fe S (b) eM 6 Fe–methionine coordination O– Fe 7 Fe–tyrosine coordination (like springs) and are important in converting processes that occur at the metal site into conformational changes; by contrast, a b-sheet region confers rigidity to support a preorganized coordination sphere suited to a particular metal ion (Sections 7.14 and 11.16). The secondary structure is largely determined by the sequence of amino acids: thus the helix is favoured by chains containing alanine and lysine but is destabilized by glycine and proline. A protein that lacks its cofactor (such as the metal ions required for normal activity) is called an apoprotein; an enzyme with a complete complement of cofactors is known as a holoenzyme. An important factor influencing metal-ion coordination in proteins is the energy required to locate an electrical charge inside a medium of low permittivity. To a first approximation, protein molecules may be regarded as oil drops in which the interior has a much lower relative permittivity (about 4) than water (about 78). This difference leads to a strong tendency to preserve electrical neutrality at the metal site, and hence influence the redox chemistry and Brønsted acidity of its ligands. All amino acid residues can use their peptide carbonyl (or amide-N) as a donor group, but it is the side chain that usually provides more selective coordination. By referring to Table 27.2 and from the discussion in Section 4.12, we can recognize donor groups that are either chemically hard or soft and that therefore confer a particular affinity for specific metal ions. Aspartate and glutamate each provide a hard carboxylate group, and may use one or both O atoms as donors (3). The ability of Ca2 to have a high coordination number and its preference for hard donors are such that certain Ca2-binding proteins also contain the unusual amino acids -carboxyglutamate and hydroxyaspartate (generated by post-translational modification), which provide additional functionalities to enhance binding. Histidine, which has an imidazole group with two coordination sites, the ε-N atom (more common) and the -N atom, is an important ligand for Fe, Cu(4), and Zn. Cysteine has a thiol S atom that is expected to be unprotonated (thiolate) when involved in metal coordination. It is a good ligand for Fe, Cu, and Zn (5), as well as for toxic metals such as Cd and Hg. Methionine contains a soft thioether S donor that stabilizes Fe(II)(6) and Cu(I). Tyrosine can be deprotonated to provide a phenolate O donor atom that is a good ligand for Fe(III) (7). Selenocysteine (a specially coded amino acid in which Se replaces S) has also been identified as a ligand, for example it is found as a ligand to Ni in some hydrogenases (Section 27.14). A modified form of lysine, in which the side-chain −NH2 has reacted with a molecule of CO2 to produce a carbamate, is found as a ligand to Mg in the crucial photosynthetic enzyme known as rubisco (Section 27.9) and in other enzymes such as urease, where it is a ligand for Ni(II). The organization of cells The primary and secondary structures of a polypeptide molecule can enforce unusual metal coordination geometries that are rarely encountered in small complexes. Proteininduced strain is an important possibility, for example the protein may impose a coordination geometry on the metal ion that resembles the transition state for the particular process being executed. N– N – N (c) Special ligands 729 N Key point: Metal ions may be bound in proteins by special organic ligands such as porphyrins and pterin-dithiolenes. The porphyrin group (8) was first identified in haemoglobin (Fe) and a similar macrocycle is found in chlorophyll (Mg). There are several classes of this hydrophobic macrocycle, each differing in the nature of the side chains. The corrin ligand (9) has a slightly smaller ring size and coordinates Co in cobalamin (Section 27.11). Rather than show these macrocycles in full, we shall use shorthand symbols such as (10) to show the complexes they form with metals. Almost all Mo and W enzymes have the metal coordinated by a special ligand known as molybdopterin (11). The donors to the metal are a pair of S atoms from a dithiolene group that is covalently attached to a pterin. The phosphate group is often joined to a nucleoside base X, such as guanosine 5-phosphate (GMP), resulting in the formation of a diphosphate bond. Why Mo and W are coordinated by this complex ligand is unknown, but the pterin group could provide a good electron conduit and facilitate redox reactions. 8 Porphyrin2– N N – N N (d) The structures of metal coordination sites 9 Corrin– Key point: The likelihood that a protein will coordinate a particular kind of metal centre can be inferred from the amino acid sequence and ultimately from the gene itself. The structures of metal coordination sites have been determined mainly by X-ray diffraction (now mostly by using a synchrotron, Section 8.1) and sometimes by NMR spectroscopy.3 The basic structure of the protein can be determined even if the resolution is too low to reveal details of the coordination at the metal site. The packing of amino acids in a protein is far denser than is commonly conveyed by simple representations, as may be seen by comparing the representations of the structure of the K channel in Fig. 27.3. Thus, even the substitution of an amino acid that is far from a metal centre may result in significant structural changes to its coordination shell and immediate environment. Of special interest are channels or clefts that allow a substrate selective access to the active site, pathways for long-range electron transfer (metal centres positioned less than 1.5 nm apart), pathways for long-range proton transfer (comprising chains of basic groups such as carboxylates and water molecules in close proximity, usually less than 0.3 nm apart), and tunnels for small gaseous molecules (which can be revealed by placing the crystal under Xe, an electron-rich gas). e F 10 Mo PO3-X S O S O HN O NH E X A M PL E 27. 2 Interpreting the coordination environments of metal ions HN Simple Cu(II) complexes have four to six ligands with trigonal-bipyramidal or tetragonal geometries, whereas simple complexes of Cu(I) have four or fewer ligands, and geometries that range between tetrahedral and linear. Predict how a Cu-binding protein will have evolved so that the Cu can act as an efficient electrontransfer site. Answer Here we are guided by Marcus theory (Section 21.12). An efficient electron-transfer reaction is one that is fast despite having a small driving force. The Marcus equation tells us that an efficient electrontransfer site is one for which the reorganization energy is small. The protein enforces on the Cu atom a coordination sphere that is unable to alter much between Cu(II) and Cu(I) states (see Section 27.8). Self-test 27.2 In certain chlorophyll cofactors the Mg is axially coordinated by a methionine-S ligand. Why is this unusual coordination choice achieved in a protein? 3 The atomic coordinates of proteins and other large biological molecules are stored in a public repository known as the Protein Data Bank located at Each set of coordinates corresponding to a particular structure determination is identified by its ‘pdb code’. A variety of software packages are available to construct and examine protein structures generated from these coordinates. N NH 2 11 Molybdopterin as ligand 730 27 Biological inorganic chemistry (a) (b) Figure 27.3 Illustrations of how protein structures are represented to reveal either (a) secondary structure or (b) the filling of space by nonhydrogen atoms. The example shows the four subunits of the K channel, which is found mainly embedded in the cell membrane. Other physical methods described in Chapter 8 provide less information on the overall structure but are useful for identifying ligands. Thus, EPR spectroscopy is very important for studying d-block metals, especially those engaged in redox chemistry, because at least one oxidation state usually has an unpaired electron. The use of NMR is restricted to proteins smaller than 20–30 kg mol1 because tumbling rates for larger proteins are too slow and 1H resonances are too broad to observe unless shifted away from the normal region (δ ⬇ 110) by a paramagnetic metal centre. Extended X-ray absorption finestructure spectroscopy (EXAFS, Section 8.9) can provide structural information on metal sites in amorphous solid samples, including frozen solutions. Vibrational spectroscopy (Section 8.4) is increasingly being used: IR spectroscopy is particularly useful for ligands such as CO and CN−, and resonance Raman spectroscopy is very helpful when the metal centre has strong electronic transitions, such as occur with Fe porphyrins. Mössbauer spectroscopy (Section 8.7) plays a special role in studies of Fe sites. Perhaps the greatest challenge is presented by Zn2, which has a d10 configuration that provides no useful magnetic or electronic signatures. Metal ion binding sites can often be predicted from a gene sequence. Bioinformatics, the development and use of software to analyse and compare DNA sequences, is a powerful tool because many proteins that bind metal ions or have a metal-containing cofactor occur at cellular levels below that normally detectable directly by analysis and isolation. A particularly common sequence of the human genome encodes the so-called Zn finger domain, thereby identifying proteins that are involved in DNA binding (Section 27.5). Likewise, it can be predicted whether the protein that is encoded is likely to bind Cu, Ca, an Fe-porphyrin, or different types of FeS clusters. The gene can be cloned and the protein for which it encodes can be produced in sufficiently large quantities by ‘overexpression’ in suitable hosts, such as the common gut bacterium Escherichia coli or yeast, to enable it to be characterized. Furthermore, the use of genetic engineering to alter the amino acids in a protein, the technique of site-directed mutagenesis, is a powerful principle in biological inorganic chemistry. This technique often permits identification of the ligands to particular metal ions and the participation of other residues essential to functions such as substrate binding or proton transfer. Although structural and spectroscopic studies give a good idea of the basic coordination environment of a metal centre, it is by no means certain that the same structure is retained in key stages of a catalytic cycle, in which unstable states are formed as intermediates. The most stable state of an enzyme, in which form it is usually isolated, is called the ‘resting state’. Many enzymes are catalytically inactive on isolation and must be subjected to an Transport, transfer, and transcription activation procedure that may involve reinsertion of a metal ion or other cofactor or removal of an inhibitory ligand. Intense efforts have been made to model the active sites of metalloproteins by synthesizing analogues. The models may be divided into two classes: those designed to mimic the structure and spectroscopic properties of the real site, and those synthesized with the intention of mimicking a functional activity, most obviously catalysis. Synthetic models not only illuminate the chemical principles underlying biological activity but also generate new directions for coordination chemistry. As we shall see throughout this chapter, the difficulty is that an enzyme not only imposes some strain on the coordination sphere of a metal atom (even a porphyrin ring is puckered in most cases) but also provides, at fixed distances, functional groups that provide additional coulombic and hydrogen-bonding interactions essential for binding and activating substrates. Indeed, the active site of a metalloenzyme is arguably the ultimate example of supramolecular chemistry. Transport, transfer, and transcription In this section we turn to three related aspects of the function of biological molecules containing metal ions, and see their role in the transport of ions through membranes, the transport and distribution of molecules through organisms, and the transfer of electrons. Metal ions also play an important role in the transcription of genes. 27.3 Sodium and potassium transport Key points: Transport across a membrane is active (energized) or passive (spontaneous); the flow of ions is achieved by proteins known as ion pumps (active) and channels (passive). In Chapter 11 we saw that differentiating between Na and K, two ions that are very similar except for their radii (Resource section 1), is achieved through their selective complexation by special ligands, such as crown ethers and cryptands, with dimensions appropriate for coordination to one particular kind of ion. Organisms use this principle in the molecules known as ionophores, which have hydrophobic exteriors that make them soluble in lipids. The antibiotic valinomycin (Section 11.16) is an ionophore that has a high selectivity for K, which is coordinated by six carbonyl groups. It enables K to pass through a bacterial cell membrane and thereby dissipate the electrical potential difference, so causing the bacterium’s death. At the higher end of the complexity scale are the ion channels, which are large membrane-spanning proteins that allow selective transport of K and Na (as well as Ca2 and Cl) and are responsible for electrical conduction in nervous systems as well as in coupled transport of solutes.4 Figure 27.4 shows the important structural aspects of the potentialgated K channel. Moving from the inside surface of the membrane, the enzyme has a pore (which can open and close on receipt of a signal) leading into a central cavity about 1 nm in diameter; up to this stage K ions can remain hydrated. Polypeptide helices pointing at this cavity have their partial charges directed in such a way as to favour population by cations, resulting in a local K concentration of approximately 2 m. Above the central cavity the tunnel contracts into a selectivity filter consisting of helical ladders of closely spaced peptide carbonyl-O donors that form a sequence of four cubic eightfold coordination sites. During operation of the channel these sites are occupied, at any one time, by a queue of two K ions and two H2O molecules in alternate fashion, as in …K…H2O…K…H2O…. The rate of passage of K ions through the selectivity filter is close to the limit for diffusion control. A plausible mechanism for selective K transport (Fig. 27.5) involves concerted displacement of K ions between adjacent cubic carbonyl-O sites through intermediate, unstable octahedral states in which the K ions are coordinated equatorially by four carbonyl-O donors and axially by the two intervening H2O molecules. This mechanism is not 4 Roderick MacKinnon shared the 2003 Nobel Prize for Chemistry for his elucidation of the structures and mechanisms of ion channels. 731 732 27 Biological inorganic chemistry Hydrated K+ ion Selectivity filter Pore (a) (b) (c) Figure 27.4 (a) Schematic structure of the K channel showing the different components and the transport of K ions: the blue halo represents hydration. (b) View of the enzyme from inside the cell showing the entrance pore that admits hydrated ions. (c) View looking up the selectivity filter showing how mobile dehydrated K ions are coordinated by peptide carbonyl-O atoms provided by each of the four subunits. Note the almost fourfold symmetry axis. K+ K+ Figure 27.5 Mechanism of transport of K ions through the selectivity filter of the K channel. Green spheres represent water molecules. effective for Na because the cavity is too large, which accounts for the 104-fold selectivity of the channel for K over Na. The binding is weak and fast because it is important to convey but not to trap K. Overall, the K channel acts by the mechanism illustrated in Fig. 27.6. Charged groups on the molecule move in response to a change in the membrane potential and cause the intracellular pore to open, so allowing entry of hydrated K ions. Selective binding of dehydrated K ions occurs in the filter region, a potential drop across the membrane is sensed, and the cavity closes. At this point the filter opens up to the external surface, where the K concentration is low and the K ions are hydrated and released. This release causes the protein to switch back to the original conformation and K ions again enter the filter. The Na/K pump (Na/K-ATPase), the enzyme that maintains the concentration differential of Na and K inside and outside a cell, is another example of the high discrimination between alkali metal ions that has evolved with biological ligands. The ions are pumped against their concentration gradients by coupling the process to ATP hydrolysis. The mechanism, which is outlined in Fig. 27.7, involves conformational changes induced by ATP-driven protein phosphorylation. 2K+(cyto) 3Na+(cyto) E1 ATP E1(3Na+) E2(2K+) Exterior Polarized Depolarization of membrane potential K P Mg-ATP H2O Mg-ADP E2P(2K+) E1P(3Na+) E2P K+ Cytoplasm Open Depolarized Closed Figure 27.6 Proposed mechanism of action of the K channel. The potential difference across the membrane is sensed by the protein, which causes the pore to open, allowing hydrated ions to enter the cavity. After shedding their hydration sphere, K ions pass up the selectivity filter at rates close to diffusion control. 2K+(ext) 3Na+(ext) Figure 27.7 General principle of the Na,K-ATPase (the Na pump). Release of two K ions into the cytoplasm is accompanied by binding of ATP (from the cytoplasm) and conversion of the enzyme into state 1, which binds three Na ions from the cytoplasm. A phosphate group (P) is transferred to the enzyme, which opens to the external side, expels three Na ions, then binds two K ions. Release of the phosphate group causes release of K into the cytoplasm and the cycle begins again. Transport, transfer, and transcription 733 E X A M PL E 27. 3 Assessing the role of ions in active and passive transport The toxic species Tl (radius 150 pm) is used as an NMR probe for K binding in proteins. Explain why Tl is suited for this purpose and account for its high toxicity. Answer To address this question we need to recall from Chapter 13 that Tl, in common with other heavy, post-d-block elements, displays the inert-pair effect, a preference for forming compounds in which its oxidation number is 2 less than the group oxidation number. Thallium (Group 13) thus resembles the heavy Group 1 elements (in fact, TlOH is a strong base) and Tl can replace K in complexes, with the advantage that it can be studied by NMR spectroscopy (203Tl and 205Tl have I 21 ). The similarity with K allows Tl, a toxic element, free entry into a cell because it is ‘recognized’ by the Na/K-ATPase. But once inside, more subtle differences in chemical properties, such as the tendency of Tl to form more stable complexes with soft ligands, are manifested and become lethal. Ca2+ Ca2+ Self-test 27.3 Explain why the intravenous fluid used in hospital procedures contains NaCl, not KCl. 2+ Ca 4 Ca2+ 27.4 Calcium signalling proteins Ca2+ Key point: Calcium ions are suitable for signalling because they exhibit fast ligand exchange and a large, flexible coordination geometry. Calcium ions play a crucial role in higher organisms as an intracellular messenger, providing a remarkable demonstration of how organisms have exploited the otherwise rather limited chemistry of this element. Fluxes of Ca2 trigger enzyme action in cells in response to receiving a hormonal or electrical signal from elsewhere in the organism. Calcium is particularly suited for signalling because it has fast ligand-exchange rates, intermediate binding constants, and a large, flexible coordination sphere. Calcium signalling proteins are small proteins that change their conformation depending on the binding of Ca2 at one or more sites; they are thus examples of the metal ionactivated proteins mentioned earlier. Every muscle movement we make is stimulated by Ca2 binding to a protein known as troponin C. The best-studied Ca2-regulatory protein is calmodulin (17 kg mol–1, Fig. 27.8): its roles include activating protein kinases that catalyse phosphorylation of proteins and activating NO-synthase, a Fe-containing enzyme responsible for generating the intercellular signalling molecule nitric oxide. Calmodulin has four Ca2-binding sites (one is shown as 12) with dissociation constants lying close to 10−6. The binding of Ca2 to the four sites alters the protein conformation and it is then recognized by a target enzyme.5 Calcium signalling requires special Ca2 pumps, which are large, membrane-spanning enzymes that pump Ca2 out of the cytoplasm, either out of the cell altogether or into Ca-storing organelles such as the endoplasmic reticulum or the mitochondria. As with Na/K-ATPases, the energy for Ca2 pumping comes from ATP hydrolysis. Hormones or electrical stimuli open specific channels (analogous to K channels) that release Ca2 into the cell. Because the level in the cytoplasm before the pulse is low, the influx easily raises the Ca2 concentration above that needed for Ca2-binding proteins such as calmodulin (or troponin C in muscle). The action can be short-lived, so that after a pulse of Ca2 the cell is quickly evacuated by the calcium pump. Although Ca2 is invisible to most spectroscopic methods, some Ca proteins, such as calmodulin or troponin C, are small enough to be studied by NMR. Because of their preference for large multicarboxylate ligands, the lanthanoid ions (Section 23.7) have been used as probes for Ca binding, exploiting their properties of paramagnetism (as chemical shift reagents in NMR spectroscopy) and fluorescence. Intracellular concentrations of Ca are monitored by using special fluorescent polycarboxylate ligands (13) that are introduced to the cell as their esters, which are hydrophobic and able to cross the membrane lipid barrier. Once in the cell, enzymes known as esterases hydrolyse the esters and release the ligands, which respond to changes in Ca2 concentration in the range 107–109 m. 5 The occupation of a binding site is 50 per cent when the concentration (in mol dm3) of species is equal to the dissociation constant (the reciprocal of the association constant). Figure 27.8 The binding of four Ca2 to apocalmodulin causes a change in the protein conformation, converting it to a form that is recognized by many enzymes. The high proportion of α-helix is typical of proteins that are activated by metal-ion binding. H2O Peptide carbonyl Asp Ca Asp Asp Glu 12 N(C H 2C O2– ) 2 O O N(C H 2C O2– ) 2 O N O C O2– 13 FURA-2 734 27 Biological inorganic chemistry E X A M PL E 27. 4 Explaining why calcium is suitable for signalling Why is Ca2 more suitable than Mg2 for fast signalling processes in cells? Answer To answer this question we need to refer to Section 21.1, in which we saw that the ligand-exchange rates of s-block metal ions increase down each group. The exchange of coordinated H2O molecules is 103104 times faster for Ca2 than for Mg2. The speed at which Ca2 can shed its ligands and bind to a target protein is crucial for ensuring fast signalling. As an example, rapid muscle contractions that may protect an organism against sudden attack are initiated by Ca2 binding to troponin C. Self-test 27.4 The Ca2 pump is activated by calmodulin. Explain the significance of this observation. Hint: Consider how a feedback mechanism could control Ca2 levels in the cytoplasm. 27.5 Zinc in transcription Key points: Zinc fingers are protein structural features produced by coordination of Zn to specific histidine and cysteine residues; a sequence of these fingers enables the protein to recognize and bind to precise sequences of DNA base pairs and plays a crucial role in transferring information from the gene. HO(H) Zn Zn 14 15 The roles of Zn are either catalytic, which we will deal with later, or structural and regulatory. Unlike Ca and Mg, Zn forms more stable complexes with softer donors, so it is not surprising that it is usually found coordinated in proteins through histidine and cysteine residues. Typical catalytic sites (14) commonly have three permanent protein ligands and an exchangeable ligand (H2O), whereas structural Zn sites (15) are coordinated by four ‘permanent’ protein ligands. Transcription factors are proteins that recognize certain regions of DNA and control how the genetic code is interpreted as RNA. It has been known since the 1980s that many DNA-binding proteins contain repeating domains that are folded in place by the binding of Zn and form characteristic folds known as ‘zinc fingers’ (Fig. 27.9). In a typical case, one side of the finger provides two cysteine-S donors and the other side provides two histidine-N donors and folds as an helix. Each ‘finger’ makes recognitory contacts with specific DNA bases. As shown in Fig. 27.10, the zinc fingers wrap around sequences of DNA that they are able to recognize by acting collectively. The high fidelity of transcription factors is the result of a number of such contacts being made along the DNA chain at the beginning of the sequence that is transcribed. The characteristic residue sequence for a Zn-finger motif is –(Tyr,Phe)–X–Cys–X2–4–Cys–X3–Phe–X5–Leu–X2–His–X3–5–His– where the amino acid X is variable. Aside from the ‘classical’ (Cys)2(His)2 zinc finger, others have been discovered that have (Cys)3His or (Cys)4 coordination, together with more elaborate examples having ‘Zn-thiolate clusters’, such as the so-called GAL4 transcription factor in which two Zn atoms are linked by bridging cysteine-S ligands (16). Various protein folds are produced, with faintly jocular names, such as ‘zinc knuckles’, joining an His Cys Zn2+ Zn2+ His Cys Zn Zn Z n2+ Figure 27.9 Zinc fingers are protein folds that form a sequence able to bind to DNA. A typical finger is formed by the coordination of Zn(II) to two pairs of amino acid side chains located either side of the ‘fingertip’. Figure 27.10 A pair of zinc fingers interacting with a section of DNA. Zn 16 Zn2(Cys)6 Transport, transfer, and transcription increasingly large family. Higher order Zn-thiolate clusters are found in proteins known as metallothioneins and some Zn-sensor proteins (see Section 27.16). Zinc is particularly suited for binding to proteins to hold them in a particular conformation: Zn2 is high in the Irving–Williams series (Section 20.1) and thus forms stable complexes, particularly to S and N donors. It is also redox inactive, which is an important factor because it is crucial to avoid oxidative damage to DNA. Other examples of structural zinc include insulin and alcohol dehydrogenase. The lack of good spectroscopic probes for Zn, however, has meant that even though it is tightly bound in a protein, it is difficult to confirm its binding or deduce its coordination geometry in the absence of direct structural information from X-ray diffraction or NMR. However, some elegant measurements have exploited the ability of Co2, which is coloured and paramagnetic, or Cd2, which has useful NMR properties, to substitute for and report on the Zn site. These substitutions depend on strong similarities between the metal ions: like Zn, Co2 readily forms tetrahedral complexes, whereas Cd lies directly below Zn in the periodic table. For many Zn enzymes, it is found that the surrogate metals are as active as Zn itself (Section 27.9). 27.6 Selective transport and storage of iron Key points: The uptake of Fe into organisms involves special ligands known as siderophores; transport in the circulating fluids of higher organisms requires a protein called transferrin; Fe is stored as ferritin. Iron is essential for almost all life forms; however, Fe is also difficult to obtain, yet any excess presents a serious toxic risk. Nature has at least two problems in dealing with this element. The first is the insolubility of Fe(III), which is the stable oxidation state found in most minerals. As the pH increases, hydrolysis, polymerization, and precipitation of hydrated forms of the oxide occur. Polymeric oxide-bridged Fe(III) is the thermodynamic sink of aerobic Fe chemistry (as seen in a Pourbaix diagram, Section 5.14). The insolubility of rust renders the straightforward uptake by a cell very difficult. The second problem is the toxicity of ‘free-Fe’ species, particularly through the generation of OH radicals. To prevent Fe from reacting with oxygen species in an uncontrolled manner, a protective coordination environment is required. Nature has evolved sophisticated chemical systems to execute and regulate all aspects, from the primary acquisition of Fe, to its subsequent transport, storage, and utilization in tissue. The ‘Fe cycle’ as it affects a human is summarized in Fig. 27.11. (a) Siderophores Siderophores are small polydentate ligands that have a very high affinity for Fe(III). They are secreted from many bacterial cells into the external medium, where they sequester Fe to give a soluble complex that re-enters the organism at a specific receptor. Once inside the cell, the Fe is released. Haemorrhage External medium soil, water Siderophores Red blood cells Food plants, bacteria Blood: transferrin Gut Bone marrow Other tissues cytochromes, FeS clusters Blood: transferrin Absorption Gut cells Liver ferritin Blood: transferrin Blood: transferrin Muscle Figure 27.11 The biological Fe cycle showing how Fe is taken up from the external medium and guarded carefully in its travels through organisms. 735 736 27 Biological inorganic chemistry O– HN NH HN O– HN O O O O O O O NH HN O NH O– O O – O N H O O O –O O N –O O NH N N O– O O – O O O– 18 Ferrichrome 17 Enterobactin Aside from citrate (the Fe(III) citrate complex is the simplest Fe transport species in biology) there are two main types of siderophore. The first type is based on phenolate or catecholate ligands, and is exemplified by enterobactin (17), for which the value of the association constant for Fe(III) is 1052, an affinity so great that enterobactin enables bacteria to erode steel bridges. The second type of siderophore is based on hydroxamate ligands and is exemplified by ferrichrome (18), a cyclic hexapeptide consisting of three glycine and three N-hydroxyl-l-ornithines. All Fe(III) siderophore complexes are octahedral and high spin. Because the donor atoms are hard O or N atoms and negatively charged, they have a relatively low affinity for Fe(II). Synthetic siderophores are proving to be very useful agents for the control of ‘iron overload’, a serious condition affecting large populations of the world, particularly South-East Asia (Section 27.17). (b) Iron-transport proteins in higher organisms There are several important, structurally similar Fe-transport proteins known collectively as transferrins. The best characterized examples are serum transferrin (in blood plasma), ovotransferrin (in egg-white), and lactoferrin (in milk). The apoproteins are potent antibacterial agents as they deprive microbes of their iron. Transferrins are also present in tears, serving to cleanse eyes after irritation. All these transferrins are glycoproteins (protein molecules modified by covalently bound carbohydrate) with molar masses of about 80 kg mol1 and containing two separated and essentially equivalent binding sites for Fe. Complexation of Fe(III) at each site involves simultaneous binding of HCO3 or CO32 and release of H: apo-TF Fe(III) HCO3– → TF–Fe(III)–CO32– H Tyr Tyr Fe His Carbonate Asp 19 where TF denotes transferrin. For each site, the association constant under physiological conditions (pH 7) is in the range 1022–1026. However, its value depends strongly on the pH and this dependence is the main factor controlling Fe uptake and release. Transferrin consists of two very similar parts, termed the N-lobe and the C-lobe (Fig. 27.12). The protein is a product of gene duplication because the structure of the first half of the molecule can almost be overlaid on the second half. Each half consists of two domains, 1 and 2, which together form a cleft with a binding site for Fe(III). There is a considerable proportion of helix, resulting in flexibility. Complexation with Fe(III) causes a conformational change consisting of a hinge motion involving domains 1 and 2 at each lobe. Binding of Fe(III) causes the domains to come together. In each active site (19), a single Fe atom is coordinated by widely dispersed amino acid side chains from both domains and the connecting region, hence the change in conformation that occurs. The protein ligands are carboxylate-O (Asp), two phenolate-O (Tyr), and an imidazole-N (His). Only one of the aspartate carboxylate-O atoms is coordinated. The protein ligands form part of a distorted octahedral coordination sphere. The coordination Transport, transfer, and transcription Fe 737 Fe Figure 27.12 Structure of the Fe-transport protein transferrin: the identical halves of the molecule each coordinate to a single Fe(III) atom (the black spheres) between two lobes. This coordination causes a conformational change that allows transferrin to be recognized by the transferrin receptor. is completed by bidentate binding to the exogenous carbonate, which is referred to as a synergistic ligand because Fe binding depends on its presence. In certain cases phosphate is bound instead of carbonate. As expected from the predominantly anionic ligand set, Fe(III) binds much more tightly than Fe(II). However, ions similar to Fe(III), particularly Ga(III) and Al(III), also bind tightly, so that these metals can use the same transport system to gain access to tissues. (c) Release of iron from transferrin Cells in need of Fe produce large amounts of a protein called the transferrin receptor (180 kg mol1), which is incorporated within their plasma membrane. This protein binds Fe-loaded transferrin. The most favoured mechanism for Fe uptake involves the Fe-loaded transferrin receptor complex entering the cell by a process known as endocytosis. In endocytosis, a section of the cell membrane is engulfed by the wall, along with its component membrane-bound proteins, to form a vesicle. The pH within this vesicle is then lowered by a membrane-bound H-pumping enzyme that is also swallowed by the cell. The subsequent release of Fe(III) is probably linked to the coordination of carbonate, which is synergistic in the sense that it is necessary for the binding of Fe but is unstable at low pH. Indeed, from in vitro studies it is known that Fe is released by lowering the pH to about 5 for serum transferrin and to 2–3 for lactoferrin. The vesicle then splits and the TF-receptor complex is returned to the plasma membrane by exocytosis, and Fe(III), probably now complexed by citrate, is released to the cytoplasm. (d) Ferritin, the cellular Fe store Ferritin is the principal store of non-haem Fe in animals (most Fe is occupied in haemoglobin and myoglobin) and, when fully loaded, contains 20 per cent Fe by mass. It occurs in all types of organism, from mammals to prokaryotes. In mammals, it is found particularly in the spleen and in blood. Ferritins have two components, a ‘mineral’ core that contains up to 4500 Fe atoms (mammalian ferritin) and a protein shell. Apoferritin (the protein shell devoid of Fe) can be prepared by treatment of ferritin with reducing agents and an Fe(II) chelating ligand (such as 1,10-phenanthroline or 2,2-bipyridyl). Dialysis then yields the intact shell. Apoferritins have average molar masses in the range 460 to 550 kg mol1. The protein shell (Fig. 27.13) consists of 24 subunits that link together to form a hollow sphere with twofold, threefold (as shown in the illustration), and fourfold symmetry axes. Each subunit consists of a bundle of four long and one short helices, with a loop that forms a section of sheet with a neighbouring subunit. The mineral core is composed of hydrated Fe(III) oxide with varying amounts of phosphate, which helps anchor it to the internal surface. The structure as revealed by X-ray or electron diffraction resembles that of the mineral ferrihydrite, 5Fe2O3.9 H2O, which is based on an hcp array of O2 and OH ions, with Fe(III) layered in both the octahedral and tetrahedral sites (20). The threefold and fourfold symmetry axes of apoferritin are, respectively, hydrophilic and hydrophobic pores. The threefold-axis pores are suited for the passage of ions. However, the Figure 27.13 The structure of ferritin, showing the arrangement of subunits that make up the protein shell. O Fe 20 738 27 Biological inorganic chemistry ferrihydrite core is insoluble and Fe must be mobilized. The most feasible mechanism so far proposed for the reversible incorporation of Fe in ferritin involves its transport in and out as Fe(II), perhaps as the Fe2 ion, which is soluble at neutral pH, but more likely some type of ‘chaperone’ complex. Oxidation to Fe(III) is thought to occur at specific di-iron binding sites known as ferroxidase centres, present in each of the subunits. Oxidation to Fe(III) involves the coordination of O2 and inner-sphere electron transfer: 2 Fe(II) + O2 + 2 H + → 2 Fe(III) + H 2O2 The mechanism by which Fe is released almost certainly involves its reduction back to the more mobile Fe(II). 27.7 Oxygen transport and storage Dioxygen, O2, is a special molecule that has not always been available to biology; in fact, to many life forms it is highly toxic. As the waste product of oxygenic photosynthesis that began with cyanobacteria more than 2 Ga ago, O2 is a biogenic substance, one that owes it existence to solar energy capture by living organisms. As we shall see in Section 27.10, the great thermodynamic advantage of having such a powerful oxidant available undoubtedly led to the evolution of higher organisms that now dominate Earth. Indeed, the requirement for O2 became so important as to necessitate special systems for transporting and storing it. Apart from the difficulty in supplying O2 to buried tissue, there is the problem of achieving a sufficiently high concentration in aqueous environments. This problem is overcome by special metalloproteins known as O2 carriers. In mammals and most other animals and plants, these special proteins (myoglobin and haemoglobin) contain an Fe porphyrin cofactor. Animals such as molluscs and arthropods use a Cu protein called haemocyanin, and some lower invertebrates use an alternative type of Fe protein, haemerythrin, which contains a dinuclear Fe site. (a) Myoglobin Key points: The deoxy form containing high-spin five-coordinate Fe(II) reacts rapidly and reversibly with O2 to produce low-spin six-coordinate Fe(II); a slow autooxidation reaction releases superoxide and produces Fe(III), which is inactive in binding O2. E F Figure 27.14 Structure of myoglobin, showing the Fe porphyrin group located between helices E and F. Myoglobin6 is an Fe protein (17 kg mol1, Fig. 27.14) that coordinates O2 reversibly and controls its concentration in tissue. The molecule contains several regions of helix, implying mobility, with the single Fe porphyrin group located in a cleft between helices E and F. Two propionate substituents on the porphyrin interact with solvent H2O molecules on the surface of the protein. The fifth ligand to the Fe is provided by a histidine-N from helix F, and the sixth position is the site at which O2 is coordinated. In common terminology, the side of the haem plane at which exchangeable ligands are bound is known as the distal region, while that below the haem plane is known as the proximal region. The histidine on helix F is one of two that are present in all species. Such ‘highly conserved’ amino acids are a strong indication that evolution has determined that they are essential for function. The other conserved histidine is located on helix E. Deoxymyoglobin (Mb) is bluish red and contains Fe(II); this is the oxidation state that binds O2 to give the familiar bright red oxymyoglobin (oxyMb). In some instances deoxymyoglobin becomes oxidized to Fe(III), which is called metmyoglobin (metMb) and is unable to bind O2. This oxidation may occur by a ligand substitution-induced redox reaction in which Cl ions displace bound O2 as superoxide: Fe(II)O2 + Cl − → Fe(III)Cl + O2− In healthy tissue, an enzyme (methaemoglobin reductase) is available to reduce the met form back to the Fe(II) form. 6 Myoglobin was the first protein for which the three-dimensional structure was determined by X-ray diffraction. For this achievement, John Kendrew shared the 1962 Nobel Prize for Chemistry with Max Perutz, who solved the structure of haemoglobin. 739 Transport, transfer, and transcription The Fe in deoxymyoglobin is five-coordinate, high-spin, and lies above the plane of the ring. When O2 binds it is coordinated end-on to the Fe atom, the electronic structure of which is tuned by the F helix histidine ligand (Fig. 27.15). The unbound end of the O2 molecule is fastened by a hydrogen bond to the imidazole-NH of the histidine in helix E. The coordination of O2 (a strong-field π-acceptor ligand) causes the Fe(II) to switch from highspin (equivalent to t2g4 eg2) to low-spin (t2g6) and, with no d electrons in antibonding orbitals, to shrink slightly and move into the plane of the ring. The bonding is often expressed in terms of Fe(II) coordination by singlet O2, in which the doubly occupied antibonding 2πg orbital of O2 acts as a donor and the empty 2πg orbital of O2 accepts an electron pair from the Fe (Fig. 27.16). An alternative description is often considered, in which the bonding is expressed in terms of low-spin Fe(III) coordinated by superoxide, O2−. With this model, the formation of metmyoglobin by reaction with anions is a simple ligand displacement. His His Fe O2 O2 –O2 Fe (b) Haemoglobin O2(πg) (a) 100 Mb 80 60 40 20 0 10 20 15 0 5 Partial pressure of O2, p/kPa Fe3dzx 2 (b) Figure 27.16 The orbitals used to form the FeO2 adduct of myoglobin and haemoglobin. This model considers the O2 ligand to be in a singlet state, in which the full 2πg orbital donates an electron pair and the other 2πg orbital acts as a p-electron pair acceptor. Hb at pH = 6.8 Partial pressure in lungs (13 kPa) O2(πg) Fe3dz Hb at pH = 7.6 Percentage saturation/% Haemoglobin (Hb, 68 kg mol–1, Fig. 27.17) is the O2 transport protein found in special cells known as erythrocytes (red blood cells): a litre of human blood contains about 150 g of Hb. Simplistically, Hb can be thought of as a tetramer of myoglobin-like units with a cavity in the middle. There are in fact two types of Mb-like subunits, which differ slightly in their structures, and Hb is referred to as an a2b2 tetramer. The O2 binding curves for Mb and Hb are shown in Fig. 27.18: it is highly significant that the curve for Hb is sigmoidal, which indicates that uptake and release of successive O2 molecules is cooperative. At low O2 partial pressure and greater acidity (as in venous blood and muscle tissue following aggressive exercise) Hb has low affinity for O2. This low affinity enables Hb to transfer its O2 to Mb. As the pressure increases, so does the affinity of Hb for O2 and as a result Hb can pick up O2 in the lungs. This change in affinity can be attributed to there being two conformations. The tensed state (T) has a low affinity and the relaxed state (R) has a high affinity. Deoxy-Hb is T and fully loaded oxy-Hb is R. To understand the molecular basis of cooperativity we need to refer to Fig. 27.15. Binding of the first O2 molecule to the T-state molecule is weak, but the decrease in the size of the Fe allows it to move into the plane of the porphyrin ring. This motion is particularly important for Hb because it pulls on the proximal histidine ligand and helix F moves. This movement is transmitted to the other O2 binding sites, the effect being to move the other Fe atoms closer to their respective ring planes and thereby convert the protein into the R state. The way is thereby opened for them to bind O2, which they now do with greater ease, although the statistical probability decreases as saturation is approached. Figure 27.15 Reversible binding of O2 to myoglobin: coordination by O2 causes the Fe to become low spin and move into the plane of the porphyrin ring. Partial pressure in muscle (5 kPa) Key point: Haemoglobin consists of a tetramer of myoglobin-like subunits, with four Fe sites that bind O2 cooperatively. Figure 27.17 Haemoglobin is an 22 tetramer. Its and subunits are very similar to myoglobin. Figure 27.18 Oxygen binding curves for myoglobin and haemoglobin showing how cooperativity between the four sites in haemoglobin gives rise to a sigmoidal curve. The binding of the first O2 molecule to haemoglobin is unfavourable, but it results in a greatly enhanced affinity for subsequent O2 molecules. 740 27 Biological inorganic chemistry (c) Other oxygen transport systems Key point: Arthropods and molluscs use haemocyanin and certain marine worms use haemerythrin. u C –O u C O2 2 O O Figure 27.19 Binding of O2 at the active site of haemocyanin causes the two Cu atoms to be brought closer together. The O2 complex is regarded as a binuclear Cu(II) centre in which the two Cu atoms are bridged by an 2,2-peroxide. In many organisms, such as arthropods and molluscs, O2 is transported by the Cu protein haemocyanin, which, unlike haemoglobin, is extracellular, as is common for Cu proteins. Haemocyanin is oligomeric, with each monomer containing a pair of Cu atoms in close proximity. Deoxyhaemocyanin (Cu(I)) is colourless but it becomes bright blue when O2 binds. The active site is shown in Fig. 27.19. In the deoxy state, each Cu atom is three-coordinate and bound in a pyramidal array by three histidine residues. The two Cu atoms are so far apart (460 pm) that there is no direct interaction between them. The low coordination number is typical of Cu(I), which is normally two- to four-coordinate. Rapid and reversible coordination of O2 occurs between the two Cu atoms in a bridging dihapto manner (μ-22) and the low vibrational wavenumber of the coordinated O2 molecule (750 cm1) shows it has been reduced to peroxide O22, with an accompanying lowering of the bond order from 2 to 1. To accommodate the binding of O2, the protein adjusts its conformation to bring the two Cu atoms closer together. The Cu sites become five-coordinate, which is typical of Cu(II). Haemerythrin is an example of a special class of dinuclear Fe centres that are found in a number of proteins with diverse functions, such as methane monooxygenase, and some ribonucleotide reductases and acid phosphatases. The two Fe atoms in the active site of haemerythrin (21) are each coordinated by amino acid side chains but are also linked by two bridging carboxylate groups and a small ligand. In the reduced form, which binds O2 reversibly, this small ligand is an OH ion. Coordination of O2 occurs at only one of the Fe atoms and the distal O atom forms a hydrogen bond to the H atom of the bridging hydroxide. (d) Reversible O2 binding by small-molecule analogues Key point: Proteins binding O2 reversibly do so by preventing its reduction and eventual OO bond cleavage. This protection is difficult to achieve with small molecules. Certain elaborate macrocyclic Fe(II) complexes exhibit reversible O2 binding by providing steric hindrance to attack on the coordinated O2. Much effort has been spent on synthesizing simple complexes that coordinate O2 reversibly and could be used as blood substitutes in special circumstances, such as emergency surgery. The problem is that although O2 reacts with d-block metal ions to form complexes in which the O−O bond is retained (as in superoxo and peroxo species), these products tend to undergo irreversible decomposition involving rapid O−O bond cleavage and formation of water or oxides. Overcoming this problem requires complexes designed to protect the coordinated O−O ligand, preventing it from reacting further. Sterically hindered Fe(II) complexes such as the ‘basket’ porphyrin (22) achieve this protection by preventing a second Fe(II) complex from attacking the distal O atom of the superoxo species to form a bridged peroxo intermediate. As we shall see in Section 27.10, peroxo complexes of Fe(III) tend to undergo rapid O−O bond proteolysis, resulting in formation of H2O and Fe(IV) O. O (CH2 )n (CH2 )n O O O N tBu O2 N OH Fe Fe N Fe N N N H 21 22 O t O Bu Transport, transfer, and transcription Simple Cu complexes that can coordinate O2 reversibly are also rare, but studies have revealed interesting chemistry that is particularly relevant for developing catalysts for oxygenation reactions. Analogues of the dinuclear Cu(I) centre of haemocyanin react with O2 but have a strong tendency to undergo further reactions that involve cleavage of the OO bond, an example being the rapid equilibrium between µ-η2:η2 peroxo-dicopper(II) and bis(µ-oxo)copper(III) complexes shown in Fig. 27.20. Cu(III) Cu(III) Cu O Cu O 2+ N E X A M PL E 27. 5 Identifying how biology compensates for strong competition by CO N Carbon monoxide is well known to be a strong inhibitor of O2 binding by myoglobin and haemoglobin, yet relative to O2 its binding is much weaker in the protein compared to a simple Fe-porphyrin complex. This suppression of CO binding is important as even trace levels of CO would otherwise have serious consequences for aerobes. Suggest an explanation. Answer We need to consider how CO and O2, both of which are π-acceptor ligands, differ in terms of the orbitals they use for bonding to a metal atom. The binding of O2 is nonlinear (see Figs 27.15 and 27.16) and the distal O atom is well positioned to form a hydrogen bond to the distal imidazole. By contrast, CO adopts a linear FeCO arrangement and does not participate in the additional bonding. 741 Cu N O Cu O N Cu(II) Cu(II) N N Figure 27.20 Rapid equilibrium between µ-η2η2 peroxo-dicopper(II) and bis (µ-oxo) copper(III) in a model complex for the active site of haemocyanin. Self-test 27.5 Suggest a reaction sequence accounting for why simple Fe-porphyrin complexes are unable to bind O2 reversibly, but give products that include oxo-bridged dinuclear Fe(III) porphyrin species. 27.8 Electron transfer In all but a few interesting cases, the energy for life stems ultimately from the Sun, either directly in photosynthesis or indirectly by acquiring energy-rich compounds (fuel) from photosynthesizing organisms. Energy can be acquired as a flow of electrons from fuel to oxidant. Important fuels include fats, sugars, and H2, and important biological oxidants include O2, nitrate, and even H. As estimated from Fig. 27.21, oxidation of sugars by O2 provides a lot of energy (over 4 eV per O2 molecule), and is the reason for the success of aerobic organisms over the anaerobic ones that once dominated the Earth. (a) General considerations Key points: Electron flow along electron-transport chains is coupled to chemical processes such as ion (particularly H) transfer; the simplest electron-transfer centres have evolved to optimize fast electron transfer. 7 An overall current of about 80 A flows through the mitochondrial respiratory chains in an average human. 1 Mn(IV) O2/H2O Fe(IV) NO3–/NO2– Cytochrome c Cu 0 CytoFe–S clusters chromes –0.5 Electron flow 0.5 E/V at pH = 7 In organisms, electrons are abstracted from food (fuel) and flow to an oxidant, down the potential gradient formed by the sequence of acceptors and donors known as a respiratory chain (Fig. 27.22).7 Apart from flavins and quinones, which are redox-active organic cofactors, these acceptors and donors are metal-containing electron transfer (ET) centres, which fall into three main classes, namely FeS clusters, cytochromes, and Cu sites. These enzymes are generally bound in a membrane, across which the energy from ET is used to sustain a transmembrane proton gradient: this is the basis of the chemiosmotic theory. The counterflow of H, through a rotating enzyme known as ATP synthase, drives the phosphorylation of ADP to ATP. Many membrane-bound redox enzymes are electrogenic proton pumps, which means they directly couple long-range ET to proton transfer through specific internal channels. We shall examine the properties of the three main types of ET centre. The same rules concerning outer-sphere electron-transfer that were discussed in Section 21.12 apply to metal centres in proteins, and we should note that organisms have optimized the structures and properties of these centres to achieve efficient long-range electron transfer. In the following discussion it will be useful to keep in mind that reduction potentials depend on several factors (Section 5.10). Besides ionization energy and ligand environment (strong donors stabilize high oxidation states and lower the reduction potential; weak donors, π acceptors, and protons stabilize low oxidation states and raise the reduction potential), an active site in a protein is also influenced by the relative permittivity (which NAD+/NADH H+/H2 Sugars –1 Figure 27.21 The ‘redox spectrum’ of life. 742 27 Biological inorganic chemistry A NADH dehydrogenase 8 Fe–S Flavin Quinone NADH NAD B Succinate dehydrogenase C Cytochrome c ubiquinone oxidoreductase 3 Fe–S Fe–S Flavin 2 Cyt b 2 Cyt b Cyt c Quinone Quinone Fumarate Succinate + E ATP synthase 2 Cyt a Cu, Mg, Zn H2 O MgATP MgADP H+ E O2 B Inside D Cytochrome c oxidase C D A Q Q Outside (intermembrane space) Cytochrome c H+ e– H+ H+ Figure 27.22 The mitochondrial respiratory electron transfer (ET) chain consists of several metalloenzyme molecules that use the energy of electron transport to transport protons across a membrane. The proton gradient is used to drive ATP synthesis. stabilizes centres with low overall charge), the presence of neighbouring charges, including those provided by other bound metal ions, and the availability of hydrogen-bonding interactions that will also stabilize reduced states. When we consider the kinetics of electron transfer, it will similarly be useful to keep in mind that ‘efficiency’ means that electron transfer is fast even when the reaction Gibbs energy is low and therefore that the reorganization energy of Marcus theory (Section 21.12) is low. This requirement is met by providing a ligand environment that does not alter significantly when an electron is added and by burying the site so that water molecules are excluded. Intersite distances are generally less than 1.4 nm in order to facilitate electron tunnelling, although it is still debated whether electron transfer in proteins depends mainly on distance alone or whether the protein can provide special pathways. (b) Cytochromes Met Fe His 23 Fe eo n la P f riy hn o p rin g Figure 27.23 Overlap between the t2g orbitals of the Fe and low-lying empty π orbitals on the porphyrin effectively extends the Fe orbitals out to the periphery of the ring. Key points: Cytochromes operate in the potential region 0.3 to 0.4 V; they have a combination of low reorganization energy and extended electron coupling through delocalized orbitals. Cytochromes were identified many years ago as cell pigments (hence the name). They contain an Fe porphyrin group and the term ‘cytochrome’ can refer to both an individual protein and a subunit of a larger enzyme that contains the cofactor. Cytochromes use the Fe3/Fe2 couple and are generally six-coordinate (23), with two stable axial bonds to amino acid donors, and the Fe is usually low spin in both oxidation states. This contrasts with ligand-binding Fe-porphyrin proteins such as haemoglobin, for which the sixth coordination site is either empty or occupied by an H2O molecule. A good way to consider the capability of cytochromes for fast electron transfer is to treat the d orbitals of Fe(III) and Fe(II) in terms of an octahedral ligand field and to consider the overlap between the electron-rich but nearly nonbonding t2g orbitals (the configurations are t2g5 and t2g6 in Fe(III) and Fe(II), respectively) and the orbitals of the porphyrin. The electron enters or leaves an orbital having π overlap with the π antibonding molecular orbital on the ring system. This arrangement provides enhanced electron transfer because the d orbitals of the Fe atom are effectively extended out to the edge of the porphyrin ring, so decreasing the distance over which an electron must transfer between redox partners (Fig. 27.23). The paradigm of cytochromes is mitochondrial cytochrome c (12 kg mol1, Fig. 27.24). This cytochrome is found in the mitochondrial intramembrane space, where it supplies electrons to cytochrome c oxidase, the enzyme responsible for reducing O2 to H2O at the end of the energy-transducing respiratory chain (Section 27.10). The fifth and sixth ligands to Fe in cytochrome c are histidine (imidazole-N) and methionine (thioether-S, 23). Methionine is not a common ligand in metalloproteins but because it is a neutral, soft Transport, transfer, and transcription ) (a ) (b Figure 27.24 Different views (but from the same viewpoint) of mitochondrial cytochrome c. (a) The secondary structure and the position of the haem cofactor. (b) The surface charge distribution that guides the docking with its natural redox partners (red and blue areas represent patches of negative and positive charge, respectively). donor it is expected to stabilize Fe(II) rather than Fe(III). The reduction potential of cytochrome c is 0.26 V, at the higher end of values for cytochromes in general. Cytochromes vary in the identity of the axial ligands as well as the structure of the porphyrin ligand (the notation a, b, c, d,… defines positions of absorption maxima in the visible region, but also refers to variations in the substituents on the porphyrin ring). Many cytochromes, in particular those sandwiched between membrane-spanning helices, have bis(histidine) axial ligation (24). In cytochrome c, the edge of the porphyrin ring is exposed to solvent and is the most likely site for electrons to enter or leave. Specific protein–protein interactions are important for obtaining efficient electron transfer and the region around the exposed edge of the porphyrin ring in cytochrome c provides a pattern of charges that are recognized by cytochrome c oxidase and other redox partners. One example in particular has been well studied, that of cytochrome c with yeast cytochrome c peroxidase. The driving force for electron transfer from either of the two catalytic intermediates of peroxidase (Section 27.10) to each reduced cytochrome c is approximately 0.5 V. Figure 27.25 shows the structure of a bimolecular complex formed between cytochrome and cytochrome c peroxidase. Electrostatic interactions guide the two proteins together within a distance that is favourable for fast electron tunnelling between cytochrome c and two redox centres on the peroxidase, the haem cofactor and tryptophan-191 (Section 27.10). Cytochrome c Electron transfer pathway Tryptophan Cytochrome c peroxidase Figure 27.25 The bimolecular ET complex between cytochrome c and cytochrome c peroxidase produced by co-crystallization of cytochrome c with the Zn derivative of cytochrome c peroxidase. The orientation suggests an electron transfer pathway between the haem groups of cytochrome c and cytochrome c peroxidase that includes tryptophan. Fe His His 24 743 744 27 Biological inorganic chemistry (c) Iron–sulfur clusters Key points: Iron–sulfur clusters generally operate at more negative potentials than cytochromes; they are composed of high-spin Fe(III) or Fe(II) with sulfur ligands in a mainly tetrahedral environment. Figure 27.26 A series of three FeS clusters provides a long-range electron transfer pathway to the buried active site in hydrogenases. 25 [2Fe–2S] Iron–sulfur clusters are widespread in biology, although their importance was not established as early as cytochromes on account of their lack of distinctive optical characteristics. By convention, FeS clusters are represented by square brackets showing how many Fe and non-protein S atoms are present, as in [2Fe–2S] (25), [4Fe–4S] (26), and [3Fe–4S] (27). The efficacy of FeS clusters as fast ET centres is largely due to their being able to delocalize the added electron to varying degrees, which minimizes bond length changes and decreases the reorganization energy. The presence of sulfur ligands to provide good lead-in groups is also important. Small ET proteins containing FeS clusters are known as ferredoxins, whereas in many large enzymes, FeS clusters are arranged in a relay, less than 1.5 nm apart, to link remote redox sites in the same molecule. The relay concept is illustrated in Fig. 27.26 with a class of enzymes known as hydrogenases, which we discuss further in Section 27.14. In nearly all cases, the Fe atoms are tetrahedrally coordinated by cysteine thiolate (RS) groups as the protein ligands. The overall assembly, including the protein ligands, is known as an ‘FeS centre’. Examples are known in which one or more of the Fe atoms is coordinated by non-thiolate amino acid ligands, such as carboxylate, imidazole, and alkoxyl (serine), or by an exogenous ligand such as H2O or OH, and the coordination number about the Fe subsite may be increased to six. The cubane [4Fe4S] (26) and cuboidal [3Fe4S] (27) clusters are obviously closely related, and may even interconvert within a protein by the addition or removal of Fe from one subsite. Larger clusters also occur, such as the ‘super clusters’ [8Fe7S] and [Mo–7Fe–8S–X] found in nitrogenase (Section 27.13). Despite the presence of more than one Fe atom, FeS clusters generally carry out single electron transfers and are good examples of mixed-valence systems comprising Fe(III) and Fe(II). The redox state of a cluster is commonly represented by summing the charges due to Fe (3 or 2, respectively) and S atoms (2) and the resultant overall charge, which is referred to as the oxidation level, is written as a superscript. [2Fe2S]2 e → [2Fe2S] 2 Fe(III) S 0 {Fe(III):Fe(II)} S= E O 0.1 to 0.4 V [4Fe4S]2 e → [4Fe4S] {2Fe(III):2Fe(II)} {Fe(III):3Fe(II)} S 0 1 E O 0.2 to 0.7 V 2 S= S 26 [4Fe–4S] Fe S 27 [3Fe–4S] 1 2 [3Fe4S] e → [3Fe4S] 3Fe(III) {2Fe(III):Fe(II)} S 2 1 S= Fe E O 0 to 0.4 V 2 Most FeS centres have negative reduction potentials (usually more negative than −0.2 V) so the reduced forms are good reducing agents: exceptions are [4Fe4S] clusters that operate instead between the 3 and 2 oxidation states (these are called ‘HiPIP’ centres because they were originally discovered in a protein called high-potential iron protein for which the reduction potential is 0.35 V) and so-called Rieske centres, which are [2Fe2S] clusters having one Fe subsite coordinated by two neutral imidazole ligands rather than cysteine (28). Coordination by histidine stabilizes the iron as Fe(II) and usually raises the reduction potential to a much more positive value (above 0.2 V). The half-reactions have been written to include the spin states of FeS clusters: individual Fe atoms are high spin, as expected for tetrahedral coordination by S2, and different magnetic states arise from ferromagnetic and antiferromagnetic coupling (Section 20.8). These magnetic properties are very important, as they allow the centres to be investigated by EPR (Section 8.6). A major question is how FeS centres are synthesized and inserted into the protein. This process has been studied mostly in prokaryotes, from which it is known that specific proteins are involved in the supply and transport of Fe and S atoms, their assembly into clusters, and their transfer to target proteins. Free sulfide (H2S, HS, or S2) in a cell is highly poisonous, so it is produced only when required by an enzyme called cysteine desulfurase, which breaks down cysteine to yield S2 ions and alanine. Catalytic processes 745 (d) Copper electron-transfer centres Key point: The protein overcomes the large inherent difference in preferred geometries for Cu(II) and Cu(I) by constraining Cu in a coordination environment that does not change upon electron transfer. The so-called ‘blue’ Cu centre is the active site of a number of small electron-transfer proteins as well as larger enzymes (the blue Cu oxidases) that contain, in addition, other Cu sites. Blue Cu centres have reduction potentials for the Cu(II)/Cu(I) redox couple that lie in the range 0.150.8 V and so they are generally more oxidizing than cytochromes. The name stems from the intense blue colour of pure samples in the oxidized state, which arises from ligand(thiolate)-to-metal charge transfer. In all cases, the Cu is shielded from solvent water and coordinated by a minimum of two imidazole-N and one cysteine-S in a nearly trigonal planar manner, with one or two longer bonds to axial ligands. The most studied examples are plastocyanin (Fig. 27.27), a small electron carrier protein in chloroplasts, and azurin, a bacterial electron carrier. These small proteins have a β-barrel structure, which holds the Cu coordination sphere in a rigid geometry. Indeed, the crystal structures of oxidized, reduced (29, numbers refer to bond distances in pm), and apo forms reveal that the ligands remain in essentially the same position in all cases. As a result, the blue Cu centre is well suited to undergo fast and efficient electron transfer because the reorganization energy is small. The dinuclear Cu centre known as CuA is present in cytochrome c oxidase and N2O reductase. The two Cu atoms (30) are each coordinated by two imidazole groups and a pair of cysteine thiolate ligands act as bridging ligands. In the reduced form, both Cu atoms are Cu(I). This form undergoes one-electron oxidation to give a purple, paramagnetic species in which the unpaired electron is shared between the two Cu atoms. Once again, we see how delocalization assists electron transfer because the reorganization energy is lowered. Fe S 28 Rieske [2Fe–2S] centre u C E X A M PL E 27.6 Explaining the function of ET centres The reduction potential of Rieske FeS centres is very pH dependent, unlike the standard FeS centres that have only thiolate ligation. Suggest an explanation. Answer We need to refer back to Sections 5.6 and 5.14 to see how protonation equilibria influence reduction potentials. Each of the two imidazole ligands that coordinate one of the Fe atoms in the Rieske [2Fe2S] cluster is electrically neutral at pH 7 and the proton located on the noncoordinating N atom is easily removed. The pKa depends on the oxidation level of the cluster, and there is a large region of pH in which the imidazole ligands are protonated in the reduced form but not in the oxidized form. As a result, the reduction potential depends on pH. Figure 27.27 The plastocyanin molecule. Self-test 27.6 Simple Cu(II) compounds show a large EPR hyperfine coupling to the Cu nucleus (I 32 for 65 Cu and 63Cu), whereas the EPR spectra of blue Cu proteins show a much smaller hyperfine coupling. What does this suggest about the nature of the ligand coordination at blue Cu centres? His Catalytic processes The classic role of enzymes is as highly selective catalysts for the myriad chemical reactions that take place in organisms and sustain the activities of life. In this section we view some of the most important examples in terms of the suitability of certain elements for their roles. HN Cys HN N HN Cu 210 190 Cu 210 S N 280 S 29a Oxidized plastocyanin N HN Cu 230 210 Cu 210 S N 290 S 29b Reduced plastocyanin 30 746 27 Biological inorganic chemistry 27.9 Acid–base catalysis Biological systems rarely have the extreme pH conditions under which catalysis by free H or OH can occur; indeed, the result would be indiscriminate because all hydrolysable bonds would be targets. One way that organisms have solved this problem has been to harness properties of certain metal ions and build them into protein structures designed to accomplish specific (Brønsted) acidbase reactions (Section 4.1). Organisms make extensive use of Zn for achieving acidbase catalysis, but not to the exclusion of other metals. For example, in addition to the numerous enzymes that feature Fe(II) and Fe(III), Mg(II) serves as the catalyst in pyruvate kinase (phosphate ester hydrolysis) and ribulose bisphosphate carboxylase (CO2 incorporation into organic molecules), Mn is the catalyst in arginase (for the hydrolysis of arginine, yielding urea and l-ornithine), and Ni(II) is the active metal in urease (for the hydrolysis of urea, yielding ammonia and, ultimately, carbon dioxide). Because of their importance to industry and medicine, many of these enzymes have been studied in great detail and model systems have been synthesized in efforts to reproduce catalytic properties and understand the mode of action of inhibitors. Many of these sites (including arginase-[Mn,Mn] and urease-[Ni,Ni]) contain two or more metal ions in an arrangement unique to the protein and difficult to model with simple ligands. (a) Zinc enzymes Key point: Zinc is well suited for catalysing acid–base reactions as it is abundant, redox inactive, forms strong bonds to donor groups of amino acid residues, and exogenous ligands such as H2O are exchanged rapidly. A Zn2 ion has high rates of ligand exchange and its polarizing power means that the pKa of a coordinated H2O molecule is quite low. Combined with strong binding to protein ligands, rapid ligand exchange (coordinated H2O or substrate molecules), a reasonably high electron affinity, flexibility of coordination geometry, and no complicating redox chemistry, Zn is well suited to its role in catalysing specific acid–base reactions. The large family of Zn enzymes include carbonic anhydrase, carboxypeptidases, alkaline phosphatase, -lactamase (responsible for penicillin resistance in bacteria), and alcohol dehydrogenase. Typically, the Zn is coordinated by three amino acid ligands (in contrast to zinc fingers, which have four) and one exchangeable H2O molecule (14). The mechanisms of Zn enzymes are normally discussed in terms of two limiting cases. In the Zn-hydroxide mechanism, the Zn functions by promoting deprotonation of a bound water molecule, so creating an optimally positioned OH nucleophile that can go on to attack the carbonyl C atom: HN O HO Zn In the Zn-carbonyl mechanism, the Zn ion acts directly as a Lewis acid to accept an electron pair from the carbonyl O atom, and its role is therefore analogous to H in acid catalysis: Nu O Zn Similar reactions occur with other X O groups, particularly the P O of phosphate esters. There is an obvious advantage of achieving such catalysis at Zn or some other acid species that is anchored in a stereoselective environment. The formation and transport of CO2 is a fundamental process in biology. The solubility of CO2 in water depends on its hydration and deprotonation to form HCO3. However, the uncatalysed reaction at pH 7 is very slow, the forward process occurring with a rate constant of less than 103 s1. Because turnover of CO2 by biological systems is very high, such a rate is far too slow to sustain vigorous aerobic life in a complex organism. In photosynthesis, only CO2 can be used by the enzyme known as ‘rubisco’ (Section 27.9(b)) so rapid dehydration of HCO3 is essential and is one of the first steps in the production of biomass. The CO2/ HCO3 equilibrium is also important (in addition to its role in CO2 transport) because it provides a way of regulating tissue pH. Catalytic processes In 1932 an enzyme, carbonic anhydrase (CA, or carbon dioxide dehydratase) was identified that catalyses this reaction with a remarkable rate enhancement, and it was found to contain one Zn atom per molecule. It is now known that there are several forms of CA, all of which are monomers with molar mass close to 30 kg mol1. The beststudied enzyme is CA II from red blood cells, which has a turnover frequency for CO2 hydration of about 106 s–1, making it one of the most active of all enzymes. The crystal structure of human CA II shows that the Zn atom is located in a conical cavity about 1.6 nm deep, which is lined with several histidine residues. The Zn is coordinated by three His-N ligands and one H2O molecule in a tetrahedral arrangement (Fig. 27.28). The neutral N-ligands lower the pKa of the bound H2O by about 3 units (compared to the aqua ion), so creating a high local concentration of OH– as attacking nucleophile. Other groups in the active site pocket, including noncoordinating histidines and ordered water molecules, are important for mediating proton transfer (which is the rate-limiting factor) and for binding the CO2 substrate (which does not coordinate to Zn). Carbonic anhydrases show some variation in their Zn coordination environment, with some CAs from higher plants having two cysteines and one histidine. The mechanism of action of CA (Fig. 27.29) is best described in terms of a Zn-hydroxide mechanism. Proton transfers are very fast, aided by a hydrogen-bonding network that extends from the protein surface to the active site. The key feature is the acidity of the H2O molecule coordinated to Zn, as the coordinated HO– ion that is produced after deprotonation is sufficiently nucleophilic to attack a nearby CO2 molecule bound noncovalently. This attack results in a coordinated HCO3 ion, which is then released. Small analogues of CA that have been studied, such as (31), reproduce the substrate binding and acid–base properties of the enzyme, but their turnover frequencies are orders of magnitude lower. Carboxypeptidase (CPD, 34.6 kg mol–1) is an exopeptidase, an enzyme that catalyses the hydrolysis of C-terminal amino acids containing an aromatic or bulky aliphatic side chain. There are two types of Zn-containing enzyme, and both are synthesized as inactive precursors in the pancreas for secretion into the digestive tract. The better studied is CPD A, which acts on terminal aromatic residues; whereas CPD B acts on basic residues. The X-ray structure of CPD shows that the Zn is located to one side of a groove in which the substrate is bound. It is coordinated by two histidine-N ligands, one glutamate-CO2 (bidentate), and one weakly bound H2O molecule (Fig. 27.30). Structures of the enzyme Thr199 Thr199 CO2 Glu106 O O HO C H C N HN H N H H O C H H 2O N HN H H O Glu106 Glu106 HO CH O HN N H His64 H O H H O Thr199 Thr199 C C Zn BaseH + Base O N H O O– H His64 Zn His64 HO C H O O O O– H H Glu106 O H O HO C H O O NH C H HN N H H H O His64 Zn O Zn N H O C O– HOCO2– 2 H2O Thr199 Glu106 HO CH O O C HN N His64 H H O H H O NH H O H Zn Figure 27.29 The mechanism of action of carbonic anhydrase indicating the importance of proton transfer in this very fast reaction. 747 His Thr H2O H 2O Glu Zn His Figure 27.28 The active site of carbonic anhydrase. H 2O N Zn N 31 N 748 27 Biological inorganic chemistry 8 4 2 ry T H O H O O H O O – N H H N H N O H2 N g rA5 4 1 – + H N O H N 2 O O Zn O N N H H N siH4 9 Figure 27.30 Structure of the active site of carboxypeptidase with a peptide inhibitor (red) bound to it. obtained in the presence of glycyl inhibitors show that the H2O molecule has moved away from the Zn atom, which has become coordinated instead to the carbonyl-O of the glycine, suggesting a Zn-carbonyl mechanism. The guanidinium group of a nearby arginine binds the terminal carboxyl group, while the tyrosine provides aromatic/hydrophobic recognition. Alkaline phosphatase (AP) introduces us to catalytic Zn centres that contain more than one metal atom. The enzyme catalyses the hydrolysis of phosphate monoesters: ROH + HOPO32− R − OPO32− + H 2O → Mg Zn Zn Phosphate 32 It occurs in tissues as diverse as the intestine and bone, where it is found in the membranes of osteoblasts, cells that form the sites of nucleation of hydroxyapatite crystals. AP catalyses the general breakdown of organic phosphates, including ATP, to provide the phosphate required for bone growth. As its name implies, its optimum pH is in the mild alkali region. The active site of AP contains two Zn atoms located only about 0.4 nm apart, with a Mg ion nearby. The crystal structure of the enzyme–phosphate complex (32) reveals that the phosphate ion (the product of the normal reaction) bridges the two Zn atoms. Alcohol dehydrogenase (ADH) is discussed here even though it is classed as a redox enzyme because the role of Zn once again is as a Lewis acid. The reaction catalysed is the reduction of NAD by alcohol: O NA H D O N H2N + BH H H H2N e d y lh a O B H n Z NA D N H n Z H O H h lco a The Zn activates the COH group towards transfer of the H as a hydride entity to a molecule of NAD. It is easy to visualize this reaction in the opposite direction, in which the Zn atom polarizes the carbonyl group and induces attack by the nucleophilic hydridic H atom from NADH. Alcohol dehydrogenase is an 2 dimer that contains both catalytic and structural Zn sites. Cadmium, the element below Zn in Group 12 and normally regarded as highly toxic, is now recognized as being an essential nutrient for certain organisms. In 2005, a carbonic anhydrase isolated from the marine phytoplankton Thalassiosira weissflogii was discovered to contain Cd at its active site. In contrast to cases in which Cd is simply able to substitute for Zn, this enzyme is specific for Cd2. The surface waters in which Thalassiosira weissflogii grows are extremely low in Zn2 and its growth in the laboratory is stimulated by adding Cd2. Catalytic processes (b) Magnesium enzymes Key points: The major direct catalytic function of magnesium is as the catalytic centre of ribulose bisphosphate carboxylase. The Mg2 cation confers less polarization of coordinated ligands than Zn2 (we often refer to Mg2 as being a ‘weaker acid’ than Zn2); however, compared to Zn it is much more mobile and cells contain high concentrations of uncomplexed Mg2 ions. Its major role in enzyme catalysis is as the Mg–ATP complex (1), which is the substrate in kinases, the enzymes that transfer phosphate groups thereby activating the target compound or causing it to change its conformation. Kinases are controlled by calmodulin (Section 27.4) and other proteins, so they are part of the signalling mechanism in higher organisms. An important example of an Mg enzyme in which Mg acts separately from ATP is ribulose 1,5-bisphosphate carboxylase, commonly known as ‘rubisco’. This enzyme, the most abundant in the biosphere, is responsible for the production of biomass by oxygenic photosynthetic organisms and removal of CO2 from the atmosphere (to the extent, globally, of over 1011 t of CO2 per year). Rubisco is an enzyme of the Calvin cycle, the stages of photosynthesis that can occur in the dark, in which it catalyses the incorporation of CO2 into a molecule of ribulose 1,5-bisphosphate (Fig. 27.31). The Mg2 ion is octahedrally coordinated by carboxylate groups from glutamate and aspartate residues, three coordinated H2O molecules, and a carbamate derived from a lysine residue. The carbamate is formed by a reaction between CO2 and the side-chain –NH2, in an activation process that is necessary for Mg2 to bind. In the catalytic cycle, the binding of ribulose 1,5-bisphosphate displaces two H2O molecules, and proton abstraction assisted by the carbamate results in a coordinated enolate. This intermediate reacts with CO2, forming a new C–C bond, then the product is cleaved to yield two new three-carbon species and the cycle continues. The reactive enolate will also react with O2, in which case the result is an oxidative degradation of substrate: for this reason the enzyme is often called ribulose 1,5-bisphosphate carboxylase-oxygenase. We note the difference from Zn, which would favour ligation by softer ligands and a lower coordination number. Rubisco requires a metal ion that combines good Lewis acidity with weak binding and high abundance. O O H C1 H C5 Asp Glu O C5 Asp Glu OH2 O O Mg H2O OH2 O – C4 O– O O O Mg O– O– C5 C4 O – O OH2 O H2N NH+ Lys C1 O– HO Lys O– C4 H3N HO O O Lys NH+ H C1 Lys CO2 O C Glu O O O C Asp O– C4 HO HO O Mg O– Lys O O– C5 C4 H3N Lys – OO O Mg O O C O H C1 O– + H Lys O OH2 O H O– NH+ Glu O C1 Asp NH+ H3N Lys Figure 27.31 Mechanism of action of ribulose 1,5-bisphosphate carboxylase, the enzyme responsible for removing CO2 from the atmosphere and ‘fixing’ it in organic molecules in plants. 749 750 27 Biological inorganic chemistry (c) Iron enzymes Key points: Acid phosphatases contain a dinuclear metal site containing Fe(III) in conjunction with Fe, Zn, or Mn; aconitase contains a [4Fe–4S] cluster, one subsite of which is modified to manipulate the substrates. Fe Phosphate Mn 33 Acid phosphatases, sometimes known as ‘purple’ acid phosphatases (PAPs) on account of their intense colour, occur in various mammalian organs, particularly the bovine spleen and porcine uterus. Acid phosphatases catalyse hydrolysis of phosphate esters, with optimal activity under mild acid conditions. They are involved in bone maintenance and hydrolysis of phosphorylated proteins (therefore they are important in signalling). They may also have other functions, such as Fe transport. The pink or purple colours of acid phosphatases are due to a tyrosinate → Fe(III) charge-transfer transition at 510–550 nm (ε 4000 dm3 mol–1 cm–1). The active site contains two Fe atoms linked by ligands, similar to haemerythrin (21). Acid phosphatases are inactive in the oxidized {Fe(III)Fe(III)} state in which they are often isolated. In the active state, one Fe is reduced to Fe(II). Both Fe atoms are high spin and remain so throughout the various stages of reactions. Acid phosphatases also occur in plants, and in these enzymes the reducible Fe is replaced by Zn or Mn. The active site of an acid phosphatase from sweet potato (33) shows how phosphate becomes coordinated to both Fe(III) and Mn(II) ions. In the mechanism shown in Fig. 27.32, rapid binding of the phosphate group of the ester occurs to the M(II) subsite, then the P atom is attacked by an OH– ion that is formed at the more acidic Fe(III) subsite. The FeZn centre is also found in an important enzyme called calcineurin, which catalyses the phosphorylation of serine or threonine residues on certain protein surfaces, in particular a transcription factor involved in controlling the immune response. Calcineurin is activated by Ca2 binding, both directly and through calmodulin. Aconitase is an essential enzyme of the tricarboxylic acid cycle (also known as the Krebs cycle, or citric acid cycle), the main source of energy production in higher organisms, where it catalyses the interconversion of citrate and isocitrate in a reaction that formally involves dehydration and rehydration, and proceeds through an intermediate, aconitate, which is released in small amounts: H+ + ROPO32– H2PO4– Fe Fe O O H2O O H O M M O P O – P O O– H N+ O– O O– H H N+ H N+ N H R N+ N H N H N H ROH O Fe O M Fe O O O P O– H N+ M O O P R O– OH H H N+ N NH NH HO O R H N+ NH NH Figure 27.32 Proposed mechanism of action of acid phosphatase. The metal site M(II) is occupied by Fe (most common in animals) or by Mn or Zn (in plants). 751 Catalytic processes H HO − H – CO2 H O2C citrate H H – CO2 – CO2 – CO2 – CO2 aconitate − – CO2 – H CO2 O2C OH Citrate isocitrate H2O/OH– The active form of the enzyme contains a [4Fe–4S] cluster, which degrades to [3Fe–4S] when the enzyme is exposed to air. The site of catalysis is the Fe atom that is lost on oxidation. The structure shows that this unique subsite is not coordinated by a protein ligand but by an H2O molecule, which explains why this Fe is more readily removed. A plausible mechanism for the action of aconitase, based on structural, kinetic, and spectroscopic evidence, involves the binding of citrate to the active Fe subsite, which increases its coordination number to 6. An intermediate in the catalytic cycle is ‘captured’ for X-ray diffraction investigation, using a site-directed mutant that can bind the citrate but cannot complete the reaction (34). The Fe atom polarizes a C–O bond and OH is abstracted, while a nearby base accepts a proton. The substrate now swings round, and the OH and H are reinserted onto different positions. A form of aconitase that is found in cytoplasm has another intriguing role, that of an Fe sensor (Section 27.15). Fe S 34 27.10 Enzymes dealing with H2O2 and O2 In Section 27.7 we saw how organisms have evolved systems that transport O2 reversibly and deliver it unchanged to where it is required. In this section we describe how O2 is reduced catalytically, either for production of energy or synthesis of oxygenated organic molecules. We start by considering a simpler case, that of the reduction of hydrogen peroxide, as this discussion introduces Fe(IV) as a key intermediate in so many biological processes. We end by completing a remarkable cycle, the production of O2 from H2O, catalysed by a unique Mn/Ca cluster. (a) Peroxidases Key points: Peroxidases catalyse reduction of hydrogen peroxide; they provide important examples of Fe(IV) intermediates that can be isolated and characterized. Haem-containing peroxidases, as exemplified by horseradish peroxidase (HRP) and cytochrome c peroxidase (CcP), catalyse the reduction of hydrogen peroxide: H 2O2 (aq) + 2 e − + 2 H + (aq) → 2 H 2O(l) The intense chemical interest in these enzymes lies in the fact that they are the best examples of Fe(IV) in chemistry. Iron(IV) is an important catalytic intermediate in numerous biological processes involving oxygen. Catalase, which catalyses the thermodynamically favourable disproportionation of H2O2 and is one of the most active enzymes known, is also a peroxidase. The active site of yeast cytochrome c peroxidase shown in Fig. 27.33 indicates how the substrate is manipulated during the catalytic cycle. The proximal ligand is the imidazole side chain of a histidine and the distal pocket, like myoglobin, also contains an imidazole side chain, but there is also a guanidinium group from arginine. The catalytic cycle shown in Fig. 27.34 starts from the Fe(III) form. A molecule of H2O2 coordinates to Fe(III) and the distal histidine mediates proton transfer so that both H atoms are placed on the remote O atom. The simultaneous bond polarization by the guanidinium side chain results in heterolytic cleavage of the O–O bond: one half leaves as H2O and the other remains bound to the Fe atom to produce a highly oxidizing intermediate. Although it is instructive to regard this system as a trapped O atom (or an O2 ion bound to Fe(V)), detailed measurements by EPR and Mössbauer spectroscopy show that this highly oxidizing intermediate (which is known historically as ‘Compound I’) is in fact Fe(IV) and an organic cation radical. In HRP the radical is located on the porphyrin ring whereas in cytochrome peroxidase it is located on nearby peptide residue tryptophan-191. Descriptions of the FeO bonding range from Fe(IV) O (‘ferryl’) to Fe(IV)O...H, in which the O atom is either protonated or linked by a hydrogen bond to a donor group. His-5 2 rTp 1 -5 rg A 8 -4 H H rT9 1p - Figure 27.33 The active site of yeast cytochrome c peroxidase showing amino acids essential for activity and indicating how peroxide is bound in the distal pocket. 752 27 Biological inorganic chemistry His H H2O2 O Arg H O Fe R + His Fe(III) Arg His O +Arg O– Resting state Fe H H Fe R R Fe(III) Fe(III) H2O + + – 2H + e His Arg His O Fe Compound II Arg O H2O Fe R e– .R+ Fe(IV) Compound I Fe(IV) Figure 27.34 The catalytic cycle of haem-containing peroxidases. Compound I is reduced back to the resting Fe(III) state by two one-electron transfers from either organic substrates or cytochrome c (Fig. 27.25). (b) Oxidases Key points: Oxidases are enzymes that catalyse the reduction of O2 to water or hydrogen peroxide without incorporation of O atoms into the oxidizable substrate; they include cytochrome c oxidase, the enzyme that is a basis for all higher life forms. Cytochrome c oxidase is a membrane-bound enzyme that catalyses the four-electron reduction of O2 to water, using cytochrome c as the electron donor. The potential difference between the two half-cell reactions is over 0.5 V but this value does not reflect the true thermodynamics because the actual reaction catalysed by cytochrome c oxidase is: 2 H 2O(l) + 4 H + (outside) O2 (g) + 4 e − + 8 H + (inside) → Fe Cu O2 His-Tyr 35 This reaction includes four H that are not consumed chemically but are ‘pumped’ across the membrane against a concentration gradient. Such an enzyme is called an electrogenic ion pump (or proton pump). In eukaryotes, cytochrome oxidase is located in the inner membrane of mitochondria and has many subunits (Fig. 27.35), although a simpler enzyme is produced by some bacteria. It contains three Cu atoms and two haem-Fe atoms, as well as a Mg atom and a Zn atom. The Cu and Fe atoms are arranged in three main sites. The active site for O2 reduction consists of a myoglobin-like Fe-porphyrin (haem-a3) that is situated close to a ‘semi-haemocyanin-like’ Cu (known as CuB) coordinated by three histidine ligands (35). One of the histidine imidazole ligands to the Cu is modified by formation of a covalent bond to an adjacent tyrosine. Electrons are supplied to the dinuclear site by a second Fe porphyrin (haem-a) that is six-coordinate, as expected for an electrontransfer centre. These centres are located in subunit 1. Subunit 2 contains the dinuclear CuA centre that was described in Section 27.8, which is believed to be the immediate acceptor of the electron arriving from cytochrome c. The electron transfer sequence is therefore Cu A → haem a → binuclear site Cytochrome c → The enzyme contains two proton-transfer channels, one of which is used to supply the protons needed for H2O production while the other is used for protons that are being pumped across the membrane. Figure 27.36 shows the proposed catalytic cycle. 753 Catalytic processes Figure 27.35 The structure of cytochrome c oxidase as it occurs in the membrane, showing the locations of the redox centres and the sites for reaction with O2 and cytochrome c. See (33) for a more detailed view of the active site. O2 Me Me N O C O2 N O N C N Fe(II) Cu(I)–Y 2H +2e OH N N ‘oxy’ (A) H+ – HN N O OH Fe(III)–OH Cu(II)–Y Fe(IV)=O Cu(II)–Y. N N O e– OH H+ + e– N Fe N R 2 H2O OH O HN N Fe(II)–O Cu(I)–Y C Cu NH NH O N N P N Fe(IV)=O Cu(II)–Y F Figure 27.36 The catalytic cycle of cytochrome c oxidase. The intermediates are labelled according to current convention. Electrons are provided from the other haem and CuA. During the cycle, an additional four H are pumped across the membrane. Starting from the state in which the active site is Fe(II)Cu(I), O2 binds to give an intermediate (oxy) that resembles oxymyoglobin. However, unlike oxymyoglobin, this intermediate takes up the other electron that is immediately available, producing a peroxy species that quickly breaks down to give an intermediate known as P. Species P has been trapped and studied by optical and EPR spectroscopy, which show that it contains Fe(IV) and an organic radical that may be located on the unusual HisTyr pair (Y.). The oxido-Fe(IV) (ferryl) group is formed by heterolytic cleavage of O2, producing a water molecule. The role of a cation radical is again noted: without this radical, the Fe would have to be assigned as Fe(V). It is vital that intermediates such as peroxide are not released during conversion of O2 to water. Studies with an elaborate model complex (36) which can be attached to an electrode show that the presence of the phenol is crucial because it allows all four electrons necessary for the reduction of the O2 to be provided rapidly without relying on long-range electron transfer, which is slow through the long-chain aliphatic linker. If the phenolic OH group is replaced by OCH3, hydrogen peroxide is released during O2 reduction because the methoxy derivative is unable to form an oxidized radical. The blue Cu oxidases contain a blue Cu centre that removes an electron from a substrate and passes it to a trinuclear Cu site that catalyses the reduction of O2 to H2O. Two examples, ascorbate oxidase and a larger class known as laccases, are well characterized, whereas another protein, ceruloplasmin, occurs in mammalian tissue and is the least well understood. Ascorbate oxidase occurs in the skins of fruit such as squash. Its role may be twofold: to protect the flesh of the fruit from O2 and to oxidize phenolic substrates to intermediates that will form the skin of the fruit. Laccases are widely distributed, particularly S S S S S S S S S S S S Au electrode 36 754 27 Biological inorganic chemistry in plants and fungi, from which they are secreted to catalyse the oxidation of phenolic substrates. The active site at which O2 is reduced (37) is well buried. It contains a pair of Cu atoms linked in the oxidized form by a bridging O atom, with a third Cu atom situated very close by, completing an almost triangular arrangement. Amine oxidases catalyse the oxidation of amines to aldehydes by using just a single Cu atom that shuttles between Cu(II) and Cu(I), yet the enzyme carries out a two-electron reduction of O2, producing a molecule of H2O2. The problem is overcome because, like cytochrome c peroxidase and cytochrome c oxidase, amine oxidases have an additional oxidizing source located near to the metal, in this case a special cofactor called topaquinone (TPQ), which is formed by post-translational oxidation of tyrosine (38). H2O Cu Cu O Cu E X A M PL E 27.7 Interpreting reduction potentials The four-electron reduction potential for O2 is 0.82 V at pH 7. Cytochrome c, the electron donor to cytochrome c oxidase, has a reduction potential of 0.26 V, whereas the organic substrates of fungal laccases often have values as high as 0.7 V. What is the significance of these data in terms of energy conservation? 37 TPQ Met H2O H2O Answer Although cytochrome c oxidase and laccase both catalyse the efficient four-electron reduction of O2, we need to consider their different biological functions. Laccases are efficient catalysts of phenol oxidation, the driving force being small. Cytochrome oxidase is a proton pump and approximately 2 eV (4 × 0.56 eV) of Gibbs energy is available from oxidation of cytochrome c to drive proton transfer across the mitochondrial inner membrane. Self-test 27.7 Before the discovery of the unusual active site structures in amine oxidase and another Cu enzyme called galactose oxidase, Cu(III) was proposed as a catalytic intermediate. What properties would be expected of this state ? Cu (c) Oxygenases 38 Key points: Oxygenases catalyse the insertion of one or both O atoms derived from O2 into an organic substrate; monooxygenases catalyse insertion of one O atom while the other O atom is reduced to H2O; dioxygenases catalyse the incorporation of both O atoms. Oxygenases catalyse the insertion of one or both O atoms of O2 into substrates, whereas with the Fe and Cu containing oxidases both O atoms end up as H2O. Oxygenases are often referred to as hydroxylases when the O atom is inserted into a CH bond. Most oxygenases contain Fe, the rest contain Cu or flavin, an organic cofactor. There are many variations. Monooxygenases catalyse reactions of the type R − H + O2 + 2 H + + 2 e− → R − O − H + H 2O in which electrons are supplied by an electron donor such as an FeS protein. Monoxygenases can also catalyse the epoxidation of alkenes. Dioxygenases catalyse the insertion of both atoms of O2 into substrates, and no additional electron donor is required. Two CH bonds on the same molecule may be oxygenated: HRRH O2 → HORROH The Fe enzymes are divided into two main classes, haem and non-haem. We discuss the haem enzymes first, the most important type being cytochrome P450. Cytochrome P450 (or just P450) refers to an important and widely distributed group of haem-containing monooxygenases. In eukaryotes, they are localized particularly in mitochondria, and in higher animals they are concentrated in liver tissue. They play an essential role in biosynthesis (for example steroid transformations), such as the production of progesterone. The designation ‘P450’ arises from the intense absorption band that appears at 450 nm when solutions containing the enzyme or even crude tissue extracts are treated with a reducing agent and carbon monoxide, which produces the Fe(II)CO complex. Most P450s are complex membrane-bound enzymes that are difficult to isolate. Much of what we know about them stems from studies carried out with an enzyme P450cam, which is isolated from the bacterium Pseudomonas putida. Catalytic processes H R R H OH R O 1 O H R . + Fe(IV ) H R Fe(III) –Cy S s –Cy S s 6 –Cy S s OH Fe(II) 4 O 3 O H R H R e– 2 H R 5 H2O H Fe(III) –Cy S s O Fe(II) Fe(III) O2 –Cy S s –Cy S s H++ e Figure 27.37 The catalytic cycle of cytochrome P450. – This organism uses camphor as its sole source of carbon, and the first stage is oxygenation of the 5-position: + – O2 , 2H , 2e O O H2O OH cytochrome P450 The catalytic cycle has been studied using a combination of kinetic and spectroscopic methods (Fig. 27.37). Starting from the resting enzyme, which is Fe(III), the binding of the substrate in the active site pocket (1) induces release of the coordinated H2O molecule. This step is detected as a change in spin state from low spin (S 12 ) to high spin (S 25 ) and the reduction potential increases, causing an electron to be transferred (2) from a small [2Fe2S]containing protein known as putidaredoxin. The five-coordinate Fe(II) that is formed resembles deoxymyoglobin and binds O2 (3). Unlike in myoglobin, addition of a second electron is both thermodynamically and kinetically favourable. The subsequent reactions (46) are very fast, but it is thought that an Fe(III) peroxide intermediate is formed that undergoes rapid heterolytic OO cleavage to produce a species similar to Compound I of peroxidases. In what is known as the oxygen rebound mechanism, the Fe(IV) O group abstracts an H atom from the substrate and then inserts it back as an OH radical. This process is remarkable, as it amounts to the ‘taming’ of an O atom or OH radical by its attachment to Fe. Other P450s are thought to operate by similar mechanisms but differ in the architecture of the active site pocket. That site, unlike the site in peroxidases, is predominantly hydrophobic, with specific polar groups present to orient the organic substrate so the correct RH bond is brought close to the Fe O entity. Non-haem oxygenases are widely distributed and are usually dioxygenases. Most contain a single Fe atom at the active site and are classified according to whether the active species in the protein is Fe(III) or Fe(II). In the Fe(III) enzymes, which are also (historically) known as intradiol oxygenases, the Fe atom functions as a Lewis acid catalyst and activates the organic substrate towards attack by noncoordinating O2: – – – CO2 CO2 CO2 O O O O O FeIII O O O FeIII O O O O FeIII product 755 756 27 Biological inorganic chemistry By contrast, in the Fe(II) enzymes, which are known historically as extradiol oxygenases, the Fe binds O2 directly and activates it to attack the organic substrate: O H+ O O O Fe(II) O Fe O O (II) H O product (IV) Fe O O O O The Fe(III) enzymes are exemplified by protocatechuate 3,4-dioxygenase: the Fe is high spin and tightly coordinated by a set of protein ligands that includes two His-N and two Tyr-O, the latter hard donors being particularly suitable for stabilizing Fe(III) relative to Fe(II). The Fe(III) enzymes are deep red due to an intense tyrosinate-to-Fe(III) charge transfer transition. The Fe(II) enzymes are exemplified by catechol 2,3-dioxygenase: the Fe is high spin and coordinated within the protein by a set of ligands that includes two His-N and one carboxylate group. The binding is weak, reflecting the low position of Fe(II) in the Irving–Williams series (Section 20.1). This weak binding, together with the difficulty of observing useful spectroscopic features (such as EPR spectra), has made these enzymes much more difficult to study than the Fe(III) enzymes. A particularly important class of Fe(II) oxygenases use a molecule of 2-oxoglutarate as a second substrate: O RH + O2 + – O Fe O Bound formate 39 N O N FeIV N N 40 N O O– O O ROH + – O O – + CO2 O The principle of oxo-glutarate-dependent oxygenases is that the transfer of one O atom of O2 to 2-oxoglutarate (also known as α-ketoglutarate) results in its irreversible decarboxylation, thus driving insertion of the other O atom into the primary substrate. Examples include enzymes that serve in cell signalling by modifying an amino acid in certain transcription factors (Section 27.15). Oxygenases play a crucial role in the metabolism of methane, a greenhouse gas. Of all hydrocarbons, methane contains the strongest C–H bonds and is the most difficult to activate. Methane-metabolizing bacteria produce two types of enzyme that catalyse the conversion of methane to methanol (a more useful chemical and fuel) and thus attract much industrial interest. One is a membrane-bound enzyme that contains Cu atoms. This enzyme, known as ‘particulate’ methane monooxygenase (p-mmo), is expressed when high levels of Cu are available. The other enzyme, soluble methane monooxygenase (s-mmo), contains a dinuclear Fe active site (39) that is related to haemerythrin (21) and acid phosphatase (33). The mechanisms are not established but Fig. 27.38 shows a plausible catalytic cycle for s-mmo. The intermediate Fe(IV) species that is proposed differs from those we have encountered up to now, as the O2-derived oxido ligands are bridging rather than terminal. Despite the importance of Fe(IV) as an enzyme intermediate, small Fe(IV) complexes that could provide important models for understanding the enzymes have been elusive. The easiest to prepare are haem analogues in which Fe is equatorially ligated by porphyrin and which can be formed by reacting the Fe(II) or Fe(III) forms with a peroxo acid. Small models of non-haem Fe(IV) species have now been prepared. The mononuclear complex (40) containing the pentadentate pentaaza ligand N,N-bis(2-pyridylmethyl)-Nbis(2-pyridyl)methylamine is formed by treating the Fe(II) complex with the oxo-transfer agent iodosylbenzene. It is fairly stable at room temperature and has been structurally characterized by X-ray diffraction. A powerful oxidizing agent, it can also be generated in acetonitrile solution by bulk electrolysis of the Fe(II) complex in the presence of water, and the standard potential for Fe(IV)/Fe(III) is estimated to be 0.9 V relative to the ferrocinium/ferrocene couple. Complex (40) and similar species are paramagnetic (S 1) and show characteristic absorption bands in the near-infrared. The Fe–O bond length in (40) is 164 pm, which is fully consistent with a multiple bond in which the O atom is acting as a π donor. Complex (40) is able to oxygenate C–H bonds in a variety of hydrocarbons, Catalytic processes O O O2 Fe(II) Fe(III) Fe(III) O Fe(II) Fe(II) O Fe(III) H2O 2 H+, 2e– Fe(IV) O Fe(IV) O Fe(III) Fe(III) O CH4 H .CH 3 O Fe(III) Fe(IV) O CH3OH Figure 27.38 A plausible catalytic cycle for methane monooxygenase. including cyclohexane. The bis(µ-oxo)Fe(IV) complex (41) has been proposed as a structural analogue of the reactive intermediate formed in s-mmo. Tyrosinase and catechol oxidase, two enzymes responsible for producing melanin-type pigments, each contain a strongly coupled dinuclear Cu centre that coordinates O2 in a manner similar to haemocyanin. However, unlike in haemocyanin, the ligands -to-Cu charge transfer is enhanced sufficiently to activate the coordinated O2 for electrophilic attack at a phenolic ring of the substrate. The structure of the active site of catechol oxidase complexed with the inhibitor phenolthiourea (42) shows how the phenol ring of the substrate can be oriented in close proximity to a bridging O2. Copper enzymes are also responsible for the production of important neurotransmitters and hormones, such as dopamine and noradrenaline. These enzymes contain two Cu atoms that are well separated in space and uncoupled magnetically. (d) Photosynthetic O2 production Key points: Biological solar energy capture by photoactive centres results in the generation of species with sufficiently negative reduction potentials to reduce CO2 to produce organic molecules; in higher plants and cyanobacteria, the electrons are derived from water, which is converted to O2 by a complex catalytic centre containing four Mn atoms and one Ca atom. Photosynthesis is the production of organic molecules using solar energy. It is conveniently divided into the light reactions (the processes by which electromagnetic energy is trapped) and the dark reactions (in which the energy acquired in the light reactions is used to convert CO2 MeO N N MeO OMe O N N FeIV Fe IV N O MeO Phenylthiourea N Cu N OMe Cu N OMe 41 42 757 758 27 Biological inorganic chemistry and H2O into carbohydrates). We have already mentioned the most important of the dark reactions, the incorporation of CO2 into organic molecules, which is catalysed by rubisco. In this section, we describe some of the roles that metals play in the light reactions. The basic principle of photochemical energy capture, applied in a technology to produce H2 from water, was described in Chapter 10 (Box 10.3). We can view photosynthesis in an analogous way in that H2 is ‘stored’ by reaction with CO2. In biology, photons from the Sun excite pigments present in giant membrane-bound proteins known as photosystems. The most important pigment, chlorophyll, is a Mg complex that is very similar to a porphyrin (8). Most chlorophyll is located in giant proteins known as light-harvesting antennae, the name perfectly describing their function, which is to collect photons and funnel their energy to enzymes that convert it into electrochemical energy. This energy conversion uses further chlorophyll complexes that become powerful reducing agents when excited by light. Each electron released by excited chlorophyll travels rapidly down a sequence of protein-bound acceptors, including FeS clusters, and (through the agency of ferredoxin and other redox enzymes) is eventually used to reduce CO2 to carbohydrate. Immediately after releasing an electron, the chlorophyll cation, a powerful oxidant, must be rapidly reduced by using an electron from another site to avoid wasting the energy by recombination (simple reversal of electron flow). In ‘oxygenic’ photosynthesis, which occurs in green algae, cyanobacteria, and most importantly in green plants, each such ‘restoring’ electron is provided from a water molecule, resulting in production of O2. In green plants, photosynthesis occurs in special organelles known as chloroplasts. Plant chloroplasts have two photosystems, I and II, operating in series, that allow low-energy light (approximately 680–700 nm, 1 eV) to span the large potential range ( 1 V) within which water is stable. The arrangement of proteins is depicted in Fig. 27.39. Some of the energy of the photosynthetic electron transfer chain is used to generate a transmembrane proton gradient which in turn drives the synthesis of ATP, as in mitochondria. Photosystem I lies at the low-potential end, its electron donor is the blue Cu protein plastocyanin that has been reduced using the electrons generated by photosystem II; in turn, the electron donor to photosystem II is H2O. Thus green plants dispose of the oxidizing power by converting H2O into O2. This four-electron reaction is remarkable because no intermediates are released. The catalyst, called the ‘oxygen evolving centre’ (OEC) also has a special significance because its action, commencing over 2 Ga ago, has provided essentially all the O2 we have in the atmosphere. The OEC is the only enzyme active site known to produce an A Antenna B Photosystem II MgChl (over 100) 4 MgChl 4 Mn + Ca Cyt. (b) Tyrosine 2 Quinone C Cytochrome b6f complex 2 Cyt (b) 2 Cyt (c) 1 FeS (Rieske) Quinone D Plastocyanin E Photosystem I G Ferredoxin Cu 3 FeS 6 MgChl Fe-S H Ferredoxin NADP reductase Flavin NADP+ Energy (hn ) Energy (hn ) G H F ATP synthase Sugars (CO2 fixation) NADPH Mg ADP Mg ATP Inside (stroma) C F A B H2O A Q O2 Outside (lumen) D e– H+ E D H+ Electron flow Figure 27.39 The arrangement of proteins in the photosynthetic electron-transport chain (The Mg— chlorophyll complex is represented ‘MgChl’.) A. Antenna (‘light harvesting’) complex. B. Photosystem II. C. The ‘cytochrome b6f complex’ (this is similar to complex III in the mitochondrial ET chain). D. Plastocyanin (soluble). E. Photosystem I. F. ATPase. G. Ferredoxin (FeS). H. Ferredoxin–NADP reductase (flavin). Blue arrows show transfer of energy. Note how the overall transfer of electrons is from Mn (high potential) to FeS (low potential): this apparently ‘uphill’ flow reflects the crucial input of energy at each photosystem. Catalytic processes O−O bond from two H2O molecules, and there is much interest in producing functional models of this catalyst for photochemical water splitting (Box 10.3). The OEC is a metal oxide cluster, containing four Mn atoms and one Ca atom, that is located in subunit D1 of photosystem II. Subunit D1 has long attracted interest because the cell replaces it at frequent intervals as it quickly becomes worn out by oxidative damage. X-ray diffraction data indicate that the metal atoms are arranged as a [3MnCa4O] cubane connected to a fourth ‘dangling’ Mn (43). The OEC exploits the oxidizing abilities of Mn(IV) and Mn(V), coupled with that of a nearby tyrosine residue, to oxidize H2O to O2. Successive photons received by photosystem II result in the OEC being progressively oxidized (the acceptor, an oxidized chlorophyll known as P680, has a reduction potential of approximately 1.3 V) through a series of states designated S0 to S4, as shown in Fig. 27.40. Apart from S4, which rapidly releases O2 and has not been isolated, these states are identified in kinetic studies by their characteristic spectroscopic properties, for example S2 shows a complex multi-line EPR spectrum. Note that the Mn ligands are hard O-atom donors and Mn(III) (d4), Mn(IV) (d3), and Mn(V) (d2) are hard metal ions. Based on the available structural evidence, different models have been proposed for the mechanism of O2 evolution from two H2O molecules. The overriding barrier to O2 formation lies in forming the weak peroxidic O−O bond, following which the formation of O O is energetically easy (Section 16.1 and Resource section 3). First, as the Mn sites are progressively oxidized, coordinated H2O molecules become increasingly acidic and lose protons, progressing from H2O through OH to O2. Second, computational studies suggest that Mn(V) O is best regarded as Mn(IV)−Ou, in which the oxido ligand is electron deficient and has appreciable radical character. An H2O or OH ligand coordinated at one Mn subsite (or the Ca) could attack such an electron-deficient O-ligand, perhaps formed at the ‘dangling’ Mn that is not part of the cubane. Such an attack by coordinated H2O or OH on Mn(IV)Ou would result in a Mn(III)-peroxide species that is easily converted to O2 by using the reservoir of oxidizing power that has accumulated on the cluster. The presence of the Ca2 is essential, and the only metal ion that can be substituted is Sr2. A possible role for the Ca is that it provides a site that will remain permanently in the 2 oxidation state and provide a rapid and stable binding site for incoming H2O, whereas if this subsite were occupied by a fifth Mn atom, the latter would certainly become oxidized and the advantage would be lost. 27.11 The reactions of cobalt-containing enzymes Key points: Nature uses cobalt in the form of complexes with a macrocyclic ligand, known as corrin. Complexes in which the fifth ligand is a benzimidazole that is covalently linked to the corrin ring are known as cobalamins. Cobalamin enzymes catalyse methyl transfer and dehalogenation. e– hn [Mn(IV)Mn(III)3] [Mn(IV)2Mn(III)2] 0 S 1 S hn 2H2O e– O2 [Mn(V)Mn(IV)3] 4 S 2 S [Mn(IV)3Mn(III)] hn 3 S e– hn e– [Mn(IV)4] Figure 27.40 The S-cycle for evolution of O2 by successive one-electron oxidations of the [4MnCa4O] cluster of photosystem II. The formal oxidation numbers of the Mn components are indicated, but H transfers are omitted. In chloroplasts that have become adapted to dark conditions, the cycle ‘rests’ in the S1 state. 759 H2O Ca O H2O O O Mn O2C-Asp Asp-CO2 Mn Mn H2O Mn O O2C-Glu Glu-CO2 N-His Glu-CO 2 43 760 27 Biological inorganic chemistry In Coenzyme B12 the sixth ligand is deoxyadenosine, which is coordinated through a CoC bond; enzymes containing coenzyme B12 catalyse radical-based rearrangements. Cobalt macrocycle complexes are cofactors in enzymes that catalyse methyl transfer reactions and they are also important for dehalogenation and radical-based rearrangements (for example, isomerizations). The macrocycle is a corrin ring (9), which is similar to porphyrin (8), except there is less conjugation and it has a smaller ring (15-membered instead of 16-membered). The five-coordinate complex known as cobalamin includes a fifth nitrogen donor in one of the axial positions: usually this ligand is a dimethylbenzimidazole that is covalently linked to the corrin ring through a nucleotide, but a histidine residue is also commonly encountered. The more elaborate structure known as coenzyme B12 (44) is an important enzyme cofactor for radical rearrangements: the sixth ligand, R, is 5-deoxyadenosine, which is bonded to the Co atom through the CH2 group, making coenzyme B12 a rare example of a naturally occurring organometallic compound.8 The sixth ligand is exchangeable and the complex is ingested in the form of species such as aquacobalamin, hydroxocobalamin, or cyanocobalamin, known generally as vitamin B12. Cobalamin is essential for higher organisms (the human requirement is only a few milligrams per day) but it is synthesized only by microorganisms. Like Fe porphyrins, the Co corrins are enzyme cofactors and exert their activities when bound within a protein. The Co atom can exist in three oxidation states under physiological conditions, Co(III), Co(II), and Co(I), all of which are low spin. The electronic structure of Co is crucial to its biological activity. As expected, the Co(III) form (d6) is an 18-electron, six-coordinate species (45). The Co(II) form (46) is 17-electron, five-coordinate and has its unpaired electron in the dz2 orbital. These species are termed ‘base-on’ forms because the fifth nitrogen ligand is coordinated. The Co(I) form (47) is a classic 16-electron, four-coordinate squareplanar species, due to dissociation of both axial ligands. The square-planar structure is a ‘base-off’ form. HO N HO O CH2 O N N NH 2 N O H 2N NH 2 O H2 N O N N Co O HN O O P O HO O – N NH 2 Co Co N O NH2 B B 45 46 N N Co O CH 2OH 44 Coenzyme B12 B 47 8 Coenzyme B12 was one of the earliest molecules to be structurally characterised by X-ray diffraction methods (Dorothy Crowfoot Hodgkin, Nobel Prize for Chemistry, 1964). In 1973, Robert Woodward and Albert Eschenmoser published the total synthesis of B12, the most complex natural product to be synthesized at that time, involving almost 100 steps. Catalytic processes Methyl transfer reactions of cobalamins exploit the high nucleophilicity of square-planar Co(I). A particularly important example is methionine synthase, which is responsible for the biosynthesis of methionine. Methionine is produced by transferring a CH3 group, derived from the methyl carrier methyl hydrofolate, to homocysteine. Not only is methionine an essential amino acid, but also accumulation of homocysteine (which occurs if activity is impaired) is associated with serious medical problems. The mechanism involves a ‘baseon/base-off’ cycle in which Co(I) abstracts an electrophilic CH3 group (effectively CH3) from a quaternary N atom on N5-tetrahydrofolate to produce methylcobalamin, which then transfers CH3 to homocysteine (Fig. 27.41). Methylcobalamin is the methyl-transferring cofactor for a wide variety of biosynthetic pathways, including the production of antibiotics. In anaerobic microbes, methylcobalamin is involved in the synthesis of acetyl coenzyme A, an essential metabolite, and in production of methane by methanogens. Radical-based rearrangements catalysed by coenzyme B12 (but see Box 27.1) include isomerizations (mutases) and dehydration or deamination (lyases). The generic reaction is X H R1 H X R4 R1 R 2 R3 R4 R 2 R3 Dehydration and deamination occur after two OH or OH and NH2 become placed on the same carbon atom and hence are triggered by isomerization: HO H R1 H OH OH R1 R2 R3 H OH NH2 R1 R2 R3 NH2 H O R2 R3 H O R2 R3 H2O R1 R2 R3 HO H R1 OH NH3 R1 R2 R3 Radical-based rearrangements occur by a mechanism involving initiation of radical formation that begins with enzyme-induced weakening of the CoC (adenosine) bond. In the free state, the CoC bond dissociation energy is about 130 kJ mol1, but when bound in the O H HN N H2N O CH3 + N NHAr N H N HN N H2N NHAr N H CH3 Co Co Co (I) B H2N Co (I) H2N CH3 SH S HO HO O O Figure 27.41 The mechanism of methionine synthase. Co(I) is a strong nucleophile and attacks the electrophilic quaternary-CH3 group on the methyl carrier tetrahydrofolate. The resulting Co(III) methyl complex transfers CH3 to homocysteine. 761 762 27 Biological inorganic chemistry B OX 27.1 Iron–sulfur enzymes in radical reactions The discovery in 1970 of an enzyme, lysine 2,3-aminomutase, which catalyses the radical-based rearrangement of an amino acid without any involvement of coenzyme B12, initiated development of a whole new area of biochemistry that has led to B12 being relegated to second place in regard to catalyzing these rearrangements. Lysine 2,3-aminomutase belongs to a large class of FeS enzymes now known as the radical S-adenosylmethionine (SAM) superfamily. Radical SAM enzymes include those responsible for synthesis of essential vitamins, such as vitamin H (biotin), vitamin B1 (thiamine), haem, and molybdopterin (Section 27.12), as well as those that undertake routine repair of DNA. Interconversion of L-lysine and L-β-lysine involves migration of the α-amino group to the β-carbon atom. L-β-lysine is required by certain bacteria for antibiotic synthesis. H NH3 + + L-lysine H H + CO 2– H3 N attacking the unique Fe subsite (a) or a cluster S atom (b). In either case this attack is followed by rapid electron transfer and bond cleavage. The amino acid part sequence CxxxCxxC is the characteristic coordination motif for the [4Fe–4S] cluster in a radical SAM protein. More than 2000 proteins have now been identified by searching for the equivalent base sequence occurring in genes. CO 2– H3 N + H H L -lysine Site of cleavage H 3N H L -β-lysine As with B12 enzymes, this reaction involves a 5-deoxyadenosyl radical, but in radical SAM enzymes the radical is generated by reductive cleavage of the S-adenosylmethionine cation using a special [4Fe–4S] cluster. The reaction sequence begins with reduction of the [4Fe–4S] cluster: [4Fe–4S] is a powerful reductant and SAM is reductively cleaved at the tertiary S to produce methionine, which remains coordinated to the special Fe of the cluster, and the deoxyadenosyl radical, which now abstracts a hydrogen atom from lysine and induces rearrangement. The X-ray structure of the precursor (Fig. B27.1) reveals how the different groups are arranged in space. Based on various lines of spectroscopic evidence, likely mechanisms for forming a 5-deoxyadenosyl radical (Fig. B27.2) involve the tertiary S Deoxyadenosine [4Fe–4S] Figure B27.1 Structure of the active site of lysine 2,3-aminomutase, showing the arrangement of the precursor state in which S-adenosylmethionine is coordinated to the [4Fe–4S] cluster. Lysine, the substrate, is held close by. Ado Ado H2 N Fe Fe O C + 2+ Fe Fe Fe C C O– H2 N Fe S O O O O– H 2C S S C Ado Ado H 2C S + (a) Fe Methionine O– H2 N Fe Fe Fe (b) Fe O– H2 N 2+ Fe Fe Fe Fe Fe Figure B27.2 Two possible mechanisms by which the reactive 5’-deoxyadenosyl radical is generated by the [4Fe–4S]2 cluster in radical SAM enzymes. enzyme the bond is substantially weakened, resulting in homolytic cleavage of the CoCH2R bond. This step results in five-coordinate low-spin Co(II) and a CH2R radical, which gives rise to controlled radical chemistry in the enzyme active site pocket (Fig. 27.42). Important examples are methylmalonyl CoA mutase and diol dehydratases. In 1970, an alternative system for catalyzing radical-based rearrangements was discovered that does not depend on Co. As described in Box 27.1, the enzymes involved use instead a [4Fe4S] cluster to generate the active deoxyadenosyl radical by reductive cleavage of its methionyl derivative. Catalytic processes H H R H2C H R 763 OH OH H O H Co N H N N N H OH H2C R H H2N H Co (I) OH O H H N Co H OH R N H N H2 C H OH O N H H H2N Co (I) N H OH H2C R H lR ica d t m e g n ra Co H N OH O H H2N H N N Co N N N H-a m to n strcio b a Co (I) N H2N Co (I) Figure 27.42 The principle of radical-based rearrangements by coenzyme B12. Homolytic cleavage of the CoC bond results in low-spin Co(II) (dz12) and a carbon radical that abstracts an H atom from the substrate RH. The substrate radical is retained in the active site and undergoes rearrangement before the hydrogen atom transfers back. E X A M PL E 27. 8 Identifying the significance of the d-electron configuration of cobalamin Why is a Co-based macrocyclic complex (rather than an Fe complex like haem) well suited for radical-based rearrangements? Answer To answer this question, we need to consider the electron configuration of Co(II) complexes in which there is a strong equatorial ligand field. Radical-based rearrangements depend on homolytic cleavage of the CoC bond, which generates the adenosine radical and leaves an electron in the dz2 orbital of the Co. This is the stable configuration for a low-spin Co(II) (d7) complex, but for Fe, this configuration would require the oxidation state Fe(I), which is not normally encountered in coordination complexes. Self-test 27.8 Provide an explanation for why the toxicity of mercury is greatly increased by the action of enzymes containing cobalamin. Cys 27.12 Oxygen atom transfer by molybdenum and tungsten enzymes O Key points: Mo is used to catalyse O atom transfer in which the O atom is provided by a water molecule; related chemistry, but in more reducing environments, is displayed by W. Molybdenum and tungsten are the only heavier elements known so far to have specific functions in biology. Molybdenum is widespread across all life forms, and this section deals with its presence in enzymes other than nitrogenase (Section 27.13). By contrast, W has so far only been found in prokaryotes. Molybdenum enzymes catalyse the oxidation and reduction of small molecules, particularly inorganic species. Reactions include oxidation of sulfite, arsenite, xanthine, aldehydes, and carbon monoxide, and reduction of nitrate and dimethyl sulfoxide (DMSO). Both Mo and W are found in combination with an unusual class of cofactors (11), at which the metal is coordinated by a dithiolene group. In higher organisms, Mo is coordinated by one pterin dithiolene along with other ligands that often include cysteine. This is illustrated by the active site of sulfite oxidase, where we see also how the pterin ligand is twisted (48). In prokaryotes, Mo enzymes have two pterin cofactors coordinated to the metal atom and although the role of such an elaborate ligand is not entirely clear, it is potentially redox active and may mediate long-range electron transfer. Coordination of the Mo Dithiolene Pterin 48 O 764 27 Biological inorganic chemistry SO32– O S O S O S S S SO42– O S OH S S EPR detectable S Cys (d1) H+ + e– Mo(IV) Figure 27.43 Oxidation of sulfite to sulfate by sulfite oxidase, illustrating the direct O-atom transfer mechanism for Mo enzymes. Haem domain Mo domain Figure 27.44 Structure of sulfite oxidase showing the Mo and haem domains. The region of polypeptide linking the two domains is highly mobile and is not resolved by crystallography. 0 Reaction enthalpy, ∆rH°/(kJ mol–1) S Cys Mo(V) –300 OH2 Mo Mo –250 Cys H2O O –200 S S Mo(VI) S –150 O Mo Cys H+ + e– –100 2– O O Mo –50 O Fe oxygenases Fe(II) → Fe(IV)O NO2– → NO3– Me2S → Me2SO CH4 → CH3OH Mo(IV)OL → Mo(VI)O2L W(IV)OL → W(VI)O2L H2 → H2O RCHO → RCOOH SO32– → SO42– CO → CO2 Figure 27.45 Scale showing relative enthalpies for O-atom transfer. Fe(IV) oxido species are powerful O-atom donors, whereas Mo(IV) and W(IV) are good O-atom acceptors. Mo is usually completed by ligands derived from H2O, specifically H2O itself, OH, and O2. Molybdenum is suited for its role because it provides a series of three stable oxidation states, Mo(IV), Mo(V), and Mo(VI), related by one-electron transfers that are coupled to proton transfer. Typically, Mo(IV) and Mo(VI) differ in the number of oxido groups they contain, and Mo enzymes are commonly considered to couple one-electron transfer reactions with O-atom transfer. In humans and other mammals, an inability to synthesize molybdenum cofactor has serious consequences. Sulfite oxidase deficiency is a rare, inherited defect (sulfite ions are very toxic) and it is often fatal. A mechanism often considered for Mo enzymes is direct O-atom transfer, which is illustrated in Fig. 27.43 for sulfite oxidase. The S atom of the sulfite ion attacks an electron-deficient O atom coordinated to Mo(VI), leading to Mo–O bond cleavage, formation of Mo(IV), and dissociation of SO42–. Reoxidation back to Mo(VI), during which a transferable O atom is regained, occurs by two one-electron transfers from an Fe-porphyrin that is located on a mobile ‘cytochrome’ domain of the enzyme (Fig. 27.44). The intermediate state containing Mo(V) (d1) is detectable by EPR spectroscopy. This kind of oxygenation reaction can be distinguished from that of the Fe and Cu enzymes described previously, because with Mo enzymes the oxido group that is transferred is not derived from molecular O2 but from water. The Mo(VI) O unit can transfer an O atom, either directly (inner sphere) or indirectly to reducing (oxophilic) substrates, such as SO32– or AsO32–, but cannot oxygenate CH bonds. Figure 27.45 shows the reaction enthalpies for O-atom transfer: we see that the highly oxidizing Fe species formed by reaction with O2 are able to oxygenate all substrates, whereas Mo(VI) oxo species are limited to more reducing substrates and Mo(IV) is able to extract an O atom from nitrate. As expected from its position below Mo in Group 6, the lower oxidation states of W are less stable than those of Mo, so W(IV) species are usually potent reducing agents. This potency is illustrated by the W-containing formate dehydrogenases present in certain primitive organisms, which catalyse the reduction of CO2 to formate, the first stage in nonphotosynthetic carbon assimilation. This reaction does not involve O-atom insertion but rather the formation of a CH bond. One mechanism that has been proposed is HO– H O W H+ (IV) O C O 2e move OH + H + O W(VI) + H C O –O Microbial oxidation of CO to CO2 is important for removing more than 100 Mt of this toxic gas from the atmosphere every year. As shown in Box 27.2, this reaction is carried out by enzymes that contain either a Mo-pterin/Cu cofactor or an air-sensitive [Ni4Fe5S] cluster. Biological cycles 765 B OX 27. 2 Life on carbon monoxide Contrary to what is expected from its well-known toxicity, carbon monoxide is one of Nature’s most essential small molecules. Even at atmospheric levels (0.050.35 ppm) CO is scavenged by a diverse range of microbial organisms for which it provides a source of carbon for growth and a ‘fuel’ for energy (CO is a stronger reducing agent than H2). Two unusual enzymes, known as carbon monoxide dehydrogenases, catalyse the rapid oxidation of CO to CO2. Aerobes use an enzyme containing an unusual Mo-pterin group (Fig. B27.3). A sulfido ligand on the Mo atom is shared with a Cu atom that is coordinated to one other ligand, a cysteine-S, completing the linear arrangement that is so common for Cu(I). A possible mechanism for CO2 formation involves one of the oxo-groups of Mo(VI) attacking the C-atom of a CO that is coordinated to Cu(I). In contrast, certain anaerobes with the unique ability to live on CO as sole energy and carbon source use an enzyme that contains an unusual [Ni4Fe–5S] cluster. X-ray diffraction studies on crystals of the Ni enzyme incubated in the presence of HCO3 at different potentials have revealed the structure of an intermediate showing CO2 coordinated by Ni and Fe (Fig. B27.4). This intermediate supports a mechanism in which, during the conversion of CO to CO2, CO binds to the Ni site (which is square-planar Ni(II)) and the C-atom is attacked by a OH ion that was coordinated to the pendant Fe atom. Carbon monoxide is carried in mammals by haemoglobin, and it is estimated that about 0.6 per cent of the total haemoglobin of an average healthy human is in the carbonylated form. Free CO is produced by the action of haem oxygenase, a P450-type enzyme that catalyses the first step in haem degradation. In addition to releasing Fe and CO, the breakdown of haem produces biliverdin and bilirubin, the familiar green and yellow pigments responsible for the appearance of a bruise. Like NO, CO appears to be a cell signalling agent and, in low amounts, it has important therapeutic effects, including suppression of hypertension (high blood pressure) and protection against tissue rejection following an organ transplant. There is therefore considerable interest in developing pharmaceutical agents, such as the water-soluble complex [Ru(CO)3Cl(glycinate)], to release CO slowly and supplement the action of haem oxygenase. Mo(O)2 Histidine S Molybdopterin cytosine dinucleotide Cu Ni CO2 Lysine Glutamate Cysteine Pendant Fe Figure B27.3 The Mo-pterin cofactor of CO-dehydrogenases found in aerobes. Figure B27.4 X-ray crystallographic observation of an intermediate in the interconversion between CO and CO2, catalysed by the [Ni4Fe5S] cluster in CO-dehydrogenases from anaerobes. Biological cycles Nature is extraordinarily economical and maximizes its use of elements that have been taken up from the non-biological, geological world, often with great difficulty. We have already seen how iron is assimilated by using special ligands, and how this hard-earned resource is stored in ferritin. Thus useful species are recycled rather than returned to the environment. An important example is nitrogen, which is so hard to assimilate from its unreactive gaseous source, N2, and the elusive gas H2, rapid cycling of which is achieved by microbes in processes analogous to electrolytic and fuel cells. 27.13 The nitrogen cycle Key points: The nitrogen cycle involves enzymes containing Fe, Cu, and Mo, often in cofactors having very unusual structures; nitrogenase contains three different kinds of FeS cluster, one of which also contains Mo and a small interstitial atom. The global biological nitrogen cycle involves organisms of all types and a diverse variety of metalloenzymes (Fig. 27.46 and Box 15.2). The cycle can be divided into uptake of usable nitrogen (assimilation) from nitrate or N2 and denitrification (dissimilation). The nitrogen cycle involves many different organisms and a variety of metal-containing enzymes. 766 27 Biological inorganic chemistry Hydroxylamine oxidoreductase – 2 NO Nitrate reductases (Mo) NO3– Nitrite reductase NH2OH Proteins, nucleic acids Nitrite reductases Nitric Arginine oxide synthase Nitric oxide reductase NO NH4+ N2O Nitrogenase N2 Nitrous oxide reductase Figure 27.46 The biological nitrogen cycle. Fe S Many of the compounds are toxic or environmentally challenging. Ammonia is a crucial compound for the biosynthesis of amino acids and NO3 is used as an oxidant. Molecules such as NO are produced in small amounts to serve as cell signalling agents that play a crucial in physiology and health. Nitrous oxide, which is isoelectronic with CO2, is a potential greenhouse gas: its release to the atmosphere depends on the balance between activities and abundancies of NO reductase and N2O reductase across the biological world. The so-called ‘nitrogen-fixing’ bacteria found in soil and root nodules of certain plants contain an enzyme called nitrogenase that catalyses the reduction of N2 to ammonia in a reaction that is coupled to the hydrolysis of 16 molecules of ATP and the production of H2: 49 Nitrogenase P-cluster [8Fe-7S] R-Homocitrate Mo S Fe X 50 Nitrogenase FeMoco [Mo7Fe–8S,X] N 2 + 8 H + + 8 e− + 16 ATP → 2 NH3 + H 2 + 16 ADP + 16 PO3 (OH)2− ‘Fixed’ nitrogen is essential for the synthesis of amino acids and nucleic acids, so it is central to agricultural production. Industrial production of ammonia by the Haber process (Section 26.12) involves reaction of N2 and H2 at high pressures and high temperatures; by contrast, nitrogenase produces NH3 under normal conditions, and it is small wonder that it has attracted so much attention. Indeed, the mechanism of activation of the N2 molecule by nitrogenase has inspired coordination chemists for several decades. The process is very costly in terms of energy for a reaction that is not very unfavourable thermodynamically. However, as we saw in Section 15.6, N2 is an unreactive molecule and energy is required to overcome the high activation barrier for its reduction. Nitrogenase is a complex enzyme that consists of two types of protein: the larger of the two is called the ‘MoFe-protein’ and the smaller is the ‘Fe-protein’ (Fig. 27.47). The Fe-protein contains a single [4Fe–4S] cluster that is coordinated by two cysteine residues from each of its two subunits. The role of the Fe-protein is to transfer electrons to the MoFe-protein in a reaction that is far from understood: in particular, it is unclear why each electron transfer is accompanied by the hydrolysis of two ATP molecules, which are bound to the Fe-protein. The MoFe protein is α22, each α pair of which contains two types of supercluster. The [8Fe7S] cluster (49) is known as the ‘P-cluster’ and is thought to be an electron transfer centre, whereas the other cluster (50), formulated as [Mo7Fe8S,X] and known as ‘FeMoco’ (FeMo cofactor), is thought to be the site at which N2 is reduced to NH3. The Mo is coordinated also by an imidazole-N from histidine and two O atoms from an exogenous molecule R-homocitrate. The mechanism of N2 reduction is still unresolved, the main question being whether N2 is bound and reduced at the Mo atom or at some other site. Here it is significant that nitrogenases are known in which the Mo atom is replaced by a V or Fe atom, arguing against a specific role for Mo. The cage-like cluster provided by the six central Fe atoms also coordinates a small central atom (‘X’), Biological cycles FeMoco [Mo7Fe–8S,X] Figure 27.47 The structure of nitrogenase showing the Fe protein and the MoFe protein complexed with each other. Positions of the metal centres (black) are indicated. The MoFe protein is an 22 of different subunits (red and blue) and it contains two P-clusters and two MoFe cofactors. The Fe protein (green) has a [4Fe4S] cluster and it is also the site of binding and hydrolysis of Mg-ATP. [4Fe–4S] P-cluster [8Fe–7S] which is proposed to be C, N, or O. It was nearly ten years after the determination of the structure of the MoFeS cage that this small central atom was discovered, after improvements in resolution and detailed consideration of the X-ray interference characteristics. The six Fe atoms would otherwise be three-coordinate with a flattened trigonal-pyramidal geometry. Nitrate reductase is another example of a Mo enzyme involved in the transfer of an O atom, in this case catalysing a reduction reaction (the standard potential for the NO3 /NO2 couple corrected to pH 7 is 0.4 V, thus NO3 is quite strongly oxidizing). The other enzymes in the nitrogen cycle contain either haem or Cu as their active sites. There are two distinct classes of nitrite reductase. One is a multi-haem enzyme that can reduce nitrite all the way to NH3. The other class contains Cu and carries out one-electron transfer, producing NO: it is a trimer of identical subunits, each of which contains one ‘blue’ Cu (mediating long-range electron transfer to the electron donor, usually a small ‘blue’ Cu protein) and a Cu centre with more conventional tetragonal geometry that is thought to be the site of nitrite binding. The nitrogen cycle is notable for using some of the most unusual redox centres yet encountered as well as some of the strangest reactions. Another unusual cofactor is a [4CuS] cluster, named CuZ, that is found in N2O reductase and has the structure shown in (51). It is puzzling how this centre is able to bind and activate N2O, which is a poor ligand. Long-range electron transfer in N2O reductase is carried out by a CuA centre, the same as found for cytochrome c oxidase. Of particular importance for humans are two enzymes that manipulate NO. One of these, NO synthase, is a haem enzyme responsible for producing NO, by oxidation of l-arginine, on receipt of a signal. Its activity is controlled by calmodulin (Section 27.4). The other enzyme is guanylyl cyclase, which catalyses the formation of the important regulator cyclic guanosine monophosphate (cGMP) from guanosine triphosphate. Nitric oxide binds to the haem Fe of guanylyl cyclase, displacing a histidine ligand and activating the enzyme. Another interesting NO-binding protein is nitrophorin, which is found in some bloodsucking parasites, notably ‘kissing bugs’ (predacious bugs of the family Reduviidae). Nitrophorin binds NO tightly until it is injected into a victim, where a change in pH causes its release. The free NO causes dilation of the surrounding blood vessels, rendering the victim a more effective blood donor. S Cu E X A M PL E 27. 9 Identifying intermediates formed during the reduction of N2 to NH3 Suggest likely intermediates formed during the six-electron reduction of an N2 molecule to NH3. Answer To identify possible intermediates we need to recall from Chapter 5 that p-block elements normally undergo two-electron transfers that are accompanied by proton transfers. Because N2 has a triple bond, the 767 51 Cuz cluster [4Cu [4Cu–S] S] 768 27 Biological inorganic chemistry two N atoms will remain bonded together throughout most of the six-electron reduction. We would propose diazene (N2H2) and hydrazine (N2H4) along with their deprotonated conjugate bases. Self-test 27.9 The MoFe cofactor can be extracted from nitrogenase by using dimethylformamide (DMF), although it is catalytically inactive in this state. Suggest experiments that could establish if the structure of the species in DMF solution is the same as present in the enzyme. 27.14 The hydrogen cycle Key point: The active sites of hydrogenases contain Fe or Ni, along with CO and CN ligands. OH CN CN Fe Ni CO 52 [NiFe]-hydrogenase Fe CO CN 53 [FeFe]-hydrogenase HCys Cys S S Fe Ni S S Cys 54 It has been estimated that 99 per cent of all organisms utilize H2. Even if these species are almost entirely microbes, the fact remains that almost all bacteria and archaea possess extremely active metalloenzymes, known as hydrogenases, that catalyse the interconversion of H2 and H (as water). The elusive molecule H2 is produced by some organisms (it is a waste product) and used by others as a fuel, helping to explain why so little H2 is in fact detected in the atmosphere (Box 10.1). Human breath contains measurable amounts of H2 due to the action of bacteria in the gut. Hydrogenases are very active enzymes, with turnover frequencies (molecules of substrate transformed per second per molecule of enzyme) exceeding 10 000 s1. They are therefore attracting much attention for the insight they can provide regarding clean production of H2 (Section 10.4, Box 10.3) and oxidation of H2 in fuel cells—technology that currently depends greatly on Pt (Box 5.1). There are three classes of hydrogenase, based on the structure of the active site. All contain Fe and some also contain Ni. The two best-characterized types are known as [NiFe]-hydrogenases and [FeFe]-hydrogenases, and structures of the active sites of two representative enzymes are shown as (52) and (53). The active sites contain at least one CO ligand, and provide further examples of biological organometallic active sites (in addition to coenzyme B12). Further ligation is provided by CN, cysteine (and sometimes selenocysteine), and the active site of [FeFe]-hydrogenases (the site is commonly referred to as an ‘H-cluster’) contains a [4Fe-4S] cluster linked by a bridging cysteine thiolate, and an unusual bridging bidentate ligand that is thought to be either dithiomethylamine or dithiomethylether. These fragile active sites are buried deeply within the enzyme, thus necessitating special pores and pathways to convey H2 and H, and a relay of FeS clusters for long-range electron transfer (as shown in Fig. 27.26). The [FeFe]-enzymes tend to operate in the direction of H2 production; they are usually found in strictly anaerobic organisms and are very sensitive to O2. The mechanism of catalysis is uncertain, but it involves the participation of unusual Fe(I) species: the form assigned as Fe(I)Fe(I) binds H and that assigned as Fe(II)Fe(I) binds H2, probably in an analogous manner to the dihydrogen complex shown (11) in Section 10.6. The [NiFe]-enzymes are noted for H2 oxidation and a catalytically active form assigned by EPR spectroscopy as Ni(III)–H– (a hydrido complex) has been identified (54). Single-crystal EPR studies show that the unpaired electron is coupled strongly to an H-atom nucleus lying along an axis that points towards the Fe (that is, the H is in a bridging position). The active sites of [NiFe]- and [FeFe]-hydrogenases react with O2, often irreversibly, and this sensitivity is a limiting factor in developing and exploiting renewable H2 production by microorganisms (Section 10.4 and Box 10.1). CN Sensors CN CO A number of metalloproteins are used to detect and quantify the presence of small molecules, particularly O2, NO, and CO. These proteins therefore act as sensors, alerting an organism to an excess or deficit of particular species, and triggering some kind of remedial action. Special proteins are also used to sense the levels of metals such as Cu and Zn that are otherwise always strongly complexed in a cell. Cys Sensors 27.15 Iron proteins as sensors Key point: Organisms use sophisticated regulatory systems based on Fe-containing proteins to adapt quickly to changes in cellular concentrations of Fe and O2. We have already seen how FeS clusters are used in electron transfer and catalysis (Sections 27.8 and 27.9). The coordination of an FeS cluster ties together different parts of a protein and thus controls its tertiary structure. The sensitivity of the cluster to oxygen, electrochemical potential, or Fe and S concentrations makes it able to be an important sensory device. In the presence of O2 or other potent oxidizing agents, [4Fe4S] clusters have a tendency (controlled by the protein) to degrade, producing [3Fe4S] and [2Fe2S] species. The cluster may be removed completely under some conditions (Fig. 27.48). The principle behind an organism’s exploitation of FeS clusters as sensors is that the presence or absence of a particular cluster (the structure of which is very sensitive to Fe or oxygen) alters the conformation of the protein and determines its ability to bind to nucleic acids. In higher organisms, the protein responsible for regulating Fe uptake (transferrin) and storage (ferritin) is an FeS protein known as the iron regulatory protein (IRP), which is closely related to aconitase (Section 27.9) but is found in the cytoplasm rather than in mitochondria. It acts by binding to specific regions of messenger RNA (mRNA) that carry the genetic command (transcribed from DNA) to synthesize transferrin receptor or ferritin. A specific interaction region on the RNA is known as the iron-responsive element (IRE). The principle is outlined in Fig. 27.49. When Fe levels are high, a [4Fe4S] cluster is present, and the protein does not bind to the IRE that controls translation of ferritin. In this case binding would be a ‘stop’ command, and the cell will respond by synthesizing ferritin. Simultaneously, binding of the [4Fe4S]-loaded protein to the transferrin receptor IRE destabilizes the RNA, so transferrin receptor is not made. When Fe levels are high, the opposite actions occur: a [4Fe4S] cluster is formed, ferritin synthesis is activated, and transferrin receptor synthesis is switched off (repressed). The common gut bacterium E. coli derives energy either by aerobic respiration (using a terminal oxidase related to cytochrome c oxidase, Section 27.10) or by anaerobic respiration with an oxidant such as fumaric acid, or nitrate, using the Mo enzyme nitrate Fe S (a) S Fe Fe S Fe S S Fe Fe S Fe S Fe Fe, S S Apo-protein [4Fe-4S] Low Fe Fe, S apo-IRP S Fe (b) Fe Apo-protein High Fe Binds to two different messenger RNAs S Fe, S Fe, S Oxidative damage Figure 27.48 The degradation of FeS clusters forms the basis for a sensory system The [4Fe4S] cluster cannot support a state in which all Fe are Fe(III); thus severe oxidizing conditions, including exposure to O2, causes their breakdown to [3Fe4S] or [2Fe2S] and eventually complete destruction. Degradation to [3Fe4S] (a) requires only removal of an Fe subsite whereas degradation to [2Fe2S] (b) may require rearrangement of the ligands (cysteine) and produce a significant protein conformational change. These processes link cluster status to availability of Fe as well as O2 and other oxidants and provide the basis for sensors and feedback control. apo-IRP apo-IRP Blocks translation Stabilizes mRNA that translates of ferritin transferrin receptor Figure 27.49 Interactions of iron-regulatory protein with iron-responsive elements on the RNAs responsible for synthesizing ferritin or transferrin receptor depend on whether an FeS cluster is present and form the basis for regulation of cellular Fe levels. 769 770 27 Biological inorganic chemistry [4Fe–4S]-FNR binds to DNA, activating genes needed for an anaerobic lifestyle [4Fe–4S] Low O2 [2Fe–2S] High O2 [2Fe–2S] [2Fe–2S]-FNR cannot bind to DNA Figure 27.50 The principle of operation of the fumarate nitrate regulatory system that controls aerobic versus anaerobic respiration in bacteria. Targeted for degradation Prolyl hydroxylase O2 HIFα OH HIFα Lack of O2 HIFβ HIFα Transcription of genes encoding for features enhancing O2 delivery to tissues Figure 27.51 The principle of O2 sensing by prolyl oxygenases. reductase (Sections 27.12 and 27.13). The problem the organism faces is how to sense whether O2 is present at a sufficiently low level to warrant inactivating the genes functioning in aerobic respiration and activate instead the genes producing enzymes necessary for the less efficient anaerobic respiration. This detection is achieved by an FeS protein called fumarate nitrate regulator (FNR). The principle is outlined in Fig. 27.50. In the absence of O2, FNR is a dimeric protein with one [4Fe–4S] cluster per subunit. In this form it binds to specific regions of DNA repressing transcription of the aerobic enzymes and activating transcription of enzymes such as nitrate reductase. When O2 is present, the [4Fe4S] cluster is degraded to a [2Fe2S] cluster and the dimer breaks up so that it cannot bind to DNA. The genes encoding aerobic respiratory enzymes are thus able to be transcribed, whereas those for anaerobic respiration are repressed. In higher animals, the system that regulates the ability of cells to cope with O2 shortage involves an Fe oxygenase. Prolyl hydroxylases catalyse the hydroxylation of specific proline residues in proteins, thus altering their properties. In higher animals, one such target protein is a transcription factor called hypoxia inducible factor (HIF), which mediates the expression of genes responsible for adapting cells to low-O2 conditions (hypoxia). We should bear in mind here that the internal environment of cells and cell compartments is usually quite reducing, equivalent to an electrode potential below –0.2 V, and even though we regard O2 as essential for higher organisms, its actual levels may be fairly low. When O2 levels are above a safe threshold, prolyl hydroxylases catalyse oxygenation of two conserved proline residues of HIF, causing the transcription factor to be recognized by a protein that induces its degradation by proteases. Hence genes such as those ultimately responsible for producing more red blood cells (which will help an individual to cope better when O2 supply is a problem) are not activated. The principle is outlined in Fig. 27.51. Although essential for higher organisms, O2 requires stringent control of its four-electron reduction, and increasing amounts of research are being carried out to prevent and cure malfunctions of normal O2 consumption. The term ‘oxidative stress’ is used to describe conditions in which the normal function of an organism is threatened by a build up of partially reduced O2 intermediates, such as superoxides, peroxides, and hydroxyl radicals known collectively as reactive oxygen species (ROS). Prolonged exposure to ROS is associated with premature aging and certain cancers. To avoid or minimize oxidative stress cells must first sense ROS and then produce agents to destroy them. Both sensory and attack agents are proteins with active groups such as metal ions (particularly Fe, Cu) and exposed, redox-active cysteine thiols. An underlying principle of haem sensors is that the small molecules being sensed are π acceptors that can bind strongly to the Fe and displace an indigenous ligand. This binding results in a change in conformation that alters catalytic activity or the ability of the protein to bind to DNA. Of the indigenous ligands, two classic examples are NO and CO; although we have long thought of these molecules as toxic to higher life forms, they are becoming well established as hormones. Indeed, there is evidence that the sensing of trace levels of CO is important for controlling circadian rhythms in mammals. The enzyme guanyl cyclase senses NO, a molecule that is now well established as a hormone that delivers messages between cells. Guanyl cyclase catalyses the conversion of guanidine monophosphate (GMP) to cyclic GMP (cGMP), which is important for activating many cellular processes. The catalytic activity of guanyl cyclase increases greatly (by a factor of 200) when NO binds to the haem, but (as is obviously important) by a factor of only 4 when CO is bound. An excellent, atomically defined example of CO sensing is provided by a haemcontaining a transcription factor known as CooA. This protein is found in some bacteria that are able to grow on CO as their sole energy source under anaerobic conditions. Whether growth on CO takes place depends on the ambient CO level, as an organism will not waste its resources synthesizing the necessary enzymes when the essential substrate is not present. CooA is a dimer, each subunit of which contains a single b-type cytochrome (the sensor) and a ‘helix-turn-helix’ protein fold that binds to DNA (Fig. 27.52). In the absence of CO, each Fe(II) is six-coordinate and both its axial ligands are amino acids of the protein, a histidine imidazole and, unusually, the main-chain NH2 group of a proline that is also the N-terminal residue of the other subunit. In this form, CooA cannot bind to the specific DNA sequence to transcribe the genes for synthesizing the CO-oxidizing enzymes necessary for existing on CO. When CO is present it binds to the Fe, displacing the distal Biomineralization 771 proline residue and causing CooA to adopt a conformation that will bind to the DNA. The likelihood of NO binding in place of CO to cause a false transcriptional response is prevented because NO not only displaces proline but also results in dissociation of the proximal histidine; the NO complex is thus not recognized. 27.16 Proteins that sense Cu and Zn levels Key point: Cu and Zn are sensed by proteins with binding sites specially tailored to meet the specific coordination preferences of each metal atom. The levels of Cu in cells are so strictly controlled that almost no uncomplexed Cu is present. An imbalance in Cu levels is associated with serious health problems such as Menkes disease (Cu deficiency) and Wilson’s disease (Cu accumulation). Most of what we know about how Cu levels are sensed and converted to cell signals stems from studies on the E. coli system, which involves a transcription factor called CueR (Fig. 27.53). This protein binds Cu(I) with high selectivity, although it also binds Ag(I) and Au(I). Metal coordination causes a conformational change that enables CueR to bind to DNA at a receptor site that controls transcription of an enzyme known as CopA, which is an ATP-driven Cu pump. CopA is located in the cytoplasmic membrane and exports Cu into the periplasm. In CueR, the Cu(I) is coordinated by two cysteine-S atoms arranged in a linear coordination geometry. Titrations using CN as a buffer show that Cu is bound with a dissociation constant of approximately 1021. As may be understood by reference to Section 7.3, this ligand environment leads to remarkably selective binding for d10 ions, and measurements with Ag and Au show that these ions are taken up with similar affinities. Most of our current insight about Zn sensing, as for Cu, is provided by studies of bacterial systems. The major difference with respect to Cu is that although Zn is also coordinated (mainly) by cysteine thiolates, the geometry is tetrahedral rather than linear. E. coli contains a Zn2-sensing transcription factor known as ZntR that is closely related to CueR. The factor ZntR contains two Zn-binding domains each of which coordinates a pair of Zns using cysteine and histidine ligands. The surrounding protein fold is shown in Fig. 27.53, for comparison with CueR. The extent to which these dynamic Zn-binding sites can be identified with zinc fingers remains unclear. Figure 27.52 The structure of CooA, a bacterial CO sensor and transcription factor. The molecule, a dimer of two identical subunits (represented as red and blue), has two haem-binding domains and two ‘helix– turn–helix’ domains that recognize a section of DNA. The protein ligands to the Fe atoms are a histidine from one subunit and an N-terminal proline from the other. The binding of CO and displacement of proline disrupts the assembly and allows CooA to bind to DNA. Cu E X A M PL E 27.10 Identifying links between redox chemistry and metal ion sensing Suggest a way in which the binding of Cu or Zn to their respective sensor proteins might be linked to the level of cellular O2. Answer To address this problem we recall from Chapter 16 that strong SS bonds arise from the combination of S atoms or radicals. A pair of cysteines that are coordinating a metal ion or are able to approach each other at close range can undergo oxidation by O2 or other oxidants, resulting in the formation of a disulfide bond (cystine). This reaction prevents the cysteine S atoms from acting as ligands and provides a way in which even redox inactive metals such as Zn may be involved in sensing O2. Self-test 27.10 Why might Cu sensors function to bind Cu(I) rather than Cu(II)? (a) Zn Zn Biomineralization Key points: Calcium compounds are used in exoskeletons, bones, teeth, and other devices; some organisms use crystals of magnetite, Fe3O4, as a compass; plants produce silica-based protective devices. Biominerals can be either infinite covalent networks or ionic. The former include the silicates, which occur extensively in the plant world. Leaves, even whole plants, are often covered with silica hairs or spines that offer protection against predatory herbivores. Ionic biominerals are mainly based on calcium salts, and exploit the high lattice energy and low solubilities of these compounds. Calcium carbonate (calcite or aragonite, Section 12.10) is the material present in sea shells and eggshells. These minerals persist long after the organism has died, indeed chalk is a biogenic mineral, a result of the process of (b) Figure 27.53 Comparison of (a) the Cuand (b) the Zn-binding sites in the respective transcription factors CueR and ZntR. Note how Cu(I) is recognized by a linear binding site (to two cysteines) whereas Zn is recognized by an arrangement of Cys and His ligands that binds two Zn(II) atoms together with a bridging phosphate group. 772 27 Biological inorganic chemistry Figure 27.54 The porous silica structure of a radiolarian microskeleton showing the large radial spines. (Photograph supplied by Professor S. Mann, University of Bristol.) calciferation of prehistoric organisms. Calcium phosphate (hydroxyapatite, Section 15.4) is the mineral component of bones and teeth, which are particularly good examples of how organisms fabricate ‘living’ composite materials. Indeed, the different properties of bone found among species (such as stiffness) are produced by varying the amount of organic component, mostly the fibrous protein collagen, with which hydroxyapatite is associated. High hydroxyapatite/collagen ratios are found for large marine animals, whereas low ratios are found for animals requiring agility and elasticity. Biomineralization is crystallization under biological control. There are some striking examples that have no counterparts in the laboratory. Perhaps the most familiar example is the exoskeleton of the sea urchin, which comprises large sponge-like plates containing continuous macropores 15 µm in diameter: each of these plates is a single crystal of Mgrich calcite. Large single crystals support other organisms, such as diatoms and radiolarians, with their skeletons of silica cages (Fig. 27.54). Biominerals produce some intriguing gadgetry. Figure 27.55 shows crystals of calcite that are part of the gravity sensor device in the inner ear. These crystals are located on a membrane above sensory cells and any acceleration or change of posture that causes them to move results in an electrical signal to the brain. The crystals are uniform in size and spindle-shaped so that they move evenly without becoming hooked together. The same property is seen for crystals of magnetite (Fe3O4) that are found in a variety of magnetotactic bacteria (Fig. 27.56). There are considerable variations in size and shape across different species, but within any one species, the crystals, formed in magnetosome vesicles, are uniform. Magnetotactic bacteria live in fluid sediment suspensions in marine and freshwater environments, and it is thought that their microcompasses allow them to swim always in a downwards direction to maintain their chemical environment during turbulent conditions. There is great interest in how biomaterials are formed, not least because of the inspiration they provide for nanotechnology (Chapter 25). The formation of biominerals involves the following hierarchy of control mechanisms: 1. Chemical control (solubility, supersaturation, nucleation). Figure 27.55 Our gravity sensor. Crystals of biologically formed calcite that are found in the inner ear. (Photograph supplied by Professor S. Mann, University of Bristol.) Spatial control (confinement of crystal growth by boundaries such as cells, subcompartments, and even proteins in the case of ferritin, Section 27. 6). 3. Structural control (nucleation is favoured on a specific crystal face). 4. Morphological control (growth of the crystal is limited by boundaries imposed by organic material that grows with time). 5. Constructional control (interweaving inorganic and organic materials to form a higherorder structure, such as bone). Bone is continually being dissolved and reformed; indeed it functions not only as a structural support but also as the central Ca store. Thus, during pregnancy, bones tend to be raided for their Ca in a process called demineralization, which occurs in special cells called osteoclasts. Depleted or damaged bones are restored by mineralization, which occurs in cells called osteoblasts. These processes involve phosphatases (Section 27.9). Figure 27.56 Magnetite crystals in magnetotactic bacteria. These are tiny compasses that guide these organisms to move vertically in river bed sludge. MV indicates empty vesicles (Photograph supplied by Professor S. Mann, University of Bristol.) The chemistry of elements in medicine Serendipity has played an important role in drug discovery, with many effective treatments arising from chance discoveries. There appears to be a special role for compounds containing metals that are not otherwise present in biological systems, for example Pt, Au, Ru, and Bi. A major challenge in pharmacology is to determine the mechanism of action at the molecular level, bearing in mind that the drug that is administered is unlikely to be the molecule that reacts at the target site. This is particularly true for metal complexes, which are usually more susceptible to hydrolysis than organic molecules. In general, the mechanism of action is proposed by extrapolation of in vitro studies. Orally administered drugs are highly desirable because they avoid the trauma and potential hazards of injection; however, they may not pass through the gut wall or survive hydrolytic enzyme action. Inorganic compounds are also used in the diagnosis of disease or damage, a particularly interesting example being the use of radioactive technetium. The chemistry of elements in medicine 27.17 Chelation therapy O O Key points: The treatment of Fe overload involves sequestration of Fe by ligands based on or inspired by siderophores. ‘Iron overload’ is the name given to several serious conditions that affect a large proportion of the world’s population. Here we recall that, despite its great importance, Fe is potentially a highly toxic element, particularly in its ability to produce harmful radicals by reaction with O2, and its levels are normally strictly controlled by regulatory systems. In many groups of people, a genetic disorder results in breakdown of this regulation. One kind of iron overload is caused by an inability for the body to produce sufficient porphyrin. Other problems are caused by faults in the regulation of Fe levels by ferritin or transferrin production. These disorders are treated by chelation therapy, the administration of a ligand to sequester Fe and allow it to be excreted. Desferrioxamine (‘Desferral’, 55) is a ligand that is similar to the siderophores described in Section 27.6. It is a very successful agent for iron overload, apart from the trauma of its introduction into the body, which involves it being plumbed into an intravenous supply. A special case of chelation therapy is the treatment of individuals who have been contaminated with Pu following exposure to nuclear weapons. In its common oxidation states, Pu(IV) and Pu(III) have similar charge densities to Fe(III) and Fe(II). Siderophorelike chelating ligands have been developed, such as 3,4,3-LIMACC (56), which contains four catechol groups. 27.18 Cancer treatment Key points: The complex cis-[PtCl2(NH3)2] results in the inhibition of DNA replication and prevention of cell division; other drugs cause DNA to be degraded by oxygenation. ‘Cancer’ is a term that covers a large number of different types of the disease, all characterized by the uncontrolled replication of transformed cells that overwhelm the normal operation of the body. The principle of treatment is to apply drugs that destroy these malignant cells selectively. The remarkable action of the complex cis-[PtCl2(NH3)2] (57, known as cisplatin) was discovered in 1964 while examining the effect of an electric field on the growth of bacteria. The behaviour of a colony of bacteria suspended in solution between two platinum electrodes was observed, and it was noted that the cells continued to grow in size, forming long filaments, but stopped replicating. The effect was traced to a complex that was CO2– O– O– O– – O2 C HN O O O – N O CO2– –O O– HN N O Cl –O H 3N –O CO2– 56 773 Pt Cl NH 3 57 Cisplatin N HN –O NH 2 O O – O N HN O 55 Desferral N O– 774 27 Biological inorganic chemistry O formed electrochemically by dissolution of Pt into the electrolyte, which contained NH4Cl. Since then, cisplatin has been a successful drug for the treatment of many forms of cancer, particularly testicular cancer, for which the success rate approaches 100 per cent. The other geometric isomer, trans-[PtCl2(NH3)2], is inactive. The ultimate molecular basis of the chemotherapeutic action of cisplatin and related drugs is thought to be the formation of a stable complex between Pt(II) and DNA. Cisplatin is administered into the bloodstream of the patient, where, because the plasma contains high concentrations of Cl–, it tends to remain as the neutral dichlorido species. The electrical neutrality of the dichlorido complex facilitates its passage through the cell and nuclear membranes. Once it is subjected to the lower Cl concentrations inside the cell (Table 27.1), the Cl ligands are replaced by H2O, and the resulting cationic species (with charges 1 or 2) are attracted electrostatically to DNA and form inner-sphere complexes in which the Pt(NH3)2 fragment becomes coordinated to the N atoms of the nucleotide bases. Some classic studies have shown that the preferred target is a pair of N atoms on consecutive guanine bases in the same strand. Complexes of the Pt(NH3)2 fragment with oligonucleotides have been studied by X-ray crystallography and 195Pt-NMR (Fig. 27.57). Complexation with Pt causes the helix to bend and partially unwind. It is thought that this distortion renders the DNA incapable of replication or repair. The distortion also makes the DNA recognizable by ‘high mobility group’ proteins that bind to bent DNA; the cell may thus be targeted for its own death. Despite its efficacy, cisplatin has highly undesirable side effects, in particular it causes serious damage to the kidneys before it is eventually excreted. Great efforts have been made to find Pt complexes that are effective with fewer side effects. One example in clinical use is carboplatin (58). Effective drugs may also include trinuclear Pt(II) (59) as well as Pt(IV) complexes such as satraplatin (60), which can be administered orally. Other metal complexes are being discovered that bind by intercalation within the DNA interior and offer improved efficacy over Pt drugs. They include Ru(III) complexes such as fac-[RuCl3(NH3)3], which are believed to function by providing a source of Ru(III) that is carried to cancer cells by transferrin, and the complex (61), which may be activated by reduction to Ru(II) in vivo. There is increasing interest in organometallic compounds. The Ru(II) arene complex (62), which possesses high anti-cancer activity, is believed to coordinate to guanine-N in a similar way to Pt complexes but the interaction is supplemented by intercalation of the biphenyl group within the hydrophobic DNA core as well as hydrogen bonding between guanine and the NH2 groups of the en ligand. Even common Ti compounds such as TiCp2Cl2 have undergone clinical trials. Metallo-supramolecular ‘cylinders’ formed by placing a metal cation at either end of a bundle of ligands (63) have much larger dimensions that mimic those of Zn fingers. The cylinders bind in the major groove of DNA, causing it to form small coils. O O O Pt H3 N NH 3 58 Carboplatin NH 3 Cl Pt 4+ NH 2 NH 3 NH 3 H2 N Pt NH2 NH 3 NH3 NH 3 H 2 N Pt Pt Cl anine u G NH 3 59 anine u G O O H 3N Cl Pt NH2 O 60 Cl O Figure 27.57 Structure of an adduct formed between —Pt(NH3)2 and two adjacent guanine bases on an oligonucleotide. Expanded view shows the square-planar ligand arrangement around the Pt atom. Coordination of Pt causes bending of the DNA helix. 775 The chemistry of elements in medicine N N N S Cl Ru N Cl N Ru Cl HN 3 N N Cl Cl N M M Me Me O N 61 NH2 N N N N Ru Ru N N N N H2 N 62 63 Compounds of Ga(III) are under investigation as anti-cancer drugs. Like Fe(III), Ga(III) is a hard Lewis acid and the two metal ions have similar radii, however Ga(III) is not easily reduced to Ga(II) and any redox or O2 binding proteins that have incorporated Ga in place of Fe will be inactive. It is thought that Ga(III) enters cells using the same transport systems as Fe. The target for Ga is the Fe-containing enzyme ribonucleotide reductase, which is essential for producing the bases used in DNA. Compounds undergoing trials range from simple salts like gallium nitrate to complexes such as (64) that can pass through the intestinal wall. The challenge with cancer chemotherapy is to identify complexes that select malignant cells and ignore healthy cells. Some Ru complexes undergo selective interaction with DNA, and are activated on irradiation, becoming potent oxidizing agents capable of carrying out cleavage of phosphodiester linkages. This method of treating cancers is known as phototherapy. Bleomycin (65) is representative of a class of drug that appears to function by binding to DNA and generating, on reaction with O2, an Fe(IV) (ferryl) species that oxygenates particular sites and leads to degradation. O O N Ga N N O 64 27.19 Anti-arthritis drugs Key point: Complexes of Au are effective against rheumatoid arthritis. Gold drugs are used in the treatment of rheumatoid arthritis, an inflammatory disease that affects the tissue around joints. The inflammation arises by the action of hydrolytic enzymes in cell compartments known as lysosomes that are associated with the Golgi apparatus (see Fig. 27.1). Although the mechanism of action is not established, it is known that Au accumulates in the lysosomes, and it is therefore possible that Au inhibits these hydrolytic enzymes. Another hypothesis is that Au(I) compounds deactivate singlet O2, a harmful species that can be formed by oxidation of superoxide. The mechanism for this reaction might involve promotion of intersystem crossing by the high spin-orbit coupling constant of the heavy element. Commonly administered drugs include sodium aurothiomalate (‘myochrisin’, 66), sodium aurothioglucose (‘solganol’, 67; the linkage between units is uncertain), and others, all of which feature Au(I) with linear coordination. Many are water-soluble polymers, but they cannot be administered orally because they undergo acid hydrolysis in the stomach. By contrast, the compound known as auranofin (68) can be given orally. Because Au(I) is chemically soft it is likely that it targets sulfur groups such as cysteine side chains in proteins. It is much more likely to survive in biological environments than Au(III), which is highly oxidizing. As expected, Au compounds lead to side effects, which include skin allergies as well as kidney and gastrointestinal problems. 65 Bleomycin C O 2 – u AS +a N – O2 C 66 Myochrisin H OH O H HO H O H H OH H 27.20 Imaging agents Key point: Particular organs and tissues are targeted according to the ligands that are present. a N 67 Solganol Au S n 776 27 Biological inorganic chemistry OAc PEt3 O AcO AcO Au S OAc 68 Auranofin O H 2O – O O O O N O Gd O N N O N Complexes of gadolinium(III) (f7) are used in magnetic resonance imaging (MRI), which has become an important technique in medical diagnosis. Through their effect on the relaxation time of 1H-NMR spectroscopic resonances, Gd(III) complexes are able to enhance the contrasts between different tissues and highlight details such as the abnormalities of the blood–brain barrier. A number of Gd(III) complexes are approved for clinical use, each exhibiting different degrees of rejection or retention by certain tissues, as well as stability, rates of water exchange, and magnitude of relaxation parameters. All are based on chelating ligands, particularly those having multiple carboxylate groups. One example is the complex (69) formed with the macrocyclic aminocarboxylate ligand DOTA, which is known as dotarem. Technetium is an artificial element that is produced by a nuclear reaction, but it has found an important use as an imaging agent. The active radionuclide is 99mTc (m for metastable), which decays by γ-emission and has a half-life of 6 h. Production of 99mTc involves bombarding 98Mo with neutrons and separating it as it is formed from the unstable product 99Mo: 98 69 Dotarem R N R N C C C N R Tc R N C C C N R N R 70 Cardiolyte O O N O N Tc S N O CO 2– 71 Tc-MAG-3 neutron capture -decay, 90h −emission,6 h Mo ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 99 Mo ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → 99m Tc ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ → -decay, 200ka 99 99 Tc ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ Ru ⎯ ⎯ ⎯ ⎯ ⎯ → High-energy γ-rays are less harmful to tissue than α- or β-particles. The chemistry of technetium resembles manganese except that higher oxidation states are much less oxidizing. A variety of substitution-inert Tc complexes can be made that, when injected into the patient, target particular tissues and report on their status. Complexes have been developed that target specific organs such as the heart (revealing tissue damage due to a heart attack), kidney (imaging renal function), or bone (revealing abnormalities and fracture lines). A good basis for organ targeting appears to be the charge on the complex: cationic complexes target the heart, neutral complexes target the brain, and anionic complexes target the bone and kidney. Of the different imaging agents, [Tc(CNR)6] (70) is the best established: known as cardiolyte, it is widely used as a heart imaging agent. The compound of Tc(V) with mercaptoacetyltriglycine (71), known as Tc-MAG-3, is used to image kidneys because of its rapid excretion. For imaging bone, complexes of Tc(VII) with hard diphosphonate ligands (72) are effective. Brain imaging is carried out with compounds such as ceretec (73). To produce Tc tracers, radioactive 99MoO42 is passed onto an anion exchange column, where it binds tightly until it decays to the pertechnate ion 99mTcO4 and the lower charge causes it to be eluted. The eluate is treated with a reducing agent, usually Sn(II), and the ligands required to convert it into the desired imaging agent. The resulting compound is then administered to the patient. Tracers are being developed that are much more specific for their targets. These tracers contain the metal in a stable coordination sphere that is covalently linked to a biologically active fragment. An example is the Gd contrast agent EP-210R (74), which contains four Gd3 complexes linked to a peptide that recognizes and binds to fibrin, a molecule produced by thrombi (blood clots). O HO O OH O N Gd O O P OH HO P –O OH R R O N OH2 N din in b m ro th tde e p g O O OH S S O O O O HN N O H N N H N H O O O N N O N H H N O N H O H N O O N H O N O O HN O HN NH2 O O O O O OH2 O H N O Gd N O O N H O OH OH O H N N O lC O O O O O O O H N N Gd N OH O OH2 O N N O O N 72 N O N – 5 O O Tc N N O Gd N O 74 O O N OH2 O Tc N N O O H 73 Ceretec Perspectives In this final section, we stand back from the material in the chapter and review it from a variety of different perspectives, from the point of view of individual elements and from the point of view of the contribution of bioinorganic chemistry to urgent social problems. Perspectives 27.21 The contributions of individual elements Key point: Elements are selected by Nature for their inherent useful properties and their availability. In this section we summarize the major roles of each element and correlate what we have discussed with emphasis on the element rather than the type of reaction that is involved. Na, K, and Li The ions of these elements are characterized by weak binding to hard ligands and their specificity is based on size and hydrophobicity that arises from a lower charge density. Compared to Na, K is more likely to be found coordinated within a protein and is more easily dehydrated. Both Na and K are important agents in controlling cell structure through osmotic pressure, but whereas Na is ejected from cells, K is accumulated, contributing to a sizeable potential difference across the cell membrane. This differential is maintained by ion pumps, in particular the Na,K-ATPase, also known as the Na-pump. The electrical energy is released by specific gated ion channels, of which the K channel (Section 27.3) has been studied the most. An important related issue is the widespread use of simple Li compounds (particularly Li2CO3) as psychotherapeutic agents in the treatment of mental disorders, notably bipolar disorder (manic depression). One possibility is that hydrated Li binds tightly in the Na or K channels in place of the dehydrated ions that are normally transported selectively in the filter region of these proteins. As Li ions are highly labile, we can be confident that the aqua ion itself, or a complex with an abundant ligand, is the active species. As a simple aqua ion, Li is strongly solvated, in fact the solvated radius is greater than that of Na. Mg Magnesium ions are the dominant 2 ion in cytoplasm and the only ones to occur above millimolar levels in the free, uncomplexed state. The energy currency for enzyme catalysis, ATP, is always present as its Mg2 complex. Magnesium has a special role in the light-harvesting molecule chlorophyll because it is a small 2 cation that is able to adopt octahedral geometry and can stabilize a structure without promoting energy loss by fluorescence. The Mg2 ion is a weak acid catalyst and is the active metal ion in rubisco, the highly abundant enzyme responsible for removing from the atmosphere some 100 Gt of CO2 per year. Rubisco is activated by weak binding of Mg2 to two carboxylates and a special carbamate ligand, leaving three exchangeable water molecules. Ca Calcium ions are important only in eukaryotes. The bulk of biological Ca is used for structural support and devices such as teeth. The selection of Ca for this function is due to the insolubility of Ca carbonate and phosphate salts. However, a tiny amount of Ca is used as the basis of a sophisticated intracellular signalling system. The principle of this process is that Ca is suited for rapid coordination to hard acid ligands, especially carboxylates from protein side chains, and has no preference for any particular coordination geometry. Mn Manganese has several oxidation states, most of which are very oxidizing. It is well suited as a redox catalyst for reactions involving positive reduction potentials. One reaction in particular, in which H2O is used as the electron donor in photosynthesis, is responsible for producing almost all the O2 in the Earth’s atmosphere. This reaction involves a special Mn4Ca cluster. Manganese(II) is also used as a weak acidbase catalyst in some enzymes. Spectroscopic detectability varies depending on oxidation state: EPR has been useful for Mn(II) and for particular states of the Mn cluster that constitutes the catalyst for the evolution of O2. Fe Versatile Fe is probably essential to all organisms and was certainly a very early element in biology. Three oxidation states are important, namely Fe(II), Fe(III), and Fe(IV). Active sites based on Fe catalyse a great variety of redox reactions ranging from electron transfer to oxygenation, as well as acidbase reactions that include reversible O2 binding, dehydration/hydration, and ester hydrolysis. Iron-containing active sites feature ligands ranging from soft donors such as sulfide (as in FeS clusters) to hard donors such as carboxylate. The porphyrin macrocycle is particularly important as a ligand. Iron(II) in various coordination environments is used to bind O2, either reversibly or as a prerequisite for activation. Iron(III) is a good Lewis acid, whereas the Fe(IV) O (ferryl) group may be considered as Nature’s way of managing a reactive O atom for insertion into CH bonds. Cells contain very little uncomplexed Fe(II) and extremely low levels of Fe(III). These ions are toxic, particularly in terms of their reaction with peroxides, which generates the hydroxyl radical. Primary uptake into organisms from minerals poses problems because Fe is found predominantly as Fe(III), salts of which are insoluble at neutral pH (see Fig. 5.12). Iron uptake, delivery, and storage are controlled by sophisticated transport systems, including a special storage protein known as ferritin. Iron porphyrins (as found 777 778 27 Biological inorganic chemistry in cytochromes) show intense UV–visible absorption bands and most active sites with unpaired electrons give rise to characteristic EPR spectra. Co Cobalt and nickel are among the most ancient biocatalysts. Cobalt is processed only by microorganisms and higher organisms must ingest it as vitamin B12 in which Co is complexed by a special macrocycle called corrin. Complexes in which the fifth ligand is a benzimidazole that is covalently linked to the corrin ring are known as cobalamins. Cobalamins are cofactors in enzymes that catalyse alkyl transfer reactions and many radical-based rearrangements. Alkyl transfer reactions exploit the high nucleophilicity of Co(I). In the special cofactor known as coenzyme B12, the sixth ligand to Co(III) is a carbanion donor atom from deoxyadenosine. Radical-based rearrangements involve the ability of coenzyme B12 to undergo facile homolytic cleavage of the CoC bond, producing stable low-spin Co(II) and a carbon radical that can abstract a hydrogen atom from substrates. Cobalamin-containing enzymes show strong UV–visible absorption bands; EPR spectra are observed for Co(II). Ni Nickel is important in bacterial enzymes, notably hydrogenases, where it also uses the 3 and 1 oxidation states, which are rare in conventional chemistry. A particularly remarkable enzyme, coenzyme A synthase, uses Ni to produce CO and then react it with CH3 (provided by a cobalamin enzyme) to produce a CC bond in the form of an acetyl ester. Nickel is also found in plants as the active site of urease. Urease was the first enzyme to be crystallized (in 1926), yet it was not until 1976 that it was discovered to contain Ni. Cu Unlike Fe, copper probably became important only after O2 had become established in the Earth’s atmosphere and it became available as soluble Cu(II) salts rather than insoluble sulfides (Cu2S). The main role of Cu is in electron transfer reactions at the higher end of the potential scale and catalysis of redox reactions involving O2. It is also used for reversible O2 binding. Both Cu(II) and Cu(I) are strongly bound to biological ligands, particularly soft bases. Free Cu ions are highly toxic and almost absent from cells. Zn Zinc is an excellent Lewis acid, forming stable complexes with ligands such as N and S donors, and catalysing reactions such as ester and peptide hydrolysis. The biological importance of Zn stems largely from its lack of redox chemistry, although its common adoption of di-, tri-, and tetrathiolate ligation provides a link to the redox chemistry of cysteine/cystine interconversions. Zinc is used as a structure former in enzymes and proteins that bind to DNA. A major problem has been the lack of good spectroscopic methods for studying this d10 ion. In some cases, Zn enzymes have been studied by EPR, after substituting the Zn by Co(II). Mo and W Molybdenum is an abundant element that is probably used by all organisms as a redox catalyst for the transfer of O atoms derived from H2O. In these oxo-transfer enzymes the Mo is always part of a larger pterin-containing cofactor in which it is coordinated by a special dithiolene ligand. Interconversion between Mo(IV) and Mo(VI) usually results in a change in the number of terminal oxo ligands, and recovery of the starting material occurs by single-electron transfer reactions with Mo(V) as an intermediate. Aside from oxo-transfer and related reactions, Mo has another intriguing role, that of nitrogen fixation, in which it is part of a special FeS cluster. Use of W is confined to prokaryotes, where it is also used as a redox catalyst, but in reactions where a stronger reducing agent is required. Si Silicon is often neglected among biological elements, yet its turnover in some organisms is comparable to that of carbon. Silica is an important material for the fabrication of the exoskeleton and of prickly defensive armour in plants. Pt, Au, Bi, and Ru These elements have no known deliberate biological functions and are foreign agents to biological systems, acting under normal conditions as poisons. However, used in controlled procedures, and dressed up by complexation to target a particular site, they are potent drugs, active against a range of diseases and disorders. 27.22 Future directions Key points: Biological metals and metalloproteins have important futures in medicine, energy production, green synthesis, and nanotechnology. The pioneering studies of the structures and mechanism of ion channels mentioned in Section 27.3 are providing important new leads in neurophysiology, including the rational design of drugs that can block or modify their action in some way. New functions for Ca Further reading 779 are continually emerging, and one intriguing aspect is its role in determining the left–right asymmetry of higher organisms, a prime example being the specific placements of heart and liver in the body cavity. The so-called Notch signalling pathway in embryonic cells depends on transient extracellular bursts of Ca2 that are dependent in some way on the activity of an H/K-ATPase. There is also a growing awareness of the role of Zn and Zn transport proteins in control of cellular activity, and also of neural transmission. Indeed, the term metalloneurochemistry has been coined to describe the study of metal-ion function in the brain and nervous system at the molecular level. An important challenge is to map out the distribution and flow of Zn in tissue such as brain, and advances are being made in the design of fluorescent ligands that will bind Zn selectively at cellular levels and report on its transport across different zones, for example the synaptic junctions. Metal ions are involved in protein folding, and it is believed that Cu, in particular, may have an important role in fatal neurodegenerative disorders. These roles include controlling the behaviour of prions involved in transmittable diseases such as spongiform encephalopathy (Creuzfeldt–Jakob disease, the human form of ‘mad cow’ disease) as well as amyloid peptides that are implicated in Alzheimer’s disease. In many regions of the world, rice is the staple food but this commodity is low in Fe. Thus transgenic techniques are being used to improve Fe content. The object is to produce better plant siderophores and improve Fe storage (by enhanced expression of the ferritin gene). Enzymes tend to show much higher catalytic rates and far higher selectivity than synthetic catalysts, leading naturally to greater efficiency and lower energy costs. The principal disadvantages of using enzymes as industrial catalysts are their lower thermal stability, limitations on solvent and pH conditions, and a large mass per active unit. There is much interest in achieving enzyme-like catalytic performance with small synthetic molecules, a concept that is known as ‘bioinspired catalysis’. The idea is to reproduce, using all the tools of synthetic chemistry, the properties of an enzyme trimmed down to its smallest fully functional component. Examples of bioinspired catalysts have already been described: areas of particular interest for industrial production are the conversion of methane to methanol (Section 27.10), activation of N2 to produce cheap fertilizers, and production of hydrogen. In the not so distant future, when fossil fuels have been depleted, H2 will become an important energy carrier, used either directly or indirectly (after conversion into fuels such as alcohol) to power vehicles of all kinds. One of the scientific challenges is how to obtain efficient electrolytic production of H2 from water, given that electricity will be widely available from a variety of sources. This process requires demanding conditions of temperature and overpotential (Section 10.4), or catalysts that are currently based on Pt and other precious metals. However, Nature has already shown us that rapid hydrogen cycling is possible under mild conditions by using just the common metals Fe and Ni. A related challenge is the synthesis of efficient electrocatalysts that can convert water to O2 without requiring a large overpotential, not because there is a need for O2 itself, but because it is an essential byproduct of electrolytic or photolytic H2 production (see Box 10.3). Once again, we can turn to the biosphere for inspiration because by elucidating the mechanism of the Mn catalyst, we might synthesize new catalysts that are both cheap and durable. We have seen the exquisite structures of materials that are produced by organisms. This understanding is now leading to new directions in nanotechnology (Chapter 25). For example, sponge-like single crystals of calcite, having intricate morphological features, have been produced on polymer membranes formed by templating the skeletal plates of the sea urchin. Another recent development is the production of Pd nanoclusters by hydrogenoxidizing bacteria, the action of hydrogenases making available controlled electron flow to effect the electroplating of Pd on to microscopic sites. FURTHER READING J.J.R. Frausto da Silva and R.J.P. Williams, The biological chemistry of the elements. Oxford University Press (2001). An excellent, detailed book that looks at the broader picture of the relationship between elements and life. L. Que Jr. and W.B. Tolman, Bio-coordination chemistry. Comprehensive coordination chemistry, Vol. 8. Elsevier (2004). A text providing particularly detailed insight into model compounds. R.R. Crichton, F. Lallemand, I.S.M. Psalti, and R.J. Ward, Biological inorganic chemistry. Elsevier (2007). A modern introduction to biological inorganic chemistry. E. Gouaux and R. MacKinnon, Principles of selective ion transport in channels and pumps. Science, 2005, 310, 1461. An article linking detailed three-dimensional structural data of giant proteins with physiological function and the chemistry of Group 1 and 2 metal ions and Cl−. 780 27 Biological inorganic chemistry R.K.O. Sigel and A.M. Pyle, Alternative roles for metal ions in enzyme catalysis and the implications for ribozyme chemistry. Chem. Rev., 2007, 107, 97. A review describing the role of metal ions, particularly Mg, as active centres in catalysts based on RNA instead of proteins. E. Kimura, Model studies for molecular recognition of carbonic anhydrase and carboxypeptidase. Acc. Chem. Res., 2001, 34, 171. This review describes the acid–base and catalytic properties of small Zn complexes in an effort to understand how Zn enzymes function. K.N. Ferreira, T.M. Iverson, K. Maghlaoui, J. Barber, and S. Iwata, Architecture of the photosynthetic oxygen-evolving center. Science, 2004, 303, 1831. A seminal article on the structure of the catalyst responsible for O2 in the atmosphere. L. Que, Jr., The road to non-heme oxoferryls and beyond. Acc. Chem. Res., 2007, 40, 493. A stimulating account of efforts to understand the chemistry of Fe(IV) species and produce new catalysts for organic oxygenation reactions. E.A. Lewis and W.B. Tolman, Reactivity of dioxygen–copper systems. Chem. Rev., 2004, 104, 1047. A review of small molecule analogues of Cu-containing enzymes. S.C. Wang and P.A. Frey, S-adenosylmethionine as an oxidant: the radical SAM superfamily. Trends in Biochemical Sciences, 2007, 32, 101. An authoritative account of the discovery of radical SAM enzymes. It categorises the different classes of enzymes and describes the mechanisms by which the [4Fe4S] cluster cleaves SAM to initiate radical reactions. H.B. Gray, B.G. Malmström, and R.J.P. Williams, Copper coordination in blue proteins. J. Biol. Inorg. Chem., 2000, 5, 551. An article that draws together and summarizes the theories spanning more than 30 years of research on blue Cu centres. P.J. Kiley and H. Beinert, The role of FeS proteins in sensing and regulation in bacteria. Curr. Opin. Chem. Biol., 2003, 6, 182. A review describing how FeS clusters are involved in sensing. J. Green and M.S. Paget, Bacterial redox centres. Nature Reviews, 2004, 2, 954. A general review on the mechanisms by which reactive oxygen species are sensed in cells. S. Mann, Biomineralisation: principles and concepts in bioinorganic materials chemistry. Oxford University Press (2001). M.D. Archer and J. Barber (ed.), Molecular to global photosynthesis. Imperial College Press (2004). C.W. Cady, R.H. Crabtree, and G.W. Brudvig, Functional models for the oxygen-evolving complex of photosystem II. Coord. Chem. Rev., 2008, 252, 444. A review of recent efforts to understand and mimic the chemistry of the Mn4Ca cluster that converts water to O2. M.J. Hannon, Supramolecular DNA recognition. Chem. Soc. Rev., 2007, 36, 280. An account of efforts to make large metal complexes that can recognise certain DNA sequences. M.A. Jakupec, M. Galanski, V.B. Arion, C.G. Hartinger, and B. Keppler, Antitumour metal compounds: more than theme and variations. Dalton Transactions, 2008, 2,183. P.C.A. Bruijnincx and P. J Sadler, New trends for metal complexes with anticancer activity. Curr. Opin. Chem. Biol., 2008, 12, 197. Two complementary reviews of recent developments in anti-cancer drugs based on metal complexes. P. Caravan, Strategies for increasing the sensitivity of gadolinium-based MRI contrast agents. Chem. Soc. Rev., 2006, 35, 512. A review of developments in producing selective Gd-based magnetic resonance imaging reagents. B.E. Mann and R. Motterlini, CO and NO in medicine. Chem. Commun., 2007, 4197. A review of the roles of NO and CO in biology and medicine. EXERCISES 27.2 In zinc enzymes, ‘spectroscopically silent’ Zn(II) can often be replaced by Co(II) with high retention of activity. Explain the principles by which this substitution can be exploited to obtain structural and mechanistic information. 27.3 Compare and contrast the acid–base catalytic activities of Zn(II), Fe(III), and Mg(II). 27.4 Propose physical methods that would allow you to determine whether a reactive intermediate isolated by rapid freeze quenching contains Fe(V). 27.5 Figure 27.58 shows Mössbauer spectra of a sample of ferredoxin from chloroplasts at 77 K. With reference to Section 8.7, interpret the data with regard to the oxidation states and spin states of the two Fe atoms and comment on the electron delocalization at this temperature. 27.6 The structure of the P-cluster in nitrogenase differs significantly between oxidized and reduced states. Comment on this observation in the light of proposals that it participates in long-range electron transfer. Oxidized Relative transmission 27.1 Calcium-binding proteins can be studied by using lanthanoid ions (Ln3). Compare and contrast the coordination preferences of the two types of metal ion and suggest techniques in which lanthanoid ions would be useful. Reduced –3 –2 –1 0 1 2 3 Isomer shift, δ/(mm s–1) Figure 27.58 Mössbauer spectra of a sample of ferredoxin from chloroplasts at 77 K. 27.7 Microorganisms can synthesize the acetyl group (CH3CO) by direct combination of methyl groups with CO. Make some predictions about the metals that are involved. 781 Problems PROBLEMS 27.1 With reference to details that have been discussed in Section 27.3 for the K channel, predict the properties of Na, Ca2, and Cl− binding sites that would be important for providing selectivity in their respective transmembrane ion transporters. 2– O 27.4 In justifying research into small molecule catalysts for producing NH3 from N2, it is sometimes stated that nitrogenase is an ‘efficient’ enzyme: how true is this statement? Comment critically on the argument that ‘knowing the three-dimensional structure of nitrogenase has not enlightened us as to its mechanism of action’ and discuss how this view might be valid more generally for enzymes for which a structure is known. 27.5 ‘In Mo enzymes, the bond between a terminal oxo anion and Mo(VI) is usually written as a double bond, whereas it is more correctly assigned as a triple bond.’ Discuss this statement. Suggest how a terminal oxido ligand influences the reactivity of other 2– O S O S O 27.2 Comment on the implications of discovering, by microwave detection, substantial levels of O2 on a planet in another solar system. 27.3 Apart from direct O-atom transfer (Fig. 27.43), another mechanism proposed for Mo enzymes is indirect O-atom transfer, also known as coupled electron–proton transfer. In this mechanism, as shown in Fig. 27.59 for sulfite oxidase, the O atom that is transferred originates instead from an uncoordinated H2O molecule. Propose a way to distinguish between direct and indirect O-atom transfer mechanisms. O VI Mo OH S HO S O O Mo S-cys S 2– –e – –H + O O S O S O V Mo S O IV OH S-cys S H2O H O OH S-cys –e – –H + O S O IV OH 2 Mo S S-cys Figure 27.59 The mechanism referred to in Problem 27.3. coordination sites on the Mo atom and explain how a terminal sulfido ligand (as occurs in xanthine oxidase) would alter the properties of the active site. This page intentionally left blank Resource section 1 Selected ionic radii Ionic radii are given (in picometres, pm) for the most common oxidation states and coordination geometries. The coordination number is given in parentheses. All d-block species are low-spin unless labelled with†, in which case values for high-spin are quoted. Most data are taken from R.D. Shannon, Acta Cryst., 1976, A32, 751, where values for other coordination geometries can be found. Where Shannon values are not available, Pauling ionic radii are quoted and are indicated by . 1 2 Li 59 (4) 76 (6) 92 (8) Be2 27 (4) 45 (6) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 B3 11 (4) 27 (6) C4 15 (4) 16 (6) N3 146 (4) O2 138 (4) 140 (6) 142 (8) F 131 (4) 133 (6) Ne 112 P5 29 (4) 38 (6) S2 184 (6) Cl 181 (6) Ar 154 P3 44 (6) S6 12 (4) 29 (6) Cl7 8 (4) 27 (6) N3 16 (6) Na 99 (4) 102 (6) 32 (8) Mg2 49 (4) 72 (6) 103 (8) Al3 39 (4) 53 (6) Si4 26 (4) 40 (6) S4 37 (6) K 137 (4) 138 (6) 151 (8) Ca2 100 (6) 112 (8) Sc3 75 (6) 87 (8) Ti4 42 (4) 61 (6) 74 (8) V5 36 (4) 54 (6) Cr 6 26 (4) 44 (6) Mn7 25 (4) 46 (6) Fe6 25 (4) Co4 40 (4) 53 (6)† Ni4 48 (6) Cu3 54 (6) Ti3 67 (6) V4 58 (6) 72 (8) Cr 5 49 (6) Mn6 26 (4) Fe4 58 (6) Co3 55 (6) Ni3 56 (6) Cu2 57 (4) 73 (6) Ti2 86 (6) V3 64 (6) Cr 4 41 (4) 55 (6) Mn5 33 (4) 63 (6) Fe3 49 (4)† 55 (6) 78 (8)† Co2 58 (4)† 65 (6) 90 (8) Ni2 55 (4) 69 (8) Cu 60 (4) 77 (6) V2 79 (6) Cr 3 62 (6) Mn4 37 (4) 53 (6) Fe2 63 (4)† 61 (6) 92 (8)† Cr 2 73 (6) Mn3 65 (6) Zn2 60 (4) 74 (6) 90 (8) Ga3 47 (4) 62 (6) Ge4 39 (4) 53 (6) As5 34 (4) 46 (6) Se2 198 (6) Br 196 (6) Ge2 73 (6) As3 58 (6) Se6 28 (4) 42 (6) Br7 39 (6) Kr 169 Se4 50 (6) Mn2 67 (6) 96 (8) Rb 148 (6) 160 (8) Sr2 118 (6) 126 (8) Y3 90 (6) 102 (8) Zr 4 59 (4) 72 (6) 84 (8) Nb5 48 (4) 64 (6) 74 (8) Mo6 41 (4) 59 (6) Tc7 37 (4) 56 (6) Ru8 36 (4) Rh5 55 (6) Pd4 62 (6) Ag3 67 (4) 75 (6) Nb4 68 (6) 79 (8) Mo5 46 (4) 61 (6) Tc5 60 (6) Ru7 38 (4) Rh4 60 (6) Pd3 76 (6) Ag2 79 (4) 94 (6) Nb3 72 (6) Mo4 65 (6) Tc4 66 (6) 95 (8) Ru5 71 (6) Rh3 67 (6) Pd2 64 (4) 86 (6) Ag 67 (2) 100 (4) 115 (6) Mo3 69 (6) Ru4 62 (6) Ru3 68 (6) Pd 59 (2) Cd2 78 (4) 95 (6) 110 (8) In3 62 (4) 80 (6) 92 (8) Sn4 55 (4) 69 (6) 81 (8) Sb5 60 (6) Te6 43 (4) 56 (6) I 220 (6) Xe 190 Sn2 102 (6) Sb3 76 (6) Te4 66 (4) 97 (6) I7 42 (4) 53 (6) Xe8 40 (4) 48 (6) 784 1 Cs 167 (6) 174 (8) Resource section 1 2 3 2 Ba 135 (6) 142 (8) 4 3 La 103 (6) 116 (8) 5 4 Hf 58 (4) 71 (6) 83 (8) 6 7 8 9 5 Ta 64 (6) 74 (8) 6 W 42 (4) 60 (6) 7 8 Re 38 (4) 53 (6) Os 39 (4) Ta4 68 (6) W5 62 (6) Re6 55 (6) Ta3 72 (6) W4 66 (6) 10 5 11 12 5 13 2 3 14 4 Pt 57 (6) Au 57 (6) Hg 96 (4) 102 (6) 114 (8) Tl 75 (4) 89 (6) 98 (8) Pb 65 (4) 78 (6) 94 (8) Bi 76 (6) Po 67 (6) Os7 53 (6) Ir 4 63 (6) Pt4 63 (6) Au3 68 (4) 85 (6) Hg 119 (6) Tl 150 (6) 159 (8) Pb2 119 (6) 129 (8) Bi3 103 (6) 117 (8) Po4 94 (6) 108 (8) Re5 58 (6) Os6 55 (6) Ir 3 68 (6) Pt2 60 (4) 80 (6) Au 137 (6) Re4 63 (6) Os5 58 (6) Tb4 76 (6) 88 (8) Dy3 91 (6) 103 (8) Ho3 90 (6) 102 (8) Er3 89 (6) 100 (8) Tm3 88 (6) 99 (8) Yb3 87 (6) 99 (8) Lu3 86 (6) 98 (8) Tb3 92 (6) 104 (8) Dy2 107 (6) 119 (8) Tm2 103 (6) 109 (8) Yb2 102 (6) 114 (8) Md No2 110 (6) Ra2 170 (8) Lanthanoids Ce4 87 (6) 97 (8) Pr 4 85 (6) 96 (8) Nd3 98 (6) 111 (8) Ce3 101 (6) 114 (8) Pr 3 99 (6) 113 (8) Nd2 129 (8) Pa5 78 (6) 95 (8) U6 52 (4) 73 (6) 100 (8) Pa4 90 (6) 101 (8) Pa3 104 (6) Pm3 97 (6) 109 (8) Sm3 96 (6) 108 (8) Eu3 95 (6) 107 (8) Gd3 94 (6) 105 (8) Sm2 127 (8) Eu2 117 (6) 125 (8) Np7 72 (6) Pu6 71 (6) Am4 85 (6) 95 (8) Cm4 85 (6) 95 (8) Bk4 63 (6) 93 (8) Cf 4 82 (6) 92 (8) U5 76 (6) Np6 72 (6) Pu5 74 (6) Am3 98 (6) 123 (8) Cm3 97 (6) Bk3 96 (6) Cf 3 95 (6) U4 89 (6) 100 (8) Np5 75 (6) Pu4 86 (6) 96 (8) Am2 126 (8) U3 103 (6) Np4 87 (4) 98 (6) Pu3 100 (6) Actinoids Th4 94 (6) 110 (8) Np3 101 (6) Np2 110 (6) Es Fm Lr 5 16 Ir 57 (6) Os4 63 (6) Fr 196 (6) 15 5 6 17 7 At 62 (6) 18 Resource section 2 Electronic properties of the elements Ground-state electron configurations of atoms are determined experimentally from spectroscopic and magnetic measurements. The results of these determinations are listed below. They can be rationalized in terms of the building-up principle, in which electrons are added to the available orbitals in a specific order in accord with the Pauli exclusion principle. Some variation in order is encountered in the d and f blocks to accommodate the effects of electron–electron interaction more faithfully. The closed-shell configuration 1s2 characteristic of helium is denoted [He] and likewise for the other noble-gas element configurations. The ground-state electron configurations and term symbols listed below have been taken from S. Fraga, J. Karwowski, and K.M.S. Saxena, Handbook of atomic data. Elsevier, Amsterdam (1976). The first three ionization energies of an element E are the energies required for the following processes: I1: E(g) m E+(g) + e–(g) I2: E+(g) m E2+(g) + e–(g) I3: E2+(g) m E3+(g) + e–(g) The electron affinity Eea is the energy released when an electron attaches to a gas-phase atom: Eea: E(g) + e–(g) m E–(g) The values given here are taken from various sources, particularly C.E. Moore, Atomic energy levels, NBS Circular 467, Washington (1970) and W.C. Martin, L. Hagan, J. Reader, and J. Sugar, J. Phys. Chem. Ref. Data, 1974, 3, 771. Values for the actinoids are taken from J.J. Katz, G.T. Seaborg, and L.R. Morss (ed.), The chemistry of the actinide elements. Chapman & Hall, London (1986). Electron affinities are from H. Hotop and W.C. Lineberger, J. Phys. Chem. Ref. Data, 1985, 14, 731. For conversions to kilojoules per mole and reciprocal centimetres, see the back inside cover. Atom 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Ionization energy (eV) 1s1 1s2 [He]2s1 [He]2s2 [He]2s22p1 [He]2s22p2 [He]2s22p3 [He]2s22p4 [He]2s22p5 [He]2s22p6 [Ne]3s1 [Ne]3s2 [Ne]3s23p1 [Ne]3s23p2 [Ne]3s23p3 [Ne]3s23p4 [Ne]3s23p5 [Ne]3s23p6 [Ar]4s1 [Ar]4s2 I1 I2 13.60 24.59 5.320 9.321 8.297 11.257 14.53 13.62 17.42 21.56 5.138 7.642 5.984 8.151 10.485 10.360 12.966 15.76 4.340 6.111 54.51 75.63 18.21 25.15 24.38 29.60 35.11 34.97 40.96 47.28 15.03 18.83 16.34 19.72 23.33 23.80 27.62 31.62 11.87 Electron I3 122.4 153.85 37.93 47.88 47.44 54.93 62.70 63.45 71.63 80.14 28.44 33.49 30.18 34.83 39.65 40.71 45.71 50.89 Atom Ionization energy (eV) affinity Eea (eV) +0.754 –0.5 +0.618 ≤0 +0.277 +1.263 –0.07 +1.461 +3.399 –1.2 +0.548 ≤0 +0.441 +1.385 +0.747 +2.077 +3.617 –1.0 +0.502 +0.02 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr [Ar]3d14s2 [Ar]3d24s2 [Ar]3d34s2 [Ar]3d54s1 [Ar]3d54s2 [Ar]3d64s2 [Ar]3d74s2 [Ar]3d84s2 [Ar]3d104s1 [Ar]3d104s2 [Ar]3d104s24p1 [Ar]3d104s24p2 [Ar]3d104s24p3 [Ar]3d104s24p4 [Ar]3d104s24p5 [Ar]3d104s24p6 [Kr]5s1 [Kr]5s2 [Kr]4d15s2 [Kr]4d15s2 Electron I1 I2 I3 6.54 6.82 6.74 6.764 7.435 7.869 7.876 7.635 7.725 9.393 5.998 7.898 9.814 9.751 11.814 13.998 4.177 5.695 6.38 6.84 12.80 13.58 14.65 16.50 15.64 16.18 17.06 18.17 20.29 17.96 20.51 15.93 18.63 21.18 21.80 24.35 27.28 11.03 12.24 13.13 24.76 27.48 29.31 30.96 33.67 30.65 33.50 35.16 36.84 39.72 30.71 34.22 28.34 30.82 36.27 36.95 40.42 43.63 20.52 22.99 affinity Eea (eV) +0.30 +1.2 +0.81 +2.021 +3.365 –1.0 +0.486 +0.05 786 Resource section 2 Atom 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ionization energy (eV) [Kr]4d45s1 [Kr]4d55s1 [Kr]4d55s2 [Kr]4d75s1 [Kr]4d85s1 [Kr]4d10 [Kr]4d105s1 [Kr]4d105s2 [Kr]4d105s25p1 [Kr]4d105s25p2 [Kr]4d105s25p3 [Kr]4d105s25p4 [Kr]4d105s25p5 [Kr]4d105s25p6 [Xe]6s1 [Xe]6s2 [Xe]5d16s2 [Xe]4f15d16s2 [Xe]4f36s2 [Xe]4f46s2 [Xe]4f56s2 [Xe]4f66s2 [Xe]4f76s2 [Xe]4f75d16s2 [Xe]4f96s2 [Xe]4f106s2 [Xe]4f116s2 [Xe]4f126s2 [Xe]4f136s2 [Xe]4f146s2 [Xe]4f145d16s2 [Xe]4f145d26s2 Electron I1 I2 I3 6.88 7.099 7.28 7.37 7.46 8.34 7.576 8.992 5.786 7.344 8.640 9.008 10.45 12.130 3.894 5.211 5.577 5.466 5.421 5.489 5.554 5.631 5.666 6.140 5.851 5.927 6.018 6.101 6.184 6.254 5.425 6.65 14.32 16.15 15.25 16.76 18.07 19.43 21.48 16.90 18.87 14.63 18.59 18.60 19.13 21.20 25.08 10.00 11.06 10.85 10.55 10.73 10.90 11.07 11.24 12.09 11.52 11.67 11.80 11.93 12.05 12.19 13.89 14.92 25.04 27.16 29.54 28.47 31.06 32.92 34.83 37.47 28.02 30.50 25.32 27.96 33.16 32.10 35.24 37.51 19.17 20.20 21.62 20.07 22.28 23.42 24.91 20.62 21.91 22.80 22.84 22.74 23.68 25.03 20.96 23.32 Atom Ionization energy (eV) affinity Eea (eV) +0.3 +1.2 +1.07 +1.971 +3.059 –0.8 73 Ta 74 W 75 Re 76 Os 77 Ir 78 Pt 79 Au 80 Hg 81 TI 82 Pb 83 Bi 84 Po 85 At 86 Rn 87 Fr 88 Ra 89 Ac 90 Th 91 Pa 92 U 93 Np 94 Pu 95 Am 96 Cm 97 Bk 98 Cf 99 Es 100 Fm 101 Md 102 No 103 Lr [Xe]4f145d36s2 [Xe]4f145d46s2 [Xe]4f145d56s2 [Xe]4f145d66s2 [Xe]4f145d76s2 [Xe]4f145d96s1 [Xe]4f145d106s1 [Xe]4f145d106s2 [Xe]4f145d106s26p1 [Xe]4f145d106s26p2 [Xe]4f145d106s26p3 [Xe]4f145d106s26p4 [Xe]4f145d106s26p5 [Xe]4f145d106s26p6 [Rn]7s1 [Rn]7s2 [Rn]6d17s2 [Rn]6d27s2 [Rn]5f 26d17s2 [Rn]5f36d17s2 [Rn]5f46d17s2 [Rn]5f67s2 [Rn]5f77s2 [Rn]5f76d17s2 [Rn]5f97s2 [Rn]5f107s2 [Rn]5f117s2 [Rn]5f127s2 [Rn]5f137s2 [Rn]5f147s2 [Rn]5f146d17s2 I1 I2 I3 7.89 7.89 7.88 8.71 9.12 9.02 9.22 10.44 6.107 7.415 7.289 8.42 9.64 10.75 4.15 5.278 5.17 6.08 5.89 6.19 6.27 6.06 5.99 6.02 6.23 6.30 6.42 6.50 6.58 6.65 4.6 15.55 17.62 13.06 16.58 17.41 18.56 20.52 18.76 20.43 15.03 16.69 18.66 16.58 21.76 23.84 26.01 24.87 26.95 29.02 30.05 34.20 29.83 31.94 25.56 27.98 30.06 21.76 10.15 11.87 11.89 11.7 14.9 11.7 11.7 12.0 12.4 12.3 12.5 12.6 12.7 12.8 13.0 14.8 32.13 34.20 19.69 20.50 18.8 19.1 19.4 21.8 22.4 21.2 22.3 23.6 24.1 24.4 25.4 27.0 23.0 Electron affinity Eea (eV) Resource section 3 Standard potentials The standard potentials quoted here are presented in the form of Latimer diagrams (Section 5.12) and are arranged according to the blocks of the periodic table in the order s, p, d, f. Data and species in parentheses are uncertain. Most of the data, together with occasional corrections, come from A.J. Bard, R. Parsons, and J. Jordan (eds.), Standard potentials in aqueous solution. Marcel Dekker, New York (1985). Data for the actinoids are from L.R. Morss, The chemistry of the actinide elements, Vol. 2 (eds. J.J. Katz, G.T. Seaborg, and L.R. Morss). Chapman & Hall, London (1986). The value for [Ru(bpy)3]3+/2+ is from B. Durham, J.L. Walsh, C.L. Carter, and T.J. Meyer, Inorg. Chem., 1980, 19, 860. Potentials for carbon species and some d-block elements are taken from S.G. Bratsch, J. Phys. Chem. Ref. Data, 1989, 18, 1. For further information on standard potentials of unstable radical species see D.M. Stanbury, Adv. Inorg. Chem., 1989, 33, 69. Potentials in the literature are occasionally reported relative to the standard calomel electrode (SCE) and may be converted to the H+/H2 scale by adding 0.2412 V. For a detailed discussion of other reference electrodes, see D.J.G. Ives and G.J. Janz, Reference electrodes. Academic Press, New York (1961). s Block Group 1 s Block Group 2 788 Resource section 3 pB lock roup13 G pB lock roup14 G Resource section 3 pB lock roup15 G 789 790 pB lock Resource section 3 roup16 G Resource section 3 pB lock roup17 G 791 792 Resource section 3 pB lock roup18 G dB lock roup3 G dB lock roup4 G dB lock roup5 G Resource section 3 dB lock roup5 G d Block Group 6 (Continued) 793 794 Resource section 3 dB lock roup6 G d Block Group 7 (Continued) Resource section 3 d Block Group 8 795 796 Resource section 3 dB lock roup9 G dB lock roup10 G Resource section 3 dB lock roup11 G 797 798 dB lock Resource section 3 roup12 G fB lock nth La noid a s Resource section 3 fB lock ctinoid A s 799 Resource section 4 Character tables The character tables that follow are for the most common point groups encountered in inorganic chemistry. Each one is labelled with the symbol adopted in the Schoenflies system of nomenclature (such as C3v). Point groups that qualify as crystallographic point groups (because they are also applicable to unit cells) are also labelled with the symbol adopted in the International System (or the Hermann–Mauguin system, such as 2/m). In the latter system, a number n represents an n-fold axis and a letter m represents a mirror plane. A diagonal line indicates that a mirror plane lies perpendicular to the symmetry axis and a bar over the number indicates that the rotation is combined with an inversion. The symmetry species of the p and d orbitals are shown on the right of the tables. Thus, in C2v, a px orbital (which is proportional to x) has B1 symmetry. The functions x, y, and z also show the transformation properties of translations and of the electric dipole moment. The set of functions that span a degenerate representation (such as x and y, which jointly span E in C3v) are enclosed in parentheses. The transformation properties of rotation are shown by the letters R on the right of the tables. The value of h is the order of the group. The groups C1, Cs, Ci C1 (1) E A 1 h=1 Cs = Ch (m) E Sh Aa 1 1 x, y, Rz Aq 1 –1 z, Rx, Ry Ci = S2 (1) E x2, y2, z2, xy Ag 1 1 yz, zx Au 1 –1 h=2 i h=2 Rx, Ry, Rz x2, y2, z2, xy, zx, yz x, y, z The groups Cn C2 (2) E C2 h=2 A 1 1 B 1 –1 C4 (4) E C4 C2 C43 A 1 1 1 1 B 1 –1 1 –1 E « 1 i 1 i º ® ® ¬ » 1 i 1 i ® ®¼ z, Rz x2, y2, z2, xy x, y, Rx, Ry yz, zx h=4 z, Rz x2 + y2, z2 x2 – y2, xy (x, y)(Rx, Ry) (yz, zx) C3 (3) E C3 C32 E = exp(2πi/3) A 1 1 1 z, Rz x2 + y2, z2 E «® 1 E E ¬ ® 1 E E (x, y)(Rx, Ry) (x2 – y2, xy) (yz, zx) º® » ®¼ h=3 801 Resource section 4 The groups Cnv C2v (2mm) E C2 Sv (xz) A1 A2 B1 B2 1 1 1 1 C4v (4mm) E 2C4 C2 2Sv 2Sd A1 1 1 1 A2 1 1 1 –1 –1 B1 1 –1 1 1 –1 –1 B2 1 –1 E 2 C6v (6mm) E A1 1 A2 1 0 1 –1 1 –1 1 –1 –1 1 1 1 h=4 Sva (yz) 1 z Rz x, Ry y, Rx 1 –1 1 1 (x, y) (Rx, Ry) 0 1 1 x ,y ,z xy zx yz 1 1 –1 1 –1 C3v (3m) E 2C3 3S v h=6 A1 1 1 1 z A2 1 1 –1 Rz E 2 –1 0 x2 + y2, z2 (x, y) (Rx, Ry) C5v E 2C5 2C52 A1 1 1 1 1 –1 5Sv A2 1 1 1 x2 – y2 E1 2 2 cos A 2 cos 2A 0 xy E2 2 2 cos 2A 2 cos A 0 Rz 2C6 2C3 C2 3Sv 3Sd 1 2 x2 + y2, z2 z 1 0 2 h=8 1 –1 –2 2 (x2 – y2, xy)(zx, yz) h = 10, A = 72º x2 + y2, z2 z Rz (x, y) (Rx, Ry) (zx, yz) (x2 – y2, xy) (zx, yz) C∞v h = 12 A1 (3+) x2 + y2, z2 z – A2 (3 Rz B1 1 –1 1 –1 1 –1 E1 (∏) B2 1 –1 1 –1 –1 1 E2 (∆) E1 2 1 –1 –2 0 0 E2 2 –1 –1 2 0 0 (x, y) (Rx, Ry) ) E 2CF 1 1 h=∞ ∞Sv 1 x2 + y2, z2 z 1 1 –1 Rz 2 2 cos F 0 (x, y) (Rx, Ry) 2 2 cos 2 F 0 (zx, yz) (xy, x2 – y2) (zx, yz) (x2 – y2, xy) The groups Dn D2 (222) E C2(z) C2(y) C2(x) A 1 1 1 1 B1 B2 B3 1 1 1 1 –1 –1 –1 1 –1 –1 –1 1 h=4 x2, y2, z2 z, Rz y, Ry x, Rx xy zx yz D3 (32) E 2C3 A1 1 1 3C2 h=6 x2 + y2, z2 1 A2 1 1 1 z, Rz E 2 –1 0 (x, y) (Rx, Ry) (x2 – y2, xy) (zx, yz) 802 Resource section 4 The groups Dnh D2h (mmm) E C2(z) C2(y) C2(x) i Ag 1 1 1 1 1 1 1 1 B1g B2g B3g 1 1 1 1 –1 –1 –1 1 –1 –1 –1 1 1 1 1 1 –1 –1 –1 1 –1 –1 –1 1 Au 1 1 1 1 –1 –1 –1 –1 S(xy) S(xz) h=8 S(yz) x2, y2, z2 Rz Ry Rx B1u 1 1 –1 –1 –1 –1 1 1 z B2u 1 –1 1 –1 –1 1 –1 1 y B3u 1 –1 –1 1 –1 1 1 –1 x D3h (6m2) E 2C3 3C2 Sh 2S3 3Sv Aa 1 1 1 1 1 1 Aa 1 1 –1 1 1 –1 Ea 2 –1 0 2 –1 0 A1q 1 1 1 –1 –1 –1 h = 12 x 2 + y2 , z 2 Rz (x2 – y2, xy) (x, y) A2q 1 1 –1 –1 –1 1 z Eq 2 –1 0 –2 1 0 (Rx, Ry) (zx, yz) D4h (4/mmm) E 2C4 C2 2C2a 2C2q A1g 1 1 1 1 1 1 1 A2g 1 1 1 –1 –1 1 B1g 1 –1 1 1 –1 B2g 1 –1 1 –1 1 i 2S4 2Sv 2Sd 1 1 1 1 1 –1 –1 1 –1 1 1 –1 1 –1 1 –1 1 Sh Eg 2 0 –2 0 0 2 0 –2 0 0 A1u 1 1 1 1 1 –1 –1 –1 –1 –1 A2u 1 1 1 –1 –1 –1 –1 –1 1 1 B1u 1 –1 1 1 –1 –1 1 –1 –1 1 B2u 1 –1 1 –1 1 –1 1 –1 1 –1 Eu 2 0 –2 0 0 –2 0 2 0 0 D5h E 2C5 2C52 Aa 1 1 A2q 1 Ea 2 Ea xy zx yz 2S52 5Sv 1 1 1 1 –1 2 cos A 2 cos 2A 0 2 cos 2A 2 cos A 0 5C2 Sh 2S5 1 1 1 1 1 1 –1 1 2 cos A 2 cos 2A 0 2 2 2 cos 2A 2 cos A 0 2 A1q 1 1 1 1 –1 –1 –1 –1 A2q 1 1 1 –1 –1 –1 –1 1 z E1q 2 2 cos A 2 cos 2A 0 –2 –2 cos A –2 cos 2A 0 (Rx, Ry) E2q 2 2 cos 2A 2 cos A 0 –2 –2 cos 2A –2 cos A 0 h = 16 x2 + y2, z2 Rz x2 – y2 xy (Rx, Ry) z (x, y) h = 20, A = 72º x2 + y2, z2 Rz (x, y) (x – y2, xy) (zx, yz) (zx, yz) Resource section 4 The groups Dnh (continued) D6h (6/mmm) E 2C6 2C3 A1g 1 1 1 A2g 1 1 B1g 1 –1 B2g 1 E1g 2 E2g C2 3C2a 3C2q i 2S3 2S6 1 1 3Sd 3Sv 1 1 1 Sh 1 1 1 1 1 1 –1 –1 1 1 1 1 –1 –1 1 –1 1 –1 1 –1 1 –1 1 –1 –1 1 –1 –1 1 1 –1 1 –1 –1 1 1 –1 –2 0 0 2 1 –1 –2 0 0 2 –1 –1 2 0 0 2 –1 –1 2 0 0 A1u 1 1 1 1 1 1 –1 –1 –1 –1 –1 –1 A2u 1 1 1 1 –1 –1 –1 –1 –1 –1 1 1 B1u 1 –1 1 –1 1 –1 –1 1 –1 1 –1 1 B2u 1 –1 1 –1 –1 1 –1 1 –1 1 1 –1 E1u 2 1 –1 –2 0 0 –2 –1 1 2 0 0 E2u 2 –1 –1 2 0 0 –2 1 1 –2 0 0 D∞h E ∞C2a A1g(3g+) 1 1 1 A1u(3u+) 1 –1 1 i 2CF 2SF 1 1 1 –1 1 –1 z Rz A2g(3 ) 1 –1 1 1 –1 1 A2u(3u) 1 1 1 –1 –1 –1 g h=∞ ∞Sv z2, x2 + y2 E1g(∏g) 2 0 2 cos F 2 0 –2 cos F E1u(∏u) 2 0 2 cos F –2 0 2 cos F E2g (∆g) 2 0 2 cos 2F 2 0 2 cos 2F E2u (∆u) 2 0 2 cos 2F –2 0 –2 cos 2F " " " " " " (Rx, Ry) (zx, yz) (x, y) (xy, x2 – y2) " The groups Dnd D2d = Vd (42m) E 2S4 C2 2C2a 2Sd h=8 A1 1 1 1 1 1 A2 1 1 1 –1 –1 B1 1 –1 1 1 1 B2 E 1 2 –1 0 1 –2 –1 0 1 0 i 2S6 3Sd h = 12 1 1 x2 + y2, z2 x2 + y2, z2 Rz x2 – y2 z (x, y) (Rx, Ry) D3d (3m) E 2C3 3C2 A1g 1 1 1 1 A2g 1 1 –1 1 1 –1 Eg 2 –1 0 2 –1 0 A1u 1 1 1 –1 –1 –1 A2u 1 1 –1 –1 –1 1 z Eu 2 –1 0 –2 1 0 (x, y) xy (zx, yz) Rz (Rx, Ry) (x2 – y2, xy) (zx, yz) h = 24 x2 + y2, z2 Rz (Rx, Ry) (zx, yz) (x2 – y2, xy) z (x, y) 803 804 Resource section 4 The groups Dnd (continued) 2S83 C2 4C2a 1 1 1 1 1 1 1 1 1 –1 –1 1 –1 1 –1 1 1 –1 1 –1 1 –1 1 –1 1 z E1 2 √2 0 –√2 –2 0 0 (x, y) E2 2 0 –2 0 2 0 0 E3 2 –√2 0 √2 –2 0 0 D4d E 2S8 A1 1 1 A2 1 B1 B2 2C4 h = 16 4Sd x2 + y2, z2 Rz (x2 – y2, xy) (Rx, Ry) (zx, yz) The cubic groups Td (43m) E 8C3 3C2 6S4 6Sd A1 1 1 1 1 1 A2 1 1 1 –1 –1 E 2 –1 2 0 0 T1 3 0 –1 1 –1 T2 3 0 –1 –1 1 h = 24 x2 + y2 + z2 (2z2 – x2 – y2, x2 – y2) (Rx, Ry, Rz) (x, y, z) 6C4 3C2(=C42) 1 1 1 1 –1 –1 2 –1 0 T1g 3 0 T2g 3 A1u 1 A2u Eu Oh (m3m) E 8C3 A1g 1 1 A2g 1 Eg 6C2 (xy, yz, zx) i 6S4 8S6 3Sh 6Sd 1 1 1 1 1 1 1 –1 1 1 –1 0 2 2 0 –1 2 0 –1 1 –1 3 1 0 –1 –1 0 1 –1 –1 3 –1 0 –1 1 1 1 1 1 –1 –1 –1 –1 –1 1 1 –1 –1 1 –1 1 –1 –1 1 2 –1 0 0 2 –2 0 1 –2 0 T1u 3 0 –1 1 –1 –3 –1 0 1 1 T2u 3 0 1 –1 –1 –3 1 0 1 –1 h = 48 x2 +y2 + z2 (2z2 – x2 – y2, x2 – y2) (Rx, Ry, Rz) (xy, yz, zx) (x, y, z) The icosahedral group I E 12C5 12C52 20C3 15C2 A1 1 1 1 1 1 T1 3 1 2 (1+ √5) 1 2 (1– √5) 0 –1 T2 3 1 2 (1– √5) 1 2 (1+ √5) 0 –1 G 4 –1 –1 1 0 H 5 0 0 –1 1 Further information: www.oxfordtextbooks.co.uk/orc/ichem5e h = 60 2 x + y2 + z2 (x, y, z) (Rx, Ry, Rz) (2z2 – x2 – y2, x2 – y2, xy, yz, zx) Resource section 5 Symmetry-adapted orbitals Table RS5.1 gives the symmetry classes of the s, p, and d orbitals of the central atom of an ABn molecule of the specified point group. In most cases, the z-axis is the principal axis of the molecule; in C2v the x-axis lies perpendicular to the molecular plane. The orbital diagrams that follow show the linear combinations of atomic orbitals on the peripheral atoms of ABn molecules of the specified point groups. Where a view from above is shown, the dot representing the central atom is either in the plane of the paper (for the D groups) or above the plane (for the corresponding C groups). Different phases of the atomic orbitals (+1 or –1; amplitudes) are shown by different colours. Where there is a large difference in the magnitudes of the orbital coefficients in a particular combination, the atomic orbitals have been drawn large or small to represent their relative contributions to the linear combination. In the case of degenerate linear combinations (those labelled E or T), any linearly independent combination of the degenerate pair is also of suitable symmetry. In practice, these different linear combinations look like the ones shown here, but their nodes are rotated by an arbitrary axis around the z-axis. Molecular orbitals are formed by combining an orbital of the central atom (as in Table RS5.1) with a linear combination of the same symmetry. Table RS5.1 Symmetry species of orbitals on the central atom s px py pz dz2 dx2y2 dxy dyz dzx D∞h C2v D3h C3v D4h C4v D5h C5v D6h C6v Td Oh ∑ ∏ ∏ ∑ ∑ ∆ ∆ ∏ ∏ A1 B1 B2 A1 A1 A1 A2 B2 B1 A1a Ea Ea A2aa A1a Ea Ea Eaa Eaa A1 E E A1 A1 E E E E A1g Eu Eu A2u A1g B1g B2g Eg Eg A1 E E A1 A1 B1 B2 E E A1a E1a E1a A2aa A1a E2a E2a E1aa E1aa A1 E1 E1 A1 A1 E2 E2 E1 E1 A1g E1u E1u A2u A1g E2g E2g E1g E1g A1 E1 E1 A1 A1 E2 E2 E1 E1 A1 T2 T2 T2 E E T2 T2 T2 A1g T1u T1u T1u Eg Eg T2g T2g T2g 806 D h Resource section 5 D3h C3v A1’ A1 A2’ A2 E‘ E D4h C4v A1g A1 A2g A2 B1g B1 B2g B2 Eu E A2u A1 C2v 3g A1 0g A2 0u B1 3u B2 E‘ E A1 A2’’ A1 B2 E E‘’ g E E B2u B2 Resource section 5 D6h C6v D5h C5v A1g A1 A2u A1 A2g A2 B2g B1 A1’ A1 A2’ A2 B1u B1 E1g E1 E1’ E1 E2’ E2 B2u B2 A2’’ A1 E1u E1 E1’’ E1 E2g E2 E2’’ E2 E2u E2 807 808 Resource section 5 Td A1 T2 Oh A1g Eg T1u T1u T2g T1g T2u Resource section 6 Tanabe–Sugano diagrams This section collects together the Tanabe–Sugano diagrams for octahedral complexes with electron configurations d2 to d8. The diagrams, which were introduced in Section 20.4, show the dependence of the term energies on ligand-field strength. The term energies E are expressed as the ratio E/B, where B is a Racah parameter, and the ligand-field splitting ∆O is expressed as ∆O/B. Terms of different multiplicity are included in the same diagram by making specific, plausible choices about the value of the Racah parameter C, and these choices are given for each diagram. The term energy is always measured from the lowest energy term, and so there are discontinuities of slope where a low-spin term displaces a high-spin term at sufficiently high ligand-field strengths for d4 to d8 configurations. Moreover, the noncrossing rule requires terms of the same symmetry to mix rather than to cross, and this mixing accounts for the curved rather than the straight lines in a number of cases. The term labels are those of the point group Oh. The diagrams were first introduced by Y. Tanabe and S. Sugano, J. Phys. Soc. Japan, 1954, 9, 753. They may be used to find the parameters ∆O and B by fitting the ratios of the energies of observed transitions to the lines. Alternatively, if the ligand-field parameters are known, then the ligand-field spectra may be predicted. d2 with C = 4.428B 1 A1 d3 with C = 4.502B 1 E 3 2 A2 70 70 A2 4 2 D 60 1 50 T1 T2 3 T1 40 3 60 2 A1 1 T2 1 A1 E/B S 50 2 F 40 E/B 1 4 T1 4 T2 2 T2 30 30 T1 2 1 G 20 3 P 1 D 1 E,1T2 10 10 3 F T1 E 2 G 20 4 P 2 3 0 10 20 $O/B 30 T1 4 F 4 10 20 $O/B 30 A2 810 Resource section 6 d4 with C = 4.611B d5 with C = 4.477B 3 A2 4 A2 1 A2 70 70 60 60 4 3 2 eg2 T2 t2g T1 1 A2 5 F F 50 3 40 1 E 30 G3 F 3 H 1 I 4 A 3 1 T2 50 2 3 3 G, 1I 3 A2 2 4 2 E T2 2 A2,2T1 D 40 2 G 30 A1 4 T2 3 2 A1 t2g eg 4 T1 E/B F E/B 1 A1,4E A1 4 4 1 E 20 6 20 1 E T2 3 5 eg1 E t2g 1 10 4 2g 3 T1 t 5 D 10 20 10 6 30 2 S 0 10 $O/B d6 with C = 4.808B 20 $O/B 5 T2 t2g 30 d7 with C = 4.633B A2 1A23A1 3 3 4 A2 4A2 t2g eg 3 3 E t2g eg 2 5 2 1 E 70 70 3 E A2 3 A2 60 3 F F 4 3 eg T2 t2g 4 E/B 1 60 50 1 50 5 1 T2 t2g eg 1 40 4 2 T2 t2g eg 5 1 1 T1 t2geg 3 T2 5 40 1 I G 30 F H 20 3 3 3 2 T2 2 T1 4 T1 2 F 30 3 T1 2 E/B 2 H G 4 P 10 10 6 1 eg E t2g 2 4 1 5 D A1 T1 4 0 10 20 $O/B 30 6 A1 t2g 4 F T1 0 10 20 $O/B 30 Resource section 6 d8 with C = 4.709B 1 A1 70 1 T2 1 E 3 T1 1 T1 60 1 1 S T2 3 50 T1 5 3 T2 t2g eg 3 E/B 40 1 A1 30 1 G 20 1 E 3 P 1 D 10 3 F 6 2 eg A2 t2g 3 0 10 20 $O/B 30 811 This page intentionally left blank Index 16-electron complex, 536 18-electron rule, 535 3c,2e bond, 283 3d-metal monoxides, 613 ab initio methods, 250 abalone shell, 683 absorbance, 229 absorption spectroscopy, 227 acceptor band, 105 acetic acid, 129 acetonitrile, 141 acetylides, 367 acid ionization constant. See acidity constant acid rain, 111, 169, 410 acid sites, 705 acid strength, 122 acid zeolite catalysis, 711 acid-base catalysis, 746 acidic oxide 126 acidic oxides, 269 acidic proton, 122 acidic solvents, 142 acidity constant, 113 acids, 111 aconitase, 750, 769 actinides, 579 actinium, 593 actinoid, 592 actinoids, 267, 579 activated carbon, 357 adenosine diphosphate, 129 adenosine triphosphate, 129, 391 adenosine triphosphate (ATP), 725 ADMET, 696 ADP. See adenosine diphosphate AFM. See atomic force microscopy agate, 351 agostic hydrogen ligand, 552 Ahrland, S., 138 albite, 369 alcohol dehydrogenase, 748 ALD. See atomic layer deposition alkali metal carbonates, 302 alkali metal hydrogencarbonate, 303 alkali metal hydroxides, 301 alkali metals, 73, 293 alkali metals, 133 alkalide anions, 295 alkalide ion, 305 alkaline earth metals, 309 alkaline phosphatase, 748 alkane ligand, 552 alkene hydrogenation, 696 alkene ligand, 545 alkene metathesis, 695 alkene polymerisation, 713 alkenyl ligand, 544 alkoxophosphanes, 391 alkyl ligand, 544 alkylaluminium, 347 alkylidyne complex, 466 alkyllead, 372 alkyllithiums, 307 alkyltin, 372 alkyne ligand, 545 alkynyl ligand, 544 allotropes, 326 allowed transition, 499 alloy, 75, 328 Allred, A.L., 31 Allred–Rochow electronegativity, 31 allyl ligand, 549 A cage, 369 A-helix, 727 A particle, 5 A -alumina, 329 A -hydride elimination, 575 alternation effect, 260, 326, 356, 421, 434 alumina, 680, 705 aluminium, 325 alkylaluminium hydrides, 344 alloys, 328, 331 extraction, 330 hydride, 328, 344 hydroxide, 325, 329 monohalide, 345 organoaluminium compounds, 347 oxide, 325, 346 sulfide, 346 tetrahydridoaluminate, 344 trihalides, 328, 344 uses, 330 aluminium chloride, 134 aluminium halides, 133 aluminium polymers, 128 aluminium recovery, 174 aluminium refining, 174 aluminophosphates, 632, 634 aluminosilicate, 325 aluminosilicates, 136, 632 aluminosilicates, 353, 368 alumoxane, 684 alums, 329 ambidentate ligand, 202 americium, 596 amethyst, 351 amides, 640 amine-borane, 336 amino acid radical, 753 amino acids, 726 aminoborane, 336 aminosulfuric acid, 123 ammonia, 267, 283, 377, 379, 383 ammonium salts, 383 hydrogen bonding, 283 liquid, 129, 383 production, 709, 381, 383 ammonia borane, 333, 336 ammoniacarboxyborane, 336 ammonium chloride, 79 ammonolysis, 625 amorphous silicon, 645 amphiprotic, 112 amphoteric, 321, 326, 329 amphoteric oxides, 126 amphoterism, 126 anaerobic, 399 anaerobic environments, 275, 768, 769 anaesthetic, 421,442 angular nodes, 12, 14 anhydrous oxides, 126 anhydrous sulfuric acid. See sulfuric acid, anhydrous anionic polyhalides, 423 anode, 150 ansa-metallocene, 715 anti-arthritis drugs, 775 antibonding orbital, 44 antiferromagnet, 503 antiferromagnetism, 586 antifluorite, 295 antifluorite structure, 80, 300, 311 anti-knock agent, 372 antimony, 346, 375 antimonides, 377 extraction, 379 organometallic compounds, 394 oxides, 391 oxoanions, 391 pentahalides, 386 stibane, 385 stibine, 377 trihalides, 385 uses, 381 antimony pentafluoride, 141, 143 anti-site defect, 97 apoprotein, 728 applications of IR and Raman spectroscopy, 232 aqua acid, 122 aqua regia, 378, 388 arachno structure, 329, 339 aragonite, 319 See also calcium carbonate arc discharge method, 669 area-detector diffractometer, 226 arene ligand, 548 argon clathrates, 447 coordination compounds, 446 extraction, 442 hydrogen fluoride, 447 use, 442 Arrhenius, S., 111 arsane. See arsine arsenic, 346, 375, 386 allotropes, 376 arsane, 385 arsenides, 377 arsine, 377 environmental, 379 extraction, 379 occurrence, 376 organometallic compounds, 394 oxides, 391 oxoanions, 391 pentahalides, 386 trihalides, 385 uses, 380 arsenicals, 380 arsenides, 346 arsenite oxidase, 391 arsenolite, 376 arsenopyrite, 376 arsine, 377 arsole, 395 artificial bone, 682 artificially layered materials, 673 aryl ligand, 544 asbestos, 351, 365 associative mechanism, 510 astatine radioactivity, 421 asymmetric oxidation, 701 atactic, 715 atmophiles, 262 atmosphere, 399, 440, 758 814 Index atomic absorption spectroscopy, 245 atomic force microscopy, 657 atomic layer deposition, 665 atomic number, 3 atomic orbital, 10 atomic parameters, 257 atomic radius, 22, 74, 258, 267, 326 atom-interchange, 97 ATP, 128, 380, 391, 408 See also adenosine triphosphate Aufbau principle, 18 auranofin, 776 aurothiomalate (myochrisin), 775 autoionization, 114, 121, 130, 141 autoprotolysis, 114, 121, 402, 413, 428 autoprotolysis constant, 114, 119 azides, 382 azimuthal quantum number, 10 azurin, 745 Bailar twist, 523 baking powder, 303 Balmer series, 9 band gap, 102, 655 barite, 400 barium, 309 barium meal, 313 barium sulfate, 94 Bartlett, N., 442 barycentre, 474 base hydrolysis, 522 base pairing in DNA, 285 base, 111 basic beryllium acetate, 321 basic oxide, 126, 269 basic sites, 705 basic solvent, 141 basicity constant, 114 battery, 147, 289 bauxite, 325, 330 Bayer process, 175 Beer–Lambert law, 229 benzene ligand, 548 berkelium, 593 Berry pseudorotation, 211 beryl, 309, 364 beryllium, 309, 311 beryllium alloys, 313 beryllium halides, 315 beryllium hydride, 310 beryllium oxide, 311, 316 beryllocene, 322 B decay, 6 B particle, 5 B-hydride elimination, 574 B-hydrogen elimination, 544 B-sheet, 727 bidentate ligand, 200 Big Bang, 3 BINAP, 542, 698 binary compounds, 268 binary phases, 78 binding energy, 6, 7 bioinformatics, 730 bioinorganic materials, 681, 772 biomimetics, 680, 682, 747, 753, 756, 757 biominerals, 313, 319, 636, 771 bionanocomposites, 682 biotags, 655 biphasic catalyst, 719 bis(triphenylphosphine)iminium cation, 394 bismite, 379 bismuth, 375 bismuthides, 383 occurrence, 379 organometallic compounds, 394 oxide, 391 oxoanions, 391 uses, 381 bismuthinite, 379 bismuthole, 395 bite angle, 202 black pigments, 644 blast furnace, 173 bleach, 408, 425, 433 bleomycin, 775 block, 21 block-copolymers, 675 blue copper centre, 745 body-centred, 66 body-centred cubic, 73 Bohr radius, 14 Bohr, N., 9 bond correlations, 51 bond dissociation energy, 425 bond dissociation enthalpy, 59 bond enthalpy, 51, 265, 352 bond length, 51, 58 bond length, 51 bond order, 50 bond strength. See bond enthalpy bonding orbital, 44 bone, 636, 748, 772 boranes, 327, 330 borax, 325, 330 borazine, 328 boric acid, 330, 334 Born equation, 119 Born–Haber cycle, 87, 299 Born–Landé equation, 107 Born–Mayer equation, 88, 106, 299 borohydride, 277 borohydrides, 329 boron, 325 allotropes, 326, 330 borate esters, 335 boric acid, 335 borohydrides, 338 carborane, 342 clusters, 329 hydrides, 277, 326, 327, 330, 331 hydroboration, 332 metallaboranes, 342 nitride, 328, 330, 335 nitrogen compiunds, 328 occurrence, 330 oxide, 326, 327, 335 polyborates, 334, 335 tetrahydridoborate, 332 trfluoride, 134, 327 tribromide, 327 trichloride, 327 trihalides, 133, 327, 333 triiodide, 327 boron neutron capture therapy, 338 borosilicate glass, 328, 334 borosilicate glasses, 335 Bosch, C., 383 boundary surface, 14 Brackett series, 9 Bragg equation, 223 Brandt, H., 378 brass, 75 brine, 420 bromine bromides, 429 bromite ions, 436 bromous acid, 437 dibrominium cation, 431 interhalogen compounds, 422, 430 oxides, 433 perbromate, 434, 435 production, 424 trifluoride, 141 bromite ions, 436 bromous acid, 437 Brønsted acid, 111, 113 Brønsted acid, 122 Brønsted base, 111 Brønsted, J., 111 Brønsted–Lowry theory, 121 bronze, 75 brucite, 636 Brust–Schiffrin method, 660 buckminsterfullerene, 354, 669 building-up principle, 18, 47 butadiene ligand, 546 butyllithium, 343 cadmium in enzymes, 748 cadmium selenide, 655 cadmium-chloride structure, 81 cadmium-iodide structure, 81 caesium, 293 caesium chloride, 84, 89 caesium clock (atomic clock), 298 caesium-chloride structure, 79, 299 cage complex, 208 calamitic, 650 calcite, 319 calcium, 309 calcium carbide, 79 calcium carbide type structure, 300 calcium carbonate, 92, 318, 771 calcium fluoride, 80, 313 calcium hydride, 280 calcium hydride, 143 calcium in biology, 733, 771 calcium in signalling, 733, 779 calcium oxide, 316 calcium silicates, 136 calcium sulfate, 311 calcium-oxide doped zirconia, 610 californium, 593 calmodulin, 733 Calvin cycle, 362 cancer, 773 cancer therapy, 686 carbaboranes. See carboranes carbamate (as a ligand), 728, 749 carbene Fischer, 551 ligand, 551 N-heterocyclic, 551 Schrock, 551 carbides, 295, 318, 353 carbon, 350, 351, 641 allotropes, 351, 354 amorphous, 357 carbonylhalides, 360 cyanogen, 366 fullerenes, 356 halides, 360 hydrocarbons, 352, 358 monoxide, 361 nanotubes, 356 occurrence, 351 oxides, 353 partially crystalline, 357 suboxide, 361 sulfur compounds, 363 tetrahalides, 353 carbon black, 357, 644 carbon cycle, 362 carbon dioxide, 362, 746, 749 carbon dioxide dehydratase, 746 carbon dioxide sequestration, 362 Index carbon fibres, 357 carbon monoxide, 49, 137, 170, 171, 172, 686, 765, 770 ligand, 540 carbon monoxide as a signalling agent, 765 carbon monoxide dehydrogenase, 765 carbon nanotubes, 666, 668, 670 carbonate as a ligand, 736 carbonic anhydrase, 746 carbonides, carbides, 366 carbonyl complex, 540, 553 homoleptic, 553 carbonyl ligand, 233, 768 carbonylate, metal, 556 carbonylation, 703 carboplatin, 774 carborane, 329, 342 carborundum, 353, 366 cassiterite, 81, 354, 365 catalysis, 633 catalyst, 690 biphasic, 719 heterogeneous, 694 homogeneous, 694 multiphasic, 704 tethered, 719 uniform, 704 catalyst poison, 690 catalytic converter, 694 catalytic cracking, 711 catalytic cycle, 691, 747, 749, 750, 753, 755, 757, 759, 761, 763, 764 catenand, 220 catenane, 220 catenation, 267 cathode, 150 cationic polyhalides, 423 Cativa process, 703 ccp, 69, 77, 81 CE process, 250 cell (electrochemical), 150 cell (living), 722, 723, 724 cement, 136, 314 cementite, 368 centre of inversion, 181 ceramic, 623 ceria, 686 ceria nanocomposites, 687 cesium. See caesium. CFC, 426 See also chlorofluorocarbon CFCs, 437 CFSE, 476 chain reactions, 425 chalcogenide thermoelectrics, 630 chalcogenides, 627 chalcogens. See Group 16 chalcophiles, 262 chalk, 771 character tables, 183, 184 characters, 184 charge-transfer transition, 136, 497, 530 Chatt, J., 138 chelate, 202 chelate effect, 202, 218, 335 chelation therapy, 773 chemical analysis, 245 chemical beam epitaxy, 665 chemical deposition, 604 chemical shift, 235 chemical vapour deposition (CVD), 604, 664 chemical vapour transport (CVT), 627 chemiosmotic theory, 741 chemisabsorbed ligands, 707 chemisorption, 707 Chevrel phases, 630 china clay, 369 Chinese blue, 642 chirality, 187, 210, 212 chloralkali cell, 424 chloralkali industry, 297, 298, 302 chlorides, 626 chlorinated hydrocarbons, 425 chlorine chlorate, 422 chlorides, 429 chlorinated hydrocarbons, 425 chlorine dioxide, 433 chlorite ions, 436 chlorofluorocarbons, 425, 426, 437 chlorous acid, 434, 437 hydrochlorofluorocarbons, 437 hypochlorite, 434, 436 hypochlorous acid, 436 interhalogen compounds, 422, 430 perchlorate, 433, 435, 436 production, 424 uses, 425 chlorite ions, 436 chlorodifluoromethane, 437 chlorofluorocarbons, 362, 425, 426, 437 chlorophyll, 311, 321, 758 chloroplast, 723, 758 chlorous acid, 434, 437 CHN analysis, 246 chromium monoxide, 613 cisplatin, 773 clathrate hydrates, 286, 447 clathrates, 359, 442 Claus process, 174 clay, 325, 351, 354, 636, 680 Clebsch–Gordan series, 488 clinoptilolite, 634 close packing, 68 close-packed structure, 69 closo structure, 329, 339 cluster metal, 564 naked, 470 cluster complex, 466 cluster valence electron, 565 CM, 696 CNTs. See carbon nanotubes cobalamin, 759, 760, 761 cobalt blue, 615 cobalt in enzymes, 759, 760, 761, 778 cobalt monoxide, 613 coenzyme B12, 760 coinage metals, 459 coke, 351 collagen, 772 colossal magnetoresistance, 621 colour centre, 98, 99 coloured solids, 642 compartmentalization of cells, 724 complex, 199 16-electron, 536 cage, 208 carbonyl, 540 cluster, 466 cubic, 207 half-sandwich, 563 high-spin, 476, 562 inert, 507 inner-sphere, 200 labile, 507 low-spin, 476, 562 nitrido, 466 nonlabile, 507 octahedral, 205, 211, 473 outer sphere, 200 oxido, 460, 461 piano-stool, 563 polymetallic, 208 spin-crossover, 504 square planar, 205, 480 square pyramidal, 205 square-planar, 209, 536 sulfide, 465 tetrahedral, 204, 210, 479 trigonal bipyramidal, 205 trigonal prismatic, 206 complex formation, 137 complex hydrides, 640 complex oxides, 613 complexation, 161 comproportionation, 160 computational techniques, 250 computer modelling, 250, 601, 607 concrete, 314 815 condensation reactions, 603 conduction band, 104 conductivity, 101 conjugate acid, 112 conjugate base, 112, 125 constructive interference, 10 cooperative magnetism, 502 cooperativity, 739 coordination compound, 199 coordination number, 69, 82, 200, 265, 267 copper carbonates., 642 copper in proteins, 740, 745, 753, 754, 757, 767, 771, 778 copper indium diselenide, 646 copper sensor, 771 copper(II) chromite, 644 core–shell composite nanoparticles, 655 correlation diagram, 56 correlation problem, 251 corrin, 729, 760 corundum structure, 614 Cossee–Arlman mechanism, 714 Coulombic attraction, 26 Coulombic interactions, 119 coupling, C–C, 702 coupling constant, 236 covalent halides, 270 covalent hydrides, 268 covalent nitrides, 382 covalent radius, 23, 58 cross-polarization, 238 crown ethers, 306 crown ligands, 321 cryolite, 429 crypt ligands, 321 cryptand ligands, 306 crystal-field stabilization energy, 476 crystal structure, 66 crystal-field theory, 473 crystallographic shear plane, 605 cubic close-packed, 69 cubic complex, 207 cubic groups, 183 Curie temperature, 503, 616 curium, 593 Curl, R., 354 cyanide ion, 365 cyanogen, 353, 366, 428 cycle, catalytic, 691 cyclic voltammetry, 250 cyclobutadiene ligand, 546 cycloheptatriene, 551 cycloheptatrienyl ligand, 551, 560 cyclometallation, 575 cyclooctatetraene ligand, 546 cyclopentadiene, 550 816 Index cyclopentadienide, 307 cyclopentadienyl ligand, 550, 560, 570 cyclopentadienyl compounds, 590 cyclopentadienyl ligand, 343 cysteine, 400 cytochrome, 742 cytochrome c, 743 cytochrome c oxidase, 365, 752, 753 cytochrome c peroxidase, 751, 752 cytoplasm, 722 Czochralski process, 173 d orbital, 11, 15 D4h, 205 dating rocks, 296 daughter nuclide, 7 Davies, N.R., 138 d-block elements, 287, 449 d-block oxides, 271 d–d transition, 486, 493, 500, 530 de Broglie, L., 9 decaborane(10), 342 defects, 95, 605, 612 degenerate orbitals, 18 delayed reaction, 530 D orbital, 46 ∆O, 474 D-backbonding, 548 D-bond, 467 D-hydride elimination, 575 deltahedra, 338 demineralization, 772 density, 71 density functional theory, 251 density of states, 103 dental health, 425 deoxyribonucleic acid (DNA), 725, 770, 771, 774 desferral (desferrioxamine), 773 deshielded nucleus, 235 destructive interference, 10 deuterium, 4, 275, 279, 291 Dewar–Chatt–Duncanson model, 545 diagonal relationship, 260, 312, 326 diamagnetic, 478 diamond, 351, 354 diars, 395 diborane, 283, 327, 330, 331, 340 dibrominium cation, 431 dicarbides, 367 diene, 545 differential scanning calorimetry (DSC), 248 differential thermal analysis (DTA), 248 diffraction pattern, 223 dihydrogen, 276, 280 ligand, 289, 544 diiodinium cation, 431 diiodocarborane, 343 dilithiocarboranes, 343 dimethylformamide, 141, 394 dimethylsulfane, 345 dimethylsulfoxide, 120, 141, 763 dinitrogen ligand, 552, 766 dinitrogen monoxide in biology, 767 dinitrogen tetroxide, 131 dioxygen, 398, 403 dioxygenase, 754 diphosphate, 128 dip-pen nanolithography, 658 discotic, 650 dispersion interaction, 89 displacement reaction, 137 disproportionation, 160, 429, 435, 437 dissociation constant, 216 dissociative mechanism, 509 distribution diagram, 116 disulfides, metal, 464 disulfurdinitride, 402, 417 dithiolene, 763 dithionates, 402, 415 dithionic acid, 415 dithionites, 415 dithionous acid, 415 DMSO, 120 See also dimethylsulfoxide, DNA, 285, 380 dolomite, 309, 313 donor-pair method, 539 doping, 105 double exchange, 621 doubly degenerate orbital, 53 Down’s process, 296 DPN. See dip-pen nanolithography Drago–Wayland equation, 140 dysprosium, 582 EAN rule, 535 Earth’s core, 73 EC process, 250 effective atomic number rule, 535 effective mass, 231 effective nuclear charge, 16, 258 effective proton affinity, 118 Egyptian blue, 642 Eigen–Wilkins mechanism, 517, 525 Einstein’s theory of relativity, 6 einsteinium, 593 electric quadrupole moment, 240 electrical conductivity, 101 electrides, 130, 306 electrocatalysis, 152 electrochemical extraction, 174 electrochemical sensor, 610 electrochemical series, 153 electrochemical techniques, 249 electrogenic ion pump, 741 electrolysis, 174 electromagnetic force, 3 electromagnetic radiation, 8 electromotive force, emf, 151 electron affinity, 27, 259, 419 electron configuration, 16 electron deficiency, 327 electron hopping, 674 electron impact, 243 electron paramagnetic resonance, 238 electron paramagnetic resonance (EPR), 744 electron spin, 12 electron transfer, 729, 741, 742, 743, 744, 745 electron-deficient species, 56 electronegativity, 29, 59, 85, 90, 118, 259, 267, 326, 351, 377, 419 optical, 498 electron-gain enthalpy, 27 electronic absorption spectra, 583 electronic properties of 3dmetal monoxides, 613 electronic spectra atomic, 486 complexes, 493 electronic spectra of the actinoids, 595 electronic spectroscopy, 228 electrophilic displacement, 342 electropositive, 260 electrospinning, 680 electrospray ionization (ESI), 244 electrostatic parameter, 91, 119 electrostatic potential, 251 Ellingham diagram, 170, 171, 172 enantiomer, 187, 210 enantiomeric excess, 697 enantioselective, 697 endocytosis, 737 endohedral fullerenes, 357 endoplasmic reticulum, 723 energy density, 277 energy dispersive analysis of X-rays (EDAX), 247 energy dispersive spectroscopy (EDS), 247 energy-dispersive spectroscopy, 658 enterobactin, 736 enthalpy of atomization, 260, 270 environmental nanomaterials, 686 enzyme, 690 epitaxial films, 664 heteroepitaxy, 664 homoepitaxy, 664 epsom salts, 400 equilibrium bond length, 58 equilibrium constant, 139 ESCA, 242 ESI, 244 ethanoic acid synthesis, 703 ethyne, 318 europium, 582 EXAFS (extended x-ray absorption fine structure), 243 exclusion rule, 188 exocytosis, 737 extended defects, 96, 605 extrinsic defects, 96 extrinsic point defects, 97 extrinsic semiconductor, 105 f block, 267 f orbital, 15 fac, 211 face-centred, 66 face-centred cubic, 69 Fajan’s rules, 31 Faraday balance, 249 fast atom bombardment, 243 fast ion conductor, 304 fast ion conductors, 608 faujasite, 711 faujasite (FAU, type X), 634 f block, 579 f-block electronic spectra, 584 f-block hydrides, 287 fcc, 69 F-centre, 98, 99 feldspar, 351, 369 FeMo cofactor (nitrogenase), 766 Fermi level, 103 ferredoxin, 744 ferrichrome, 736 ferrihydrite, 737 ferrimagnet, 503 ferritin, 737, 769 ferrocene, 534, 550, 560 ferroelectric, 616 ferroelectricity, 82, 672 ferromagnet, 502, 503 Index ferromagnetism, 586 ferroxidase centre, 738 ferryl, 751, 756 fertilizers, 297 FIB. See focused ion beam fireworks, 314 first ionization energy, 25, 294 first-order spectra, 236 flame photometry, 245 flame test, 294, 310 flares, 314 fluorapatite, 376 fluorescence, 501 fluoridation of water, 425 fluorides, 94, 429, 626 fluorinating agent, 430 fluorine, 123 chlorofluorocarbons, 425, 426, 437 fluorocarbons, 421, 437 hydrochlorofluorocarbons, 437 hydrofluorocarbons, 421, 425, 437 hydrogen fluoride, 428 hypofluorous acid, 436 interhalogen compounds, 422, 430 polytetrafluoroethene, 437 production, 424 uses, 421, 425 fluorite, 420, 421 fluorite structure, 80, 311, 315, 609 fluoroapatite, 425 fluorocarbons, 421, 437 fluorophiles, 351 fluorosis, 425 fluorosulfonic acid, 130, 142 fluorosulfuric acid, 123, 131 fluxional compound, 563 fluxionality, 237 focused ion beams, 658 forbidden transition, 499 force constant, 231 formal constant, overall, 216 formate dehydrogenase, 764 formation constant, 215 four-circle diffractometer, 226 Fourier transformation, 232 fractional coordinates, 68 framework electrolytes, 608 framework representation, 369 framework structures, 631 francium, 293 Franck–Condon principle, 501 Frasch process, 404 Frenkel defect, 96 fresh water, 157, 169 Friedel–Crafts acylations, 137 Friedel–Crafts alkylation, 134, 137, 143 Friedel–Crafts substitution, 341 frontier orbitals, 28 Frost diagram, 164, 165, 166, 167, 168, 387, 391, 434, 437, 594 fuel cell, 152, 277 fullerene, 70, 685 Fullerene–metal complexes, 357 fullerenes, 351, 356 fullerides, 357, 647 fumarate–nitrate regulator, 770 fuming sulfuric acid, 401 Fuoss–Eigen equation, 518 furnace, 602 fusion reactions, 5 galena, 351, 400 gallium, 8, 325, 327, 346 alloys, 328 arsenide, 346, 383, 645 extraction, 330 hydride, 278, 344 monohalide, 345 nitride, 86, 646 oxide, 325, 329, 346 sulfides, 346 tetrahydrogallate, 344 trihalides, 328, 344 uses, 330 galvanic cell, 150 G-alumina, 329 G-ray photon, 5 G-hydride elimination, 575 garnets, 588 gemstones, 98 gerade, 46 germane, 332, 359 germanite, 351 germanium, 350 dihalides, 353 halides, 360 occurrence, 351 oxides, 353, 365 production, 354 tetrahalides, 353 glass, 327, 352, 353, 364, 623 glass formation, 623 glass transition temperature, 623 Glauber’s salt, 304 gold compounds in medicine, 775 gold nanoparticles, 656, 681 Goldschmidt classification, 261 Goldschmidt radii, 74, 83 Gouy balance, 249 Grätzel cell, 643, 660 graphene, 354, 356 graphene, 356 graphite, 351, 354, 641 bisulfates, 355 fluoride, 356 intercalation compound, 355, 367 tape, 356 gravity sensor, 772 greenhouse effect, 362 greenhouse gas, 426, 766 Grignard reagent, 311, 322, 334, 347, 395 ground-state, 16, 18 Group 1, 257, 377 hydrides, 286 Group 2, 309 hydrides, 280, 286 Group 13, 325 hydrides, 327, 344 occurence, 325 occurrence and recovery, 330 organometallic compounds, 347 oxides, 327, 329 oxo compounds, 346 oxosalts, 329 sulfides, 346 trihalides, 327, 333, 344 with Group 15 elements, 346 Group 13/15 semiconductors, 645 Group 14, 350 allotropes, 351 carbides, 366 halides, 359 hydrides, 352, 358 nitrogen compounds, 365 occurrence, 351 organo compounds, 371 oxides, 353, 361 semiconductors, 644 silicides, 368 tetrahalides, 353 Group 15, 375 allotropes, 375 antimonides, 383 arsenides, 383 azides, 382 bismuth, 383 halides, 377 hydrides, 283, 377, 383 nitrides, 382 occurrence, 376, 378 organometallic compounds, 394 oxides, 390 oxoanions, 391 oxohalides, 386 pentahalides, 385 phosphides, 382 recovery, 378 trihalides, 385 uses, 379 with Group 13 elements, 383 Group 16, 398 catenation, 406 817 halides, 400, 407 hydries, 400 hydrogen bonding, 406 oxides, 410 oxohalides, 412 ring and cluster compouns, 402 Group 17, 419, 433 astatine, 419 bromine, 419 chlorine, 419 fluorine, 419 halogen oxides, 432 hydrides, 421 hypohalous acids, 436 interhalogen compounds, 422, 429, 430 iodine, 419 occurrence, 420 oxoacids and oxoanions, 422, 433 oxohalides, 422 polyhalides, 423, 431 polyhalogen cations, 431 production, 424 pseudohalogens, 427 reactivity, 420, 427 Group 18, 440 argon, 440 clathrates, 442, 447 fluorides, 441, 442, 446, 447 helium, 440 hydrides, 445 krypton, 440 neon, 440 radon, 440 xenon, 440 group theory, 179 guanine, 774 g-value, 239 gypsum, 320 Haber process, 379, 381, 383, 710, 766 Haber, F., 383 Haber–Bosch process, 383 haemerythrin, 740 haemocyanin, 740 haemoglobin, 137, 365, 399, 739 half-life, 7 half-reaction, 148 half-sandwich complex, 563 halides 269, 460 Hall–Héroult process, 174, 330 halogens, 271 See also Group 17 hapticity, 540 hard acid, 138 hard base, 138 Hartree–Fock, 251 818 Index HCFC. See hydrochlorofluorocarbons hcp, 69, 77, 81 Heck coupling, 702 hectorite, 636 Heisenberg, W., 9 helium fullerene complexes, 447 helium-II, 442 occurrence, 440, 442 use, 442 helium burning, 6 heme-. See haemheterodiamond, 330 heterogeneous, 694 heterogeneous catalysis, 704 heterolytic cleavage, 281 heteronuclear, 43 heteronuclear coupling, 236 heteronuclear diatomic molecules, 48 heteronuclear molecular orbitals, 48 heteropolyoxometallates, 463 hexafluorophosphate, 433 hexagonal, 67 hexagonal close-packing, 69 HFC, 437 See also hydrofluorocarbon high gas pressures, 603 high pressures, 603 high temperature superconductors, 317 higher oxides, 613 highest occupied molecular orbital, 48 high-spin complex, 476, 562 high-temperature superconductivity, 82, 617 HIV, 686 HMPA, 394 hole size, 84 holoenzyme, 728 HOMO, 48 homogeneous catalyst, 694 homoleptic, 290, 553 homonuclear, 43 homonuclear coupling, 236 homonuclear diatomic molecules, 40, 45 Hoppe, R., 443 Hund’s rule, 47 hybrid nanocomposites, 678 hybrid orbital, 41 hybridization, 41, 55 hydration enthalpy, 95, 582 hydrazine, 384 hydride, 268, 275, 276, 277, 314, 639 binary, 278 ligand, 289, 543 metallic, 277, 287 molecular, 276, 283 non-classical, 544 saline, 276, 286 synthesis, 290 hydroboration, 332, 344, 347, 359 hydrocarbons, 352 hydrocarbonylation, 698 hydrochlorofluorocarbons, 437 hydrofluorocarbons, 421, 425, 437 hydroformylation, 698 hydrogen, 274, 768 bond enthalpy, 276 cleavage, 281 fuel, 277 fuel cell, 328, 333 nuclear properties, 279 hydrogen bonding, 130, 267, 284, 285, 286, 421 hydrogen burning, 6 hydrogen compounds, synthesis, 291 hydrogen cyanide, 353, 365 hydrogen cycle, 275, 768 hydrogen economy, 277, 281, 282 hydrogen fluoride, 49, 130, 267, 285, 286, 288, 378, 428 acidity, 428 nonaqueous solvent, 428 reactivity, 428 hydrogen in biology, 275, 744, 768 hydrogen peroxide, 400, 406, 408, 433, 751 hydrogen production, 280, 282 hydrogen purifier, 289 hydrogen storage, 288, 295, 328, 333 hydrogen storage capacities, 640 hydrogen storage material, 288, 304, 639 hydrogen sulfide, 407 hydrogenase, 744, 768 hydrogenation, alkene, 696 hydrogenation catalysts, 709 hydrogencarbonate, 317 hydrogenic atoms, 8 hydrogenic energy levels, 10 hydrogensulfate, 402 hydrogensulfite, 125 hydrometallurgical extraction, 173 hydronium ion, 111 hydrosilylation, 359 hydrothermal techniques, 603 hydroxide, 317 hydroxoacid, 122, 125 hydroxyapatite, 376, 425, 635, 683 hydroxylamine, 384 hypercoordinate, 368 hyperfine coupling, 239 hyperfine structure of an EPR spectrum, 239 hypertetrahedral frameworks, 637 hypervalence, 54, 134, 265 hypervalence, 41 hypochlorite, 434, 436 hypochlorite bleach, 408 hypochlorous acid, 436 hypofluorous acid, 436 hypophosphite, 391 ice, 285, 286, 406 icosahedral group, 183 identity operation, 179 Ih, 183 imaging, 775, 776 imidazole, 728 improper rotation, 181 indium, 325, 327 hydride, 278 monohalide, 345 oxide, 329, 346 sulfides, 346 trihalides, 328, 344 uses, 330 indium tin oxide, 346 inert complex, 507 inert gases. See Group 18 inert pair effect, 264, 269, 326, 328, 350, 353, 377 infrared spectroscopy, 230 effect of hydrogen bonding, 285, 286 inner-sphere mechanism, 524 inorganic liquid crystals, 650 inorganic pigments, 642 inorganic–organic nanocomposites, 677 insertion 574, 629 insulator, 101 interband transitions, 655 intercalation, 295, 629, 680 intercalation compound, 355 interchange mechanism, 510 interference, 9 interhalogens, 422, 429, 430 intermetallic compounds, 76, 640 interstitial carbides, 367 interstitial mechanism, 609 interstitial nitrides, 382 interstitial solid solution, 75 intraband transitions, 655 intrinsic defects, 96 intrinsic semiconductor, 104, 645 inverse micelle, 662 inverse spinel, 83, 615 inversion, 46 inverted behaviour, 529 iodine 136 diiodinium cation, 431 interhalogen compounds, 430 interhalogens, 422 iodides, 429 oxides, 433 periodate, 434, 435 periodates, 433 polyiodide ions, 420, 431 triiodide ions, 431 iodine deficiency, 425 iodometric titrations, 415 ion channel, 731, 732 ion cyclotron resonance (ICR) mass spectrometer, 244 ion diffusion, 606 ion mobility, 606 ion pump, 732 ion-exchange, 634 ionic bonding, 60, 65, 86 ionic chlorides, 270 ionic model, 77 ionic radii, 23, 83, 84, 452, 458, 586, 594 ionic solid, 77 ionization energy, 24, 151, 258, 263, 267, 270, 310, 326 ionization enthalpy, 25, 88 ionization-based techniques, 241 ionophore, 731 IR spectra, 188, 231 iron, 73 iron cycle, 735 iron in biology, 735, 744, 750, 757, 762, 765, 769, 773, 777, 779 iron monoxide, 99, 612, 613 iron pyrites, 79 iron regulatory protein, 769 iron–sulfur cluster, 744, 750, 751, 762, 766, 769, 770 irreducible representation, 184, 190 Irving–Williams series, 482, 724, 735, 756 isocyanates, 360 isodensity surface, 251 isoelectronic, 50, 328, 332, 336, 343, 370, 381, 382 isolobal, 343, 428, 566 isomer shift, 240 isomerism cis–trans, 209 coordination, 208 fac–mer, 211 Index geometric, 199, 209 hydrate, 208 ionization, 208 linkage, 208 optical, 210, 212 isosbestic points, 230 isotactic, 715 isotope, 3 Jacobsen oxidation, 701 jadeite, 364 Jahn–Teller effect, 480 jj-coupling, 488 kaolinite, 354, 369, 636 Kapustinskii equation, 91, 271, 321 KCP, 649 kernite, 325 Ketelaar triangle, 60, 80, 85, 315 Ketelaar, J, 60 kinase, 749 kissing bug, 767 Kohn–Sham equations, 251 Koopmans’ theorem, 242 Kroto, Harold, 354 krypton clathrates, 447 coordination compounds, 446 extraction, 442 fluorides, 447 hydrides, 445 radioactivity, 442 labile, 507 laccase, 753 lamp black, 351 Landé g-factor, 586 lanthanide contraction, 22, 258, 582 lanthanoid(III) trihalides, 587 lanthanoid coordination compounds, 589 lanthanoid hydrides, 587 lanthanoid ion magnetic moments, 586 lanthanoid ion spectra, 584 lanthanoid nitrides, 587 lanthanoid organometallic compounds, 590 lanthanoid(III) oxides, 587 lanthanoids, 267, 579 Laporte selection rule, 500 laser-assisted embossing, 662 lasers, 585 quantum cascade, 673 Latimer diagram, 162, 391 lattice, 66 lattice enthalpy, 87, 88, 90, 270, 336 lattice points, 66 laughing gas, 378 Laves phases, 641 lawrencium, 593 layered materials, 627 layered sulfides, 627 LCAO approximation, 43 lead, 350 extraction, 354 halides, 361 hyride, 353 occurrence, 351 organometallic compounds, 371 oxide, 353, 365 plumbane, 359 lead chromate, 643 LED. See light emitting diode levelling. See solvent levelling Lewis acid, 131, 141, 199, 327, 331, 344, 390, 432 Lewis base, 131, 141, 199, 341 Lewis structure, 423 Lewis, G.N., 131 LFAE, 520 LFSE, 476, 479 ligand, 199 ambidentate, 202 bidentate, 200 chirality, 214, 489 monodentate, 200 polydentate, 200 spectator, 508 strong-field, 475 substitution, 507 weak-field, 475 ligand substitution, 568 ligand-field activation energy, 520 ligand-field splitting parameter, 474 ligand-field stabilization energy, 475, 476 ligand-field theory, 473, 482 ligand-field transition, 493 light emitting diode, 655 light-harvesting antenna, 758 lime, 316 limestone, 136, 313 Linde Type A (LTA), 634 linear combination of atomic orbitals, 43 liquid ammonia, 295 liquid ammonia. See ammonia, liquid liquid soaps, 297 lithium, 293 lithium alkyls and aryls, 295, 307 lithium batteries, 297 lithium cobalt oxide, 622 lithium in medicine, 777 lithium ion battery, 147 lithium nitride, 143, 304, 608 lithium oxide, 80 lithium-containing alloys, 297 lithophiles, 261 lithopone, 320 LMCT, 497 localization, 54 low energy electron diffraction (LEED), 706 Lowenstein’s rule, 632 lowest unoccupied molecular orbital, 48 Lowry, T., 111 low-spin complex, 476, 562 luminescence, 501, 585 LUMO, 48, 136, 139 lustre pigment, 644 Lyman series, 9 lysine 2,3 aminomutase, 762 lysosome, 723 macrocyclic effect, 218 Maddrell’s salt, 393 Madelung constant, 88 magic-angle spinning, 238 Magnéli phases, 606 magnesium, 309 magnesium alloys, 313 magnesium carbonate, 92 magnesium diboride, 338 magnesium hydride, 310, 639 magnesium in biology, 723, 725, 749, 777 magnesium sulfate, 94 magnetic properties, 613 magnetic quantum number, 10, 11 magnetic resonance, 233 magnetic resonance imaging (MRI), 443, 775 magnetic susceptibility, 502 magnetism, 502 magnetite, 772 magnetometry, 249 magnetoresistance, 621 magnetotactic bacteria, 772 major element, 723 MALDI, 244 manganese in biology, 723, 750, 759 manganese monoxide, 613 manganites, 621 many-electron atoms, 8 Marcus cross-relation, 529 Marcus equation, 527 Marcus theory, 729 Marsh test, 381 mass number, 3 819 mass spectrometry, 243 materials chemistry, 601 matrix isolation, 233, 432 matrix-assisted laser desorption/ionization (MALDI), 244 MBE. See molecular beam epitaxy mean bond enthalpy, 59 Meissner effect, 617 membrane, 723, 731 Mendeleev, D., 20 mer, 211 mercury, 73 mesogenic, 650 mesoporous materials, 675 mesoporous silica, 712 metal boride, 337 metal carbonylate, 556 metal cluster, 208, 564 metal disulfide, 464 metal halide, 460 metal hydride battery, 289 metal hydride, 639 metal ion activated protein, 727 metal oxide, 460 metal sulfide, 464 metal/organic chemical vapour deposition, 665 metal–ammonia solutions, 305 metallaboranes, 329, 342 metallic bonding, 60, 65, 263 metallic carbides, 367 metallic character, 263, 326 metallic conductor, 101 metallic nanoparticles, 655 metallic radius, 23, 74, 452 metallocene, 534, 560 ansa, 715 metalloenzyme, 746 metalloid, 20, 326, 398 metalloneurochemistry, 779 metallo–organic chemical vapour deposition (MOCVD), 604 metal–metal bond, 466, 467 metal–organic framework (MOF), 604, 638, 676 metal–organic polyhedra, 676 metals, 20 coinage, 459 platinum, 459 metaphosphate, 128 metathesis, 138, 331 alkene, 695 methane clathrates, 359 methanesulfonic acid, 429 methanoic acid,, 125 methanol reformer, 277 methide, 120, 367 methionine synthase, 761 methyl transfer, 761 820 Index methylberyllium, 322 methylchlorosilane, 371 methyllithium, 342 methylmalonyl CoA mutase, 762, 763 Meyer, L., 21 mica, 351, 354, 369 microporous solid, 369, 633 microstate, 487 migratory insertion 573 minimal basis set, 45 mirror plane, 180 mischmetal, 581 mitochondria, 365, 723 mixed ionic–electronic conductors, 610 mixed oxides, 613 MLCT, 497 mobile communication device, 297 MOCVD. See metal/organic chemical vapour deposition modifier, 624 MOF. See metal-organic frameworks molar absorption coefficient, 229 molar absorptivity, 229 molecular beam epitaxy, 664 molecular magnets, 649 molecular materials chemistry, 647 molecular orbital, 43, 192 molecular orbital energy level diagram, 44 molecular orbital theory, 42, 52 molecular potential energy curve, 39 molecular sieve, 712 molecular sieves, 354, 369, 633, 676 molecular vibrations, 230 molybdenum in biology, 723, 763, 764, 765, 766, 767, 768 molybdenum sulfide, 628 monazite, 580 monoclinic, 67 monodentate ligand, 200 mononuclear acids, 123 monooxygenase, 754, 755, 756 monoxides of the 3d metals, 611 Monsanto process, 703 montmorillonite, 636 MOP. See metal-organic polyhedra morphosynthesis, 666 Mössbauer effect, 240 Mössbauer spectroscopy, 240, 780 Mulliken electronegativity, 29 Mulliken, R., 29 multiphasic catalyst, 704 myoglobin, 738 naked cluster, 470 nanocomposite, 684, 685 nanofabrication, 658 top-down, 659 nanomaterials, 653 nanoparticles, 603, 654 nanoplates, 656 nanoporosity, 676, 677 nanoscience, 654 nanosized reaction vessels, 662 nanostructure building block, 667 nanotechnology, 654 nanotube, 672 nanowelding, 671 nanowire, 670, 672 NASICON, 607 natural gas, 400 natural gas clathrates, 359 natural waters, 169 NBBs. See nanostructure building blocks n-butyllithium, 307 Néel temperature, 503 nematic, 650 neodymium, 582 neodymium iron boride, 581 neon extraction, 442 fullerene complexes, 447 use, 442 nephelauxetic series, 496 neptunium, 596 Nernst equation, 154, 155 neurotransmitter, 779 neutral ligand method, 537 neutral oxide, 264 neutral solvent, 142 neutrino, 5, 8 neutron capture, 6 neutron diffraction, 226 neutron flux, 7 neutron-capture, 8 NEXAFS (near-edge X-ray absorption fine structure), 243 NHC, 551 N-heterocyclic carbene, 551 nickel in enzymes, 746, 765, 768, 778 nickel oxide, 613 nickel-arsenide structure, 80 nido structure, 339 niobium sulfide, 628 nitrate reductase, 763 nitrates, 320 nitric acid nitric acid, 379 nitride, 317, 625 nitrido complex, 466 nitrogen, 375, 377, 378, 379, 380, 383, 384, 386, 387, 389, 390 activation, 381, 766 ammonia, 377, 379, 383 azide, 382 dinitrogen oxide, 378 dinitrogen pentoxide, 389 dinitrogen tetroxide, 378 dinitrogen trioxide, 390 Haber process, 379, 381 hydrazine, 384 hydroxylamine, 384 iodine nitrogen oxide, 390 nitrate ion, 387 nitric acid, 379, 387 nitric oxide, 378 See also nitrogen(II) oxide nitrides, 377, 382 nitrogen dioxide, 378 nitrosium ion, 390 nitrous acid, 378, 390 oxides and oxoanions, 377, 387 oxohalides, 386 polycation, 382 recovery, 378 trihalides, 385 uses, 379 nitrogen cycle, 380, 766 nitrogen fixation, 766 nitrogen monoxide ligand, 552 nitrogen monoxide (NO) in biology, 767 nitrogen triiodide, 431 nitrogenase, 766 nitronium ion, 414 nitrosyl ligand, 552, 570 nitrosyl halides, 386 nitrous oxide in biology, 767 nitrous oxide reductase, 767 nitryl halides, 386 NMR intensities, 236 NMR shift reagents, 590 NMR spectrometer, 235 noble gas ligand, 552 noble gas. See Group 18 nodal plane, 14 node, 12 nonaqueous solvent, 129, 141, 402, 413, 425, 428 nonbonding orbital, 45 non-close-packed structures, 72 nondegenerate orbital, 53 nonlabile, 507 nonmetals, 20, 263 nonstoichiometric compound, 99 nonstoichiometric hydride, 268 nonstoichiometry, 99, 612 normal mode, 188, 231 normal spinel, 615 n-type semiconductivity, 105 nuclear burning, 5 nuclear charge, 16 nuclear fission, 7, 592 nuclear fusion, 7 nuclear magnetic resonance, sensitivty of, 234 nuclear power, 7 nuclear reaction, 5, 8 nuclear waste, 634 nuclear waste management, 592 nucleation heterogeneous, 662 homogeneous, 661 nucleic acid, 380 nucleon number, 4 nucleophilic discrimination factor, 513 nucleophilicity parameter, 513 nucleosynthesis, 5, 6 nuclide, 5 occupation factor, 615 occurrence of the elements, 261 octahedral complex, 205, 211, 473 octahedral hole, 70, 77 octahedron, capped, 206 octet, 132 Oh, 183, 205 oleum, 135, 401 one-dimensional metals, 648 one-equivalent process, 526 opal, 351 optical electronegativity, 498 optical isomerism, 210, 212 optically active, 187 orbital angular momentum, 12 orbital angular momentum quantum number, 10 orbital approximation, 15, 43 orbital, 11 order, group, 185 organelle, 723 organoarsenic compound, 395 organoborane, 347 organolithium, 343, 372 organomagnesiation, 323 organometallic compounds of Group I, 307 organosilicon compound, 371 organotin, 372 Orgel diagram, 495 orpiment, 376, 642 orthoclase, 369 orthophosphate, 128 Index orthorhombic, 67 oscillating reaction, 434 osmium tetraoxide, 264 osteoblast, 748, 772 osteoclast, 772 Ostwald process, 379, 387 Ostwald ripening, 659 outer-sphere mechanism, 524, 527 overpotential, 175 overtone, 231 oxidase, 752, 753, 754, 755, 757, 763, 764, 781 oxidation, 147, 741 asymmetric, 701 Wacker, 700 oxidation number, 61, 264 oxidation state, 61, 264 oxidative addition, 290, 544, 571 oxide glasses, 623 oxides of the elements, 268 oxido complex, 460, 461 oxoacid, 122, 123 oxophile, 351 oxygen, 398, 755, 758, 759, 764 allotropes, 403 difluoride, 422, 432 dioxygen, 398, 403 dioxygen difluoride, 433 halides, 407 hydrogen peroxide, 400, 408, 433 hyrogen peroxide, 406 oxides, 400 ozone, 399, 403 peroxides, 400 superoxides, 400 uses, 403 water, 400, 406 oxygen sensor, 610, 770 oxygen rebound mechanism, 755 oxygenase, 754, 755, 756, 757 ozone, 399, 403 ozone depletion, 422, 437 ozone hole, 426 ozonide, 301, 404 p band, 102 p orbital, 11 P450 (cytochrome P450), 754, 755 pairing energy, 476 palladium nanocluster, 779 paramagnetic, 478 Paschen series, 9 Pauli exclusion principle, 16, 44 Pauling electronegativity, 30, 59 Pauling, L., 29, 59, 125 Pauling’s rules, 124, 422, 433 Pauson–Khand reaction, 712 P-cluster (nitrogenase), 766 PDMS. See polymdimethylsiloxane Pearson, R.G., 138 Peierls distortion, 648 penetration, 16, 258 pentaborane(11), 331 pentagonal bipyramid, 206 pentathionate, 402, 415 perbromate, 434, 435 perchlorate, 433, 435, 436 perchloric acid, 131 period, 21 periodate, 433, 435 periodic table, 20, 257 periodicity, 257, 270 permanent hardness, 320 permanganate, 264 perovskite, 81, 100, 317, 429, 58, 602, 616 artificially layered structures, 673 peroxidase, 751, 752 peroxide, 295, 316 peroxide ion, 300 peroxodisulfate, 415 peroxodisulfuric acid, 415 perxenate, 444 petroleum refining, 634 Philip’s catalyst, 713 phosgene, 360 phosphane. See phosphine phosphatase, 748, 750, 772 phosphate polymers, 128 phosphates, 632, 635 phosphazene, 393 phosphide, 346 phosphine, 377, 380 ligand, 542 phosphite, 391 phosphonic acid, 124 phosphonic acid, 124 phosphor, 313, 317, 585, 642 phosphorescence, 501 phosphorus, 346, 375, 385, 391 allotropes, 376 condensed phosphates, 379, 391 extraction, 378, 379 hydrogenphosphates, 392 nitrogen compounds, 394 oxides, 390 oxoanions, 391 oxohalides, 386 pentahalides, 385 phosphates, 392 phosphazenes, 393 phosphides, 382 phosphine, 377, 380, 385 phosphonic acid, 390 phosphoric aci, 391 phosphoric acid, 379 polyphosphates, 393 trihalides, 385 uses, 379 phosphorus pentafluoride, 135 phosphorus(V) oxide, 128 phosphoryl halides, 386 photocatalyst, 624 photochemical dissociation, 425 photochromic, 624 photoelectron spectra, 54, 423 photoelectron spectroscopy, 241 photoionization, 54 photosynthesis, 363, 746, 749, 757, 758, 759 photosystem, 758 phthalocyanine, 370 physisorption, 641, 707 π-acid, 139, 265 π-back donation, 290 π bonds, 40 π orbitals, 46 π-acceptor ligand, 485 π-backbonding, 541, 542 π-donor ligand, 485 piano-stool complex, 563 piezoelectricity, 82, 616 pigment, 615 pillaring of clays, 636 pitchblende, 580 plaster of Paris, 320 plastocyanin, 745 platinum, 648, 773 platinum metals, 459 PLD. See pulsed laser deposition plutonium, 596, 773 PMMA. See polymethylmethacrylate p-n junctions, 646 PNCs. See polymer nanocomposites pnictogens. See Group 14 point defects, 96 point-group, 179 polarizability, 31, 139 polarizable, 31 polarizable ions, 90 polarizing ability, 31 polonium, 73, 74, 398 dioxide, 402, 411 halies, 400 occurrence, 400 toxicity, 406 trioxide, 402 poly(tetrafluoroethene), 360 poly(vinyl chloride), 371 polyalkene, 713 polyarsane, 395 polyatomic molecular orbitals, 52 polyatomic molecules, 40 polyatomic molecules, 52 821 polycation, 127 polychlorotrifluoroethylene, 130 polydentate ligand, 200 polydimethylsiloxane, 656 polyelectron atoms, 8 polyene, 545 polyethene, 713 polyhalide ions, 136 polyhalides, 423, 431 polyhalogen cations, 431 polyiodides, 431 polymer nanocomposites, 679 polymerization, 127 polymers, 371 polymetallic complex, 208 polymethylmethacrylate, 680 polymorphism, 73 polyoxoanion, 127 polyoxometallates, 462 polyphosphates, 370, 391, 725 polypropene, 713 polyprotic acid, 115, 125 polyselenides, 416 polysiloxanes, 371 polystyrene, 713 polysulfides, 295, 301, 402, 415 polytellurides, 416 polytetrafluoroethene, 130, 421, 427, 437 polythionic acids, 402, 415 polytype, 69 polytypism, 72 porosity, 704 porphyrin, 321, 729 positive holes, 104 post-translational modification, 726 potassium, 293 potassium channel, 731, 732 potassium chloride, 88 potassium in biology, 723, 731, 732, 777 potassium nitrate, 91 Pourbaix diagram, 168 powder diffractometer, 224 powder method, 224 powder X-ray diffraction, 299 praseodymium, 582 precursor, 603 primitive, 66 primitive cubic, 73 principal axis, 180 principal quantum number, 10 probability density, 10 probability distribution, 43 projection operators, 196 projections, 68 prokaryotic cell, 722 promotion, 41 promotion energy, 42 prompt reaction, 530 822 Index protein folding, 779 protein structure, 285, 728 proton affinity, 116 proton exchange membrane fuel cell (PEMFC), 152 proton pump, 752 proton transfer, 112, 156 proton-gain enthalpy, 116 Prussian blue, 643 pseudohalide ion, 366, 428 pseudohalogen, 366, 427 pterin, 729, 763 PTFE. See polytetrafluoroethene p-type semiconductivity, 105 pulsed laser deposition, 664 purification, 634 PVC. See poly(vinyl chloride) PVD. See physical vapour deposition Pyrex, 335, 624 pyrophyllite, 369 quadruple bond, 468 quantization, 10 quantum confinement, 654 quantum dots, 654 quantum mechanics, 9 quantum numbers, 10 quantum well, 672 quartz, 351 quicklime, 313, 316 quinol, 447 quintuple bond, 468 Racah parameter, 492 radial distribution function, 13 radial node, 12 radial orbital, 339 radical anion, 307 radical rearrangement, 761, 762, 763 radical SAM enzyme, 762 radio imaging, 775, 776 radiolarian, 772 radionuclides, 634 radium, 309, 313 radius ratio, 84 radon fluorides, 446 occurrence, 440, 442 Raman, C.V., 232 Raman spectra, 188 Raman spectroscopy, 230 rare gases. See Group 18 Raschig process, 384 rate-determining step, 509 ratio of tetrahedral and octahedral holes, 71 rationalization of structures, 83 Ray–Dutt twist, 523 RCM, 696 realgar, 376 rearrangement reaction, 712 rechargeable battery, 147, 289, 295 rechargeable battery materials, 622 red giant, 8 red lead, 365 redox, 741 redox reaction, 147, 524, 741 redox stability, 147, 156, 157, 158, 159, 160, 161, 162 reducible representation, 190 reduction, 147, 741 reduction potential, 147 reductive elimination, 571 reflection, 224 refractory ceramics, 625 relative permittivity, 121, 129, 130, 406, 421 relativistic effects, 24 renewable energy, 281, 282 representation, reduction of, 194 respiratory chain, 742 rhenium trioxide, 614 rhombohedral, 67 ribonucleic acid (RNA), 769 ribulose bisphosphate carboxylase (rubisco), 749 Rieske centre, 745 Rietveld method, 224 RNA, 380 Rochow process, 371 Rochow, E., 31 rock salt, 420 rocket fuel, 384 rock-salt structure, 78, 299 ROM, 696 ROMP, 696 rotation, improper, 181 rotation axis, 179 rubidium, 293 Rubisco, 362 ruby, 97, 614 Ruddlesden–Popper phases, 617 Russell–Saunders coupling, 488, 583 ruthenium compounds as anti-cancer drugs, 774 rutile structure, 80 Rydberg constant, 9 Rydberg, J., 9 s band., 102 s orbital, 11 SALC, 191, 192, 196 generation of, 196 saline carbides, 367 saline hydrides, 268 saline nitrides, 382 samarium, 582 SAMs. See self-assembled monolayers sand, 351 sandwich compound, 534, 560 saponification, 111 sapphire, 97 scanning electron microscopy, 658 scanning near-field optical microscopy, 658 scanning probe microscopy, 657 scanning tunnelling microscopy (STM), 654, 657, 706 Schlenk equilibria, 322 Schoenflies symbol, 182, 183 Schottky defect, 96 Schrödinger equation, 9 Schrödinger, E., 9 seashell, 771 second ionization energy, 25 selection rules, 499 selectivity filter, 731, 732 selenium, 398 dioxide, 402, 411 extraction, 405 halides, 400 hyride, 407 occurrence, 400 oxides, 411 oxohalides, 412 polycations, 416 polymorphs, 405 polyselenides, 402, 416 selenides, 400 trioxide, 402, 411 uses, 405 selenium in biology, 728 self-assembled monolayers, 658 self-assembled nanostructures, 665, 685 self-assembly, 666 self-consistent field, 251 SEM. See scanning electron microscopy semiconducting nanoparticles, 654 semiconductor, 101, 346, 352, 355, 359, 377, 383 semiconductor chemistry, 644 semi-empirical methods, 251 semimetal, 20, 104 separation, 634 Sharpless epoxidation, 701 shear plane, 605 shell, 11 shielded nucleus, 235 shielding, 16, 258 shielding constant, 16 siderophiles, 262 S orbital, 45 S-bond metathesis, 572 silica, 353, 704, 772 silicates, 353, 635 silicides, 368 silicon, 350 aluminosilcates, 368 carbide, 366 halides, 360 halosilanes, 361 hydride, 326 hydrosilylation, 359 nitride, 353 nitrogen compounds, 366 occurrence, 351 organo compounds, 371 oxide, 326 oxo compounds, 353 polymers, 371 production, 354 silane, 352, 358 silicides, 368 simple oxo compounds, 364 tetrahalides, 353 silicon carbide, 80 silicon in biology, 771, 772, 778 silicon sulfide, 68 silicones, 371 siloxanes, 371 silver chloride, 96 single-crystal X-ray diffraction, 224, 225 single-molecule molecular magnet, 649 singly occupied molecular orbital, 48 sintered, 624 skeletal nickel, 709 skeleton, 772 Smalley, R., 354 smoke detector, 597 SNFOM. See scanning nearfield optical microscopy sodalite cage, 369 sodides, 306 sodium, 293 sodium B-alumina, 607 sodium bicarbonate, 303 sodium chloride, 78, 84, 89, 96 sodium hydrogencarbonate, 303 sodium in biology, 723, 731, 732 sodium perborate, 335 sodium phosphate, 379 sodium sulfite, 414 sodium–sulfur battery, 301, 302 soft acid, 138 soft base, 138 solar cell, 645 Index solganol, 775 sol–gel process, 624 solid acids, 143 solid anionic electrolytes, 609 solid cationic electrolytes, 607 solid electrolytes, 607 solid oxide fuel cell (SOFC), 152, 611 solid solution, 100 solid-state chemistry, 601 solid-state NMR, 237 solubility, 94 solubility product, 162 solution synthesis, nanoparticles, 659 solvated electron, 130 solvatochromism, 497 Solvay process, 313 solvent levelling, 119 solvent system, 121, 141 solvothermal reactions, 604 spallation, 227 specific enthalpy, 277 spectrochemical series, 475 spectrophotometry, 230 spectroscopic term, 487 sphalerite, 346 sphalerite structure, 79, 311 spin correlation, 19, 264, 265 spin magnetic quantum number, 12 spin pairing, 39 spin-allowed transition, 500 spin-crossover complex, 504 spinel, 317, 615, 622 spinel structure, 82 spin-forbidden transition, 500 spin-only paramagnetism, 478 spin–orbit coupling, 488 spin–spin coupling, 236 spintronics, 622 SPM. See scanning probe microscopy spontaneity, 149 square antiprism, 206 square planar complex, 205, 480 square pyramidal complex, 205 square-planar complex, 209, 536 stabilities of oxidation states, 94 stabilization using large cations, 93 stabilizers, 660 standard potential, 149, 150, 151, 582 steam reforming, 277, 280, 281 steel, 76 stibane. See stibine stibnides, 346 stibole, 395 Stille coupling, 702 STM. See scanning tunnelling microscopy Stock, A., 338 stopped-flow, 230 street lamps, 297 strong acid, 115 strong base, 115 strong force, 3 strong-field case, 476 strong-field ligand, 475 strontium, 309 structure map, 85 structure prediction, 85 subatomic particles, 3 suboxides, 301 subshell, 11 substitution reaction, 138 substitutional solid solution, 75 sulfide complex, 465 sulfides, 301, 317 metal, 464 sulfur, 271, 398 allotropes, 400 dioxide, 401, 410 disulfides, 400 disulfurous acid, 414 dithionates, 415 dithionic acid, 415 dithionites, 402, 415 dithionous acid, 415 extraction, 404 halides, 400, 405, 408 hydride, 407 hydrogensulfates, 402 hyrogensulfites, 402 nitrogen compounds, 416 occurrence, 400 oxoacids and oxoanions, 412 oxohalides, 410 pentathionate, 402, 415 peroxodisulfates, 415 peroxodisulfuric acid, 415 peroxomonosulfuric acid, 415 polycations, 416 polymorphs, 404 polysulfides, 402, 415 polythionic acids, 402, 415 sulfates, 402 sulfides, 400 sulfuric acid, 401, 402, 413 sulfurous aci, 402 sulfurous acid, 414 tetrathionate, 402, 415 thiosulfate ion, 415 thiosulfuric acid, 414 trioxide, 401, 410 uses, 405 sulfur cycle, 408 sulfur dichloride, 271 sulfur difluoride, 271 sulfur dioxide, 135, 142 sulfur dioxide oxidation, 710 sulfur hexafluoride, 271 sulfur trioxide, 135 sulfuric acid, 125, 401, 402, 413 sulfuric acid, anhydrous, 130 sulfurous acid, 125, 402 sulfuryl dihalides, 412 superacid, 135 superacids, 142 superbases, 142 superconducting elements, 619 superconducting oxides, 620 superconducting quantum interference device (SQUID), 249 superconductivity, 630, 672 superconductor, 101, 338, 356, 417, 429 superexchange, 613, 649 superfluid, 442 superfluid carbon dioxide, 363 superlattice, 672 supernova, 7 superoxides, 295, 300 supertetrahedral clusters, 638 supramolecular assembly, 676 supramolecular chemistry, 666 surface acid, 143 surface area, 704 surface metal sites, 706 surface migration, 708 surface plasmons, 655 surface sites, 705, 708 Suzuki coupling, 702 symmetric cleavage, 331 symmetry adapted linear combination. See SALC symmetry element, 179 symmetry operation, 179 symmetry species, 185 synchrotron radiation, 226 syndiotactic, 715 synthesis of solids, 602 synthetic diamonds, 355 synthetic element, 7 tacticity, 715 talc, 354, 369 Tanabe–Sugano diagram, 495 tangential orbitals, 339 taxol, 421 Td, 183, 204, 479 technetium, 7 technetium in medicine, 776 teflate, 412 Teflon, 437 tellurite, 411 tellurium, 398 dioxide, 402, 411 extraction, 405 823 halides, 400 hyride, 407 occurrence, 400 oxides, 411 oxohalides, 412 polycations, 416 polytellirides, 402 polytellurides, 416 trioxide, 402, 412 TEM. See transmission electron microscopy templated synthesis, physical vapour deposition, 663 temporary hardness, 319 terbium, 582 term symbol, 490 ternary phase, 81 tethered catalyst, 719 tetraborane(10), 331, 341 tetracarbonylnickel, 534 tetracyanoplatinate(II) complexes, 648 tetraethyllead, 372 tetrafluorethane, 421 tetrafluoridoborate, 334 tetrafluoridonickelate, 617 tetrafluoroborate, 433 tetrafluoroethene, 437 tetragonal, 67 tetrahalomethanes, 353, 360 tetrahedral complex, 204, 210, 479 tetrahedral hole, 71, 77 tetrahedral oxoanions, 631 tetrahydridoalanates, 640 tetrahydridoborate, 327, 332, 640 tetramethylammonium ion, 431 tetraphenylborate, 347 tetrasulfurtetranitride, 402, 416 tetrathionate, 402, 415 thallium, 325, 327, 329, 345, 733 extraction, 330 monohalide, 345 monohalides, 328 peroxide, 346 sulfides, 346 trihalides, 344 uses, 330 thermal analysis, 247 thermal decomposition, 92 thermal stability, 92 thermodynamic acidity parameters, 140 thermoelectric device applications, 631 thermoelectric, 630 thermogravimetric analysis (TGA), 247, 303 thionyl dihalides, 412 thiosulfate, 124 824 Index thiosulfate ion, 415 thiosulfuric acid, 414 thorium, 595 thorocene, 596 three-dimensional nanostructures, 675 thulium, 582 tight-binding approximation, 102 time-of-flight (TOF) mass spectrometer, 244 tin, 350 allotropes, 351 halides, 361 hydride, 353 occurrence, 351 organometallic compounds, 371 oxide, 352, 353 oxides, 365 stannane, 359 titanium (IV) oxide, 225 titanium compounds as anti-cancer drugs, 774 titanium dioxide, 643, 672 titanium monoxide, 613 titanium(IV) oxide, 80 titanosilicates, 637 Tolman cone angle, 520 tooth decay, 425 topaquinone, 754 topotactic reactions, 629 trace elements, 723 trans effect, 514, 569 trans influence, 514 transcription factor, 734, 770, 771 transferrin, 736, 737 transition allowed, 499 charge-transfer, 497 forbidden, 499 transition element, 449 transition state effect, 514 translational symmetry, 66 transmetallation reaction, 307 transmission electron microscopy, 658 transporters, 731 tricarboxylic acid cycle, 751 trichloromethane, 136 triclinic, 67 trifluoromethanesulfonate, 433 trifluoromethanesulfonic acid, 429 trifluoromethylsulfonic acid, 124 trigonal bipyramidal complex, 205 trigonal prism, 206 trigonal prismatic complex, 206 triiodide ion, 136, 431 trimethylaluminium, 342 trimethylamine, 342, 410 triply degenerate orbital, 53 tritium, 4 tube furnace, 603 tungsten, 68, 73 tungsten carbide, 76, 368 tungsten in biology, 763, 764, 778 tunnelling, 742 turnover frequency, 693, 747 turnover number, 693 two-equivalent process, 526 Type I superconductor, 618 Type II superconductor, 618 type-A zeolite, 370 tyrosinase, 757 ultramarine, 643 ultraviolet photoelectron spectroscopy, 45, 242 ultraviolet–visible spectroscopy, 228 uncertainty principle, 9 ungerade, 46 uniform catalyst, 704 unit cell, 66 unit cell parameters, 66 unsymmetrical cleavage, 332 uranium, 7, 595 uranium(VI) hexafluoride, 429 uranocene, 596 uranyl ion, 595 UV spectroscopy, 228 valence band, 104 valence electron configurations, 257 valence shell, 21 valence-bond theory, 39 valinomycin, 306, 731 van Arkel, A., 60 van Arkel–Ketelaar triangle, 60 van der Waals complexes, 447 van der Waals diameter, 370 van der Waals force, 355 van der Waals gap, 355 van der Waals interaction, 89 vanadium monoxide, 613 vapour-phase synthesis nanoparticles, 661 plasma synthesis, 661 vaterite, 319 Vegard’s rule, 99 vermilion, 642 vibrating sample magnetometer (VSM), 249 vitamin B12, 760 vitreous, 623 volcano diagram, 708 VSEPR, 286, 419, 423, 430, 443 Wackenroder’s solution, 415 Wacker process, 700 Wade’s rules, 338 Wade–Mingos–Lauher rules, 565 Wadsley defects, 605 Walsh diagram, 56 water, 400, 406 as a reductant, 157 as an oxidant, 158 stability field of, 158 structure of ice, 285 water softener, 297, 303 water splitting, 282 water treatment, 686 wavefunction, 9, 43 wave–particle duality, 9 weak acid, 115 weak base, 115 weak-field case, 476 weak-field ligand, 475 white pigment, 643 Wilkinson’s catalyst, 696 Wurtz coupling, 323 wurtzite, 346 wurtzite structure, 79, 311 XANES (X-ray near edge structure), 243 xenodeborylation, 445 xenon clathrates, 442, 447 coordination compounds, 446 extraction, 442 fluorides, 441, 442 hydrides, 442, 445 insertion compounds, 445 magnetic resonance imaging, 443 nitrogen compounds, 444 occurrence, 440 organoxenon compounds, 445 oxides, 441 oxofluorides, 441, 445 perxenates, 444 radioactivity, 442 use, 442 xenodeborylation, 445 xenon compounds, 90 xenotime, 580 XMET, 696 X-ray absorption spectroscopy, 242 X-ray diffraction, 58, 71, 77, 83, 223, 623 X-ray fluorescence elemental analysis method, 247 X-ray fluoresence (XRF), 247 X-ray photoelectron spectroscopy, 242 ytterbium, 582 yttrium barium copper oxide (YBCO), 620 yttrium stabilized zirconia (YSZ), 609 Y-type zeolite, 370 Zachariasen rules, 623 Zeise’s salt, 534 zeolite, 354, 353, 369, 632, 641, 710 zeolite-Y, 711 zeotypes, 637 zero-point energy, 231 Ziegler–Natta catalyst, 713 Ziegler–Natta polymerization, 347, 591 zinc, 8 zinc blende, 346 zinc carbonyl mechanism, 746 zinc finger, 734 zinc hydroxide, 746 zinc in biology, 734, 735, 746, 748, 771, 778, 779 zinc oxide, 80, 106 zinc sensor, 771 zinc sulfide, 84 zinc-blende structure, 79 Zintl phase, 76, 301, 305, 330, 347 ZSM-5, 369, 711
4675	https://www.quora.com/What-are-some-of-the-disadvantages-of-systematic-sampling	Something went wrong. Wait a moment and try again. Research Methods Disadvantages of Non Prob... Data Collection Systematic Random Samplin... Population Counting Metho... Reasons for Sampling Low Sampling Efficiency Probability Sampling Meth... Research Methodology 5 What are some of the disadvantages of systematic sampling? S D Roemerman CEO at Lone Star Analysis · Author has 727 answers and 414.6K answer views · 6y The simple forms of systematic sampling do have risks. Suppose we want to establish the distribution of an enzyme in the blood of 8 year old children. We have a class of 50 kids, and we estimate that 25 of them is enough for our purposes. Using systematic sampling, we want every nth child, which is every second child. But the teacher lines the kiddos up boy-girl-boy-girl. So we only get one gender. The point is that systematic sampling does have the risk of sample bias because there may be features in the sample population we don’t understand. In fact, there are almost always features we don’t kn The simple forms of systematic sampling do have risks. Suppose we want to establish the distribution of an enzyme in the blood of 8 year old children. We have a class of 50 kids, and we estimate that 25 of them is enough for our purposes. Using systematic sampling, we want every nth child, which is every second child. But the teacher lines the kiddos up boy-girl-boy-girl. So we only get one gender. The point is that systematic sampling does have the risk of sample bias because there may be features in the sample population we don’t understand. In fact, there are almost always features we don’t know that we don’t know. So, we don’t know how our sampling system might amplify a bias. An alternative would be tossing a coin for each kid. Heads we do a blood draw, tails we don’t. We are still sampling about half the population but the sample is less likely to have subtle bias from however the kids were arranged or how they arranged themselves. Ananthanarayana Sharma Practical experience in using design of experiment techniques in agriculture · Author has 510 answers and 508.2K answer views · 5y Usually in systematic sampling, we first order the data, in what is termed a “sampling frame”: which is a list of all possible data points/frequencies/observations of the population one wishes to study? Suppose we term the population size N: and the desire sample size n? If we divide N/n = we get a number (say k) which we can term the sampling interval? Suppose we start from any random point in the “sampling frame”: and choose every kth data point/frequency/observation, that would be the sampling frame? For example if one is surveying the yield of some fruit in each tree, in an orchard - and if Usually in systematic sampling, we first order the data, in what is termed a “sampling frame”: which is a list of all possible data points/frequencies/observations of the population one wishes to study? Suppose we term the population size N: and the desire sample size n? If we divide N/n = we get a number (say k) which we can term the sampling interval? Suppose we start from any random point in the “sampling frame”: and choose every kth data point/frequency/observation, that would be the sampling frame? For example if one is surveying the yield of some fruit in each tree, in an orchard - and if we wish for a sample size of 50 from a total number of 1000 trees, the sampling interval would be 1000/50 =20? So if we start from any random tree, and the select every 20th tree from this random tree, this would be systematic sampling? In market research, if we are studying consumer behaviour is a super market, we can say sample every 5thor 10th or 20th customer buying behaviour? This sampling method breaks down, if the data is already in some pattern? Suppose some different agronomical/silvicultural treatments were given: for different sets of trees, in some pattern? We may inadvertently mis select? Similarly, if there is some pattern in the queing behaviour of customer at the billing counter- that also can upset the selection? The ideal method is random sampling? Chose the samples in a completely random fashion? If you can specify the sort of data you are handling, a more precise answer can be attempted? Satish Khode Lives in Bengaluru, Karnataka, India (2005–present) · 8y Related What is systematic sampling? What are the pros and cons? In some instances the most practical way of sampling is to select every nth item in the list. Such type of sampling is known as systematic sampling. An element of randomness is introduced by introducing random numbers to pick up the unit with which to start. For e.g. if we have select 4 items from the queue of 100 (i.e. 4 per cent sample is desired) the first item be selected randomly from the first twenty five and thereafter every 25th item would automatically be included in the sample. Thus in systematic sampling only the first item is selected randomly and thereafter all the items are selec In some instances the most practical way of sampling is to select every nth item in the list. Such type of sampling is known as systematic sampling. An element of randomness is introduced by introducing random numbers to pick up the unit with which to start. For e.g. if we have select 4 items from the queue of 100 (i.e. 4 per cent sample is desired) the first item be selected randomly from the first twenty five and thereafter every 25th item would automatically be included in the sample. Thus in systematic sampling only the first item is selected randomly and thereafter all the items are selected after regular/ fixed intervals. Although a systematic sample is not a random sample in the strict sense, but its often considered reasonable to treat systematic sample as if it were a random sample. Systematic sampling has few advantages as it can be taken as an improvement over a simple random sample in as much as the systematic sample is spread more evenly over the entire population. It is easier and less costlier method and can be conveniently used in case of large populations. But the most significant disadvantage of this type of sampling is the redundancy pattern. If there is hidden periodicity in the population this technique will prove inefficient. For instance every 25th item produced is defective in a production process and if we were to select 4% sample of the items of this process in systematic manner, we would either get all the defective items or all good items based on the random starting position of sampling. If the list of the population is in random order, systematic sampling is considered equivalent to random sampling. But if this is not so then the results of the sampling is not very reliable. Advantages: Simple and convenient: The results obtained by this method are generally satisfactory and it is a simple and convenient method. Gives similar results: The results obtained by proportionate stratified random sampling are similar to the results given by systematic sampling when the population size is large. Independent: The sample obtained is the true representative sample as this method of selection is dependent on the property of universe under study. Little chance of bias: sample is free from any kind of bias. Helps in random selection: The popularity of this method is in the selection of the candidate contesting in a tie position and in random drawing of the prizes. Disadvantages: High chances of sampling error: If there is hidden periodicity pattern in the population there are very high chances of error. Works only for random population: If the population list is on random order then this technique is almost as random sampling and if not then sampling is not reliable. May not be suitable for large population: Because it is very difficult to create a list of all the names. Michael Lamar Dealt With This Too Often · Author has 3.7K answers and 17.5M answer views · Updated Sep 15 Related What are 3 disadvantages of simple random sampling? That’s such an oddly specific number… It almost sounds like you are parroting a question from an assignment rather than asking a question with the desire to learn. Here are a few disadvantages. Pick your favorite three, or think of some others. To conduct simple random sampling, you must have a way to randomly sample from the entire population. This task, simply put, is typically not possible even for a well-defined static population. You cannot, for example, compel your fellow students to participate in your experiment for your psychology PhD dissertation so whatever data you can get, it will n That’s such an oddly specific number… It almost sounds like you are parroting a question from an assignment rather than asking a question with the desire to learn. Here are a few disadvantages. Pick your favorite three, or think of some others. To conduct simple random sampling, you must have a way to randomly sample from the entire population. This task, simply put, is typically not possible even for a well-defined static population. You cannot, for example, compel your fellow students to participate in your experiment for your psychology PhD dissertation so whatever data you can get, it will never be a simple random sample. Simple random sampling has larger variance than sampling without replacement. For example, in a simple random sample of size n drawn from a population of size N>n, the variance of a sample mean is equal to the variance of the population divided by the sample size. If the sample is drawn without replacement thereby preventing the same individual from possibly appearing multiple times in the sample, the variance of the sample mean is reduced. It decreases by a factor of (1−n−1N−1) as I recall. When N>>n, this factor is negligible, but for other cases, it can be significant. Even when it is possible, simple random sampling can be far more time consuming and expensive than other sampling methods. Inference from simple random samples is not as accurate as using census data. Although population size does not typically play a part in the inference conducted from a simple random sample, it is often important to know the population size in order to insure that you are actually sampling from all of it. Many populations are so large and are ever-changing in number, so you cannot know their size nor be sure you are sampling from all of it. Sponsored by Grammarly 92% of professionals who use Grammarly say it has saved them time Reclaim your workday all year with clear, effective communication at your fingertips. Try Grammarly now! Related questions What is systematic sampling and what are some disadvantages? What is systematic sampling? What are the pros and cons? What are the advantages and disadvantages of systematic sampling in statistics? What are some of the main and most important advantages and disadvantages of systematic sampling? What are the benefits of systematic sampling? Assistant Bot · Sep 6 Systematic sampling—selecting every k-th element from an ordered list after a random start—is simple and efficient, but it has several important disadvantages to consider: Risk of periodicity bias If the population has a hidden periodic pattern that coincides with the sampling interval k, the sample can be unrepresentative (over- or under-sampling specific cycles), producing biased estimates. Dependence on ordering Results depend on the sequence/order of the sampling frame. Different orderings can yield different samples and estimates; improper ordering can introduce structure-related bias. Not str Systematic sampling—selecting every k-th element from an ordered list after a random start—is simple and efficient, but it has several important disadvantages to consider: Risk of periodicity bias If the population has a hidden periodic pattern that coincides with the sampling interval k, the sample can be unrepresentative (over- or under-sampling specific cycles), producing biased estimates. Dependence on ordering Results depend on the sequence/order of the sampling frame. Different orderings can yield different samples and estimates; improper ordering can introduce structure-related bias. Not strictly probabilistic for all units While every element has the same long-run chance when the frame is random, individual selection probabilities are not independent and unequal inclusion probabilities across positions can arise if the frame length is not an exact multiple of k. Issues with variance estimation Standard variance formulas for simple random sampling (SRS) generally do not apply. Accurate variance estimation requires accounting for the systematic design; common practice often underestimates true variance if periodicity exists. Sensitivity to frame errors and omissions Missing, duplicated, or misordered frame entries can disproportionately affect the sample because selection follows a fixed interval. Less flexible with stratification and complex designs Combining systematic selection with stratification or clustering is possible but more complicated than with SRS; ensuring correct inclusion probabilities and variance estimation becomes harder. Potential for reduced randomness and representativeness in small samples In small populations or with large k, the sample may cover only limited parts of the population structure, lowering representativeness. Practical mitigations: randomize ordering where feasible, check for known periodicities before choosing k, use smaller intervals or multiple random starts (or use random-start systematic+replication), and apply design-based variance estimators or switch to SRS/stratified sampling when periodicity or frame problems are suspected. Michael Lamar PhD in Applied Mathematics · Author has 3.7K answers and 17.5M answer views · 1y Related What are the advantages and disadvantages of non-probability sampling? The primary disadvantage is that you lose the mathematical underpinning that allows you to quantify how well you understand what your data is teaching you. Random sampling allows you to construct confidence intervals with know confidence levels and allows you to calculate attained significance (aka p-values). The primary advantage of non-probability sampling is that you can actually do it in real life. In most circumstances, it’s simply impossible to generate a large number of a random samples from a real-world population. You can learn quite a lot about your population from a large sample that The primary disadvantage is that you lose the mathematical underpinning that allows you to quantify how well you understand what your data is teaching you. Random sampling allows you to construct confidence intervals with know confidence levels and allows you to calculate attained significance (aka p-values). The primary advantage of non-probability sampling is that you can actually do it in real life. In most circumstances, it’s simply impossible to generate a large number of a random samples from a real-world population. You can learn quite a lot about your population from a large sample that is reasonably representative which is often plenty good enough for real-world practice. Jose Renato Kitahara Corporate & Academic career. PhD,MSc,MBA,Engineer,Professor · Author has 732 answers and 595.8K answer views · 5y Related What are the disadvantages and advantages of statistical sampling? Advantage: saves money, time and resources when we need to know how the world works without ask it in full. Disadvantage: requires high quality to plan, design, build, develop and perform the sampling, without bias, acceptable error and enough level of confidence. When you fail in the sampling proccess (in any step), all following process are lost it doesn’t matter how powerfull and knowlegeable people, software and computers you are using. Sponsored by Project-Management Quickly find the Project Management software you need. Enhance team collaboration and streamline your projects with powerful Project Management tools. Eric Hortop Union Steward at Professional Institute of the Public Service of Canada (2013–present) · Author has 112 answers and 178.9K answer views · 5y Related Why would you use systematic sampling? When you don't have a frame in advance but you can crawl along the population… every nth dwelling along a predetermined route in a household survey with personal interviews and no resources for listing a frame (this is less common now but not many censuses ago in Canada this was more or less how it worked for selecting long form respondents), for example, or every hundredth widget on an assembly line. Or when you can order your population by some magnitude variable that you want to assure a reasonably representative sample of. You can even use the cumulative total of that magnitude variable ra When you don't have a frame in advance but you can crawl along the population… every nth dwelling along a predetermined route in a household survey with personal interviews and no resources for listing a frame (this is less common now but not many censuses ago in Canada this was more or less how it worked for selecting long form respondents), for example, or every hundredth widget on an assembly line. Or when you can order your population by some magnitude variable that you want to assure a reasonably representative sample of. You can even use the cumulative total of that magnitude variable rather than just counting list items if you want a systematic sample with probability proportional to size (that's how sites [clusters] were selected for the Canadian Health Measures Survey). Another reason you might use a systematic sample is to control overlap between cycles of a repeated survey… move the starting point and keep the sampling fraction (this is most relevant with large sampling fractions) and you will keep the probability of being chosen twice down. Related questions What are the advantages of cluster sampling? What are the disadvantages? What are the disadvantages and advantages of probability sampling? What are the disadvantages and advantages of statistical sampling? What is systematic sampling? Why would you use systematic sampling? Stefano Marchetti Ph.D. in Statistics (academic discipline), University of Florence (Graduated 2009) · Author has 67 answers and 329.3K answer views · 7y Related What are the disadvantages and advantages of probability sampling? Probability sampling, which means that we know the inclusion probability for each unit in the target population (if this probability is equal for all the units, then is the case of simple random sampling), allow to infer on target parameters, like means/averages, proportions, totals, etc. Infer means that using a sample (a sub-set of the population) we can draw conclusions on the whole population, for instance by confidence intervals and test of hypothesis. The only way we know to make inference is by means of probability sampling. This is the main advantage. The random sampling (the same as pr Probability sampling, which means that we know the inclusion probability for each unit in the target population (if this probability is equal for all the units, then is the case of simple random sampling), allow to infer on target parameters, like means/averages, proportions, totals, etc. Infer means that using a sample (a sub-set of the population) we can draw conclusions on the whole population, for instance by confidence intervals and test of hypothesis. The only way we know to make inference is by means of probability sampling. This is the main advantage. The random sampling (the same as probability sampling) designs allow the sample to be representative of the population. E.g. if I want to know the average height of persons living in Italy, I have two choices: 1. measure the height of 60 millions persons, 2. draw a random sample from which to make inference on the height of the persons living in Italy. Assuming no measurement errors and no non-responses the option 1 allows to know the “true” average height. However, this solution, also called Census, is unfeasible because is too much expansive and time consuming. On the contrary, the option 2 is feasible, but doesn’t allow to know the true average. Nevertheless, one can draw some conclusions about the average height using the sample observations, the true unknown average is included in an interval with a specified confidence (let ¯x be the sample average, in general we can say P(¯x−1.96√σ2/n<μ<¯x−1.96√σ2/n)=0.95). If the sample is not a random sample may be it is not representative of the population, for example the selected persons are all basket players and the inference will be totally wrong. The main disadvantage of the probability sampling is that it is some time difficult, or impossible, to know a priori the probability of each unit in the population to be part of the sample (that is draw a probability sample). This is particularly true in finite populations. If I would like to draw a simple random sample in the above example, I need a list of the 60 millions persons living in Italy from which to select randomly a sample of n persons. This list, called frame, is not always available or accessible. This is one of the reason why there exists many random sample designs (not only the simple random sample), such as stratified sampling, two stage sampling, multi-stage sampling, etc. Of course, increasing the complexity of the design increases the complexity of the estimators of mean and its standard error. Sponsored by MEDvi Prescription Weight Loss Why does Tirzepatide shed fat 2x as fast as Ozempic? Ozempic is made for diabetes. Tirzepatide is specifically for weight loss. It starts at $179/month. Lisa B. 55 years old · Author has 1.6K answers and 2.6M answer views · 2y Related What are some disadvantages of a sample survey? You're not necessarily getting the complete makeup o of opinions within a group. Chandrasekaran Avanavadi Sundaram Iyer Specialised in TQM , Behavioural Sciences and Education MGt · Author has 3.6K answers and 7.3M answer views · 3y Related What are the disadvantages of time sampling? Let us understand Time sampling in direct observation, a data collection strategy that involves noting and recording the occurrence of a target behavior whenever it is seen during a stated time interval. The process may involve fixed time periods (e.g., every 5 minutes) or random time intervals What is the purpose of time sampling? time samples are a useful way to collect and present observation data over a long period of time. time samples are repeated short focused snapshots of child development used to collect precise data. time samples can be used to observe a child's behaviour to identify possib Let us understand Time sampling in direct observation, a data collection strategy that involves noting and recording the occurrence of a target behavior whenever it is seen during a stated time interval. The process may involve fixed time periods (e.g., every 5 minutes) or random time intervals What is the purpose of time sampling? time samples are a useful way to collect and present observation data over a long period of time. time samples are repeated short focused snapshots of child development used to collect precise data. time samples can be used to observe a child's behaviour to identify possible concerns Time sampling Advantages observer has time to record what they have seen Disadvantages some behaviours will be missed outside the intervals – observations may not be representative. In the area of Education Why is time sampling an advantage? A major advantage of the momentary time sample recording process is that a teacher does not need to be attending to a student's behavior all of the time. Momentary time sampling provides an estimate of behavior rather than the documentation of every occurrence and can be fairly easy to implement in class… Another possible disadvantage can be Teacher daily emotional cycle versus child emotional cycle should not conflict.Then biased data will get compiled. Single teacher observing and analysing such data should ensure personal bias of the observer gets into the observed data In other words A major disadvantage of this measurement strategy is that it can underestimate a student's behavior since the student may engage in a behavior throughout an interval but stop right before the end of the interval. In this case, momentary time sampling would not capture a good estimate of the occurrence of a behavior. Nonetheless Time sampling as a tool is Helpful to understand the ' mutiple intelligence trats ’ in a child over a time period to get the child focused on it's strength Also helps school and parents to take counter measures to come out of undesirable traits in a child Blessings and best wishes Terry Moore PhD in statistics · Author has 16.5K answers and 29.3M answer views · 4y Related Why is systematic sampling sometimes referred to as a non-probability sampling method? In statistics Why is systematic sampling sometimes referred to as a non-probability sampling method? In statistics Because sometimes it is a non-probability sampling method. It is a probability sampling method if a random start is used, if not it is a non-probability sampling method. However it is tricky to analyse even with a random start. There are unknown correlations between observations. In the best case, sampling errors are reduced, in the worst case they increase. One case when sampling errors can be reduced is if there is a trend and the sample steps along a route that goes in both directions over the Why is systematic sampling sometimes referred to as a non-probability sampling method? In statistics Because sometimes it is a non-probability sampling method. It is a probability sampling method if a random start is used, if not it is a non-probability sampling method. However it is tricky to analyse even with a random start. There are unknown correlations between observations. In the best case, sampling errors are reduced, in the worst case they increase. One case when sampling errors can be reduced is if there is a trend and the sample steps along a route that goes in both directions over the region where the trend is. Stuart Banks Self Employed Electrician at Your Electrical Solution (2010–present) · Author has 7.3K answers and 8M answer views · 5y Related What are the different sampling techniques? “Sampling Methods \| Types and Techniques Explained” “To draw valid conclusions from your results, you have to carefully decide how you will select a sample that is representative of the group as a whole. There are two types of sampling methods: Probability sampling involves random selection, allowing you to make statistical inferences about the whole group. Non-probability sampling involves non-random selection based on convenience or other criteria, allowing you to easily collect initial data. Probability sampling methods Probability sampling means that every member of the population has a chance “Sampling Methods \| Types and Techniques Explained” “To draw valid conclusions from your results, you have to carefully decide how you will select a sample that is representative of the group as a whole. There are two types of sampling methods: Probability sampling involves random selection, allowing you to make statistical inferences about the whole group. Non-probability sampling involves non-random selection based on convenience or other criteria, allowing you to easily collect initial data. Probability sampling methods Probability sampling means that every member of the population has a chance of being selected. It is mainly used in quantitative research. If you want to produce results that are representative of the whole population, you need to use a probability sampling technique. There are four main types of probability sampling technique Simple random sampling In a simple random sample, every member of the population has an equal chance of being selected. Your sampling frame should include the whole population. To conduct this type of sampling, you can use tools like random number generators or other techniques that are based entirely on chance. You want to select a simple random sample of 100 employees of Company X. You assign a number to every employee in the company database from 1 to 1000, and use a random number generator to select 100 numbers. Systematic sampling Systematic sampling is similar to simple random sampling, but it is usually slightly easier to conduct. Every member of the population is listed with a number, but instead of randomly generating numbers, individuals are chosen at regular intervals. Example All employees of the company are listed in alphabetical order. From the first 10 numbers, you randomly select a starting point: number 6. From number 6 onwards, every 10th person on the list is selected (6, 16, 26, 36, and so on), and you end up with a sample of 100 people. If you use this technique, it is important to make sure that there is no hidden pattern in the list that might skew the sample. For example, if the HR database groups employees by team, and team members are listed in order of seniority, there is a risk that your interval might skip over people in junior roles, resulting in a sample that is skewed towards senior employees. Stratified sampling This sampling method is appropriate when the population has mixed characteristics, and you want to ensure that every characteristic is proportionally represented in the sample. You divide the population into subgroups (called strata) based on the relevant characteristic (e.g. gender, age range, income bracket, job role). From the overall proportions of the population, you calculate how many people should be sampled from each subgroup. Then you use random or systematic sampling to select a sample from each subgroup. Example The company has 800 female employees and 200 male employees. You want to ensure that the sample reflects the gender balance of the company, so you sort the population into two strata based on gender. Then you use random sampling on each group, selecting 80 women and 20 men, which gives you a representative sample of 100 people. Cluster sampling Cluster sampling also involves dividing the population into subgroups, but each subgroup should have similar characteristics to the whole sample. Instead of sampling individuals from each subgroup, you randomly select entire subgroups. If it is practically possible, you might include every individual from each sampled cluster. If the clusters themselves are large, you can also sample individuals from within each cluster using one of the techniques above. This method is good for dealing with large and dispersed populations, but there is more risk of error in the sample, as there could be substantial differences between clusters. It’s difficult to guarantee that the sampled clusters are really representative of the whole population. Example The company has offices in 10 cities across the country (all with roughly the same number of employees in similar roles). You don’t have the capacity to travel to every office to collect your data, so you use random sampling to select 3 offices – these are your clusters. Non-probability sampling methods In a non-probability sample, individuals are selected based on non-random criteria, and not every individual has a chance of being included. This type of sample is easier and cheaper to access, but it has a higher risk of sampling bias, and you can’t use it to make valid statistical inferences about the whole population. Non-probability sampling techniques are often appropriate for exploratory and qualitative research. In these types of research, the aim is not to test a hypothesis about a broad population, but to develop an initial understanding of a small or under-researched population. Convenience sampling A convenience sample simply includes the individuals who happen to be most accessible to the researcher. This is an easy and inexpensive way to gather initial data, but there is no way to tell if the sample is representative of the population, so it can’t produce generalizable results. Example You are researching opinions about student support services in your university, so after each of your classes, you ask your fellow students to complete a survey on the topic. This is a convenient way to gather data, but as you only surveyed students taking the same classes as you at the same level, the sample is not representative of all the students at your university. Voluntary response sampling Similar to a convenience sample, a voluntary response sample is mainly based on ease of access. Instead of the researcher choosing participants and directly contacting them, people volunteer themselves (e.g. by responding to a public online survey). Voluntary response samples are always at least somewhat biased, as some people will inherently be more likely to volunteer than others. Example You send out the survey to all students at your university and a lot of students decide to complete it. This can certainly give you some insight into the topic, but the people who responded are more likely to be those who have strong opinions about the student support services, so you can’t be sure that their opinions are representative of all students. Purposive sampling This type of sampling involves the researcher using their judgement to select a sample that is most useful to the purposes of the research. It is often used in qualitative research, where the researcher wants to gain detailed knowledge about a specific phenomenon rather than make statistical inferences. An effective purposive sample must have clear criteria and rationale for inclusion. Example You want to know more about the opinions and experiences of disabled students at your university, so you purposefully select a number of students with different support needs in order to gather a varied range of data on their experiences with student services. Snowball sampling If the population is hard to access, snowball sampling can be used to recruit participants via other participants. The number of people you have access to “snowballs” as you get in contact with more people. Example You are researching experiences of homelessness in your city. Since there is no list of all homeless people in the city, probability sampling isn’t possible. You meet one person who agrees to participate in the research, and she puts you in contact with other homeless people that she knows in the area.” Related questions What is systematic sampling and what are some disadvantages? What is systematic sampling? What are the pros and cons? What are the advantages and disadvantages of systematic sampling in statistics? What are some of the main and most important advantages and disadvantages of systematic sampling? What are the benefits of systematic sampling? What are the advantages of cluster sampling? What are the disadvantages? What are the disadvantages and advantages of probability sampling? What are the disadvantages and advantages of statistical sampling? What is systematic sampling? Why would you use systematic sampling? What are some examples of cluster sampling and systematic sampling? What is the difference between the cluster and systematic sampling? What is an example of systematic sampling? What is the disadvantage of non-probability sampling? What are the disadvantages of sampling? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
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Write the first five terms of a_n, and find \lim_{n \rightarrow \infty} a_n. If the sequence diverges. enter "divergent" in the answer box for... View AnswerThe 31st term of a sequence is 123 and the 32nd is 1107. What is the 35th term if the sequence is: a) geometric b) arithmetic View AnswerThe recursive formula for a sequence is given below. A1 = -29 An = An-1 + 13 Which function produces the same sequence? Explain your answer. A) f(n) = -42 + 13n B) f(n) = -29 + 13n C) f(n) = -42n +... View AnswerUse iteration to solve the recurrence relation a_n = a_{n-1} + n with a_0 = 4. View AnswerDetermine the recursive formula for the sequence 4, 8, 16, 32, 64, 128, ... ? View AnswerFind the value of the 8th term of the sequence below. 3, 6, 7, -2 View AnswerGiven below the first five terms of sequence C. -3, -3, -1, 3, 9 Find nth term of sequence C in its simplest form. View AnswerWhat is the next number? 2 ,7, 8,3, 12, 9 View AnswerFind the sum of the first 200 terms of the arithmetic sequence 12, 9, 6, . View AnswerA brick wall contains 52 bricks in its bottom row and 49 bricks in the next row up from the bottom row. Each subsequent row contains 3 fewer bricks than the row immediately below it. If the wall co... View AnswerWrite 11 + 14 + 17 + 20 + + 38 in the summation form by using sigma notation. View AnswerFind the common difference and the three terms in the sequence after the last one given. 1/5, 23/15, 43/15, 21/5, ... View AnswerDifferentiate an arithmetic series from a geometric series. (Include relevant examples of both types of series in your explanation.) View AnswerFind the forty-first term of the arithmetic sequence: 2, 6, 10, 14, 18, View AnswerVerify that x_n = 2n + 1 is a solution of the difference equation x_n + 1 = x_n + 2, x_0 = 1. View AnswerPredict the next term in the sequence of numbers. 1, 9, 17, 3, 11, 19, 5, 13, 21, .... a. 15 b. 7 c. 19 d. 25 View AnswerDecide whether it is possible to fill in the blanks to form an arithmetic sequence. If so, find a recursion formula for the sequence. Explain how you found your answers. (a) (b) (c) (d) (e) View AnswerFind the common difference and the three terms in the sequence after the last one given. -39, -33, -27, -21, ... View AnswerGiven the progression 12, 17, 22 .... Determine the 40th term and the sum of the first 20 terms. View AnswerAn infinite geometric series has first term 100. When all its terms are added together, the sum is 80. Write the expression for the infinite sum in summation (sigma) notation, and write out the fir... View AnswerFind the first 40 terms of the sequence defined by a_(n+1) = (1/2) a_n if a_n is an even number and a_(n+1) = 3(a_n) + 1 if a_n is an odd number; and a_1 = 11. Do the same if a_1 = 25. Make a conje... View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = (n^2)/(sqrt(n^3 + 4n)). View AnswerFinish the sequence: 1, 2, 4, ... a. 8 b. 16 c. 4 d. 32 View AnswerWe can make a_n as close to L as we want for all suffciently large n. What can we say about the sequence (a_n)? a) The sequence (a_n) is monotone increasing to L. b) The sequence (a_n) converges... View AnswerFind the formula for the nth term of the arithmetic sequence. 10, 6.5, 3, . . . View AnswerIs the sequence an = sin(n)/n converging or diverging? Find the limit if the sequence converges. View AnswerThe limit of the sequence (sqrt(n^2 + 5n) - n)_n = 1 to inf is a. 5 b. 1/5 c. 5/3 d. None of these e. 5/2 View AnswerIf two positive sequences curly bracket open a_n curly bracket close and curly bracket open b_n curly bracket close converge, then the sequence curly bracket open a_n by b_n curly bracket close alw... View AnswerThe nth term of the following sequence 1, -4, 9, -16, 25, ... is given by the following formula. Select one. a. a_n = (-n)^2 b. a_n = (-1)^n (Square root of n) c. a_n = (-1)^(n-1) n^2 d. a_n = (-1)... View AnswerIf Summation (a_n) is divergent and Summation (b_n) is divergent, then Summation (a_n + b_n) is divergent. Select one: a. True b. False. View AnswerFind the first 5 terms of the following sequence. a_n = 1 / 4 n + 1 View AnswerState true or false. If a_n is a sequence and limit (n tends to infinity) a_n = infinity, then the sequence diverges. View AnswerIf two positive sequence curly bracket open a_n curly bracket close and curly bracket open b_n curly bracket close converges, then the sequence curly bracket open a_n by b_n curly bracket close con... View AnswerFor the given sequence 1,5,25,... a. Classify the sequences as arithmetic, geometric, Fibonacci, or none of these. b. If arithmetic, give d; if geometric, give r; if Fibonacci's give the first two... View AnswerFor the given sequence 5,15,25,... a. Classify the sequences as arithmetic, geometric, Fibonacci, or none of these. b. If arithmetic, give d; if geometric, give r; if Fibonacci's give the first two... View AnswerFor the given sequence 2,4,6,8,... a. Classify the sequences as arithmetic, geometric, Fibonacci, or none of these. b. If arithmetic, give d; if geometric, give r; if Fibonacci's give the first two... View AnswerIf a_n is a sequence and limit (n tends to infinity) a_n = infinity, then the sequence diverges. Select one. a. True b. false. View AnswerSummation (n = 1 to infinity) (-1)^(n-1) by (2n - 1) = Pi by 4. True or false? View AnswerTrue or false? If {a_n} is decreasing and a_n greater than 0 for all n, then {a_n} is convergent. View AnswerTrue or false? If \lim_{n \to x} a_n = L, then \lim_{n \to x} a_{2n + 1} = L. View AnswerDetermine whether each sequence is arithmetic or not if yes find the next three terms. a. 17, 12, 7, 2,......... b. -129, -98, -67,......... View AnswerDetermine whether each sequence is arithmetic or not if yes find the next three terms. a. -29, -2, 25,......... b. -4.5, -9.5, -14.5,......... View AnswerDetermine whether each sequence is arithmetic or not if yes find the next three terms. a. 1/4, 2/6, 3/8, 4/10,...... b. 12, 24, 36, 48,......... View AnswerDetermine whether each sequence is arithmetic or not if yes find the next three terms. -129, -98, -67 .... View AnswerIf la_n\| converges, then a_n converges. True or false? View AnswerThe nth term of the following sequence is given by the following formula 1, -4, 9, -16, 25, ... Select one: a. a_n = (-n)^2 b. a_n = (-1)"n c. a_n = ((-1)^(n-1))(n^2) d. a_n =(-1)^n square root of n. View AnswerFind the 4th term of the recursively defined sequence. a_1 = 4, a_(n + 1) = 2a_n - 2. View AnswerConsider the following sequence 15, - 150, 1500, - 15000, 150000, ... Find the 27th term. Write the result in scientific notation N x 10^k, with N rounded to three decimal places. View AnswerList the next term of the sequence 3,5,7,9..... View AnswerList the next term of the sequence 9, 11, 13, 15,... View Answer(a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples. View AnswerFind a formula for the general term a_n of the sequence, assuming that the pattern of the first few terms continues. {1, 0, - 1, 0, 1, 0, -1, 0, \dots} View Answer(a) What is a sequence? (b) What does it mean to say that \displaystyle \lim_{n \to \infty} a_n = 8? (c) What does it mean to say that \displaystyle \lim_{n \to \infty} a_n = \infty? View AnswerDetermine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? a_n = (-2)^{n + 1} View AnswerA deposit of $3000 is made in an account that earns 2% interest compounded quarterly. The balance in the account after n quarters is given by (a) Compute the first eight terms of this sequence. (b)... View AnswerA deposit of $5000 is made in an account that earns 3% interest compounded quarterly. The balance in the account after n quarters is given by (a) Compute the first eight terms of this sequence. (b)... View AnswerIf you deposit $100 at the end of every month into an account that pays 3% interest per year compounded monthly, the amount of interest accumulated after n months is given by the sequence I_R = 10... View AnswerSuppose that lim_n a_n = L. Prove that lim_n \|a_n\| = \|L\|. View AnswerUse the formal definition of the limit of a sequence to prove that the sequence {a_n} converges, where a_n = 5^n + pi. View AnswerFind a formula for the nth term of the following sequence. 1, - \frac{1}{4}, \frac{1}{9}, - \frac{1}{16}, \frac{1}{25}, \cdots (a) a_n = \frac{(-1)^n}{n^2} (b) a_n = \frac{(-1)^{2n + 1}}{n^2} (c) a... View AnswerFind the 66th term in the following arithmetic sequence. Hint: Write a formula to help you. 1st term + common difference (desired term - 1). -92, -85, -78, -71,... View AnswerWhat is the 12th term in the following sequence? 1,\, 4,\, 7,\, 10\, \dots View AnswerProve that if \displaystyle \lim_{n \to \infty} a_n = 0 and {b_n} is bounded, then \displaystyle \lim_{n \to \infty} a_nb_n = 0. View AnswerTrue or false? If {a_n} and {b_n} are divergent, then {a_n + b_n} is divergent. View AnswerList the first five terms of the sequence. a_1 = 2, \enspace a_{n + 1} = \dfrac{a_n}{1 + a_n} View AnswerFind a formula for the general term a_n of the sequence, assuming that the pattern of the first few terms continues. \left{1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \dots \right} View AnswerProve that the sequence a_n =1/n is bounded. View AnswerThe first six terms of a sequence are 1, 1, 2, 3, 5, 8. Explain why the formula for this sequence may be given by a_1 = 1 a_2 =1 a_n = a_{n-1} + a_{n-2}, n ge 3. That is, the first two terms of the... View AnswerAn arithmetic sequence is defined as consecutive terms that have a common difference. a. True b. False View AnswerDetermine if the following sequence is monotone or strictly monotone. \dfrac{n}{3n + 1} View AnswerFind the seventh term of the sequence. 1, 3, \frac{9}{2}, \frac{9}{2}, \frac{27}{8}, \frac{81}{40}, ... (A) \frac{77}{80} \(B) \frac{79}{80} \(C) \frac{81}{80} \(D) \frac{83}{80} \(E) \frac{87}... View AnswerFind a formula for the nth term of the sequence in terms of n. 1, 0, 1, 0, 1, \dots View AnswerCompute the sum: \sum_{i \in S} \left(i^2 + 1\right) where S = {1, 3, 5, 7} View AnswerWrite the first four terms of an = 2n + 3. View AnswerFind the formula for this pattern. 1.5, 2.5, 3.5, 4.5, ... (Hint: You are starting with x = 1.) View AnswerFind a formula for the general term a_n of the sequence, assuming that the pattern of the first few terms continues. (Assume that n begins with 1.) \left{\frac{1}{4}, -\frac{4}{5}, \frac{9}{6}, -... View AnswerFind the sum of the first 600 terms. 3, 5, 7, 9, . . . View AnswerList the first five terms of the sequence. a_n = ((-1)^n n)/(factorial of (n) + 1). View AnswerFind the limit of the following sequence: x_n = \left(1 - \frac{1}{n^2}\right)^n. View AnswerDetermine if the sequence {a_n} converges, and if it does, find its limit when a_n = dfrac{6n+(-1)^n}{4n+2}. a) the sequence converges with limit = dfrac{7}{4} b) the sequence converges with lim... View AnswerHow many positive integers between 22 and 121, inclusive, are divisible by 4? View AnswerTrue or false? \sum_{n = 0}^\infty \frac{2^n + 3^n}{5^{n + 1}} = \frac{5}{6} View AnswerFind the sum. sum_{k=1}^8 (7k-6) View AnswerFind the sum of the even integers from 20 to 60. View AnswerFind the value of sum of 4absolute of (-3 - i^2) from i = -1 to 1. View AnswerGiven the following arithmetic sequence: 7, -1, -9, -17, ... Find: (i) The general term of the sequence a_n. (ii) The 9th term (a_9) of the sequence. (iii) The sum to infinity of the sequence. View AnswerConsider the following sequence: 1000, 100, 10, 1 a) Is the sequence an arithmetic sequence, why or why not? b) Is the sequence a geometric sequence, why or why not? c) Write down the series of thi... View AnswerFind the formula for the nth term of the following sequence 1, -4, 9, -16, 25, ... View AnswerDetermine whether the sequence is bounded. n over n + 1 View AnswerFind the formula for the nth term of the sequence below. -1, 1, -1, 1, -1, ... View AnswerWrite the first three terms of the sequence. a_n = (2n - 1)/(n^2 + 4). View AnswerWrite a recursive formula for the following sequence. 8, 17, 26, 35, 44, ... View AnswerFind the first five terms of the sequence. a_n = (1 over 2)^n (n) View AnswerDetermine if the following sequence is monotone or strictly monotone. A) n - 2^n B) n - n^2 View AnswerDetermine whether the sequence converges or diverges. a_n = (1 + 4n^2)/(n + n^2). If it converges, give the limit as your answer. If it diverges, give divergent as your answer. View AnswerDetermine whether the sequence converges or diverges. If it converges, enter the limit as your answer. If it diverges, enter divergent as your answer. an = n^3e^-n View AnswerIn an Arithmetic Progression, the 9th term is 2 times the 4th term and the 12th term is 78. What is the sum of the first twenty terms? View AnswerIn this sequence arithmetic, geometric, or neither? 45, 50, 65, 70, 85, dots View AnswerThe graph of an arithmetic sequence is shown. What is the value of the fifth term? A. 0.5 B. 1 C. 6.5 D. 7 View AnswerLet S = 1 + 2 + 3 + . . . + n be the length of the sides of the square in the figure. Mark off segments of lengths 1, 2, 3, . . . , n along two adjacent sides. (a) Show that the area A of the squar... View AnswerWrite the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume n begins with 0.) View AnswerIn your own words, describe the characteristics of an arithmetic sequence. Give an example of a sequence that is arithmetic and a sequence that is not arithmetic. View AnswerRewrite the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume n begins with 1.) View AnswerWrite the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume n begins with 1.) View AnswerWrite the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume n begins with 0.) a_n = ((-1)^2n)/(2n)! View AnswerWrite the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume n begins with 0.) a_n = 1/(n + 1)! View AnswerWrite the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume n begins with 1.) a_n = (2n - 1)(2n + 1). View AnswerWrite the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, explain why. Assume n begins with 1. a_n = (2/n)(n + (2/n)(n(n - 1)/2 - n)). View AnswerRewrite the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. a1 = 8, d = -2 View AnswerWrite the first five terms of the sequence defined recursively. a_1 = 49, a_{k+1} = a_k + 6 View AnswerLet a_n = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \sqrt{5}} be a sequence with nth term an. Use the table feature of a graphing utility to find the first five terms of the sequence. View AnswerCreate a scatter plot of the terms of the sequence. Determine whether the sequence converges or diverges. When it converges, estimate its limit. a_n=3(1-(1.5)^n)/(1-1.5) View AnswerCreate a scatter plot of the terms of the sequence. Determine whether the sequence converges or diverges. When it converges, estimate its limit. a_n=4(2/3)^n View AnswerFind the next number in the pattern below. 7, 12, 17, 22, 27 View AnswerDetermine whether the sequence -1/2, 1/2, 3/2, 5/2, 7/2, ..., is arithmetic, geometric, or neither. View AnswerWrite the first five terms of the sequence. (Assume n begins with 1.) a_n = (2^n)/(2^n + 1) View AnswerWrite an expression for the apparent nth term of the sequence. (Assume n begins with 1.) 50, 48, 46, 44, 42,... View AnswerWrite the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, then explain why. Assume n begins with 1. a_n=1/2n^2 [3-2n(n+1)] View AnswerWhat is the next number in the sequence? 5, 15, 35, 75, _____ View AnswerA sequence is called a _ sequence when the ratios of consecutive terms are the same. This ratio is called the _ ratio. View AnswerThe function values a1, a2, a3, a4, . . . are called the __ of a sequence. View AnswerWrite the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. a_2 = 14, a_6 = 22 View AnswerWrite the first five terms of the arithmetic sequence. a_8 = 26, a_{12} = 42 View AnswerWrite the first five terms of the sequence. (Assume n begins with 1.) a_n = \frac{n}{n + 1} View AnswerWrite the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume n begins with 1.) a_n = \frac{1 + (-1)^n}{2n} View AnswerWrite the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume n begins with 1.) a_n = \frac{1 + (-1)^n}{n} View AnswerUse the table feature of a graphing utility to find the first 10 terms of the sequence. (Assume n begins with 1.) a_n = 1 + \frac{n + 1}{n} View AnswerWrite an expression for the apparent nth term of the sequence. (Assume that n begins with 1.) 1,2,\frac{2^2}{2}, \frac{2^3}{6},\frac{2^4}{24},\frac{2^5}{120},... View AnswerWrite an expression for the apparent nth term of the sequence. (Assume n begins with 1.) \frac{2}{3}, \frac{3}{4},\frac{4}{5}, \frac{5}{6}, \frac{6}{7}, ... View AnswerWrite the first five terms of the sequence. (Assume that n begins with 1.) a_n = 2n + 5 View AnswerFind a formula for a_n for the arithmetic sequence. a_1 = 100, d = -8 View AnswerFind a formula for a_n for the arithmetic sequence. a_1 = 15, d = 4 View AnswerWrite the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, then explain why. Assume n begins with 1. a_n = ((-1)^(n+1))/n^2 View AnswerWrite the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, then explain why. Assume n begins with 1. a_n = ((-1)^n)/n View AnswerWrite the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, then explain why. Assume n begins with 1. a_n = (n+1)/(n^2+1) View AnswerWrite the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, then explain why. Assume n begins with 1. a_n = n/(n^2+1) View AnswerWrite the first five terms of the sequence. a_1 = 12 and a_(k+1)= a_k + 4 View AnswerFind the indicated term of the sequence. a_n = \frac{2^{n+1}}{2^n +1}. a_7 = View AnswerFind the indicated term of the sequence. a_n = (-1)^{n-1} (n(n - 1)). a_{16} = View AnswerUse a graphing utility to graph the first 10 terms of the sequence. (Assume n begins with 1.) a_n = \frac{2n}{n + 1} View AnswerUse a graphing utility to graph the first 10 terms of the sequence. (Assume n begins with 1.) a_n = 8(0.75)^{n-1} View AnswerThe sum of the first 20 terms of an arithmetic sequence with a common difference of 3 is 650. Find the first term. View AnswerFind the indicated nth partial sum of the arithmetic sequence. a_1 = 100, a_{25} = 220, n = 25 View AnswerWrite the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, explain why. Assume n begins with 1. a_n = (2n-3)/(5n+4) View AnswerWrite the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, then find the common difference. (Assume that n begins with 1.) a_n = 2^{n-1} View AnswerWrite the first five terms of the sequence. Determine whether or not the sequence is arithmetic. If it is, find the common difference. (Assume n begins with 1.) a_n = 2^n + n View AnswerWrite the first five terms of each sequence an. (Assume that n begins with 1.) (a_n = (-1)^(n+1)/(2n+3) View AnswerWrite an expression for the apparent nth term (a_n) of the sequence. (Assume that n begins with 1.) 3, 7, 11, 15, 19, ... View AnswerWrite an expression for the apparent nth term (a_n) of the sequence. (Assume that n begins with 1.) 1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, ... View AnswerWrite an expression for the apparent nth term (a_n) of the sequence. (Assume that n begins with 1. 1, (1/2), (1/6), (1/24), (1/120)... View AnswerWrite the first five terms of the sequence. Then find an expression for the nth partial sum. a_n = \frac{1}{n +1} - \frac{1}{n +2} View AnswerUse the table feature of a graphing utility to find the first 10 terms of the sequence. (Assume n begins with 1.) a_n = 20 - 3/4 n View AnswerDetermine whether or not the sequence is arithmetic. If it is, find the common difference. \frac{1}{3}, \frac{2}{3}, \frac{4}{3}, \frac{8}{3}, \frac{16}{3}, ... View AnswerDetermine whether the sequence is arithmetic. If so, then find the common difference. 4, 9, 14, 19, 24, ... View AnswerWrite the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, then explain why. Assume n begins with 1. a_n = \frac{(-1)^n}{n^3} View AnswerUse a graphing utility to graph the first 10 terms of the sequence. a_n = 10 (-1.2)^{n-1} View AnswerWrite the first five terms of the sequence defined recursively. Use the pattern to write the nth term of the sequence as a function of n. a_1=81, a_k+1 = 1/3 a_k View AnswerWrite the first five terms of the sequence. (Assume n begins with 1.) a_n= (n+1)/n View AnswerFind the next two terms of the given sequence. 2, 0, -18, -64, -5 View AnswerFind the next two terms of the given sequence. 2, 7, -3, 2, -8 View AnswerWrite the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, then explain why. Assume n begins with 1. a_n = \frac{n^2 + 3n - 4}{2n^2 +... View AnswerWrite the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, then explain why. Assume n begins with 1. a_n = (1 + (-1)^n)/n View AnswerFind the first five terms of the sequence. {a_n} = {{{x^n}} \over {n!}} View AnswerWrite the first five terms of the sequence. {a_n} = {{{{\left( { - 1} \right)}^{n + 1}}{{\left( {x + 1} \right)}^n}} \over {n!}} View AnswerFind the first 10 terms of the sequence. a_1 = -y, d = 5y View AnswerFind the first 10 terms of the sequence. a_1 = x, d = 2x View AnswerFind the nth term of the sequence 1 / 3, 1 / 7, 1 / 11, 1 / 15, . . . . View AnswerFind the first five terms of the sequence a_n = (-\frac{1}{5})^n. View AnswerWrite out the first five terms (beginning with n = 1) of the sequence given. View AnswerList the first four terms of the sequence whose nth term is a_n = (-1)^n + 1 / n. View AnswerSolve the recurrence relation a_n = 2a_n-1 + 8a_n-2 with initial conditions a_0 = 1, a_1 = 4. View AnswerFind a formula for the nth term of the sequence. 1, -1 / 4 , 1 / 9, -1 / 16, 1 / 25, . . . View AnswerWrite the first five terms of the given sequence where the nth term is given. a_n = 1 - n / n^2 View AnswerThe following list shows the first six terms of a sequence. 1, 1, 2, 3, 5, 8, ... Which of the following formulas can be used to find the terms of the sequence? View AnswerFind the limit of the following sequence: c_n = \left ( \dfrac{n^2 + n - 6}{n^2 - 2n - 2} \right )^{5n+2}. View AnswerFind a formula for the general term a_n of the sequence \displaystyle{ {a_n}_{n=1}^\infty = \left{1, \dfrac{ 5}{2}, \dfrac{ 25}{4}, \dfrac{ 125}{8}, \dots \right} } as... View AnswerFind the limit of the sequence whose terms are given by a_n = (n^2) (1 - cos (1.8 / n)). View AnswerCompare the differences between the sequence with Alu and the sequence without Alu in PCR. View AnswerFind the nth term (and the general formula) for the following sequence; 1, 3, 15, 61, 213 View AnswerClassify the following sequence as arithmetic, geometric, or other. If arithmetic or geometric, find t(n). 7, 8, 10, 13, ... View AnswerClassify the following sequence as arithmetic, geometric or other. If arithmetic or geometric, find t(n). 5, 9, 13, 17, ... View AnswerIn the sequence -1, -5, -9, -13, ... (a) Is -745 a term? If so, what term is it? (b) What is the 1000th term? View AnswerIf the sequence is arithmetic or geometric, write the explicit equation for the sequence. If the sequence is not arithmetic or geometric, describe the pattern. 1, 2, 3, 4, 5, 6, ... View AnswerIf the sequence is arithmetic or geometric, write the explicit equation for the sequence. If the sequence is not arithmetic or geometric, describe the pattern. -1, -5, -9, -13, ... View AnswerIf the sequence is arithmetic or geometric, write the explicit equation for the sequence. If the sequence is not arithmetic or geometric, describe the pattern. 5, 7, 11, 17, 25, ... View AnswerList the first four terms of the sequence. tn=40n-15 View AnswerPredict the product from the reaction of substance (reddish-brown = Br) with Br_2, FeBr_3. View AnswerFind out whether the sequence is increasing ,decreasing or not monotonic or is the sequence bounded {n-n^{2} / n + 1} View AnswerFind the sum of all the positive integers from 1 to 300 that are not divisible by 3. View AnswerWhat is the 18th term of the following arithmetic sequence? -13, -9, -5, -1, 3, ... View AnswerIn a sequence that begins 25, 23, 21, 19, 17, ..., what is the term number for the term with a value of -11? View AnswerFind the limit of s(n) as n to infinity. s (n) = 1 / {n^2} ({n (n + 1)} / 2) View AnswerAn employee has a starting salary of $40,000 and will get a $3,000 raise every year for the first 10 years. What will be the employee's total earned income over the 10 years? View AnswerAn employee has a starting salary of $40,000 and will get a $3,000 raise every year for the first 10 years. How much will the employee make in year 6? View AnswerDetermine the sum of the following arithmetic series. 7 + 14 + 21 + ... + 98 View AnswerDetermine the sum of the following arithmetic series. 5 + 8 + 11 + ... + 53 View AnswerWrite the first four terms of the arithmetic sequence with a first term of 5 and a common difference of 3. View AnswerHow many terms are in the following sequence? 2, 5, 8, ..., 20 View AnswerWhat is the sum of the first seven terms of the following arithmetic sequence? -7, -4, -1, ... View AnswerWhat is the 7th term of the following arithmetic sequence? -7, -4, -1, ... View AnswerIn a sequence, the first term is 82 and the common difference is -21. Find the fourth term of this sequence. View AnswerIn a sequence, the first term is 4 and the common difference is 3. Find the fifth term of this sequence. View AnswerFind the common difference in the following arithmetic sequence. x + 1, x + 4, x + 7, x + 10, ... View AnswerWhat is the sum of the first 10 terms of the following arithmetic sequence? 1, 3, 5, ... View AnswerWhat is the sum of the 2nd, 7th, and 10th terms for the following arithmetic sequence? -10, -6, -2, ... View AnswerWhat is the sum of the next five terms of the following arithmetic sequence? 2, 4, 6, ... View AnswerSuppose you gave your friend a total of $630 over the course of seven days. Each day, you gave him $10 more than the previous day. The increase in money per day stayed constant. How much money did... View AnswerIs the following sequence arithmetic, geometric, or neither? 2, 8, 14, 20 View AnswerIf the 2nd term of an arithmetic sequence is -15 and the 7th term is 10, find the 4th term. View AnswerDetermine which type of sequence is given below: arithmetic, geometric, or neither. 1, 8, 27, 64 View AnswerFind term 21 of the following sequence. -6, -13, -20, -27, ... View AnswerFind the next four terms in the arithmetic sequence. -15, -7, 1, ... View AnswerList the first five terms of the sequence. a_1 = 1, a_(n + 1) = 5a_n - 3. View AnswerList the first five terms of the sequence. a_1 = 6, a_(n + 1) = (a_n)/n. View AnswerList the first five terms of the sequence. a_n = 1/(n + 1)! View AnswerCalculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not,... View AnswerList the first five terms of the sequence. a_1 = 2, a_2 = 1, a_(n + 1) = a_n - a_(n - 1). View AnswerList the first five terms of the sequence. a_1 = 2, a_(n + 1) = (a_n)/(1 + a_n). View AnswerDetermine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? a_n = n^3 - 3n + 3. View AnswerList the first five terms of the sequence. {a_n} = {{{2^n}} \over {2n + 1}} View AnswerWhat is the sequence of 7, 14, 28, 56, 112... called? View AnswerWhat is the rule for the sequence corresponding to this series? -4 + -7 + -10 + -13 View AnswerWhat is the common difference of the sequence 1, 5, 9, 13, . . .? View AnswerWhat is a recursive rule for -6, 12, -24, 48, -96, ...? View AnswerWrite the first three terms (a_1, a_2, a_3) of the sequence whose general term is a_n = (3n)!. View AnswerWrite the first five terms of the sequence whose general term is a_n = \frac{3^n}{n}. View AnswerDetermine whether the following is true or false: The sequence a_n = ne^{-4n} is monotone. View AnswerThe first term of a sequence along with a recursion formula for the remaining terms is given below. Write out the first ten terms of the sequence. a_1 = 1, a_{n + 1} = {n a_n} / {n + 3} View AnswerDetermine the limit of the following sequence: \left{ \sqrt{n^2 - n +4} - n + 3 \right}_{n=1}^{\infty}. View AnswerFind the recursive formula of the ODE y'' + y = 0. View AnswerThe speed range of an electric motor vehicle is divided into 5 equal divisions between 0 and 1,500 rpm. Identify the common difference on the scale of the speedometer. View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = (1 + 2 / n)^{2 n} lim_{n to infinity} a_n View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = cos (n / 7) View AnswerWrite an equation for the nth term of the arithmetic sequence. Then find a_{10}. 13, 9, 5, 1, ... View AnswerThe sum of the 2nd term and the 9th term of an arithmetic sequence is -6. The. Sum of the 4th and the 6th terms of the same sequence is 4. Find the first term, the common difference, the general fo... View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = {7 + 2 n^2} / {n + 7 n^2} View AnswerDetermine if the given sequence converges or diverges. If it converges what is its limit? {(-1)^n}_{n = 0}^infinity View AnswerHow do you find the nth term rule for 1, 5, 9, 13, ? View AnswerLet a1 3, a2 4 and for n 3, an 2an 1 an 2 5, express an in terms of n. View AnswerLet, a1 3 and for n 2, an 2an 1 1, express an in terms of n. View AnswerWhat is the 100th term of the sequence 2, 3, 5, 8, 12, 17, 23,...? View AnswerFind the first five terms given a_1 = 4, a_2 = -3, a_{(n + 2)} = a_{(n+1)} + 2a_n. View AnswerThe sequence a1, a2, a3,..., an is an arithmetic sequence with a4 = -a6. What is a5? View AnswerWhat is the sum of a finite arithmetic sequence from n = 1 to n = 10, using the the expression 3n - 8 for the nth term of the sequence? View AnswerWhat is the nth term for the sequence 1, 4, 9, 16, 25, ...? View AnswerWhat is the next number in the pattern: 4, 9, 16, 25, ...? View AnswerWhat is the rule for the sequence 3, 4, 7, 12? View AnswerWhat is the sum of the sequence 5, 10, 15, 20, 25, 30, 35, 40, 45, 50? View AnswerWhat are the next two terms in the sequence 3, 6, 5, 10, 9, 18, 17, ...? View AnswerThe terms of a sequence are -2, -6, -10, -14, -18. What is the common difference, and what are the explicit and recursive formulas for the sequence? View AnswerAn architect designs a theater with 15 seats in the first row, 18 in the second, 21 in the third, and so on. If the theater is to have a seating capacity of 870, how many rows must the architect us... View AnswerFind the nth term of the sequence: 1 / 2, 1 / 4, 1 / 4, 3 / 8, . . View AnswerHow do you write the first five terms of the sequence a_n=3n+1? View AnswerThe nth term of a sequence is 2n^2. What is the 4th term of the sequence? View AnswerWhat recursive formula can be used to generate the sequence 5, -1, -7, -13, -19, where f(1) = 5 and n is greater than 1? View AnswerGiven that the nth term of a sequence is given by the formula 4n+5, what are the first three terms of the sequence? View AnswerWhat is the rule for the sequence 3, 5, 8, 13, 21,...? View AnswerCompute the first five terms of the sequence using the format for a dynamical system defined by a difference equation: Delta t_n = 1.5(100 - t_n), t_0 = 200 View AnswerFill in the blank so that the resulting statement is true. In a certain year, 35% of adults in a certain country viewed a college education as essential for success. For the following ten-year peri... View AnswerFind the nth term of an of a sequence whose first four terms are given. \ -\dfrac{4}{9},\ -\dfrac{5}{18},\ -\dfrac{6}{27},\ -\dfrac{7}{36} View AnswerFind the first five terms in sequences with the following n^{th} terms. a) 2n-1 b) 7n-2 c) 4n+1 d) 2n^2-1 View AnswerWrite a formula for the general term (the nth term) of this arithmetic sequence. Do not use a recursion formula. Then use the formula for a_n, to find a_{20}, the 20th term of the sequence. a_1 =... View AnswerWhat is the 5^{th} term in the sequence? b(n) = -1(2)^{n - 1} View AnswerWhat is the 4th term in the sequence? \begin{cases} b(1) = -54 \b(n) = b(n - 1) \cdot \frac{4}{3}\end{cases} View Answer\left{\begin{matrix} a(1)=-11\ a(n)=a(n-1)\cdot 10 \end{matrix}\right. What is the 4^{th} term in the sequence? View AnswerComplete the recursive formula of the arithmetic sequence 1, 15, 29, 43, .... a(1) = _ a(n) = a(n - 1)+ _ View AnswerComplete the recursive formula of the arithmetic sequence 14, 30, 46, 62, .... d(1) = _ d(n) = d(n - 1)+ _ View AnswerComplete the recursive formula of the arithmetic sequence -15, -11, -7, -3, .... (a) c(1) = _ (b) c(n) = c(n - 1) + _ View AnswerOn day one, a scientist (using a microscope) observes 5 cells in a sample. On day two, the scientist observes 11 cells in the sample. On day three, the scientist observes 17 cells in the sample and... View AnswerWrite the first six terms of the arithmetic sequence. (Type an integer or simplified fraction.) a1 = 11/2 , d = 1/2 View AnswerA sequence of numbers is formed by adding together corresponding terms of an arithmetic progression and a geometric progression with a common ratio of 2.The 1st term is 48, the 2nd term is 73, and... View AnswerLet \left { x_n \right } be a non-stochastic sequence of scalars and \left { \epsilon_n \right } be a sequence of i.i.d. centered random scalars with finite variance. Is \left { x_n\epsilon_n... View AnswerWhat are the first five terms of the sequence an = \text{n}^{2} + {2}? View AnswerMark is building a pyramid out of blocks. The top of his pyramid has 1 block, the second layer has 4 blocks, the third layer has 9 blocks, the fourth layer has 16 blocks, and the fifth layer has 25... View AnswerA rock, dropped into a well, falls 4 and 9/10 meters in the first second, and at every next second after that it falls 9 and 4/5 meters more than the preceding second. Find the depth of the well if... View AnswerConsider the following sequence: a_1 = 3, \; a_{n+1} = \dfrac{4}{5} -a_n. Find the second and the third element in the sequence. View AnswerConsider a sequence of numbers given by the definition c_1 = 2, c_i = c_i -1\cdot 3, how do you write out the first 4 terms, and how do you find the value of c_4 - c_2? View AnswerIf (an) is an increasing sequence and (bn) is a sequence of positive real numbers, then (an.bn) is an increasing sequence. - True - False View AnswerShow directly from the definition that the sequence \left ( \frac{n + 1}{n} \right ) is a Cauchy sequence. View AnswerFind the limit of the sequence {square root {3}, square root {3 square root {3}}, square root {3 square root {3 square root {3}}}, ...} View AnswerFind a formula for the general term a_n of the sequence. For example, answer n^2 if given the sequence: {1, 4, 9, 16, 25, 36,...}. Assume that n starts at 1. {1/4, 2/9, 3/16, 4/25,...} View AnswerThe first term of a sequence along with a recursion formula for the remaining terms is given below. Write out the first ten terms of the sequence. a_1 =5, a_{n+1}=frac{na_n}{n+2} View AnswerWrite the first or next four terms of the sequence and make a conjecture about its limit if it converges, or explain why if it diverges. a_n = 1 - 10^(-n), n = 1, 2, 3, ... View AnswerWrite the first or next four terms of the following sequences. a_(n + 1) = (a_n)^2 - 1; a_1 = 1. (find a_2 through a_5) View AnswerIf S_n = \overset{n}{\underset{i = 1}{\Sigma}} \left(\dfrac{1}{9}\right)^i, then list the first five terms of the sequence S_n. View AnswerDetermine whether the sequence is decreasing, increasing, or neither. a_n = (n^2)/(n^3 + 1). View AnswerDetermine whether the following sequence converges or diverges. If converge, compute the limit. a_n = n(2^(1/n) - 1) View AnswerDetermine if the series will converges or diverges or neither if the series converges then find the limit: a_n = cos ^2n/2^n View AnswerDetermine if the series will converges or diverges or neither if the series converges then find the limit: a_n = (-1)^n/2 square root{n} = lim_{n to infinty} a_n= View AnswerDetermine whether the following sequence converges or diverges. If it converges, find the limit. a_n = (-1)^n(1.001)^n View AnswerDetermine whether the following sequence converges or diverges. If it converges, find the limit. a_n = \dfrac{5+2n}{n^2} View AnswerThe number which best completes the sequence below is: 3, 9, 4, 5, 25, 20, 21, 441, . . . A. 436 B. 442 C. 430 D. 439 E. 454. View AnswerWhich of the following DNA sequences most likely represents the recognition sequence of a restriction endonuclease? A. c a g g a c B. c t g c a g C. t a g g t a D. c c t c c t View AnswerDetermine if the sequence is convergent or divergent. a_n = (5(-1)^n + 3)((n + 1)/n). View AnswerFor the sequence bn = \frac{3n^4 + 2n^3 - n^2 + 8}{3n + 2n^4}, tell whether it converges or diverges. View AnswerShow that, for every real number y, there is a sequence of rational numbers which converges to y. View AnswerGive an example of each of the following or argue that such a request is impossible: 1) A Cauchy sequence that is not monotone. 2) A monotone sequence that is not Cauchy. 3) A Cauchy sequence wit... View AnswerFind the first four terms of the sequence given, a=5, for a_n=3a+5 for x geq 2. View AnswerFor the following sequence, decide whether it converges. If it does, compute its limit. a_n = \dfrac{5 -2n}{3n -7} View AnswerA definite relationship exists among the numbers in the series. Extend the series below through combinations of addition, subtraction, multiplication and division. 260, 130, 120, 60, , View AnswerA definite relationship exists among the numbers in the series. Extend the series below through combinations of addition, subtraction, multiplication and division. 100, 400, 200, 800, , View AnswerA definite relationship exists among the numbers in the. Extend the series below through combinations of addition, subtraction, multiplication and division. 200, 100, 500, 250, 1,250, , View AnswerWhich one of the numbers does not belong in the following sequence; 2, - 3, - 6, - 7, - 8, - 14, - 15, - 30? A. THREE B. SEVEN C. EIGHT D. FIFTEEN E. THIRTY View AnswerSuppose a_n is an always positive sequence and that lim_{n to infinity} a_n diverges. Then lim_{n to infinity} a_n = infinity. View AnswerSuppose a_n is an always increasing sequence. Then the sequence b_n = 8-3a_n is an always decreasing sequence. View AnswerExplain that every monotonic sequence converges. View AnswerFind an equation for the nth term of the arithmetic sequence. -15, -6, 3, 12, ... View AnswerExplain arithmetic progression and geometric progression. View AnswerShow step-by-step solution and briefly explain each step: Let Sn be an increasing sequence of positive numbers and define Prove that sigma n s an increasing sequence. View AnswerList the first five terms of the sequence. a_n = {(a - 1)^{n - 1}} / {6 n} View AnswerFind the next two apparent terms of the sequence. Describe the pattern you used to find these terms. 1, -\frac{1}{8}, \frac{1}{27}, -\frac{1}{64},... View AnswerWrite the first five terms of the sequence. a_n = \left(-\frac{3}{4}\right)^n, n \geq 1 View AnswerFind the limit of the sequence. a_n = (2n) / (sqrt(n^2+5)) View AnswerCompute the limit of the following sequence as ''n'' approaches infinity: \: log(1+7^{1/n}). View AnswerDetermine whether the sequence is increasing, decreasing, or not monotonic. {a_n} = {1 \over {3n - 1}} View AnswerWhat is the formula for the nth term of the sequence 15, 13, 11, 9, ...? View AnswerWhat is the next term in the series 2a, 4b, 6c, 8d, ? View AnswerDetermine whether the sequence converges or diverges. a_n = \dfrac{n^2 + 7}{n + 6} a. converges to 0 b. converges to 1 c. converges to \frac{7}{6} d. diverges View AnswerFind a formula for the general term a_n of the sequence, assuming that the pattern of the first few terms continues. You must state if n starts at 0 or 1. 0, -1/3, 2/5, -3/7, 4/9, -5/11, 6/13, ... View AnswerWhat is the 100th term of the sequence a_n = \dfrac{8}{n+1}? View AnswerEvaluate: lim k k 3 + 1 k 2 View AnswerA sequence of numbers a_1, a_2, a_3, ... is defined by a_{n + 1} = \frac{k(a_n + 2)}{a_n}; n \in \mathbb{N} where k is a constant. Given that: View AnswerConsider the sequence: \begin{Bmatrix} \dfrac{k}{k^2 + 2k +2 } \end{Bmatrix}. Determine whether the sequence is monotonic or eventually monotonic, and whether the sequence is bounded above and/or... View AnswerWhat is ith or xi from this sentence "Take n number of measurements: x1, x2, x3, etc., where the ith measurement is called xi and the last measurement is called xn"? View AnswerFind the limit of the following sequence. a_n = square root {n + square root {n + 1}} - square root n View AnswerFind the limits of the following sequence as n . a n = ( 1 2 n ) n View AnswerFind the limits of the following sequence as n . a n = ( e n 3 n + 2 n ) View AnswerFind the limits of the following sequence as n . a n = n 3 + n 2 + 1 2 n 3 2 n + 2 View AnswerConsider the sequence { 2 n 5 n } n = 1 : Find a function f such that a n = f ( n ) . View AnswerConsider the sequence { n 2 + 2 n + 3 3 n 2 + 4 n 5 } n = 1 : Find a function f such that a n = f ( n ) . View AnswerCalculate the first 10 terms (starting with n=1) of the sequence a_1=-2, \ a_2=2, and for n \geq 3, \ a_n=a_{n-1}-2a_{n-2} View AnswerFind the recursive rule for the nth term of the following sequence: 1, 4/3, 5/3, 2, ... View AnswerA potentially infinite process: a. is, in fact, continued on and on without end. b. is mere potentiality, without reality. c. could, in principle, be continued on and on without end. d. could, in p... View AnswerDetermine whether the sequence is (eventually) decreasing, (eventually) increasing, or neither. Explain. {\frac{n!}{3^n}} View AnswerWhat is the fifth term of the following sequence? a_n = (-(1/2))^(n - 1) View AnswerWhat is the fifth term of the following sequence? a_1 = 48, a_n = (1/2) a_(n-1) - 8 View AnswerConsider a fish population that increases by 8\% each month and from which 300 fish are harvested each month. formulate a difference equation model (ie. a recursion statement) that describes the po... View AnswerExpress the following integral as an infinite series. Integral of ((1-cos x)/x) dx from 0 to 0.25, and approximate its sum to five decimal places. View AnswerIn an arithmetic sequence, a17 = -40 and a28 = -73. Write a recursive formula for this sequence. View AnswerAn arithmetic sequence has a common difference of 9 and a(41) = 25. Find a rule for this arithmetic sequence. View AnswerSketch the following sequence. x ( n ) = 2 ( n + 3 ) 0.5 ( n + 1 ) 4 ( n 5 ) View AnswerSketch a graph that represents the sequence: 7, 5.5, 4, 2.5, 1. View AnswerLet's play three-yard football (the games are shown on Thursday afternoon between 4:45 and 5 on the SASN Short Attention Span Network). The home team starts with the ball on the 1-yard line. The t... View AnswerWrite a formula for the general term or nth term for the sequence. Then find the indicated term. 1,3,5,7,9, ; a10 View AnswerFind the cardinal number for the following sets. A = {111, 112, 113, 114,....., 169} B = {111, 113, 115,....., 411} View AnswerUse the passage below to answer the question. Though he gained fame as a magician and escape artist, View AnswerFor the sequence below, find a closed formula for the general term, an. -2, -8, -18, -32, -50, ...,an=.... View AnswerFor the following sequence, find a closed formula for the general term, an. 0,3,8,15,24,..., an=.... View AnswerFor the following sequence, find a closed formula for the general term, an. -2,-8,-18,-32,-50,...,an=.... View AnswerFind a formula for the general term, A_n, given the following sequence. {1, 4, 9, 16, 25, 36...} View AnswerConsider the sequence 67, 63, 59, 55... Is 85 a member of the sequence? View AnswerConsider the sequence 67, 63, 59, 55... Show that the sequence is arithmetic. View AnswerAn arithmetic sequence is defined by U_n=11n-7. What is U_1 and d? View AnswerFind k given that k-1, 13, and 3k+3 are consecutive terms of an arithmetic sequence. View AnswerFind a formula for the general term an of the sequence starting with a1: 4/10, 16/15, 64/20, 256/25,.... View AnswerFind a formula for the general term, a_n. Assume that the first term in the sequence is a_1: {\frac{3}{4}, \frac{4}{9}, \frac{5}{16}, \frac{6}{25},............... } View AnswerFind a closed formula for the general term, a_n. 0, 3, 8, 15, 24, ............... View AnswerEach term is the term number times the next term number. 1 2 3 4 5 6 7 8 9 _ _ _ _ _ _ _ _ 90 View AnswerFind the first 4 terms and the 100^{th} term of the sequence whose n^{th} the term is given. a_n = (-1)^{n + 1} \frac{n}{n + 1} View AnswerFind the first four terms of the sequence with a recursive formula. \displaystyle u_1=3, \; u_n = 2 \times u_{n-1}-1,\; n \geq 2 View AnswerDescribe the sequence 5, 8, 11, 14, 17, 20,................ using: a. word b. a recursive formula View AnswerMike walks at a rate of 3 miles per hour. If he needs to walk 26.2 miles, how long will his trip last? View AnswerLook at the sequence in this table Which function represents the sequence? A) a_n = a_{n - 1} + 1 B) a_n = a_{n - 1} + 2 C) a_n = 2a_{n - 1} -1 D) a_n = 2a_{n - 1} - 3 View AnswerFor the sequences shown: i) Find the next 2 numbers in the sequence ii) Write the rule to explain the link between consecutive terms in the form [{MathJax fullWidth='false' a_{n+1}=f(a_n) }] iii)... View AnswerFind a formula for the general term and of the sequence, assuming that the pattern of the first few terms continues. { \frac{1}{4}, \frac{-2}{9}, \frac{3}{16}, \frac{-4}{25}, ...} View AnswerFind a formula for the general term and of the sequence, assuming that the pattern of the first few terms continues. {1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, ...} View AnswerFor the sequences shown: i. Write the next 2 numbers in the sequence ii. Write the rule for finding consecutive terms in the form a_{n+1}=f(a_n) iii. Write the nth term in the sequence in the f... View Answer12^{10},\; 12^{10} + 12^{10},\; 12^{10} + 12^{10} + 12^{10}... In the sequence above, the first term is 12^{10} and each term after the first is 12^{10} more than the preceding term. Which term in... View AnswerWhat woud be the 41st term of the sequence 2, 5, 8, 11, 14, 17, . . .? View AnswerWrite the first four terms of the sequence whose general term is given by: an = 4n + 1 a1 = ____? View AnswerWrite out the first five terms of the sequence with, [(1-5/n+1)^n]_{n=1}^{infinity}, determine whether the sequence converge and if so find its limit. View AnswerDetermine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit. If it is divergent, indicate if it diverge to infinity, to negative infinity, or to neither: limi... View AnswerDetermine whether the sequence converges or diverges, and, if it converges, find \displaystyle \lim_{n \to \infty} a_n. a_n = \ln(4n - 4) - \ln(3n -1) View AnswerWhat is the recursive rule for a_n = 2n + 11? - a_1 = 2; a_n = a_{n-1} + 11 - a_1 = 11; a_n = a_{n-1} + 2 - a_1 = 13; a_n = a_{n-1} + 11 - a_1 = 13; a_n = a_{n-1} + 2 View AnswerFind a formula for a_n, n greater than equal to 1. 1/2, -4/3, 9/4, -16/5, 25/6, cdots View AnswerFind the limit of the sequence or state if it diverges. \left { \frac{\sin^3n}{3^n} \right } View AnswerDetermine whether the sequence converges or diverges. an = n!/2n View AnswerFind the limit of the sequence or determine that the limit does not exist. a_n = n - square root{n^2 - 17n} View AnswerFind the limit of the sequence or determine that the limit does not exist. a_n = ln (5n - 4) - ln (4n + 7) View AnswerFind the limit of the sequence or determine that the limit does not exist. a_n = (1+3/n)^n View AnswerGiven that \frac{1}{1 - x} = \sum\limits_{n = 0}^{\infty}x^n if -1 less than x less than 1, find the sum of the series \sum\limits_{n = 1}^{\infty}\frac{n^2}{ - \pi^n}. View AnswerWrite complete solutions for all the following questions. Show all your work/steps. Determine if the sequence n^2 e^(-n) converges or diverges. If it converges, find the limit. View AnswerFind an expression for the n^{th} term of the sequence. Assume that the pattern continues. \Bigg{ \frac{2}{5},\frac{4}{25}, \frac{6}{125},\frac{8}{625},...\Bigg} View AnswerFind an expression for the nth term of the sequence. Assume that the pattern continues. {2/5, 4/25, 6/125, 8/625, ...} View AnswerCalculate the first four-term of the sequence, starting with n = 1. a_1 = 2, a_{n+1} = 2a_{n}^2-2 View AnswerThe sum of the first n terms of an infinite sequence is 3n2 + 5n 2 for all n belongs to Z+. a) Find the nth term. b) Prove that the sequence is arithmetic. View AnswerFind the limit of the sequence: a_n = 2n/(3n + 1). View AnswerThe sequence \left {a_n = \frac{1}{n} \right } is Cauchy because _____. View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = \frac {\cos^2 (n)}{2^n} View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = n^2e^{-n} View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = \frac {4 + 4n^2}{n + 2n^2} View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = \frac {(-1)^n}{9\sqrt n} View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = \sqrt [n] {2^{7 + 3n}} View AnswerDetermine whether the following sequence converges or diverges. If it converges, find the limit. a_n = tan^(-1)(ln 1/n). View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = {\cos^2 (n)}/{3^n} View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = \ln (n + 1) - \ln (n) View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = \frac {(-1)^n}{6\sqrt n} View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = \frac {2 + 3n^2}{n + 8n^2} View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = (1 + \frac 5n)^n View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. a_n = \frac {\ln (4n)}{\ln (12n)} View AnswerDetermine whether each sequence converges or diverges. If it converges, find the limit. View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, specify.) a n = ( 1 ) n 8 n View AnswerFind the limit of the following sequence or determine that the limit does not exist please. a n = 1 + 8 n n View AnswerFind a formula for the sum of n terms. Use the formula to find the limit as n \to \infty. lim_{n \to \infty} sum_{i=1}^{n} \bigg ( 1 + \dfrac{2i}{n} \bigg )^n \bigg ( \dfrac{2}{n} \bigg ) View AnswerDetermine whether the sequence converges or diverges. a_n = (1 + 7 / n)^n View AnswerFind the sum of the infinite geometric series: a) \sum\limits_{n=0}^\infty \left(\frac{1}{2} \right) ^n . b) \sum\limits_{n=0}^\infty 2 \left(\frac{3}{4} \right)^n . View AnswerGraph the first 10 terms of the sequence: a) a_n = 15 \frac{3}{2} n . b) a_n = 5 + 2n . c) a_n = 0.2 n +3 . d) a_n = 0.3n + 8 . View AnswerFind the sum of the infinite geometric series. \frac{1}{9} - \frac{1}{3} + 1 - 3\; +\; . . . View AnswerFind the sum of the infinite geometric series. \sum_{n = 0}^{\infty}\left ( -\frac{1}{2} \right )^n View AnswerDetermine if the following sequence converges or diverges. If it converges, what does it converge to? a_n = ((n + 1)/n)^n. View AnswerDetermine whether the sequence converges or diverges. a_n = cot ({n pi} / {2 n + 3}) View AnswerGiven the sequence b^1 = 5. B^n = 2b(n -1) when n>1. Write an explicit definition of the sequence and use it to find the 12th term. View AnswerFind the first 6 terms of the sequence b^1 = 5. B^n = 2b(n -1) when n>1. View AnswerDetermine whether the sequence converges or diverges. a n = cot n 2 n + 3 View AnswerList the first three terms of each sequence. a n = n n + 1 2 View AnswerDetermine if the following sequence converges or diverges: an = (n + 1) n n. If the sequence converges, find its limit. View AnswerComplete the next two equations of this sequence: 1 = 1 \1 - 4 = 3 \1 - 4 + 9 = 6 \1 - 4 + 9 - 16 = - 10 View AnswerDetermine the convergence or divergence of the sequence an = 8n + 5 4n. If the sequence converges, find its limit. View AnswerDetermine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. an = 3rd root of n / 3rd root of n + 5 View AnswerLet V be the set of sequences of real numbers. For {a, n}, {bn} belongs to V and any real number t, define {an} + {bn} = {an + bn} and t{an} = {tan}. Prove that, with these operations, V is a vecto... View AnswerDetermine whether each sequence converges or diverges a) a_n = (1 + 7/n)^n b) b_n = 2^{n - 1}/7^n View AnswerFor the geometric sequence 5 / 3, -5 / 6, 5 / {12}, -5 / {24}, . . . a. Give the first term and the common ratio for the given geometric sequence. b. Find the general term, a_n, for the given seque... View AnswerWrite the first five terms of the sequence: c_1 = 5, c_n = -2c_{n - 1} + 1 View AnswerWrite the first six terms of the sequence defined by a_1= -2, a_2 = 3, a_n = -2 + a_{n - 1} for n \geq 3. View AnswerGiven the sequence defined by b_n= (-1)^{n-1}n , which terms are positive and which are negative? View AnswerWrite the first five terms of the sequence. a_n=\frac{(n+1)!}{n!} View AnswerFind a formula for the nth term, an, of the sequence assuming that the indicated pattern continues. {1/5, -4/11, 9/17, -16/23,... } View AnswerA sales person working for a heating and air-conditioning company earns an annual base salary of $30,000 plus $500 on every new system he sells. Suppose that { a_n} is a sequence representing the... View AnswerA retirement account initially has $500,000 and grows by 5% per year. Furthermore, the account owner adds $12,000 to the account each year after the first. Let a_1 represent the original amount in... View AnswerFind the nth term of a sequence whose first four terms are given. 1, -8, 27, -64.... View AnswerThe nth term of a sequence is given. Find the indicated term. d_n = 6n + 7 Find d_{204}. View AnswerIf the nth term of a sequence is (-1)^n n^2, which terms are positive and which are negative? View AnswerWhat is the difference between a sequence and a series? View AnswerA _ sequence is a function whose domain is the set of the first n positive integers. View AnswerAn infinite _____ is a function whose domain is the set of positive integers. View AnswerDetermine whether the sequence converges or diverges by finding the limit. d_n = (-4)^n/3^n View AnswerDetermine whether the sequence converges or diverges by finding the limit. b_n = (- 1)^n (2n + 1)/n - 2 View AnswerFind the 5th term of: -f(n) = 3(n - 3) . View AnswerFind the first six terms of the sequence: a1=1, a,-4-an-1 0, 4, 4, 8, 12, 16 4, 16, 64, 256, 1024, 4096 1, 4, 8, 12, 16, 20 1, 4, 16, 64, 256, 1024 View AnswerFind the first 4 terms and the 8th term of the recursively-defined sequence. Choose the letter of the correct answer v1 = 0.75 and vn = (-2)vk-1 for n > 1 A. 0.75, 1.5, -3, 6; -192 B. -1.5, -3, -6,... View AnswerDetermine whether the sequence converges or diverges. an = cot(n pi)(2n + 3) View AnswerList the first three terms of the sequence. an = ( n n + 1)2. View AnswerWrite the first five terms of the sequence defined recursively. a_1=6, a_n=-\frac{1}{a_{n-1}-1} View AnswerGiven: a_n= \frac{(-1)^{n + 1} 6n}{2n + 7} Determine the convergence or divergence of the sequence. If the sequence converges state the value to which it converges. View AnswerDetermine if the following sequence converges or diverges. a_n = (frac{n + 1}{n})^n If the sequence converges, find its limit. View Answerf(0) = 2 and f(n + 1) = 3f(n) -1. Find f(4). View AnswerFind a formula for the nth term of the sequence. 0,0,2,2,0,0,2,2 (\text{ alternating 0's and 2's in pairs}) View AnswerSeveral terms of a sequence {a_n}_{n = 1}^{infty} are given below. {3, frac{3}{2}, frac{3}{4}, frac{3}{8}, frac{3}{16}, ...} A. Find the next two terms of the sequence. B. Find a recurrence relatio... View AnswerDetermine whether the given sequence converges or diverges. {(1 + 3 / n)^n} View AnswerDetermine whether the given sequence converges or diverges. {arctan ({2 n} / {2 n + 1})} View AnswerFind the missing term in the sequence: 2,4,8,16,30,,,,186, View AnswerFind the next term in the sequence. 1 / 2, 2 / 3, 1, 8 / 5, 8 / 3,... View AnswerLet f(x) = x^2 - 4x - 3, what is f(a + 2)? View AnswerThe limit of the sequence b_n = \dfrac{2 + n}{1 + 3n} is a. 1 b. 2 c. 2/3 d. 1/3 e. none of the above View AnswerUse calculus to determine whether the sequence a_n = \frac{2}{n+3} is increasing or decreasing. View AnswerFind a formula for the nth term of the sequence. 1,-\frac{1}{4},\frac{1}{9},-\frac{1}{16},\frac{1}{25} View AnswerLet a_1=1 and then define a_{n+1}=\frac{7.a_n}{n+1}. List the terms a_n and n=1,2,3,4,5 Write a formula for a_n =? View AnswerFind the first 8 terms to four decimal places, and find the limit: a_{1} = 3, a_{n+1} = 8 - a_{n} View AnswerFind an equation for the nth term of the sequence. View AnswerFind the first six terms of the following sequence. View AnswerUse the Alternating test to determine if the series converges or diverges. \sum^\infty_{n=1} \frac {(-1)^n n}{In \ n} View AnswerDetermine the limit of the sequence. \frac{3n+5}{4n-1} View AnswerCompute the first six terms of the sequence (start with n=1): c_n=n+(n+1)+(n+2)+...+(2n) View AnswerCompute the first four terms of the sequence {a_n} = {\left( { - 1} \right)^n}{{n!} \over {n + 1}}, \quad \left( {n \ge 1} \right). View AnswerFind at least three nonzero terms including a0, and at least two cosine terms and two sine terms if they are not all zero) of the Fourier series for the given function. f(x) = \begin{cases} 5, -&p... View AnswerWrite the first four terms of the recursive sequence. a_1 = -6,\ a_n = a_{n - 1} + 9\ text{ for } n \geq 2 View AnswerFind the sum of the infinite geometric series: 2 - 1/2 + 1/8 - 1/32 + ... View AnswerFind the next values of the sequence on the basis of the values a(0), a(1), ...., a(4). { (-1) , (1/4) (-1/9) , (1/16) , (-1/25) ... }. View AnswerShow that: a) if x_n converges, then \frac{x_n}{n} also converges. View AnswerLet a_n = \frac {1}{2n-1} for n = 1, 2, 3, ....Write out the the first three terms of the following sequences bn =a_{n +1} \b_1=? \b_2=? \b_3=? View AnswerA ball dropped from a height of h = 23 ft begins to bounce. Each time it strikes the ground, it returns to 1/5 of its previous height (a = 1/5). What is the total distance traveled by the ball if i... View AnswerFind the first three terms of the sequence a_n = (- 3)^(sin((2n + 1) n/2))/(n + 1)!. View AnswerFind the next three terms of the sequence a_(n + 2) = (a_n)/(a_(n + 1)); a_0 = - 6; a_1 = 3. View AnswerFind the first three terms (a0, a1, a2) of the sequence a_n = (5 + n + 3n^2)/(2 + n + n^2). View AnswerFind the first three terms (a0, a1, a2) of the sequence a_n = (-2)^(sin((2n + 1) n/2))/(n + 1)!. View AnswerFind the first 5 terms of the sequence given by a1=2, an=3a(n-1) - 1. View AnswerFind the limit of the following sequence or verify that it is divergent. ((ln n)/(n^2 - 1)) from n = 2 to infinity. View AnswerWrite an expression for the nth term of the sequence. a) \ -3, 0, 5, 12. 21, 32. 45, 60, 77, \ b) \ 1, 2, 2, \frac {8}{6}, \frac {16}{24}, \frac {32}{120}, \frac {64}{720}, View AnswerFind the first 100 numbers in the set by using arithmetic series: "Every other odd integer starting with 1." View AnswerFind a formula for the nth term of the sequence. 1, -\dfrac{1}{16}, \dfrac{1}{81}, - \dfrac{1}{256}, \dfrac{1}{625}, ... View AnswerFind a formula for the nth term of the sequence a_n is calculated directly from the value of n. The sequence -9, 9, -9, 9, -9, ...... View AnswerA ball is dropped from a height of 10 feet and bounces. Each bounce is 3/4 of the height of the bounce before. Thus, after the ball hits the floor for the first time, the ball rises to a height of... View AnswerA deposit of $1000 is made in as account that earns 8% interest compounded quarters. The balance in the account after n quarters is given by the following sequence. Find the balance in the account... View AnswerFind the limit of the sequence n^2(1-cos(1/n)). View AnswerDetermine if the sequence {a_n} converges or diverges. Find the limit if the sequence converges. a=\sqrt[n]{6^nn} View AnswerFind the limit of the following sequence or determine that the limit does not exist. \left{ {{{\left( {1 + {{13} \over n}} \right)}^n}} \right} Select the correct choice below. a. The sequence... View AnswerFind the 100th and the nth term for each of the following sequences. a. 1, 4, 7, 10, ... b. 50, 90, 130, ... c. 1, 3, 9, ... d. 10, View AnswerFind the first five terms in sequences with the following nth terms. View AnswerFind the sum of the series if possible: \sum\limits_{i = 1}^{\infty} (4/7)^i. View AnswerLet fn: a, b rightarrow 0, +infty, n = 1, 2, be g-integrable on a, b, where g: a, b rightarrow R is an increasing function. Suppose that f:= lim inf (fn) is g-integrable on a, b. Prove that int a b... View AnswerTell whether the following list is a series or a sequence. Then tell whether it is finite or infinite. 40, 20, 10, 5, 2.5, 1.25, ... View AnswerTell whether the following list is a series or a sequence. Then tell whether it is finite or infinite. 8, 8.2, 8.4, 8.6, 8.8, 9.0, ... View AnswerIs an=(2n+1)/1-3(n^1/2) divergent or convergent? If it is convergent, find the limit. View AnswerWrite out the first five terms of the sequence: a_n = \frac{2}{n^2} . View AnswerDetermine if the sequence {a_n} converges or diverges. Find the limit if the sequence converges. a_n = 3+(0.7)^n. Select the correct choice below and, if necessary, fill in the answer box to comple... View AnswerWrite the first four terms of the sequence: a_1 = 9 and a_n = 2/(3a_(n - 1)), n greater than or equal to 2. View AnswerThe (generalized) Fibonacci sequence is defined by setting a(0) = 1 and then choosing some arbitrary number for a(1). All other terms are now defined by the recursive relation a(n) = a(n - 1) + a(n... View AnswerGive an example of a monotonic sequence that diverges. View AnswerGive an example of a bounded sequence that diverges. View AnswerFind the sum S_n of the first n terms of the following sequence: {a_n} : \ 3, \ 7, \ 13, \ 21, \ 31 , \ \dots . View AnswerDetermine whether the following converges or diverges. If it converges, find the limit: \left{ {{k_n}} \right} = \left{ {{{\left( {n - 2} \right)!} \over {\left( {n + 1} \right)!}}} \right}_{n... View AnswerWrite out the first 10 terms of the sequence: a_1=7, \; a_{n+1}=(-1)^n a_{n}/2 View AnswerFind the sequence of partial sums S1, S2, S3, S4, and S5 View AnswerThe first few terms of a sequence are given. Find an expression for the nth term of the sequence, assuming the indicated patterns continues for all n. (Assume the sequence begins with n=1. \frac{1}... View AnswerSuppose {x_{n}}\rightarrow0 and each {x_{n}}\rightarrow0 Let y_{n}=\frac{1}{x^{n}}. Prove that y_{n} diverges. View AnswerDetermine if the following sequences converge or diverge. Explain or show work to justify your answer. If the sequence converges, show all the work to get the limit. View AnswerDetermine whether the sequence converges or diverges. If it converges, find the limit. {a_n} = {{{{( - 1)}^n}} \over {{n^2} - 1}} View AnswerFind the upper and lower limits of the sequence \left { s_n \right } defined by s_1=0; s_{2m}=\frac{s_{2m-1}}{2}; s_{2m+1}=\frac{1}{2}+s_{2m}. View AnswerWrite the first five terms of the sequence. a_n = {(3 n)!} / {(n - 1)!} View AnswerWrite the first five terms of the sequence. a_n = 6n + 2 View AnswerFind the next term in the series: 2 3 6 1 8 6 8 4 8 4 3 2 3 2 3. View AnswerWrite out the first 10 terms of the given sequence. View AnswerThe sequence { a_k } must be convergent if: a. { a_k } is real for all k greater than or equal 1. b. { a_k } is monotone increasing and a_k greater than 0 for all k greater than or equal 1. c. {a_k... View AnswerFind the first five terms in the sequence whose nth term is given. n2 + 2. a) 2, 8, 18, 32, 50 b) 4, 6, 8, 10, 12 c) 3, 6, 11, 14, 22 d) 3, 6, 11, 18, 27 View AnswerLet a_n = (-1)^n (n^2 + 2^n) / 3^n. Determine if the sequence converges or diverges and what value does it converges or diverges to? Explain in detail. View AnswerUse the properties of limits to evaluate the limit, if it exists, of each of the following sequences {a_i}. If a sequence is divergent, explain why. a_i = \frac{3^{2i-1} + 7}{9^i - 4} View AnswerDetermine the convergence or divergence of the sequence with the given nth term. (1) a_n = {3n - 1} / n^2 (2) a_n = 2n^2 / {5n^2 + 1} View AnswerThe population of deer in Dearborn County is initially 7900 deer. Each year the population decreases by 30% due to hunting and development, so a gaming commission introduces 200 new deer each year.... View AnswerWhat are the first 4 terms of the sequence for the problem a_n = n/(n + 2)? View AnswerGenerate the first five terms in the sequence using the explicit formula. Cn= 12n - 11 View AnswerDetermine whether the sequence is increasing, decreasing, or not monotonic. a_n = {cos (n)} / {5^n} View AnswerDetermine whether the sequence is increasing, decreasing, or not monotonic. a_n = {n - 5} / {n + 5} View AnswerDetermine whether the sequence is increasing, decreasing, or not monotonic. a_n = 1 / {5 n + 7} View AnswerWrite the first five terms of the sequence {a_n} whose nth term is given. a_n = (n + 3)/(2n - 1) View AnswerA multistory building has a floor area of 2000 square feet on the ground floor, 1995 square feet on the 1st floor, 1990 square feet on the 2nd floor, and so on. If the areas of the higher floors fo... View Answert_n = t_{n-1} -2 Using the recursive form above and given that t_1 = 35 , find t_6 . View AnswerWhat are the first three terms of the sequence represented by the expression n(n-1)-4n View AnswerWrite the first three terms of the following sequence whose n^{th} term is given by: z_n = {(-1)^n n (n + 2)} / {4}. View AnswerFind the sum of 3 + 6 + 9 + ... + 96. View AnswerWhat is a recursive formula? Explain it with an example. View AnswerCalculate the 10th term for the following sequence. a_n = a_{n - 1} + 2, a_1 = 9. View AnswerWrite an explicit formula for the sequence 1/2, 3/7, 1/3, 5/19, 3/14. View AnswerIf T_n =2n- 3, which term of sequence is, (a) - 43 (b) - 82 (c) 91 View AnswerUse mathematical induction to prove that the statement holds for all positive integers. Also, label the basis, hypothesis, and induction step. 1 + 5 + 9 + \dots + (4n 3)= n(2n 1) View AnswerFor the following sequences whose nth terms are given, write down the first six terms of each. (a) 4n-1 (b) n^3 (c) 5-2n (d) (-1)N n (e) (n +1)^2 (f) n ( n+2) View AnswerDevise a recursive algorithm for computing n^2 where n is a non-negative integer, using the fact that (n + 1)^2 = n^2 + 2n + 1. Then prove that this algorithm is correct. View AnswerFind the missing term in 4 , \ 8 , \ 12 ,\ 20 ,\ x , \ 52 . View AnswerShow that the sequence is geometric. Then find the common ratio and write the first four terms. {e^n} = {2^n/3} View AnswerFind the general term for the sequence whose first five terms are shown. e^3, e^4, e^5, e^6, e^7, ... View AnswerWhat is the next number in the following series? 16, 21, 26, 26, 12, 5, .... (hint: just because numbers are used doesn't always mean the relationship between them is mathematical. these numbers... View AnswerWrite a rule for the pattern: -3, -9, -14, -47, -141, ... View AnswerGiven (1,12) (2,76) (3,108) (4,124) (5,132) (6,136) (7,138) (8,139) (9,139.5), what is the rule or equation? View AnswerWhat do the symbols below a multiple sequence alignment represent? View AnswerFind the second and fifth terms of A(n)= -2 + (n - 1)(-2). View AnswerGiven f(11) = 11 and f(x + 3) = (f(x) - 1)/(f(x) + 1) for all x , find f(2000) . View AnswerThe first 4 rows of an array of consecutive integers is shown below. The first row has 1 entry. Every other row has 2 more entries than the row directly above it. What is the exact value of the 201... View AnswerWrite the first five terms of the sequence defined by the explicit formula a_{n} = 150(\frac{1}{5})^{n - 1}. View AnswerWhat is the units digit of 7^218 + 4? View AnswerWhat are the next three letters in this combination O, T, T, F, F, S, S, . . .? View AnswerFind the next number in the number series 2, 6, 12, 20, 30, 42, 56, . . .? View AnswerThe new school has exactly 1000 lockers and 1000 students. On the first day of school, the students meet outside the building and agree on the following plan: the first student will enter the schoo... View AnswerWrite out the first five terms of the given sequence \left { 2 + (-1)^n \right }. View AnswerWhat number comes next in the sequence: 1, 2, 4, 7, 11, 16, . . .? View AnswerWhat is the nth term of the arithmetic sequence 7, 5, 3, 1, . . .? View AnswerFind a formula for the \text{n}^\text{th} term of the following sequences: a) \frac{1}{1},\ \frac{1}{4}, \ \frac{1}{9}, \ \dots . b) \frac{4}{7},\ \frac{5}{8},\ \frac{6}{9}, ... View AnswerWhat are the next three letters in this combination: OTTFFS? View AnswerWhat sequence is -1, 2, -4, . . .? View AnswerWhat comes next in the sequence 3, 4, 7, 16, 43, . . .? View AnswerFind the nth term of the sequence: 1, 8/7, 5/4, 4/3,... Assume obvious pattern continues. View AnswerFind the general term for the sequence whose first five terms are given below. \frac{-1}{8}, \frac{1}{27}, \frac{-1}{64}, \frac{1}{25}, \frac{-1}{216},... View AnswerA superball is dropped from rest from a height of 2.0 m. It bounces repeatedly from the floor, as superballs are prone to do. After each bounce the ball dissipates some energy, so eventually it com... View AnswerThe following function is not continuous at 0. Construct a sequence \left { x_n \right } such that x_n\rightarrow 0\ and\ f(x_n)\nrightarrow 0. f(x)= \left{\begin{matrix} \frac{1}{x^2 + 2}, &x l... View AnswerThe following function is not continuous at 0. Construct a sequence \left { x_n \right } such that x_n\rightarrow 0\ and\ f(x_n)\nrightarrow 0. f(x)= \left{\begin{matrix} \sin(\frac{1}{x}),&x ... View AnswerA radioactive substance decreases in the amount of grams by one third each year. If the starting amount of the substance in a rock is 1,452 g, write a recursive formula for a sequence that models t... View AnswerWhat comes next in sequence 7, 11, 16, 22, 26, 31. . .? View AnswerWhat number comes next in the following series? 6, 1, 6, 2, 6, 4, 6, 8, 6, . . . View AnswerWhat is the summation notation of 2 + 6 + 12 + 20 + 30 + 42? View AnswerWhat is the ones digit of 3^2019? View AnswerWhat is next number in the sequence: 0, 2, 6, 12, 20, 30. . .? View AnswerIn a sequence, each term after the second is the sum of the two preceding terms. The first and fifth term are each 3. What is the second term in the sequence? View AnswerWhat is the recursive formula for the given sequence below? 15, 26, 48, 92, . . . View AnswerWhat is the recursive formula for the sequence? 8, 10, 12, 14, . . . View AnswerWhat is the recursive formula for the following sequence 15, 26, 48, 92, ...? View AnswerLet a_i = 2^i 20 . Find a_{i+1} a_i \text{ for } i=1 . View AnswerA store had 50 bottles of olive oil. Each week, 40% of the olive oil bottles were sold and 20 new bottles arrived in shipments. Which recursive function best represents the number of bottles in the... View AnswerUse the method of finite differences to create a new sequence of numbers for the following sequence of square numbers. 1, 4, 9, 16, 25, 36, 49, 64, 81 .What kind of a sequence do you obtain? View AnswerA tower of blocks has 1 block in the top row, 4 blocks in the next row, 7 blocks in the next row, an so on. There are 5 rows in all. How many blocks are in the tower? View AnswerYou have just landed your first summer job and your boss has told you that you can choose from one of 3 pay structures for the first week (7 days) that you work there. And after the first week, you... View AnswerWhich is larger, the 10th term of an arithmetic sequence that begins with the terms 0 and 100 or the 10th term of a geometric sequence that begins with the terms 5 and 10? View AnswerConsider the sequence 1, 3, 7, 13, .... Is this sequence arithmetic, geometric, or neither? View AnswerWhat is the formula for the nth term of the sequence is 2, 4, 7, 11. . . ? View AnswerWhat are the first five terms of the sequence an = 4^n/(2n)? View AnswerDetermine whether 447 is a term of each sequence(s). A. t(n)= 5n - 3 B. t(n)= 24 - 5n C. t(n)= -6 + 3(n - 1) D. t(n)= 14 - 3n E. t(n)= -8 - 7(n - 1) View Answerwhat are the first four terms of the sequence given by the formula below an=-2n+4 View AnswerWhat is the 8th term of the sequence given by a_{n} = 3n - 5. View AnswerFind whether the sequence converge or diverge. If they converge, find the limit. a_{n} = \cos(2/n) View AnswerFive points lie on a straight line. Alex finds the distances between every possible pair of points. He obtains, in increasing order 2,5,6,8,9,k,15,17,20,22.what is the value of k? View AnswerUsan has 10 pockets and 44 dollar bills. She wants to arrange the money so that there are a different number of dollars in each pocket. Can she do it? Explain. View Answerwhat are the first five terms of the sequence given by the formula an=6n+1? View AnswerWhat is the sum of (100 - 97)^2 + (97 - 94)^2 + (94 - 94)^2 +.............+ (4 - 1)^2? View AnswerTo find the 87th term in a sequence, which formula would be given? a) analog form b) digital form c) explicit form d) recursive form View AnswerRead the numbers and decide what the next number should be? 3, 4, 7, 13, 14, 17, 23........ View AnswerFor the following sequence, find an explicit and a recursive description of the pattern. Assume a sequence starts from n = 1. -2, 0, 0, -2, -6... View AnswerGenerate the first 5 terms of this sequence: f(1) = 2 and f(2) = 3, f(n) = f(n - 1) + f(n - 2), for n greater than 2. View AnswerWrite the first five terms of the sequence defined by the recursive formula a_n = 2(an - 1) + 3 with a_1 = -2. View AnswerWhat are the first four terms of the sequence represented by the expression n(n - 1) - 5? View AnswerFormulate the recursive formula for the following geometric sequence. {-16, 4, -1, ..........} View AnswerDetermine the convergence or divergence: A_n = (-1)^n(\frac{n}{(n+1)}). View AnswerWhich is not a counterexample to the formula 1^2 + 3^2 + 5^2 + + (2n - 1)^2 = n(2n + 1)/3? (a) n = 3 (b) n = 2 (c) n = 1 (d) n = 4 View AnswerWhich of the following methods can be used to construct a pattern of 10 numbers for which it would be mathematically impossible to predict the 11th term? (Select all that apply.) A. Make a pattern,... View Answer Create an account to browse all assets today Create an account Concepts related to Sequences More general Algebra Calculus × Concepts related to Sequences More general Algebra Calculus © Copyright 2025 Homework.Study.com. All other trademarks and copyrights are the property of their respective owners. All rights reserved. Resources and Guides About Us Terms of Use Privacy Policy DMCA Notice ADA Compliance Honor Code For Students Support
4677	https://teachy.ai/en/lesson-plan/high-school/10th-grade/mathematics-en/expository-methodology-or-function-even-or-odd-or-lesson-plan	Lesson plan of Expository Methodology \| Function: Even or Odd \| Lesson Plan We use cookies Teachy uses cookies to enhance your browsing experience, analyze site traffic, and improve the overall performance of our website. You can manage your preferences or accept all cookies. Manage preferences Accept all TeachersSchoolsStudents Teaching Materials EN Log In Teachy> Lesson plans> Mathematics> 10th grade> Function: Even or Odd Lesson plan of Function: Even or Odd Lara from Teachy Subject Mathematics Mathematics Source Original Teachy Original Teachy Topic Function: Even or Odd Function: Even or Odd Objectives (5 - 7 minutes) Understand the definition of even and odd functions, recognizing their distinct characteristics. Identify the even function as the one where f(x) = f(-x) for any value of 'x' in the function's domain. Identify the odd function as the one where f(-x) = -f(x) for any value of 'x' in the function's domain. Apply the acquired knowledge in the analysis of function graphs, identifying if they are even, odd, or neither. Develop the ability to recognize even or odd symmetry in a graph. Solve practical problems involving even and odd functions, using the properties of these functions to facilitate the resolution. Develop the ability to apply the concept of even and odd functions in everyday situations. Secondary Objectives: Promote students' logical and analytical reasoning skills, encouraging structured problem-solving. Stimulate teamwork and active participation of students during discussions and proposed activities in the lesson. Introduction (10 - 15 minutes) Review of Previous Content: The teacher should start the lesson by reviewing important concepts that are prerequisites for understanding the current topic, such as the concept of function, domain and range, and the analysis of function graphs. (2 - 3 minutes) Problem Situations: Present two problem situations that will be solved throughout the lesson, both involving even and odd functions: Situation 1: 'Imagine you are drawing the profile of a roller coaster on a graph, where the x-axis represents time and the y-axis represents height. If the roller coaster is symmetrical with respect to the y-axis, what type of function does it represent?' Situation 2: 'If we have the function f(x) = x^3 - 3x, how can we determine if it is even, odd, or neither?' (5 - 7 minutes) Contextualization: Explain the importance of studying even and odd functions, emphasizing their applicability in various areas, such as physics (when studying the parity of physical quantities), economics (in cost and revenue analysis), and even in cryptography (where they are used in public key encryption algorithms). (2 - 3 minutes) Engaging Students' Attention: Curiosity 1: 'Did you know that even and odd functions are related to the symmetry of our body? If we draw a vertical line in the middle of our body, we realize that our arms and legs are symmetrical, meaning that if we invert the sign of one coordinate, the other also inverts. This is a characteristic of even functions! On the other hand, our face, which is not symmetrical, is a characteristic of odd functions!' Curiosity 2: 'Did you know that even and odd functions are also present in nature? The symmetry of the leaves of many plants, such as ferns, is an example of an even function. On the other hand, the shape of ocean waves, which are not symmetrical, is an example of an odd function!' (3 - 5 minutes) Development (20 - 25 minutes) Theory: Definition and Characteristics of Even and Odd Functions (8 - 10 minutes) The teacher should explain clearly and objectively what an even and odd function is, using mathematical notation and practical examples. Definition of Even Function: f(x) = f(-x) for any value of 'x' in the function's domain. Definition of Odd Function: f(-x) = -f(x) for any value of 'x' in the function's domain. Characteristics of Even and Odd Functions: Symmetry with respect to the y-axis (even function) and origin (odd function). Examples: f(x) = x^2 (even function), f(x) = x^3 (odd function), f(x) = x (neither even nor odd). Reflection Activity: Connection with the Real World (5 - 7 minutes) The teacher should lead a classroom discussion, questioning students about possible applications of even and odd functions in everyday situations or in other disciplines. For example, students can be asked how even and odd functions can be useful in areas such as physics, economics, or cryptography. Students should be encouraged to share their ideas and think critically about the relevance of the subject. Theory: Identification of Even and Odd Functions in Graphs (5 - 7 minutes) The teacher should explain how to identify if a function is even or odd by looking at its graph. Even Function: the function is symmetric with respect to the y-axis, meaning that if we reflect the graph with respect to this axis, we obtain exactly the same graph. Odd Function: the function is symmetric with respect to the origin (0, 0), meaning that if we reflect the graph with respect to this point, we obtain exactly the same graph. Examples: Graphs of the functions f(x) = x^2 (even), f(x) = x^3 (odd), f(x) = x (neither even nor odd). Practical Activity: Analysis of Graphs (2 - 3 minutes) The teacher should provide students with a series of function graphs and ask them to identify if each function is even, odd, or neither. Students should work in groups to solve the activity, discussing their answers and justifying their conclusions. The teacher should move around the classroom, guiding students and clarifying doubts. Theory: Solving Practical Problems with Even and Odd Functions (5 - 7 minutes) The teacher should explain how to use the properties of even and odd functions to solve practical problems. Examples of Problems: Determine if a function is even, odd, or neither based on its algebraic expression. Determine if a function is even, odd, or neither based on its graph. The teacher should solve some of these problems on the board, step by step, so that students can follow and understand the process. Practical Activity: Problem Solving (2 - 3 minutes) The teacher should provide students with a series of problems involving even and odd functions and ask them to solve them. Students should work in groups to solve the problems, discussing their solutions and justifying their reasoning. The teacher should move around the classroom, guiding students and clarifying doubts. Return (10 - 15 minutes) Group Discussion (5 - 7 minutes) The teacher should promote a classroom discussion where each group shares their solutions or conclusions from the practical activities carried out. Each group will have a maximum of 3 minutes to present their conclusions. During the presentations, other groups and the teacher can ask questions or make comments. The teacher should guide the discussion, highlighting the main points and correcting misconceptions, if necessary. It is important for the teacher to encourage the participation of all students, valuing their contributions. Connection with Theory (3 - 5 minutes) After the presentations, the teacher should summarize the main ideas discussed, connecting them with the theory presented at the beginning of the lesson. The teacher should emphasize how the theory of even and odd functions was applied in solving the proposed problems, reinforcing the importance of theoretical knowledge for solving practical problems. The teacher can ask targeted questions to students to stimulate reflection and verify their understanding of the content. For example: 'How did you apply the definition of even and odd functions to solve problem X?' or 'Can what we learned today be useful in everyday situations or in other disciplines?'. Individual Reflection (2 - 3 minutes) The teacher should suggest that students make a brief individual reflection on what was learned in the lesson. Students should think for a minute and then share aloud an idea or concept they found most interesting or challenging. The teacher should value students' contributions, reinforcing the importance of reflection and self-awareness in the learning process. Feedback and Closure (1 - 2 minutes) The teacher should provide overall feedback on the lesson, highlighting the positive points and areas that can be improved. The teacher should thank the students for their participation and effort, encouraging them to continue studying and dedicating themselves. The teacher should remind students about the content of the next lesson and any homework that may be necessary to prepare them for the next topic. Conclusion (5 - 7 minutes) Summary and Recapitulation (2 - 3 minutes) The teacher should give a brief summary of the main points covered during the lesson, reiterating the definition of even and odd functions, their characteristics, and how to identify them in graphs. For example, the teacher can review the difference between even and odd functions, highlighting even symmetry with respect to the y-axis and odd symmetry with respect to the origin (0, 0). The teacher should reinforce the importance of understanding these concepts for solving practical problems and their applicability in various contexts, from cryptography to physics. Connection between Theory, Practice, and Applications (1 - 2 minutes) The teacher should explain how the lesson connected the theory, practice, and applications of the concept of even and odd functions. The teacher can mention how the theory was used to solve practical problems and how these problems have applications in everyday life and other disciplines. The teacher should emphasize that understanding the theory is essential to be able to apply it correctly and solve problems effectively. Extra Materials (1 - 2 minutes) The teacher should suggest some extra materials for students who wish to deepen their knowledge of even and odd functions. Extra materials may include explanatory videos, interactive math websites, additional exercises, and textbooks. The teacher should encourage students to explore these materials on their own, emphasizing that learning is not limited to what is taught in the classroom. Importance of the Subject (1 minute) Finally, the teacher should summarize the importance of the subject presented for daily life and other disciplines. For example, the teacher can mention how the ability to identify even and odd functions in real-world problems can be useful in various professions, from engineering to economics. The teacher should end the lesson by reiterating the relevance of the subject and motivating students to continue studying and making an effort. Need more materials to teach this subject? 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4678	https://www.chilimath.com/lessons/advanced-algebra/inverse-of-a-2x2-matrix/	Published Time: 2017-03-04T18:19:38+00:00 Inverse of a 2x2 Matrix \| ChiliMath Skip to content Calculators Unit Converter Math Lessons Basic Math Introductory Algebra Intermediate Algebra Advanced Algebra Word Problems Geometry Math Proofs Number Theory Trigonometry Quizzes Math Solver Worksheets New Inverse of a 2×2 Matrix ×search Web Image Sort by:RelevanceRelevanceDate Inverse of a 2×2 Matrix Formula In this lesson, we are only going to deal with 2×2 square matrices. I have prepared five (5) worked examples to illustrate the procedure on how to solve or find the inverse matrix using the Formula Method. Just to provide you with the general idea, two matrices are inverses of each other if their product is the identity matrix. An identity matrix with a dimension of 2×2 is a matrix with zeros everywhere but with 1’s in the diagonal. It looks like this. It is important to know how a matrix and its inverse are related by the result of their product. So then, LATEST VIDEOS next stay CC SettingsOffArabicChineseEnglishFrenchGermanHindiPortugueseSpanish Font ColorwhiteFont Opacity100%Font Size100%Font FamilyArialText ShadownoneBackground ColorblackBackground Opacity50%Window ColorblackWindow Opacity0% WhiteBlackRedGreenBlueYellowMagentaCyan 100%75%50%25% 200%175%150%125%100%75%50% ArialGeorgiaGaramondCourier NewTahomaTimes New RomanTrebuchet MSVerdana NoneRaisedDepressedUniformDrop Shadow WhiteBlackRedGreenBlueYellowMagentaCyan 100%75%50%25%0% WhiteBlackRedGreenBlueYellowMagentaCyan 100%75%50%25%0% If a 2×2 matrix A is invertible and is multiplied by its inverse (denoted by the symbol A−1), the resulting product is the Identity matrix which is denoted by III. To illustrate this concept, see the diagram below. In fact, I can switch the order or direction of multiplication between matrices A and A−1, and I would still get the Identity matrix III. That means invertible matrices are commutative. How do we find the inverse of a matrix? The formula is rather simple. As long as you follow it, there shouldn’t be any problem. Here we go. The Formula to Find the Inverse of a 2×2 Matrix Given the matrix A Its inverse is calculated using the formula where det A\color{red}{\rm{det }}\,AdetA is read as the determinant of matrix A. A few observations about the formula: Entries a\color{blue}aa and d\color{blue}dd from matrix A are swapped or interchanged in terms of position in the formula. Entries b\color{blue}bb and c\color{blue}cc from matrix A remain in their current positions, however, the signs are reversed. In other words, put negative symbols in front of entries bbb and ccc. Since det A\color{red}{\rm{det }}\,AdetA is just a number, then 1detA\large{1 \over {{\rm{det }}A}}detA1 is also a number that would serve as the scalar multiplier to the matrix See my separate lesson on scalar multiplication of matrices. Examples of How to Find the Inverse of a 2×2 Matrix Example 1: Find the inverse of the 2×2 matrix below, if it exists. The formula requires us to find the determinant of the given matrix. Do you remember how to do that? If not, that’s okay. Review the formula below on how to solve for the determinant of a 2×2 matrix. So then, the determinant of matrix A is To find the inverse, I just need to substitute the value of detA\=–1{\rm{det }}A = – 1detA\=–1 into the formula and perform some “reorganization” of the entries, and finally, perform scalar multiplication. Here goes again the formula to find the inverse of a 2×2 matrix. Now, let’s find the inverse of matrix A. Let’s then check if our inverse matrix is correct by performing matrix multiplication of A and A−1 in two ways, and see if we’re getting the Identity matrix. ![Image 24: this shows that matrix A when multiplied by its inverse A^-1 results to the identity matrix I. in the same manner, when the inverse of matrix A when multiplied to matrix A also gives the identity matrix which means that they are commutative under the operation of matrix multiplication. Therefore, AA^-1 = [5,2;7,-3] 3,2;-7,-5} = [1,0;0,1]. Since multiplying both ways generate the Identity matrix, then we are guaranteed that the inverse matrix obtained using the formula is the correct answer! Example 2: Find the inverse of the 2×2 matrix below, if it exists. First, find the determinant of matrix B. Secondly, substitute the value of det B = 1 into the formula, and then reorganize the entries of matrix B to conform with the formula. I will leave it to you to verify that In other words, the matrix product of B and B−1 in either direction yields the Identity matrix. Example 3: Find the inverse of the matrix below, if it exists. This is a great example because the determinant is neither +1+1+1 nor −1−1−1 which usually results in an inverse matrix having rational or fractional entries. I must admit that the majority of problems given by teachers to students about the inverse of a 2×2 matrix is similar to this. Step 1: Find the determinant of matrix C. The formula to find the determinant Below is the animated solution to calculate the determinant of matrix C Step 2: The determinant of matrix C is equal to −2−2−2. Plug the value in the formula then simplify to get the inverse of matrix C. Step 3: Check if the computed inverse matrix is correct by performing left and right matrix multiplication to get the Identity matrix. Yep, matrix multiplication works in both cases as shown below. First case: Second case: Example 4: Find the inverse of the matrix below, if it exists. In our previous three examples, we were successful in finding the inverse of the given 2×22 \times 22×2 matrices. I don’t want to give you the impression that all 2×22 \times 22×2 matrices have inverses. In this example, I want to illustrate when a given 2×22 \times 22×2 matrix fails to have an inverse. How does that happen? If we review the formula again, it is obvious that this situation can occur when the determinant of the given matrix is zero because 1 divided by zero is undefined. And so, an undefined term distributed into each entry of the matrix does not make any sense. Let’s go back to the problem to find the determinant of matrix D. Therefore, the inverse of matrix D does not exist because the determinant of D equals zero. This is our final answer! Example 5: Find the inverse of the matrix below, if it exists. Step 1: Find the determinant of matrix E. Step 2: Reorganize the entries of matrix E to conform with the formula, and substitute the solved value of the determinant of matrix E. Distribute the value of 1detE\large{1 \over {{\rm{det }}E}}detE1 to the entries of matrix E then simplify, if possible. Step 3: Verify your answer by checking that you get the Identity matrix in both scenarios. First scenario: Second scenario: You might also like these tutorials: Inverse of Absolute Value Function Inverse of Constant Function Inverse of Exponential Function Inverse of Linear Function Inverse of Logarithmic Function Inverse of Quadratic Function Inverse of Rational Function Inverse of Square Root Function Tags: Advanced Algebra, Lessons ABOUT About Us Sitemap Contact Us Cookie Policy Privacy Policy Terms and Conditions Accessibility MATH SUBJECTS Introductory Algebra Intermediate Algebra Advanced Algebra Word Problems Geometry Trigonometry Intro to Number Theory Math Proofs FEATURES Worksheets Quizzes Mathway Unit Converter Useful Math Websites © 2025 ChiliMath.com Calculators Unit Converter Math LessonsToggle child menu Basic Math Introductory Algebra Intermediate Algebra Advanced Algebra Word Problems Geometry Math Proofs Number Theory Trigonometry Quizzes Math Solver Worksheets New ✕ Do not sell or share my personal information. 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4679	https://wordsinasentence.com/hackneyed-in-a-sentence/	WORDS IN A SENTENCE Hackneyed in a Sentence 🔊 Definition of Hackneyed repeated too often; overused Examples of Hackneyed in a sentence Too often used by young girls, the word “like” has become hackneyed. 🔊 Every time my internet goes down, the cable company gives me a hackneyed explanation. 🔊 Although I like him as a talk show host, his hackneyed catchphrase is starting to get on my nerves! 🔊 The impersonator’s performance was made worse by his hackneyed impressions. 🔊 Eventually, the phrase became so hackneyed that people stopped saying it. 🔊 Because I understand the meaning of the word "love", I do not want it to be overused to the point where it becomes hackneyed. 🔊 Since dyeing your hair purple has become a hackneyed style, I am dyeing my hair green! 🔊 When it comes to writing, teachers frown against students using hackneyed phrases. 🔊 “A hackneyed version of her previous works” is how the critics described the author’s new novel. 🔊 Even though the movie’s plot has been hackneyed to death, I still enjoyed watching my favorite actor. 🔊 Other words in the Boring category: Most Searched Words (with Video) Voracious: In a Sentence Verbose: In a Sentence Vainglorious: In a Sentence Pseudonym: In a Sentence Propinquity: In a Sentence Orotund: In a Sentence Magnanimous: In a Sentence Inquisitive: In a Sentence Epoch: In a Sentence Aberrant: In a Sentence Apprehensive: In a Sentence Obdurate: In a Sentence Heresy: In a Sentence Gambit: In a Sentence Pneumonia: In a Sentence Otiose: In a Sentence
4680	https://www.apollopharmacy.in/salt/Satranidazole?srsltid=AfmBOoq_35sDwIAD26fD4fdaguoB65ycAoHwDxJskVnilSNMekt3p_IE	Select Address Personal Care Baby Care OTC Vitamins & Supplements Food & Drink Health Accessories Weight Management Diapering Diaper By Weight Baby Food Baby Skin Care Baby Food By Age Baby Hair Care Baby Bath Baby Oral Care Baby Gift Sets Feeding Bottles & Accessories Breast Feeding Mother Care Baby Health & Safety Nutritional Drinks Sports Nutrition Vitamins & Supplements Minerals Omega & Fish Oil Speciality Supplements Weight Management Immunity Boosters Sexual Health Supplements Feminine Hygiene Women Supplements Gyno Care Pregnancy Grooming Skin Care Hair Care Oral Care Mens Grooming Sexual Wellness Fragrances Adult Diapers Health Concerns Herbs Herbs Herbal Juices Chyawanprash Honey BP Monitors Glucometers & Test Strips Covid Test Kits Thermometers Pulse Oximeters Pregnancy test Kit Heating Belts Weighing Machine Nebulizer Supports & Splints Health Accessories Insect Repellents Antiseptic Liquids Room Freshners Cleaning Essentials Batteries Pet Food Mental Wellness Liver Care Diabetic Pain Relief Cardiac Kidney Care Stomach Care Respiratory Eye care Cold & Cough Wound Care Sleep Aids Bone, Joint & Muscle Satranidazole Satranidazole About Satranidazole Satranidazole belongs to a group of medicines known as antibiotics used to treat various infections caused by bacteria and parasites (amoeba). It is prescribed for the treatment of diarrhea. Additionally, it also treats vaginal and other bacterial infections. A bacterial infection is a condition in which harmful bacteria enter, multiply, and infect our bodies. It can target any body part and multiple very quickly. Amoebic dysentery can be caused due to Entamoeba histolytica, Giardia lamblia, or Clostridium difficile. Satranidazole contains ‘Satranidazole’ that kills the harmful bacteria and parasites (amoeba) by damaging its DNA and preventing these bacteria growth spread hence relieving the symptoms. It is bactericidal and works by killing bacteria that cause infections. It prevents the division of bacterial cells and inhibits the repairing of bacterial cells. Altogether it kills the bacteria. Satranidazole does not treat a viral infection like flu or a common cold. Take Satranidazole as prescribed. You are advised to take Satranidazole for as long as your doctor has prescribed it for you depending on your medical condition. In some cases, you may experience certain common side effects such as nausea, vomiting, headache, weakness, dizziness, dry mouth, and metallic taste. Most of these side effects of Satranidazole do not require medical attention and gradually resolve over time. However, you are advised to talk to your doctor if you experience these side effects persistently. If you are allergic to Satranidazole or any other medicines, please tell your doctor. If you are pregnant or breastfeeding, please inform your doctor before using Satranidazole. Satranidazole should not be used in children unless prescribed by a doctor. It is not advisable to suddenly stop this medicine to avoid unpleasant side effects and sudden withdrawal symptoms. Intake of probiotics, prebiotics, and plenty of fluids is recommended if you are suffering from dysentery and diarrhea. Do not drive or operate heavy machinery after taking Satranidazole as it may cause dizziness, drowsiness, and make you less alert. You should not take any alcoholic beverages during your treatment as it can cause flushing and a general feeling of being unwell. Uses of Satranidazole Medicinal Benefits Satranidazole contains 'Satranidazole’ is a broad-spectrum antibiotic that is effective against a wide range of gram-positive and gram-negative bacteria. Kills the harmful anaerobic bacteria and certain anaerobic protozoa (Trichomonas vaginalis, Entamoeba histolytica, and Giardia lamblia) causing diarrhea. Satranidazole; being bactericidal in nature kills the harmful bacteria by damaging its DNA and preventing these bacteria growth spread hence relieving the symptoms. Satranidazole is two times more active than other nitroimidazoles like metronidazole against amoebiasis and giardiasis and anaerobes. Directions for Use Storage Side Effects of Satranidazole Drug Warnings If you are allergic to Satranidazole or any other antibacterial or anti-amoebic agents, please inform your doctor to avoid any unwanted side effects like swallowing or breathing problems, and swelling of your lips, face, throat, or tongue. Let your doctor know if you use any prescription and non-prescription medications you are taking, including vitamins, and herbal supplements before starting Satranidazole. Do not take Satranidazole on your own as self-medication may lead to antibiotic resistance in which antibiotics fail to act against specific bacterial infections. If you are pregnant or breastfeeding, please inform your doctor before using Satranidazole. Do not drink alcohol with Satranidazole as it may cause flushing and drowsiness which can alter your mental alertness. Drug Interactions Drug-Drug Interaction: Satranidazole have interactions with medicines to stop your blood clotting (coumarins, warfarin), medicines used to treat alcohol addiction (disulfiram). Drug-Food Interaction: The consumption of alcoholic beverages should be avoided as it may lead to flushing, drowsiness, and dizziness. Drug-Disease Interaction: If you have liver problems, muscle problems (myasthenia gravis), heart rhythm disorder (arrhythmia), kidney problems, please inform your doctor before taking Satranidazole. Drug-Drug Interactions Checker List: Safety Advice Alcohol You are recommended not to consume alcohol along with Satranidazole to avoid unpleasant side-effects. Pregnancy The safety of Satranidazole in pregnant women is unknown. Therefore, it is given to pregnant women only if the doctor thinks the benefits outweigh the risks. Breast Feeding It is unknown whether Satranidazole is excreted in human milk. It is given to breastfeeding mothers only if the doctor thinks the benefits are greater than the risks. Driving Satranidazole can cause dizziness in some patients. If you feel dizzy, please do not operate the car. Liver Satranidazole to be taken with caution, especially if you have a history of liver diseases/conditions. The dose may have to be adjusted by your doctor. Kidney Satranidazole to be taken with caution, especially if you have a history of kidney diseases/conditions. The dose may have to be adjusted by your doctor. Children Satranidazole should not be used in children unless prescribed by a doctor. Habit Forming Diet & Lifestyle Advise Special Advise Do not stop taking the tablets your infection may come back, Talk to your doctor before you stop taking the tablets and follow their advice. Patients Concern Disease/Condition Glossary Bacterial infection: A bacterial infection is a condition in which harmful bacteria enter, multiply, and infect our body. It can target any body part and multiple very quickly. When you get infected with bacteria, you can experience generalized symptoms, like fevers, chills, and fatigue. Bacteria are of various forms comprising commonly spherical, rod, and spiral-shaped. Bacterial infections vary from minor illnesses like sore throat and ear infections to severe brain infections like meningitis and encephalitis. A few harmful bacteria that cause infections include Streptococcus, Staphylococcus, and E. coli. Anyone can become infected with a bacterial infection. But, people with weak immune systems or taking immunosuppressive medicine can make you more prone to bacterial infection. Amoebic dysentery: It is an intestinal (bowel) illness caused by a parasite (amoeba) known as Entamoeba histolytica spread through human feces (poop). Usually, there are no symptoms, but, sometimes it causes diarrhea (loose stool/poop), nausea (a feeling of sickness in the stomach), and weight loss. FAQs What is the use of Satranidazole? Satranidazole is used to treat various infections caused by bacteria and parasites (amoeba). It is prescribed for the treatment of diarrhea. Additionally, it also treats vaginal and other bacterial infections. Can I drive while taking Satranidazole? Satranidazole causes dizziness. so, drive only if you are alert and omit driving or operating machinery if you feel dizzy. Can I stop taking Satranidazole on my own? Satranidazole is an antibiotic medication. You need to complete the full course even if you start to feel better to prevent infection return. Can Satranidazole cause dry mouth? Satranidazole is known to cause dry mouth, especially when you start taking Satranidazole. If you feel excessively thirsty, do frequent mouth rinses, and increase fluid intake. What should I do if I forget a dose? If you miss a dose of Satranidazole, take the missed dose as soon as you remember it. However, if it's almost time for the next dose, do not take a double dose to make up for a missed one. Can I take a higher than the recommended dose of Satranidazole? No, you should not take more than the recommended dose of Satranidazole as it may not increase the efficacy of the treatment and can increase the risk of side effects. It's important to follow your doctor's instructions regarding the dosage and duration of the medication. Is Satranidazole safe and effective? Yes, Satranidazole is generally considered safe and effective when taken as prescribed by your doctor. However, it's important to follow your doctor's instructions regarding dosage and duration to ensure the best results and to prevent the development of antibiotic resistance. Is Satranidazole safe to use in elderly people? Satranidazole should be used in elderly people only if advised by a doctor. Older adults may have other medical conditions or be taking other medications that could interact with Satranidazole. It's important to consult your doctor before starting this medication to ensure it is safe and appropriate for your specific condition. For how long should I avoid alcohol when on Satranidazole? You should avoid alcohol while taking Satranidazole and for at least 3 days after completing the course. Consuming alcohol while on treatment with Satranidazole can lead to disulfiram-like reactions associated with symptoms such as dizziness, flushing, headache, and abdominal discomfort. However consult your doctor, if you have any concerns. Does Satranidazole cause metallic taste? Yes, metallic taste is a known side effect of Satranidazole. Chewing sugar-free gum or mints, brushing your teeth after meals, and drinking plenty of water can all help reduce this metallic taste. If this side effect persist, consult your doctor. What is the dose of Satranidazole? The typical dose of Satranidazole depends on the condition being treated. It's important to follow your doctor's instructions regarding the dosage and duration of the medication. What are the side effects of Satranidazole? The common side effects of Satranidazole are nausea, vomiting, headache, weakness, dizziness, dry mouth, and metallic taste. If any of these side effects persist or worsen, please consult your doctor. Can Satranidazole cause diarrhoea? Yes, Satranidazole may cause diarrhoea as a side effect in some people. Intake of probiotics, prebiotics, and plenty of fluids can help you to manage diarrhoea. If you experience severe or persistent diarrhoea, consult your doctor. Is Satranidazole effective for stomach infection? Yes, Satranidazole is effective for treating stomach infections caused by bacteria and parasites. It works by stopping the growth of the bacteria and parasites, which helps to clear the infection. However take the advice of your doctor before taking Satranidazole. Is Satranidazole a strong antibiotic? Satranidazole is considered a strong antibiotic because it contains satranidazole, which is effective against a wide range of bacterial and parasitic infections. It is important to follow the dosage prescribed by your doctor. Can we use Satranidazole for colitis? Satranidazole may be used to manage certain types of colitis caused by bacterial or parasitic infections. However, consult your doctor before taking Satranidazole. Should Satranidazole be taken before or after food? Take Satranidazole with or without food as prescribed by your doctor but take it at the same time each day to get the most benefit. Swallow it whole with a glass of water. Do not chew, crush, or break it. Does Satranidazole cause constipation? Satranidazole does not cause constipation as a side effect commonly. However If you experience constipation while taking this Satranidazole, it's important to drink plenty of water, eat a high-fibre diet, and exercise regularly. 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4681	https://chem.libretexts.org/Courses/University_of_South_Carolina__Upstate/CHEM_U109%3A_Chemistry_of_Living_Things_-_Mueller/01%3A_Chemistry_Matter_and_Measurement/1.7%3A_Converting_Units	1.7: Converting Units - Chemistry LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 1: Chemistry, Matter, and Measurement CHEM U109: Chemistry of Living Things - 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Home 2. Campus Bookshelves 3. University of South Carolina - Upstate 4. CHEM U109: Chemistry of Living Things - Mueller 5. 1: Chemistry, Matter, and Measurement 6. 1.7: Converting Units Expand/collapse global location 1.7: Converting Units Last updated Aug 26, 2020 Save as PDF 1.6: The International System of Units 1.8: Chapter Summary Page ID 83044 ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Conversion Factors 2. Conversion Factors With Prefix Multipliers (Exponents) 3. Significant Figures in Conversions 4. Multiple Conversions 5. Squared or Cubed Units 6. Concept Review Exercises 7. Answers 8. Key Takeaway 9. Contributors Skills to Develop To convert a value reported in one unit to a corresponding value in a different unit. The ability to convert from one unit to another is an important skill. For example, a nurse with 50 mg aspirin tablets who must administer 0.2 g of aspirin to a patient needs to know that 0.2 g equals 200 mg, so 4 tablets are needed. Fortunately, there is a simple way to convert from one unit to another. Conversion Factors If you review the SI unitsdescribed earlier, you can find that one kilogram (kg)is about equal to 2.20 pounds (lbs). (1.7.1)1⁢k⁢g=2.20⁢l⁢b⁢s Suppose we divide both sides of the equation by 1 kg(both the number and the unit): (1.7.2)1 kg 1 kg=2.20 lbs 1 kg As long as we perform the same operation on both sides of the equals sign, the expression remains an equality. Look at the left side of the equation; it now has the same quantity in the numerator as it has in the denominator. Any fraction that has the same quantity on the top and on the bottom has a value of 1: (1.7.3)1 kg 1 kg=1 and 2.20 lbs 1 kg=1 We know that one kilogram is equal to 2.20 pounds, so we have the same quantity on the top and the bottom of our fraction, although it is expressed in different units. A fraction that has equivalent quantities in the numerator and the denominator but expressed in different units is called a conversion factor. Instead of dividing both sides by 1 kg, as above, we could have obtained a conversion factor if we divided both sides by 2.20 kg. (1.7.4)1⁢k⁢g=2.20⁢l⁢b⁢s will become (1.7.5)1 kg 2.20 lbs=2.20 lbs 2.20 lbs Again, because we performed the same operation on both sides of the equals sign, the expression remains an equality (=1). (1.7.6)1 kg 2.20 lbs=1 and 2.20 lbs 2.20 lbs=1 This means that every conversion factor will have two forms,both equal to one, but reciprocated (flipped). For 1 kg = 2.20 lbs, the two conversion factors would be: (1.7.7)1⁢k⁢g 2.20⁢l⁢b⁢s or 2.20⁢l⁢b⁢s 1⁢k⁢g Here is a simple example of how the conversion factor might be used. You are about to return from a European vacation with a suitcase full of gifts. The airline does not allow bags heavier than 55 pounds, but the scale in your hotel only reads in kilograms. How many kg are equal to 55 lbs? To solve the problem with the conversion factor used earlier, we first write the quantity we are given, 55 lbs. Then we multiply this quantity by a conversion factor, which is the same as multiplying it by 1. We can write 1 as either version of the conversion factor,1 kg 2.20 lbs or 2.20 lbs 1 kg. Here we will use 1 kg 2.20 lbs and multiply so that the units we started with (kg) cancel and we end with the units we want (lbs). (1.7.8)55⁢l⁢b⁢s×1⁢k⁢g 2.20⁢l⁢b⁢s The 55 kg can be thought of as a fraction with a 1 in the denominator (though it is often not shown). Because kg, the abbreviation for kilograms, occurs in both the numerator (top)and the denominator (bottom) of our expression, they cancel out. The 55 on top is divided by 2.20 on bottom and the only units that did not cancel are kg. (1.7.9)55 l⁢b⁢s 1×1⁢k⁢g 2.20 l⁢b⁢s=25⁢k⁢g In the final answer, we omit the 1 in the denominator. We find that 55 lbs equals 25 kg. A generalized description of this process is as follows: quantity (in old units)×conversion factor=quantity (in new units) This involved conversion only involved one step. We will see that the same basic principle applies for conversions with multiple steps.If you can master the technique of applying conversion factors, you will be able to solve a large variety of problems. In the previous example, if we had used the other possible reciprocated conversion factor, 2.20 lbs 1 kg, the original unit would not have canceled, and the result would have been meaningless. Here is the incorrect we would have gotten: (1.7.10)55⁢l⁢b⁢s×2.20⁢l⁢b⁢s 1⁢k⁢g=0.0355⁢l⁢b⁢s 2 k⁢g For the answer to be meaningful, we have to construct the conversion factor in a form that causes the original unit to cancel out. Conversion Factors With Prefix Multipliers (Exponents) If you learned theprefixes described earlier in this chapter, then you know that the prefix centi(c) is 10−2. The prefixes come in front of some other base unit, like meters (m). The base units must be included on both sides of the equality, so if c= 10−2, then cm = 10−2 m. As before, this can be made into these two different reciprocal conversion factors: (1.7.11)1⁢c⁢m 10−2 m or 10−2 m 1⁢c⁢m If you wanted to find out how many centimeters in 3.55 meters, we again follow the process: quantity (in old units)×conversion factor=quantity (in new units) (1.7.12)3.55 m 1×1⁢c⁢m 10−2 m=355⁢c⁢m Pull out your calculator and ensure that you get 355 as your answer when you divide 3.55 by 10−2. If not, get help with how to properly use your calculator. (For example, in the Casio fx-260, 10−2 might be entered as , then [EXP],[+/-], and .) To avoid using exponents, some instead use that 1 cm is 1/100th of a meter(because centi=10−2= 1/10 2= 1/100). (1.7.13)1⁢c⁢m=1 100 m Since fractions of a unit can make conversions difficult later, both sides can be multiplied by 100 to cancel the 1/100 and get this version. (1.7.14)100⁢c⁢m=1 m The two versions of the conversion factor are therefore: (1.7.15)100⁢c⁢m 1 m or 1 m 100⁢c⁢m To convert 3.55 meters into centimeters this way, we first write the quantity we are given, 3.55 m. Then we multiply this quantity by a conversion factor, which is the same as multiplying it by 1. We can write 1 as 100 cm 1 m and multiply: (1.7.16)3.55 m 1×100⁢c⁢m 1 m=355⁢c⁢m Notice that we get the same answer, 355 cm, either way. Using the exponent is most direct and makes things easier with large exponents, like micro= 10−6 or giga = 10 9. But for those who have trouble with exponential notation, using micro= 1/1,000,000 or giga = 1,000,000,000 is also an option. Significant Figures in Conversions How do conversion factors affect the determination of significant figures? Numbers in conversion factors based on prefix changes, such as kilograms to grams, are not considered in the determination of significant figures in a calculation because the numbers in such conversion factors are exact. Exact numbers are defined or counted numbers, not measured numbers, and can be considered as having an infinite number of significant figures. (In other words, 1 kg is exactly 1,000 g, by the definition of kilo-.) Counted numbers are also exact. If there are 16 students in a classroom, the number 16 is exact. In contrast, conversions that do not involve just a prefix change, like those between different systems of units, are usually not exact. For example, kilograms are S.I. units, but pounds come from an older system. If given 1 kg = 2.20 lbs, you sould assume three significant figures (but can be found to six or more figures, 1 kg = 2.20462 lbs).Conversion factors that come from measurements (such as density, as we will see shortly) or are approximations have a limited number of significant figures and should be considered in determining the significant figures of the final answer. Example 1.7.1 The average volume of blood in an adult male is 4.7 L. What is this volume in milliliters? A hummingbird can flap its wings once in 18 ms. How many seconds are in 18 ms? SOLUTION We start with what we are given, 4.7 L. We want to change the unit from liters to milliliters. The prefix milli is 10−3. If m =10−3,adding the liters to both sides, makes it 1 mL =10−3 L.From this relationship, we can construct two conversion factors: (1.7.17)10−3 L 1⁢m⁢L or 1⁢m⁢L 10−3 L We use the conversion factor that will cancel out the original unit, liters, and introduce the unit we are converting to, which is milliliters. The conversion factor that does this is the one on the right. (1.7.18)4.7⁢L×1⁢m⁢L 10−3 L=4,700⁢m⁢L Because the numbers in the conversion factor are exact, we do not consider them when determining the number of significant figures in the final answer. Thus, we report two significant figures in the final answer. We can construct two conversion factors from the relationships between milliseconds and seconds, 1 ms =10−3 s: (1.7.19)1⁢m⁢s 10−3 s or 10−3 s 1⁢m⁢s To convert 18 ms to seconds, we choose the conversion factor that will cancel out milliseconds and introduce seconds. The conversion factor on the right is the appropriate one. We set up the conversion as follows: (1.7.20)18 m⁢s×10−3 s 1 m⁢s=0.018 s The conversion factor’s numerical values do not affect our determination of the number of significant figures in the final answer. (If you prefer the method without exponents, you could instead use that milli is 1/1,000. If you do it that way, ensure that you get the same answers.) Example 1.7.2 Perform each conversion. 101,000 ns to seconds 32.08 kg to grams SOLUTION The prefix nano is 10−9. If n=10−9,adding the seconds to both sides, makes it 1 ns=10−9 s.From this relationship, we can construct two conversion factors: (1.7.21)10−9 s 1⁢n⁢s or 1⁢n⁢s 10−9 s We use the conversion factor that will cancel out the original unit, nanoseconds, and introduce the unit we are converting to, which is seconds. The conversion factor that does this is the one on the left. (1.7.22)101,000⁢n⁢s×10−9 s 1 n⁢s=0.000101 s We can construct two conversion factors from the relationships between kilograms and grams, 1 kg=10 3 g: (1.7.23)1⁢k⁢g 10 3 g or 10 3 g 1⁢k⁢g To convert 32.08 kg to grams, we choose the conversion factor that will cancel out kilograms and introduce grams. The conversion factor on the left is the appropriate one. We set up the conversion as follows: (1.7.24)32.08⁢k⁢g×10 3 g 1 k⁢g=32080 g To more clearly show that the answer has four significant figures, it can be written in scientific notation as 3.208x10 4 g. Conversion factors can also be constructed for converting between different kinds of units. For example, density can be used to convert between the mass and the volume of a substance. Consider mercury, which is a liquid at room temperature and has a density of 13.6 g/mL. The density tells us that 13.6 g of mercury have a volume of 1 mL. We can write that relationship as follows: 13.6 g mercury = 1 mL mercury This relationship can be used to construct two conversion factors: (1.7.25)13.6 g 1 mL and 1 mL 13.6 g Which one do we use? It depends, as usual, on the units we need to cancel and introduce. For example, suppose we want to know the mass of 16 mL of mercury. We would use the conversion factor that has milliliters on the bottom (so that the milliliter unit cancels) and grams on top so that our final answer has a unit of mass: (1.7.26)16 mL×13.6 g 1 mL=217.6 g=220 g In the last step, we limit our final answer to two significant figures because the volume quantity has only two significant figures; the 1 in the volume unit is considered an exact number, so it does not affect the number of significant figures. The other conversion factor would be useful if we were given a mass and asked to find volume, as the following example illustrates. Density can be used as a conversion factor between mass and volume. Example 1.7.3: Mercury Thermometer A mercury thermometer for measuring a patient’s temperature contains 0.750 g of mercury. What is the volume of this mass of mercury? SOLUTION Because we are starting with grams, we want to use the conversion factor that has grams in the denominator (bottom). The gram unit will cancel algebraically, and milliliters will be introduced in the numerator. (1.7.27)0.750 g×1⁢m⁢L 13.6 g=0.055147...m⁢L≈0.0551⁢m⁢L We have limited the final answer to three significant figures. You could instead rearrange the density formula, d = m/V, to solve for volume, V = m/d =(0.750g) / (13.6 g/mL) = 0.0551 mL, to arrive at the same answer. Looking Closer: Density and the Body The densities of many components and products of the body have a bearing on our health. Bones. Bone density is important because bone tissue of lower-than-normal density is mechanically weaker and susceptible to breaking. The density of bone is, in part, related to the amount of calcium in one’s diet; people who have a diet deficient in calcium, which is an important component of bones, tend to have weaker bones. Dietary supplements or adding dairy products to the diet seems to help strengthen bones. As a group, women experience a decrease in bone density as they age. It has been estimated that fully half of women over age 50 suffer from excessive bone loss, a condition known as osteoporosis. Exact bone densities vary within the body, but for a healthy 30-year-old female, it is about 0.95–1.05 g/cm 3. Osteoporosis is diagnosed if the bone density is below 0.6–0.7 g/cm 3. Urine. The density of urine can be affected by a variety of medical conditions. Sufferers of diabetes produce an abnormally large volume of urine with a relatively low density. In another form of diabetes, called diabetes mellitus, there is excess glucose dissolved in the urine, so that the density of urine is abnormally high. The density of urine may also be abnormally high because of excess protein in the urine, which can be caused by congestive heart failure or certain renal (kidney) problems. Thus, a urine density test can provide clues to various kinds of health problems. The density of urine is commonly expressed as a specific gravity, which is a unitless quantity defined as (1.7.28)density of some material density of water Normal values for the specific gravity of urine range from 1.002 to 1.028. Body Fat. The overall density of the body is one indicator of a person’s total body fat. Fat is less dense than muscle and other tissues, so as it accumulates, the overall density of the body decreases. Measurements of a person’s weight and volume provide the overall body density, which can then be correlated to the percentage of body fat. (The body’s volume can be measured by immersion in a large tank of water. The amount of water displaced is equal to the volume of the body.) Multiple Conversions Sometimes you will have to perform more than one conversion to obtain the desired unit. For example, suppose you want to convert 54.7 km into millimeters. You can either memorize the relationship between kilometers and millimeters, or you can do the conversion in steps. Most people prefer to convert in steps. To do a stepwise conversion, we first convert the given amount to the base unit. In this example, the base unit is meters. We know that there are 10 3 m in 1 km: (1.7.29)54.7 k⁢m×10 3 m 1 k⁢m=54,700 m Then we take the result (54,700 m) and convert it to millimeters, remembering that there are 1⁢m⁢m for every 10−3 m: (1.7.30)54,700 m×1⁢m⁢m 10−3 m=54,700,000⁢m⁢m=5.47×10 7⁢m⁢m Expressing the final answer in scientific notation more clearly shows the three significant figures. As a shortcut, both steps in the conversion can be combined into a single, multistep expression: (1.7.31)54.7 k⁢m×10 3 m 1 k⁢m×1⁢m⁢m 10−3 m=54,700,000⁢m⁢m=5.47×10 7⁢m⁢m NOTE: When using the prefixes and exponents, the prefix should always be on one side and the exponent that it equals should be on the other side. (In the first step here, since kilo is on bottom, the 10 3 must be on top. Then in the other step, because milli is on the top, the 10-3 should be on bottom.) Either method—one step at a time or all the steps together—is acceptable. If you do all the steps together, the restriction for the proper number of significant figures should be done after the last step. As long as the math is performed correctly, you should get the same answer no matter which method you use. Example 1.7.4 Convert 58.2 ms to megaseconds in one multistep calculation. SOLUTION The milli prefix abbreviatioin is lower case,m = 10-3, so with base units of seconds it would be, ms = 10-3 s. The mega abbreviation is upper case, M =10 6, so Ms = 10 6 s. First, convert the given unit to the base unit—in this case, seconds—and then convert seconds to the final unit, megaseconds. (1.7.32)58.2 m⁢s×10−3⁢s 1 m⁢s×1⁢M⁢s 10 6 s=0.0000000582⁢M⁢s=5.82×10−8⁢M⁢s Neither conversion factor affects the number of significant figures in the final answer. Example 1.7.5 Convert 43.00 ng to kilograms in one multistep calculation. SOLUTION For nano, n= 10-9, so with base units of grams it would be, ng= 10-9 g. For kilo, k=10 3, so kg= 10 3 g. First, cancel ng and convert the given unit to the base unit (g), and then convert grams to the final unit, kilograms. (1.7.33)43.00 n⁢g×10−9⁢g 1 n⁢g×1⁢k⁢g 10 3 g=0.00000000004300⁢k⁢g=4.300×10−11⁢k⁢g Neither conversion factor affects the number of significant figures (four)in the final answer. Squared or Cubed Units Area is often found by multiplying two lengths, and thus has squared units. For example length in feet, times width in feet,gives area in square feet,ft 2. Volume is often found by multiplying three lengths, and thus has cubed units. For a box, length in centimeters, times width in centimeters, times depth in centimeters, gives volume in cubic centimeters, cm 3. A length conversion factor must be used twice for squared area and three times for cubed volume. Assume we find that the area of notebook page is 93.5 square inches and we want to convert that to square centimeters.Most often, you will only be able to find the relationship between inches and centimeters, 1 in = 2.54 cm. The conversion factor will need to be used twice here (so that both inches in in 2 cancel and so that the two centimeters on top leave you with cm 2). (1.7.34)93.5 i⁢n 2×2.54⁢c⁢m 1 i⁢n×2.54⁢c⁢m 1 i⁢n=603⁢c⁢m 2 Example 1.7.6 Tire pressure is often measured in pounds per square inch, abbreviated psi or lbs/in 2. Convert 32.0 psi to units of kilograms per square centimeter. SOLUTION The pressures have weight units (lbs or kg) on top and area units (in 2 or cm 2) on bottom. We will use 1 kg = 2.20 lbs and 1 in = 2.54 cm to create our conversion factors. (1.7.35)32.0⁢l⁢b⁢s i⁢n 2×1⁢k⁢g 2.20 l⁢b⁢s×1⁢i⁢n 2.54⁢c⁢m×1⁢i⁢n 2.54⁢c⁢m=2.25⁢k⁢g c⁢m 2 Since 32.0 lbs/in 2 had inches twice on the bottom, 1 in 2.54 cm was used twice. The final answer is rounded to three significant figures. Career Focus: Pharmacist A pharmacist dispenses drugs that have been prescribed by a doctor. Although that may sound straightforward, pharmacists in the United States must hold a doctorate in pharmacy and be licensed by the state in which they work. Most pharmacy programs require four years of education in a specialty pharmacy school. Pharmacists must know a lot of chemistry and biology so they can understand the effects that drugs (which are chemicals, after all) have on the body. Pharmacists can advise physicians on the selection, dosage, interactions, and side effects of drugs. They can also advise patients on the proper use of their medications, including when and how to take specific drugs properly. Pharmacists can be found in drugstores, hospitals, and other medical facilities. Curiously, an outdated name for pharmacist is chemist, which was used when pharmacists formerly did a lot of drug preparation, or compounding. In modern times, pharmacists rarely compound their own drugs, but their knowledge of the sciences, including chemistry, helps them provide valuable services in support of everyone’s health. Concept Review Exercises How do you determine which quantity in a conversion factor goes in the denominator of the fraction? State the guidelines for determining significant figures when using a conversion factor. Answers The unit you want to cancel from the numerator goes in the denominator of the conversion factor. Exact numbers that appear in many conversion factors do not affect the number of significant figures; otherwise, the normal rules of multiplication and division for significant figures apply. Key Takeaway A unit can be converted to another unit of the same type with a conversion factor. There are plenty more conversion exercises to practice in Section 1.E at the end of this chapter. Contributors Anonymous 1.7: Converting Units is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts. Back to top 1.6: The International System of Units 1.8: Chapter Summary Was this article helpful? Yes No Recommended articles 1.1: What Is Chemistry?Chemistry is the study of matter and how it behaves. The scientific method is the general process by which we learn about the natural universe. 1.2: The Classification of MatterMatter can be described with both physical properties and chemical properties. Matter can be identified as an element, a compound, or a mixture. 1.3: MeasurementsIdentify a quantity properly with a number and a unit. 1.4: Expressing Numbers: Scientific NotationLarge or small numbers are expressed in scientific notation, which use powers of 10. 1.5: Expressing Numbers: Significant FiguresSignificant figures properly report the number of measured and estimated digits in a measurement. There are rules for applying significant figures in ... Article typeSection or PageLicenseCC BY-NC-SALicense Version4.0Show Page TOCno on page Tags This page has no tags. © Copyright 2025 Chemistry LibreTexts Powered by CXone Expert ® ? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 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4682	https://www.interviewbit.com/blog/climbing-stairs-problem/	Climbing Stairs Problem Problem Statement Given a staircase of N steps and you can either climb 1 or 2 steps at a given time. The task is to return the count of distinct ways to climb to the top. Note: The order of the steps taken matters. Examples: Input: N = 3 Output: 3 Explanation: Confused about your next job? Expand in New Tab There are three distinct ways of climbing a staircase of 3 steps : Input: N = 2 Output: 2 Explanation: There are two distinct ways of climbing a staircase of 3 steps : Brute Force (Recursive) Approach The approach is to consider all possible combination steps i.e. 1 and 2, at every step. To reach the Nth stair, one can jump from either (N – 1)th or from (N – 2)th stair. Hence, for each step, total ways would be the summation of (N – 1)th stair + (N – 2)th stair. The recursive function would be: ClimbStairs(N) = ClimbStairs(N – 1) + ClimbStairs(N – 2). If we observe carefully, the expression is nothing but the Fibonacci Sequence. SampleRecursive Tree for N = 5: Algorithm: C++ Implementation of Recursive Approach Java Implementation of Recursive Approach Python Implementation of Recursive Approach Bottom-up Approach In the above approach, observe the recursion tree. It can be clearly seen that some of the subproblems are repeating. Hence, it is unnecessary to calculate those again and again. Therefore, we can store the result of those subproblems and retrieve the solution of those in O(1) time. Since the problem contains an optimal substructure and has overlapping subproblems, it can be solved using dynamic programming. One can reach the ith step in one of the two ways : Algorithm C++ Implementation of DP Approach Java Implementation of DP Approach Python Implementation of DP Approach Fibonacci Number Approach In the above approach, the dp array is just storing the value of the previous two steps from the current ith position i.e. (i – 1)th and (i – 2)th position. The space complexity can be further optimized, since we just have to find an Nth number of the Fibonacci series having 1 and 2 as their first and second term respectively, i.e. Fib(1) = 1 and Fib(2) = 2. Algorithm C++ Implementation Java Implementation Python Implementation Practice Question Stairs Problem FAQs What is the most efficient approach to solving the Climbing stairs problem? The approach to finding the Nth Fibonacci number is the most efficient approach since its time complexity is O(N) and space complexity is O(1). How to solve this problem if it’s given that one can climb up to K steps at a time?If one can climb K steps at a time, try to find all possible combinations from each step from 1 to K. The recursive function would be : climbStairs(N, K) = climbStairs(N – 1, K) + climbStairs(N – 2, K) + … + climbStairs(N – K , K). Previous Post Activity Selection Problem Next Post Job Sequencing With Deadlines Categories
4683	https://www.youtube.com/watch?v=Df4b1ofA4IQ	The complete set of values of x in the domain of function f (x)= sqrt(log _(x+2{x})([x] ^(2)-5[x... Doubtnut 3940000 subscribers 12 likes Description 1171 views Posted: 2 Nov 2021 The complete set of values of x in the domain of function f (x)= sqrt(log _(x+2{x})([x] ^(2)-5[x] +7))where [.] denote greatest integer functioon and {.} denote fraction pert function) is : Class: 12 Subject: MATHS Chapter: FUNCTION Board:IIT JEE You can ask any doubt from class 6-12, JEE, NEET, Teaching, SSC, Defense and Banking exam on Doubtnut App or You can Whatsapp us at - 8400400400 Link - Join our courses to improve your performance and Clear your concepts from basic for Class 6-12 School and Competitive exams (JEE/NEET) - Contact Us: 👉 Have Any Query? Ask Us. 🤙 Call: 01247158250 💬 WhatsApp: 8400400400 📧 Email: info@doubtnut.com 🌐 Website: Welcome to Doubtnut. 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Exams: Follow Us: 🔔 Facebook: 🔔 Instagram: 🔔 Telegram: 🔔 Twitter: 🔔 LinkedIn: Our Telegram Pages: 🔔 Doubtnut Official: 🔔 Doubtnut IIT JEE: 🔔 Doubtnut NEET: 🔔 Doubtnut CBSE Boards: 🔔 Doubtnut UP Boards: 🔔 Doubtnut Bihar Boards: 🔔 Doubtnut Government Exams: class 12 class 12 physics class 12 chemistry class 12 english class 12 maths class 12 biology cbse class 12 result class 12 economics class 12 accountancy class 12 syllabus physics cbse class 12 class 12 english grammar class 12 syllabus cbse class 12 history class 12 geography class 12 ncert class 12 syllabus cbse class 12 maths cbse class 12 english cbse class 12 physics cbse class 12 chemistry class 12 grammar cbse class 12 biology cbse class 12 commerce cbse class 12 accountancy class 12 syllabus chemistry class 12 grammar syllabus class 12 syllabus maths class 12 latest syllabus class 12 syllabus english class 12 syllabus biology class 12 syllabus of physics class 12 syllabus of chemistry class 12 syllabus science class 12 syllabus ncert class 12 syllabus of english class 12 syllabus commerce 1 comments Transcript: कि आज डाउनलोड करें डॉक्टर पर होगा आपके सभी मैच केमिस्ट्री फिजिक्स बायोलॉजी डाइट का सफाया बस में पशुओं की फोटो खींचो उसे करो करो और तुरंत वीडियो सलूशन पाव डाउनलोड नाउ है इस कोशिश में हमें क्या दिया गया है कि कंपलीट सेट आफ वैल्यूज आफ एक सिद्ध भूमि नॉट फंक्शन फैक्स इक्वल्स टू और विकसित किया गया और बोला है कि इसके जो है डोमेन निकालना हमें तो ठीक है तो देखिए फैसले लेते हैं पर सबसे पहले फिक्स कर दिया गया है फर्क हमें दिया गया है रूट ओवर कुछ लॉग आफ एक्स प्लस टू इन टूर्स आफ टॉपिक्स और ब्रैकेट में ऑपरेट एस टीचर ऑफ एक्स स्क्वायर माइनस 5.2 ग्रेटेस्ट इंटिगर ऑफ एक्स प्लस 7 तो देखिए यहां पर सबसे पहले तो देखिए पूरी लोग वाली टाइम जो है यह हमेशा जो यह पॉजिटिव होनी चाहिए यह कंस्ट्रेंट आ जाएगा सेकंड कंस्ट्रेंट होगा इस देश के लिए ठीक है तो बेस होगा इधर देखिए सबसे पहले लेता हूं मैं हनी यह जो एक्स प्लस है तू फ्रेक्शनल पार्ट ऑफिसर इस ग्रेटर 1002 नॉट इक्वल तो एक पुस्तक यह धोखे से लिखना हम है और सेकंड कि इसमें क्या यह वाला जो अंदर वाला टाइम है यह इस ग्रेट डेंजर हो तो मोड ऑफ और सिगरेट या फंक्शन ऑफ एक्स स्क्वायर प्लस सॉरी - यह और सुनाइए नस फाइव क्रिटिसिजम फ्रॉम एक्स प्लस 01 होना चाहिए अब देखिए यह जो है और फिर इस लुक की वैल्यू जो है कभी नेगेटिव अपनी होनी चाहिए यह तीन अगर हमें कंटेंट तो देखिए अगर हम ध्यान से देखें तो देखिए यह क्या है तय को रिलेटेड क्वेश्चन है अब इसको ट्रैफिक क्वेश्चन कि अगर हम देखिए बी स्क्वायर माइनस फॉर एसी देखें ठीक है तो बी स्क्वेयर - पूरे अगर देखें इस क्वाड्रेटिक इक्वेशन का तुझे यह क्या हो रहा है यह जाएगा के बीच कर हो जाएगा ट्वेंटी फाइव से - फॉर इनटू ए क्या है वन 7 देखा जाएगा थ्री लिए रिमाइंड स्त्री का यह एपिसोड - फॉर इसीइ ए स्ट्रेंज टो हम आप व्हिस्कर - कोई ऐसी लेसेस तो देखिए ग्राफ की वैल्यू के होगी हमेशा पॉजिटिव ठीक है सूर्य कि अगर हमेशा पॉजिटिव होगी इसकी वैल्यू तो देखिए ग्रेटेस्ट टीचर फंक्शन जो है यह कभी नेगेटिव नहीं होगा क्योंकि इसकी वैल्यू कभी जी क्वेश्चन होगी ठीक है इस तरीके से इसकी वैल्यू रोशनी अच्छी नहीं होगी तो देखिए ग्रेटेस्ट टीचर ऑफ एक्स य हम 100 से ऊपर होगा तो हमारा ऑप्शन यह ऑप्शन हो गया ठीक है है और यह हमारा दिखे एग्जाम शून्य से नीचे गिर जाए या से नीचे नहीं होगी ग्रेटेस्ट टीचर फंक्शन की वैल्यू क्लिक यह मिनिमम क्या हो सकता है 0 - 07 उपलब्धियां की वजह से हमेशा फंक्शन ग्रेटर लायंस रहेगा तो ओनली कंट्री जो हमारे पास बचती है वह यह वाली होगी चुकी है एक्स प्लस 2 एफ़ फ्रेक्शनल पार्ट आफ इट्स ग्रेट डेन जीरो एंड नॉट इक्वल 2100 स्टूडेंट को अगर हम देखते जाए आगे तो देखिए इसे इसको क्या लिख सकते हम एक को लिखते लिख सकता हूं ग्रेटेस्ट टीचर फंक्शन प्लस फ्रेक्शनल स्टाफ के किसी भी नंबर को हम उसके इन टीचर और ट्रैक्टर पार्ट्स में डिवाइड कर सकते हैं प्लस 2 फ्रेक्शनल पार्ट ऑफिसर ग्रेट एंड जीरो का सलूशन चाहिए हमें और देखिए इसको अगर हम देखें तो यह क्या है ग्रेटेस्ट इंटिगर एक्स प्लस थ्री टाइम्स आफ क्राइसिस आफ टॉपिक्स ब्रिटेन जीरो अब देखिए तो इससे रिलेटिड क्लीनसोल कर सकते या इसका ग्राफ थ्रो कर सकते हैं ठीक है अब लिखे कि आपका एडिशन है तो देखिए मैं दोनों डिफरेंट खिलाफ विद्रोह कर देता हूं सबसे पहले क्या लिखे ग्रेटेस्ट टीचर सोंग्स ऑफ एक्स तो इसका ग्राफ कुछ ऐसा होगा का यह इस तरह से स्टेप फंक्शन होता है ग्रेटेस्ट लीडर्स ऑप्शन है कि सर ठीक है तो कुछ वैल्यूज खिलाफ विद्रोह कर लेते हैं वो अपने समझने के लिए उन और सेकंड फंक्शन किया है थ्री टू फ्रंट पार्ट ऑफिसर ठीक है इस समुद्री जीव वृद्धि होगी अच्छा ठीक है तो देखिए अभी फ्रेक्शनल पर टो फंक्शन टो सैक्रिफिस यह जाएगा का यह की व्यवस्था होगी तो यह थी कि यह ट्रेडिशनल और ऐसे इधर-उधर है दोनों तरफ कि एक तरफ नीचे नीचे वाला पॉइंट यह ऊपर जो है डिस्कंटीन्यूटी आ जाएगी ठीक है तो देखिए दो कप बना रहे अब हमें इन दोनों ग्राहक एडिशन चाहिए रिजल्ट एंड ग्राफ में तो देखिए ध्यान से समझते का यह देखिए अगर मैं 1 बनाता हूं कि ऐश्वर्या व्यक्ति चाहिए अगर और अगर मैं एक्सिस बनाता हूं तो कि हमें वर्सिटीज के बीच में देखना होगा कि क्रिकेट इसके बीच में चेंज अरे यू सुरे कि हम लेंगे यह 1234 सिमिलरली माइनस वन माइनस टू तक देखेंगे तो देखिए अगर फर्स्ट यूज करने में ग्रेटेस्ट टीचर प्लस यह सिर्फ एक ग्राफ तो लेकिन जीरो सेवन के बीच में देखे ठीक है 102 ए340 123 इस तरह से ग्राहक हमारा 207 के बीच में देखें तो इसकी वैल्यू कार्य0 और यह ग्रेड थ्री लेटेस्ट न्यूज आफ फंक्शंस ऐसे ही आ जाएगा तो जीरो प्लस ग्राहक यह ऐसे ही हमें ग्राफ मिलेगा का या अच्छा ठीक है फिर देखिए इस जीरो सेवन के बीच में इस तरह से हमें ग्राफ मिलकर सिमिलरली व्वे टू के बीच में ग्रेटेस्ट टीचर फंक्शन की वैल्यू के देखिए यह क्या 123 के सेंटर में यह फंक्शन चाहिए वनप्लस 3 सब्सक्राइब करने का यह देखिए इस तरह से हम ग्राफ का या कि यह देखिए वैल्यू क्या है अच्छी तरह के 123 लिए कि यह क्या का यह जाएगी 4 सिमिलरली अगर इधर होगा लेकिन नेक्स्ट में अगर टू थ्री के बीच में अगर हम देखें तो रिक्वेस्ट दूसरी के बीच में ग्रेटेस्ट टीचर फंक्शन की वैल्यू के हो गई टू पर ग्राहक हो जाएगा टू प्लस 382 ग्रेटेस्ट टीचर फंक्शन एक्स ठीक है तो अधिक इन नेक्स्ट वाले इंटरवल में हमें ग्राफ इस तरह मिलेगा अच्छा ठीक है अब सिमिलरली अगर हम नेगेटिव साइड में देखें इधर तो नेगेटिव साइड में दिखे इधर जो है बिटवीन - 1080 फंक्शन की वैल्यू के माइनस वन के बजाय माइनस वन प्लस थ्री लेयर चैनल पर टॉपिक्स तो यह जो ग्राफ यह जस्ट एक यूनिट नीचे से स्टार्ट होगा का या और इस तरह से हमें मिलेगा अच्छा ठीक है अब देखिए हम एक सलूशन क्या चाहिए था ग्रेटर 1001 नॉट इक्वल टो दो ए ग्रेटर देन जीरो कहां कहां पर वायर फंक्शन की रैली हो रही है तो देखिए अगर हम देखें तो वह यहां होगी ठीक है तो ए ग्रेटर 1004 यह सब वैल्यू उसके लिए है और लेकिन नॉट इक्वल टू वन बोला है तो मतलब हमें एक डिस्कंटीन्यूटी वाला चीज मिलेगी नॉट इक्वल टू वन अगर हम देखें तो देखिए वन कहां पर इक्वल और है यहां पर और यहां पर ठीक है सूर्य कि हमारा बेस की वैल्यू जो आगे हमें वह क्या गई इस देश को हमें एग्री केयर अच्छी क्वालिटी की होगी तो अब हम गैस बंद कर देंगे कि मेघनाद होना चाहिए कि उन्हें ग्रेटर थन वन लख ठीक है तो देखिए यह अगर हम इन क्वालिटी देखिए 1 लीटर ग्रेड वन तो देखिए हमें जो है ए ग्रेटर थन वन करने से क्या-क्या सलूशन मिल रहे हैं यह वाला जो है 3X ग्रेटर थन वन यह वाली चीज आ गई ठीक है - वनप्लस 3T इससे यह देखिए - वंश के बीच में मैंने इसलिए चैनल सब्सक्राइब बटन टू थ्री यह देखिए यह जो है यह फ्रंट पार्ट और पार्ट है यह देखिए तो यह थ्री लेना है यह 1313 इंटरनेशनल चाहिए और इधर शीघ्र होगी इस ग्राफ यह यह - में 153 आ गया सुन - 1.3 लाख mar0 यह फर्स्ट टाइम टेबल आ गया जो कि यह हमें देखिए मैं ग्रीन से ऐड करूंगा वह हमारा सलूशन होगा का यह सब वैल्यूज हमारी सलूशन है ठीक है तो देखिए इस - 13040 नॉट इंक्लूडेड यह फर्स्ट टाइम टेबल आ गया सेकंड में देखिए यह वाला पॉइंट इंक्लूडेड नहीं होगा क्योंकि देखिए इस पर बने और वे कैन नॉट इक्वल टू वन ग्रेट दैनिक 121 चाहिए कि क्या हो गया 153 लकमा 101.1 नॉटेड ठाट ऑपरेट्स यहां पर और फिर वह सिगरेट तो यह वाला पॉइंट तो हम इसे इस तरीके से - 1304 यूनियन 130 यूनियन कि 12158 ठीक है तो ज्यादा हमें फाइनल सलूशन तो गर्म ऑप्शंस देखें तो ऑप्शन फोर हमारा सही ऑप्शन आ जाएगा प्रशिक्षण लेकर नीचे की तरफ आ आ आ आ आ आ आ आ आ आ
4684	https://bio.libretexts.org/Bookshelves/Microbiology/Microbiology_(Boundless)/14%3A_Pathogenicity/14.02%3A_Overview_of_Microbe-Host_Interactions/14.2B%3A_Opportunistic_Microorganisms	Skip to main content 14.2B: Opportunistic Microorganisms Last updated : Nov 23, 2024 Save as PDF 14.2A: Normal Microbiota and Host Relationships 14.2C: Cooperation Among Microorganisms Page ID : 11984 ( \newcommand{\kernel}{\mathrm{null}\,}) Learning Objectives Describe the traits of an opportunistic microorganism In the general realm of biology, an opportunist is an organism that is able sustain its life from a number of different sources, but when favorable conditions arise, the organism immediately takes advantage of the opportunity to thrive. When the focus is turned more specifically to microbiology, scientists call organisms that behave this way opportunistic microorganisms. A microorganism is a microscopic organism that can either be a single cell, cell cluster, or multicellular. Microorganisms are very diverse and include bacteria, fungi, algae, and protozoa. Opportunistic microorganisms are typically non-pathogenic microorganisms that act as a pathogen in certain circumstances. They lay dormant for long periods of time until the hosts’ immune system is suppressed and then they seize the opportunity to attack. Patients with Human Immunodeficiency Virus (HIV) are particularly susceptible to opportunistic infections. HIV can develop into Acquired Immune Deficiency Syndrome ( AIDS ), which infects and destroys helper T cells (specifically CD4+ T cells). When the number of CD4+ T cell numbers fall below a critical level, cell-mediated immunity is lost. When immunity is lost, the opportunistic microorganisms can easily infect the AIDS patient without being destroyed by the immune system. These opportunistic pathogens thrive while the human body slowly deteriorates. An example of an opportunistic microorganism is Haemophilus ducreyi. This microorganism infects its host through broken skin or epidermis. In other words, without an open wound, this sexually transmitted disease would be unable to use the human body as a host. It takes advantage of the opportunity to infect the lymphocytes, macrophages and granulocytes as soon as it enters the area of broken skin. Key Points A microorganism is a microscopic organism that can either be a single cell, cell cluster, or multicellular. Microorganisms are very diverse and include bacteria, fungi, algae, and protozoa. Opportunistic microorganisms are typically non-pathogenic microorganisms that act as a pathogen in certain circumstances. They lay dormant for long periods of time until the host ‘s immune system is suppressed and then they take that opportunity to attack. Haemophilus ducreyi, a microorganism, infects its host through broken skin or epidermis. Without the open wound, this sexually transmitted disease would be unable to use the human body as a host. Key Terms microorganism: An organism that is too small to be seen by the unaided eye, especially a single-celled organism, such as a bacterium. Opportunistic: Taking advantage of situations that arise. immunodeficiency: A depletion in the body’s natural immune system, or in some component of it. 14.2A: Normal Microbiota and Host Relationships 14.2C: Cooperation Among Microorganisms
4685	https://focustutoring.com/gmat-blog/gmat-rate-problems-combined-work-problems/	GMAT Rate Problems: Combined Work Problems by Elijah Mize \| Apr 23, 2025 \| GMAT Elijah Mize Welcome back to our series on rate problems. In the first article, we introduced the relationship between the variables of rate, work, and time – or speed, distance, and time. This article and the next will teach you how to approach combined work problems which involve two or three machines working together to complete a job. Almost all of these problems ask for time: if one machine takes a hours to do the job alone and the other machine takes b hours, how many hours do they take to do the job when they work together? Or, less frequently, if one machine takes a hours to do the job alone and the two machines take c hours to do the job when they work together, how many hours does the other machine take to do the job alone? GMAT Combined work Problems These scenarios are all governed by the same formula. Using a, b, and c for the independent and combined times for the machines as in the example above: (1 / c) = (1 / a) + (1 / b) Recall from our first article that rate and time are inversely related. Since this is true, then when a is the amount of time to complete a given job, the reciprocal (1 / a) is the fraction of the job that is completed in a single unit of time. If a is expressed in hours, then (1 / a) is the fraction of the job completed by machine A in a single hour. The fractions (1 / a) and (1 / b) represent the rates for machines A and B in a unit we might call “jobs per hour.” GMAT rates problems sometimes use another unit of time like days or seconds, but hours are common. Since this is true, then the equation above simply says this: the combined rate of the machines, (1 / c), is equal to the sum of their independent rates, (1 / a) and (1 / b). Rates are “stackable” or addable! If in a single hour, you can paint ⅙ of a room and your friend can paint ⅛ of a room, then together you can ⅙ + ⅛ of the room in that hour. Simple enough, right? As you can see, these problems often involve finding a common denominator in order to add the fractions. But in the next article, we’ll learn a workaround to help you avoid that step. Remember that the sum of your fractions is still in the rates unit of “jobs per hour.” In order to get back to the “hours per job” figure that most problems ask for, simply flip the fraction. In the example above, you and your friend paint at a combined rate of ⅙ + ⅛ = 4/24 + 3/24 = 7/24 “jobs per hour.” How many hours will it take you and your friend to paint the room? Flip the fraction to find 24/7, or 3 3/7 hours. Official GMAT problems Let’s try this out on an official GMAT problem: Printing press X can print an edition of a newspaper in 12 hours, whereas press Y can print the same edition in 18 hours. What is the total number of hours that it will take the two presses, working together but independently of one another, to print the same edition? (A) 15 (B) 7.4 (C) 7.2 (D) 7.0 (E) 6.8 Press X takes 12 hours to print the edition, meaning that it prints at a rate of 1/12 editions per hour. Press Y takes 18 hours to print the edition, meaning that it prints at a rate of 1/18 editions per hour. (1 / c) = (1 / a) + (1 / b) a = 12 b = 18 1 / c = 1/12 + 1/18 = 3/36 + 2/36 = 5/36 editions per hour c = 36/5 hours per edition = 7.2 hours per edition And the correct answer is C. All you have to do is plug in the values, add the fractions, and flip the sum. Here’s another official problem: Three printing presses, R, S, and T, working together at their respective constant rates, can do a certain printing job in 4 hours. S and T, working together at their respective constant rates, can do the same job in 5 hours. How many hours would it take R, working alone at its constant rate, to do the same job? (A) 8 (B) 10 (C) 12 (D) 15 (E) 20 I guess the GMAT really likes printing presses. This time there are three of them, but don’t let that scare you! The way the problem is set up, presses S and T can be treated like a single printing press. We have the combined time (for all three presses) of 4 hours and the “press ST” time of 5 hours. We can plug these right into our “add the rates” equation to find the time for press ST. (1 / c) = (1 / a) + (1 / b) c = 4 a = 5 (1 / 4) = (1 / 5) + (1 / b) (5/20) = (4/20) + (1/20) b = 20 And the correct answer is E. Data sufficiency problems Let’s move into some data sufficiency problems. We won’t have to solve for anything, but the same formula will govern our thinking. Machine R and machine S work at their respective constant rates. How much time does it take machine R, working alone, to complete a job? (A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. Statement 1 provides a factor of relationship between the times for machine S and machine R to complete the job. On its own, this isn’t sufficient. It is relative information comparing the two machines, but we were asked the absolute question of how long a single machine takes. There is nothing here to help “scale” our times. Maybe machine R takes 4 minutes, while machine S takes 3 minutes. Maybe machine R takes 4 days, while machine S takes 3 days. We don’t know. Statement 2 provides the combined time for the machines working together. On its own, this is insufficient, because we don’t know how the machines relate to each other. Maybe S is much faster than R and basically doing all the work itself. Or maybe R is much faster than S. Or maybe they work at the same rate. The combined time of 12 minutes simply doesn’t tell us how the machines are splitting the work. Together, the statements are sufficient. We can prove it with the formula. Let’s keep the variable c for combined time but use r and s instead of a and b. (1 / c) = (1 / r) + (1 / s) s = (¾ r) (statement 1) c = 12 (statement 2) (1 / 12) = (1 / r) + [1 / (¾ r)] Since r is the only variable left in the equation, the two statements together are sufficient to solve for r, the time for machine R alone to complete the job. Here’s a similar data sufficiency problem: Working together, Rafael and Salvador can tabulate a certain set of data in 2 hours. In how many hours can Rafael tabulate the data working alone? (A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. We can again use the variables r and s for the “working alone” times. But this problem, unlike the last, provides the combined time of 2 hours in the question stem. This means that any statement putting Rafael’s independent time r in terms of Salvador’s independent time s – or vice versa – is sufficient. Statement 1 says that Rafael lakes 3 hours less than Salvador. r = s – 3, or s = r + 3. This is sufficient. (1 / c) = [1 / (s – 3)] + (1 / s) Statement 2 says that Rafael’s time r is ½ of Salvador’s time s. r = s/2, or s = 2r. Again, this is sufficient. (1 / c) = (1 / r) + [1 / (2r)] The correct answer is D: each statement on its own is sufficient. Data sufficiency problem: Combined work Here’s a final data sufficiency problem for this first combined work article: If two copying machines work simultaneously at their respective constant rates, how many copies do they produce in 5 minutes? (A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. We’ve moved on from printing presses but are still stuck in the early 2000s with copying machines. This question also differs from the previous ones in that it did not ask how much time it takes the machines to complete a given amount of work; it asked how much work the machines can complete in a given amount of time. Since work = rate time, we are looking for statements that supply us with the rates at which the machines work. Statement 1 is a good start, but it can’t be sufficient on its own because it only provides a rate for one of the two machines. Statement 2 can’t be sufficient on its own because it only tells us that one machine works twice as fast as the other – we know nothing about the speed of the machines in absolute terms. The trap answer on this problem is C. Combining the statements, we have a lot of info. We know the rate of one of the machines, and we know that one machine is twice as fast as the other. But we don’t know whether the rate given of 250 copies per minute is for the fast machine or the slow machine! If it is for the fast machine, then the other machine produces 125 copies per minute. If it is for the slow machine, then the other machine produces 500 copies per minute. In the former case, the combined rate of the machines is 375 copies per minute. In the latter case, their combined rate is 750 copies per minute. Therefore the correct answer is E: both statements together are still insufficient. Remember this distinction between the fast machine and the slow machine in a combined work problem. It will be a major focus of the next article and a key to solving these problems very quickly. Are there any questions that you still have about the combined work problems on the GMAT? Register now for a free 30-minute free consultation with one of our top tutors. personalized, like no other. Contact Us US Clients: +1 (917) 819-5945International Clients: +44 79 1468 1083 Quick Links Resources © APEX LEARNING SERVICES 2025 Apex Learning Services is not affiliated with the GMAC. All trademarks and brands mentioned, including GMAT, are the sole property of their trademark owners.
4686	https://nhssomerset.nhs.uk/prescribing-and-medicines-management/prescribing-guidelines-by-clinical-area/menopause-and-hormone-replacement-therapy/	Skip to content Menopause and Hormone Replacement Therapy This formulary page covers diagnosis and management of menopause following NICE guideline [NG23] Menopause: diagnosis and management. Back to Prescribing Guidelines by Clinical Area Prescribing Guidelines by Clinical Area Cardiovascular System Gender Identity Hypnotics and Anxiolytics Infant Feeding Menopause and HRT Medicines in pregnancy, children and lactation Mental Health Nutrition and hydration Pain Management Reproductive Health Thyroid Conditions Wound Care The guidance aims to support in the diagnosis and management of menopause including people with premature ovarian insufficiency to support GPs so they can optimise their patient’s treatment choices with them. Please expand the topics below to see more detail. There are several resources also under Resources below, some of these will be useful to use with patients, but they may also support you in CPD. The HRT formulary is linked, however specific detail is shared as treatment pathways in each of the further pages under transdermal, oral or topical options. Menopause Guidance Active Serious Shortage Protocols HRT PPC (Pre-Payment) Certificate Incretin (GLP-1) Therapies and HRT Testosterone Formulary Information Transdermal HRT Oral HRT Local (Vaginal) Estrogen HRT Quick Reference Table NHS Somerset netFormulary Recurrent UTI Guidance BMS Guidance, Stock Availability and Safety Epilepsy and HRT? - Consider interactions - access BNF interactions See the SPS supply availability page for current stocks- Log in to access this page NHSBSA Serious Shortage Protocols for current serious shortage protocols Learn about the Somerset NHS Menopause Service. BMS Consensus Statements - British Menopause Society including: Advice from the British Menopause Society- Statement regarding progestogens BMS Statement on Testosterone - British Menopause Society News - British Menopause Society The British Menopause Society (BMS) have provided BMS Tools for Clinicians: The British Menopause Society (BMS) have provided BMS Tools for Clinicians: NEW Measurement of serum estradiol in the menopause transition - BMS July 2025 What is the menopause? Menopause practice standards NICE: Menopause, Diagnosis and Management – from Guideline to Practice Guideline Summary Menopause: Guidance for Practice Top Ten Tips HRT Guide HRT – Practical prescribing HRT preparations and equivalent alternatives -Useful when looking at stock equivalents during supply issues. Progestogens and endometrial protection Management of menopause for women with cardiovascular disease HRT after myocardial infarction Fast Facts: HRT and breast cancer risk Prescribable alternatives to HRT Testosterone replacement in menopause Cognitive Behaviour Therapy (CBT) for Menopausal Symptoms Menopause: Nutrition and Weight Gain Menopause: Nutrition and Weight Gain Top Ten Tips Surgical Menopause Induced menopause in women with endometriosis Migraine and HRT HIV and the menopause Menopause in ethnic minority women Menopause and the Workplace Guidance: what to consider Back to top Under Review December 2024 Resources NHS Somerset netFormulary: Hormone Replacement Therapy Early Menopause British Menopause Society – Tools for clinicians MHRA Summary of HRT risks and benefits [NG23] Menopause: diagnosis and management British Menopause Society: HRT Guide July 2020 NHS Employers: Menopause and the workplace The Somerset Emotional Wellbeing Podcast: Demystifying the Menopause – Hosted by Dr Andrew Tresidder featuring Dr Kathryn Patrick Long Term Benefits and Risks of HRT- Discussion Aid HRT and the likelihood of some medical conditions A discussion aid for healthcare professionals and patients from NICE guideline NG23- Menopause: identification and management updated 07 November 2024. Venous thromboembolism •the risk of venous thromboembolism (VTE) is increased by oral HRT compared with baseline population risk •the risk of VTE associated with HRT is greater for oral than transdermal preparations •the risk associated with transdermal HRT given at standard therapeutic doses is no greater than baseline population risk. •Consider transdermal rather than oral HRT for menopausal people who are at increased risk of VTE, including those with a BMI over 30 kg/m2 •Consider referring menopausal people at high risk of VTE to a haematologist for assessment before considering HRT. Cardiovascular disease •HRT does not increase cardiovascular disease risk when started under 60 years •HRT does not affect the risk of dying from cardiovascular disease •the presence of cardiovascular risk factors is not a contraindication to HRT as long as they are optimally managed •HRT with oestrogen alone is associated with no, or reduced, risk of coronary heart disease •HRT with oestrogen and progestogen is associated with little or no increase in the risk of coronary heart disease •Explain that taking oral (but not transdermal) oestrogen is associated with a small increase in the risk of stroke and the baseline population risk of stroke in women aged under 60 years is very low. Osteoporosis Give advice on bone health and discuss these issues at review appointments. Explain that the baseline population risk of fragility fracture for women around menopausal age in the UK is low and varies from one woman to another. Explain to women that their risk of fragility fracture is decreased while taking HRT and that this benefit: •is maintained during treatment but decreases once treatment stops •may continue for longer in women who take HRT for longer. Review patients for risk of osteoporosis and fragility fractures in line with NICE Clinical Knowledge Summary: Osteoporosis – Prevention of fragility fractures: Assessment. Check FRAX score and consider bisphosphonate treatment if indicated. See formulary chapter 6.6. Ensure patients are taking vitamin D as self-care in line with [PH56] Vitamin D: supplement use in specific population groups and ensure adequate dietary intake of calcium. Breast Cancer History or at Risk of Breast Cancer People with or at High Risk of, Breast Cancer Offer menopausal women and people with, or at high risk of breast cancer: -Information on all available treatment options -Information that the SSRIs paroxetine, fluoxetine, or sertraline should notbe offered to people with breast cancer who are taking tamoxifen -Referral to a healthcare professional with expertise in menopause. See the Somerset NHS Menopause Service for referral criteria and information. Breast Cancer Risks Using the MHRA risk table, explain to people around the age of natural menopause that: -the baseline risk of breast cancer for people around menopausal age varies from one person to another according to the presence of underlying risk factors -HRT with oestrogen alone is associated with little or no change in the risk of breast cancer -HRT with oestrogen and progestogen can be associated with an increase in the risk of breast cancer -any increase in the risk of breast cancer is related to treatment duration and reduces after stopping HRT. Menopausal Symptoms with Breast Cancer -Information and counselling should be offered about the possibility of early menopause and menopausal symptoms associated with breast cancer treatment. -Stop systemic hormone replacement therapy (HRT) in people who are diagnosed with breast cancer. -Do not routinely offer HRT (including oestrogen/progestogen combination) routinely to people with menopausal symptoms and a history of breast cancer. -In exceptional circumstances, offer HRT to people with severe menopausal symptoms and with whom the associated risks have been discussed in conjunction with the oncologist (unlicensed use- HRT is contraindicated with a history of breast cancer). ALL alternatives should be trialled before taking this step, the patient should be fully aware of the associated risks of this treatment option, with input from a menopause specialist. -Consider selective serotonin reuptake inhibitor (SSRI) antidepressants for people with breast cancer for relieving menopausal symptoms, particularly hot flushes, but not for those taking tamoxifen. However, there is no clear evidence for use of SSRIs or SNRIs. The BMS guidance on prescribable alternatives to HRT may be useful. Do not offer soy (isoflavone), red clover, black cohosh, vitamin E or magnetic devices to treat menopausal symptoms in people with breast cancer. Hormone Replacement Therapy Risk Reduction Strategies For people who have been assessed as having a moderate to high risk due to their family history. For those with a family history of breast cancer who are considering taking, or already taking HRT, should be informed of the increase in breast cancer risk with type and duration of HRT. Advice to individuals on the use of HRT should vary according to the individual clinical circumstances (such as asymptomatic menopausal symptoms, age, severity of menopausal symptoms, or osteoporosis). HRT usage when at familial risk should be restricted to as short a duration and as low a dose as possible. If having an early (natural or artificial) menopause, patients should be informed of the risks and benefits of HRT, but generally HRT usage should be confined to patients younger than age 50 years if at moderate or high risk. Alternatives to HRT should be considered for specific symptoms such as osteoporosis or menopausal symptoms Consideration should be given to the type of HRT if it is being considered for use in conjunction with risk-reducing gynaecological surgery. Breastfeeding and Menopause See CCG webpage Breastfeeding and Medicines for more information on prescribing medicine in breastfeeding patients. Breastfeeding and HRT – Oestrogen: Breastfeeding and Oestrogen cream or pessary Breastfeeding and HRT – Progesterone: Mirena® 20 micrograms/24 hours intrauterine delivery system can be used as protection from endometrial hyperplasia during oestrogen replacement therapy. See SPC for detail on length of treatment. Mirena is effective for 5 years in the indication of contraception, but 4 years for the progesterone component of HRT. Breastfeeding and Medication – The Menopause and Breastfeeding Contraception and Menopause See CCG contraception webpage FSRH Clinical Guideline: Contraception for Women Aged over 40 Years People with premature ovarian insufficiency will be offered sex steroid replacement with a choice of HRT or a combined hormonal contraceptive (CHC). CHC (unless contraindicated) has the added benefit of providing contraceptive cover to people with premature ovarian insufficiency. For people with a uterus requiring protection from endometrial hyperplasia during oestrogen replacement therapy as well as contraception, Mirena® 20 micrograms/24 hours intrauterine delivery system can be used. See SPC for detail on length of treatment. Mirena is effective for 5 years in the indication of contraception, but 4 years for the progesterone component of HRT. For people who are perimenopausal, or menopausal, see the FSRH guidance above. Diagnosis of Menopause Diagnose the following without laboratory tests in otherwise healthy women or people assigned female at birth (AFAB), aged over 45 years with menopausal symptoms: -perimenopause based on vasomotor symptoms and irregular periods -menopause in women and people AFAB who have not had a period for at least 12 months and are not using hormonal contraception -menopause based on symptoms in women and people AFAB without a uterus. Follow NICE guidance for further information Consider using a FSH test to diagnose menopause only: -in women aged 40 to 45 years with menopausal symptoms, including a change in their menstrual cycle -in women aged under 40 years in whom menopause is suspected Information and advice -explain the stages of menopause -common symptoms and diagnosis -lifestyle changes and interventions that could help general health and wellbeing -benefits and risks of treatments -long-term health implications of menopause. Education Explain as well as changes in menstrual cycle, other symptoms may include: -vasomotor symptoms (for example, hot flushes and sweats) -musculoskeletal symptoms (for example, joint and muscle pain) -effects on mood (for example, low mood) -urogenital symptoms (for example, vaginal dryness) -sexual difficulties (for example, low sexual desire). Information on treatments -hormonal, for example hormone replacement therapy (HRT). See formulary and above for further information. -non-hormonal, for example clonidine -non-pharmaceutical, for example cognitive behavioural therapy (CBT). – Give information on contraception Diagnosing Premature Ovarian Insufficiency Diagnose premature ovarian insufficiency in women aged under 40 years based on: menopausal symptoms, including no or infrequent periods (taking into account whether the woman has a uterus) and elevated FSH levels on 2 blood samples taken 4–6 weeks apart. If there is doubt about the diagnosis of premature ovarian insufficiency, refer the woman to a specialist with expertise in menopause or reproductive medicine. Do not diagnose premature ovarian insufficiency on the basis of a single blood test. Do notroutinely use anti-Müllerian hormone testing to diagnose premature ovarian insufficiency. NICE Clinical Knowledge Summary: Diagnosis of premature ovarian insufficiency See below for information on managing premature ovarian insufficiency. Managing Premature Ovarian Insufficiency Offer sex steroid replacement with a choice of HRT or a combined hormonal contraceptive to women with premature ovarian insufficiency, unless contraindicated (for example, in women with hormone-sensitive cancer). Give women with premature ovarian insufficiency and contraindications to hormonal treatments advice, including on bone and cardiovascular health, and symptom management. Consider referring women with premature ovarian insufficiency to healthcare professionals who have the relevant experience to help them manage all aspects of physical and psychosocial health related to their condition. Explain to women with premature ovarian insufficiency: the importance of starting hormonal treatment either with HRT or a combined hormonal contraceptive and continuing treatment until at least the age of natural menopause (unless contraindicated) that the baseline population risk of diseases such as breast cancer and cardiovascular disease increases with age and is very low in women aged under 40 that HRT may have a beneficial effect on blood pressure when compared with a combined oral contraceptive that both HRT and combined oral contraceptives offer bone protection that HRT is not a contraceptive. NICE Clinical Knowledge Summary: Osteoporosis – Prevention of fragility fractures: Assessment Review patients with premature ovarian insufficiency for risk of osteoporosis and fragility fractures. Check FRAX score and consider bisphosphonate treatment if indicated. See formulary chapter 6.6. Ensure patients are taking vitamin D as self-care in line with [PH56] Vitamin D: supplement use in specific population groups and ensure adequate dietary intake of calcium. Gender Language and Definitions The Equality Act 2010 gives legal protection to people who reassign their gender away from the sex they were assigned at birth. Trans people legally are to be treated in accordance with their legal gender identity. The Equality Act describes gender reassignment as those who ‘propose to undergo, are undergoing or have undergone a process or a part of a process’to reassign their gender away from the sex assigned at birth. Not all Trans people will have surgery, not all Trans people will be under the care of the Gender identity clinic (GIC) and not all Trans people will be taking hormones to affirm their gender. The Gender Recognition Act 2005 provides Trans people with the option to apply for a Gender Recognition Certificate (GRC). This enables a person to legally change the sex detailed on their birth certificate and other legal documents. It is not a requirement to change the sex recorded on other records, such as NHS records, bank accounts, etc. Not all Trans people will choose to apply for a GRC. A GRC is not required in order to live and be afforded the protection under the Equality Act 2010, as detailed above. It is neither appropriate nor lawful to ask a person if they have a GRC. Gender Identity Research and Education Society has information on terminology which you may find useful and more in depth than this short section. For the purposes of the menopause page, we include women and people assigned female at birth (AFAB) who are perimenopausal or menopausal. For more resources, please see the Equality and Diversity page. HRT Options Chart HRT Formulary Options HRT Pre-payment Certificate (HRT PPC) As of 1st April 2023, a new NHS hormone replacement therapy prescription pre-payment certificate (HRT PPC) has been introduced. Before purchasing a HRT PPC, patients should check if they are entitled to free NHS prescriptions using the eligibility checker. The HRT PPC covers certain HRT medicines, regardless of why they are prescribed, for twice the single prescription charge and is valid for 12 months. It will save the patient money if they need to pay for 3 or more HRT prescription charges within 12 months. If patients get prescriptions for items other than HRT medicines, they may save more with a 3 or 12 month PPC that covers all NHS prescriptions. The HRT PPC does not cover all HRT medicines. A list of eligible HRT medicines can be found on the NHSBSA website. Due to the potential complexities which may arise when a patient only has a HRT PPC but is prescribed another medication, all prescribers are requested to ensure that HRT prescriptions have no other medication prescribed on the same form. For example, a patient has HRT and an urgent antibiotic – please ensure that two separate prescriptions are issued, a patient has HRT and a repeat statin – please ensure that two separate prescription forms are issued, etc. If prescriptions have mixed HRT and another item on the same form but the patient only has a HRT PPC then the pharmacy may refuse to dispense some or all of the items and refer patient back to the prescriber, which would obviously cause inconvenience and additional workload to all. N.B. Patients over 60 will be automatically exempt so can have multiple items on same prescription as HRT. For further information see the Department of Health and Social Care (DHSC) Guidance on the Hormone Replacement Therapy prescription prepayment certificate: Handling NHS HRT prescriptions. PSNC have produced some guidance and FAQs to support pharmacy teams in implementing the DHSC guidance. Osteoporosis Give advice on bone health and discuss these issues at review appointments. Explain that the baseline population risk of fragility fracture for women around menopausal age in the UK is low and varies from one woman to another. Explain to women that their risk of fragility fracture is decreased while taking HRT and that this benefit: •is maintained during treatment but decreases once treatment stops •may continue for longer in women who take HRT for longer. Review patients for risk of osteoporosis and fragility fractures in line with NICE Clinical Knowledge Summary: Osteoporosis – Prevention of fragility fractures: Assessment. Check FRAX score and consider bisphosphonate treatment if indicated. See formulary chapter 6.6. Ensure patients are taking vitamin D as self-care in line with [PH56] Vitamin D: supplement use in specific population groups and ensure adequate dietary intake of calcium. Psychological Symptoms and Mental Health in Menopause Consider HRT for low mood associated with menopause. Consider CBT for low mood or anxiety associated with menopause. There is no clear evidence for SSRIs or SNRIs. See our mental health prescribing page for further mental health resources. Starting and Stopping HRT Explain to women and people with a uterus that unscheduled vaginal bleeding is a common side effect of HRT within the first 3 months of treatment but should be reported at the 3-month review appointment, or promptly if it occurs after the first 3 months Offer women and people who are stopping HRT a choice of gradually reducing or immediately stopping treatment. Explain that: -gradually reducing HRT may limit recurrence of symptoms in the short term -gradually reducing or immediately stopping HRT makes no difference to their symptoms in the longer term. Summary of HRT Risks Summary of HRT risks and benefits during current use and current use plus post-treatment from age of menopause up to age 69 years, per 1000 women with 5 years or 10 years use of HRT Testosterone NICE indicates the use of testosterone supplementation for menopausal women with low sexual desire if HRT alone is not effective. Currently this is an off-licence indication. PAMM approved the use of testosterone within guidelines in October 2021, agreeing the Traffic Light Status as GREEN ‘on the advice of a GP with additional training in menopause and hormone replacement therapy or a suitable specialist’. For detailed information see British Menopause Society – Tools for Clinicians:Testosterone replacement in menopause BMS Statement on Testosterone - British Menopause Society July 2024 BMS Testosterone Patient Information Leaflet First line– Testogel sachets 2.5g sachets contain 40.5mg testosterone. The 5g Sachets containing 50mg testosterone are now no longer being made. Switch patients to the new 2.5g sachets containing 40.5mg and follow directions below – Testogel Sachets have changed. The new formulation will be 2.5g sachets containing 40.5mg Testosterone. These sachets should last 8 days and will need to be prescribed with new instructions: Apply 1/8th of a sachet daily. Apply a small pea sized amount once daily to lower abdomen, buttock or outer thigh. Rotate site of application daily. Use at the same time each day. One 2.5g sachet should last 8 days, seal with a clip between uses. Second line– Tostran pump (2% testosterone gel) – (60g metered dose pump) Apply one metered dose (10mg)three times a week or on alternate days to lower abdomen, buttock, or outer thigh. Rotate site of application daily. Use at the same time each day. 60g pump container delivers 120 metered doses, each cannister should last 240 days. Check stock availability before prescribing. Do not prescribe the pump version of Testogel or the Testavan Pump as they are unsuitable for HRT in menopause PRACTICAL TIPS ON PRESCRIBING AND USING TESTOSTERONE Testosterone gel should be applied to clean, dry skin of the lower abdomen, buttock, or outer thigh, rotating the site of application to avoid hair growth in one area. It should be allowed to dry before dressing. Skin contact with partners and children should be avoided until dry and hands should be washed immediately after application. The area should not be washed for 3 hours after application. The patient should be on transdermal oestrogen at a sufficient dose to relieve most symptoms before considering if testosterone supplementation is needed. Blood tests for testosterone, and sex hormone binding globulin (SHBG) levels are now suggested before starting testosterone. If testosterone levels are very low, the lab will not need to process SHBG. Blood tests are also needed 2 – 3 months after initiation or any dose change and then at least annually once stable. This is to make sure the levels are still within the female physiological range. It can take at least 3 months to notice any difference in symptoms. If no improvement after 6 months of use, then we would suggest stopping it. Common side-effects when starting testosterone are greasier hair and skin, spots, increased irritability, frontal hair thinning and weight gain. If these symptoms don’t settle the dose can be reduced before the 3-month review (but patients must never increase the dose without medical advice). Serious and possibly irreversible side-effects can develop if the dose is too high. These include male pattern hair loss, deepening of the voice, increased body and facial hair, and very rarely an enlarged clitoris. Please warn patients that the patient information leaflet only relates to male use and give them the patient information leaflet from either the British Menopause Society or Women’s Health Concern. Patient information With Testogel sachets it can take a while for the patient to work out how much to apply each day – suggest starting with a (small) pea-sized amount and then they can adjust up or down depending on how long the first tube/sachet lasts. The gel should be applied to clean, dry skin and allowed to dry before dressing. Skin contact with partners and children should be avoided until dry and hands should be washed immediately after application. The area should not be washed for 3 hours after application. Please warn the patients that the patient information leaflet only relates to male use and give them the BMS information leaflet as above. BLOOD TESTS Blood tests for testosterone and sex hormone binding globulin (SHBG) are now suggested before starting testosterone, if the testosterone level is low, the lab won't need to process the test for SHBG. If testosterone gel is normally applied in the morning, the dose that day should be omitted until after the blood test. Please make sure the patient has the blood results at her appointment if not available at the time of the referral. Calculating the Free Androgen Index is no longer considered helpful. Unscheduled and Vaginal Bleeding Problems The latest guidelines on the Management of Unscheduled Bleeding from the British Menopause Service provides a comprehensive guide to support your patients while they're taking HRT, there is also guidance on optimising progestogen dosing to protect the endometrium which has been reviewed at the Medicines Programme Board May 2024 and approved for use in Somerset. BMS GUIDELINE Management of unscheduled bleeding HRT JULY 2024 links to the BMS website to access the document. The BMS have produced a frequently asked questions document for the management of unscheduled bleeding December 2024. Content below is under review and will be updated in due course. June 2024 Explain to women and people with a uterus that unscheduled vaginal bleeding is a common side effect of HRT within the first 3 months of treatment but should be reported at the 3-month review appointment, or promptly if it occurs after the first 3 months (see recommendations on endometrial cancer in the NICE guideline on suspected cancer). Unscheduled bleeding after 6 months of starting HRT should be reviewed by a GP, with examination where suitable. Patients may be reluctant to stop HRT to see if bleeding settles due to the benefits of treatment for menopause symptoms which previously affected quality of life. Topical oestrogen may be useful in patients with vaginal atrophy, see formulary guidance below on vaginal atrophy. Vaginal bleeding problems– A detailed history: Unscheduled vaginal bleeding is a common adverse effect of HRT within the first 3 months of treatment. Monthly cyclical regimens should produce regular withdrawal bleeding towards the end of the progestogen phase. Continuous combined HRT commonly produces irregular breakthrough bleeding or spotting in the first 4–6 months of treatment. If bleeding persists beyond 6 months, becomes heavier, or occurs after a spell of amenorrhoea, endometrial pathology should be excluded. See the CKS topic on Gynaecological cancers – recognition and referral for more information. Unpredictable or unexpected bleeding may also be due to non-adherence with treatment, drug interactions, or a gastrointestinal disorder (which may affect drug absorption). If serious gynaecological pathology has been excluded, altering the progestogen part of the regimen may improve bleeding problems. Options include: Increasing the duration or dose of the progestogen, or changing the type of progestogen, for example switching to the levonorgestrel-releasing intrauterine system (LNG-IUS, Mirena®) combined with an oral or transdermal oestrogen preparation. Hormone replacement therapy (HRT) \| Prescribing information \| Menopause \| CKS \| NICE Lichen sclerosus is a chronic inflammatory skin condition which can affect any part of the skin, but it most often affects the genital skin (vulva) and the skin around the anus. It can start in childhood or adulthood, most commonly after menopause. For more information see the NHS Somerset Dermatology formulary page. For more information, see: Hormone replacement therapy (HRT) \| Prescribing information \| Menopause \| CKS \| NICE Recommendations \| Menopause: diagnosis and management \| Guidance \| NICE Gynaecological cancers – recognition and referral \| Health topics A to Z \| CKS \| NICE Urogenital Atrophy For detailed information on vulval and urinary symptoms that often develop with low estrogen levels, known as urogenital atrophy, or genitourinary syndrome of the menopause (GSM) please see the Local Estrogen - NHS Somerset ICB page. See the Antimicrobial - NHS Somerset ICB guidance for information on the management of recurrent UTI's. Urogenital Symptoms: •Painful intercourse •Vaginal bleeding – Must be investigated •Vulvo-vaginal Discomfort •Infection and discharge •Itch – the skin around the vulva is more sensitive and more likely to itch in some women. This produces a tendency to scratch which then makes the skin more likely to itch. An itch/scratch cycle follows which can be both difficult to break and quite distressing. •Urinary problems (frequency/urgency to pass urine) Urinary symptoms that may occur include one or more of the following: – Passing water too often (frequency) Not being able to hold on (urgency) Pain when passing urine (dysuria) See formulary chapter 7.4.2 for more information on drugs for urinary frequency. INVESTIGATE unexplained bleeding People suffering from vaginal dryness can use genital moisturisers and lubricants, these can be used alone or in addition to local vaginal estrogen. Genital moisturisers and lubricants are suitable for self-care. People should be made aware that barrier methods for contraception may be damaged by local estrogen preparations. Vasomotor symptoms Offer HRT for vasomotor symptoms after discussing with the patient the short-term (up to 5 years) and longer-term benefits and risks. Offer a choice of preparations: oestrogen and progestogen to women with a uterus oestrogen alone to women without a uterus. There is some evidence that isoflavones or black cohosh may relieve vasomotor symptoms (suitable for self-care). NOTE: Health food supplements may vary, their safety isn’t regulated as medicines are and interactions may be possible. Healthcare professionals are reminded to be vigilant for suspected adverse reactions associated with the use of herbal and homeopathic medicines and interactions with other medicines and report suspicions to the MHRA’s Yellow Card scheme. Do not routinely offer selective serotonin reuptake inhibitors (SSRIs), serotonin and norepinephrine reuptake inhibitors (SNRIs) or clonidine as first-line treatment for vasomotor symptoms alone. Clonidine is licensed for and may be considered for hot flushes in people where HRT is not suitable or compatible. Consideration should be taken for side effects and review efficacy after initiation. Working with Menopause Menopause and work: why it’s so important CIPD provides printable resources to help break the stigma around the menopause in workplaces: The menopause at work: printable resources NHS Employers: Menopause and the workplace Back to top Manage Cookie Consent To provide the best experiences, we use technologies like cookies to store and/or access device information. Consenting to these technologies will allow us to process data such as browsing behaviour or unique IDs on this site. Not consenting or withdrawing consent, may adversely affect certain features and functions. Functional Always active The technical storage or access is strictly necessary for the legitimate purpose of enabling the use of a specific service explicitly requested by the subscriber or user, or for the sole purpose of carrying out the transmission of a communication over an electronic communications network. 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4687	https://en.wikipedia.org/wiki/Coenzyme_A	Jump to content Search Contents (Top) 1 Discovery of structure 2 Biosynthesis 2.1 Commercial production 3 Function 3.1 Fatty acid synthesis 3.2 Energy production 3.3 Regulation 3.4 Antioxidant function and regulation 4 Use in biological research 5 Non-exhaustive list of coenzyme A-activated acyl groups 6 References 7 Bibliography Coenzyme A العربية تۆرکجه Български Bosanski Català Čeština Deutsch Eesti Español Euskara فارسی Français Galego 한국어 Hrvatski Bahasa Indonesia Italiano עברית Lietuvių Magyar Македонски Bahasa Melayu 日本語 Occitan Polski Português Romnă Русский Simple English Slovenčina Српски / srpski Srpskohrvatski / српскохрватски Suomi Svenska Türkçe Українська Tiếng Việt 中文 Edit links Article Talk Read Edit View history Tools Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikidata item Appearance From Wikipedia, the free encyclopedia Coenzyme, notable for its synthesis and oxidation role Coenzyme A \| \| \| \| \| \| \| Names \| \| Systematic IUPAC name [(2R,3S,4R,5R)-5-(6-Amino-9H-purin-9-yl)-4-hydroxy-3-(phosphonooxy)tetrahydro-2-furanyl]methyl (3R)-3-hydroxy-2,2-dimethyl-4-oxo-4-({3-oxo-3-[(2-sulfanylethyl)amino]propyl}amino)butyl dihydrogen diphosphate \| \| Identifiers \| \| CAS Number \| 85-61-0 (free acid)Y 55672-92-9 (sodium salt hydrate)N 18439-24-2 (lithium salt)N \| \| 3D model (JSmol) \| Interactive image \| \| ChEBI \| CHEBI:15346N \| \| ChEMBL \| ChEMBL1213327N \| \| ChemSpider \| 6557Y \| \| DrugBank \| DB01992Y \| \| ECHA InfoCard \| 100.001.472 \| \| KEGG \| C00010Y \| \| MeSH \| Coenzyme+A \| \| PubChem CID \| 6816 \| \| UNII \| SAA04E81UXY \| \| InChI InChI=1S/C21H36N7O16P3S/c1-21(2,16(31)19(32)24-4-3-12(29)23-5-6-48)8-41-47(38,39)44-46(36,37)40-7-11-15(43-45(33,34)35)14(30)20(42-11)28-10-27-13-17(22)25-9-26-18(13)28/h9-11,14-16,20,30-31,48H,3-8H2,1-2H3,(H,23,29)(H,24,32)(H,36,37)(H,38,39)(H2,22,25,26)(H2,33,34,35)/t11-,14-,15-,16?,20-/m1/s1Y Key: RGJOEKWQDUBAIZ-DRCCLKDXSA-NY InChI=1/C21H36N7O16P3S/c1-21(2,16(31)19(32)24-4-3-12(29)23-5-6-48)8-41-47(38,39)44-46(36,37)40-7-11-15(43-45(33,34)35)14(30)20(42-11)28-10-27-13-17(22)25-9-26-18(13)28/h9-11,14-16,20,30-31,48H,3-8H2,1-2H3,(H,23,29)(H,24,32)(H,36,37)(H,38,39)(H2,22,25,26)(H2,33,34,35)/t11-,14-,15-,16?,20-/m1/s1 Key: RGJOEKWQDUBAIZ-DRCCLKDXBU \| \| SMILES O=C(NCCS)CCNC(=O)C(O)C(C)(C)COP(=O)(O)OP(=O)(O)OC[C@H]3OC@@HC@H[C@@H]3OP(=O)(O)O \| \| Properties \| \| Chemical formula \| C21H36N7O16P3S \| \| Molar mass \| 767.535 \| \| UV-vis (λmax) \| 259.5 nm \| \| Absorbance \| ε259 = 16.8 mM−1 cm−1 \| \| Except where otherwise noted, data are given for materials in their standard state (at 25 °C [77 °F], 100 kPa). N verify (what is YN ?) Infobox references \| Chemical compound Coenzyme A (CoA, SHCoA, CoASH) is a coenzyme, notable for its role in the synthesis and oxidation of fatty acids, and the oxidation of pyruvate in the citric acid cycle. All genomes sequenced to date encode enzymes that use coenzyme A as a substrate, and around 4% of cellular enzymes use it (or a thioester) as a substrate. In humans, CoA biosynthesis requires cysteine, pantothenate (vitamin B5), and adenosine triphosphate (ATP). In its acetyl form, coenzyme A is a highly versatile molecule, serving metabolic functions in both the anabolic and catabolic pathways. Acetyl-CoA is utilised in the post-translational regulation and allosteric regulation of pyruvate dehydrogenase and carboxylase to maintain and support the partition of pyruvate synthesis and degradation. Discovery of structure [edit] Coenzyme A was identified by Fritz Lipmann in 1946, who also later gave it its name. Its structure was determined during the early 1950s at the Lister Institute, London, together by Lipmann and other workers at Harvard Medical School and Massachusetts General Hospital. Lipmann initially intended to study acetyl transfer in animals, and from these experiments he noticed a unique factor that was not present in enzyme extracts but was evident in all organs of the animals. He was able to isolate and purify the factor from pig liver and discovered that its function was related to a coenzyme that was active in choline acetylation. Work with Beverly Guirard, Nathan Kaplan, and others determined that pantothenic acid was a central component of coenzyme A. The coenzyme was named coenzyme A to stand for "activation of acetate". In 1953, Fritz Lipmann won the Nobel Prize in Physiology or Medicine "for his discovery of co-enzyme A and its importance for intermediary metabolism". Biosynthesis [edit] Coenzyme A is naturally synthesized from pantothenate (vitamin B5), which is found in food such as meat, vegetables, cereal grains, legumes, eggs, and milk. In humans and most living organisms, pantothenate is an essential vitamin that has a variety of functions. In some plants and bacteria, including Escherichia coli, pantothenate can be synthesised de novo and is therefore not considered essential. These bacteria synthesize pantothenate from the amino acid aspartate and a metabolite in valine biosynthesis. In all living organisms, coenzyme A is synthesized in a five-step process that requires four molecules of ATP, pantothenate and cysteine (see figure): Pantothenate (vitamin B5) is phosphorylated to 4′-phosphopantothenate by the enzyme pantothenate kinase (PanK; CoaA; CoaX). This is the committed step in CoA biosynthesis and requires ATP. A cysteine is added to 4′-phosphopantothenate by the enzyme phosphopantothenoylcysteine synthetase (PPCS; CoaB) to form 4'-phospho-N-pantothenoylcysteine (PPC). This step is coupled with ATP hydrolysis. PPC is decarboxylated to 4′-phosphopantetheine by phosphopantothenoylcysteine decarboxylase (PPC-DC; CoaC) 4′-phosphopantetheine is adenylated (or more properly, AMPylated) to form dephospho-CoA by the enzyme phosphopantetheine adenylyl transferase (COASY; PPAT; CoaD) Finally, dephospho-CoA is phosphorylated to coenzyme A by the enzyme dephosphocoenzyme A kinase (COASY, DPCK; CoaE). This final step requires ATP. Enzyme nomenclature abbreviations in parentheses represent mammalian, other eukaryotic, and prokaryotic enzymes respectively. In mammals steps 4 and 5 are catalyzed by a bifunctional enzyme called COASY. This pathway is regulated by product inhibition. CoA is a competitive inhibitor for Pantothenate Kinase, which normally binds ATP. Coenzyme A, three ADP, one monophosphate, and one diphosphate are harvested from biosynthesis. Coenzyme A can be synthesized through alternate routes when intracellular coenzyme A level are reduced and the de novo pathway is impaired. In these pathways, coenzyme A needs to be provided from an external source, such as food, in order to produce 4′-phosphopantetheine. Ectonucleotide pyrophosphates (ENPP) degrade coenzyme A to 4′-phosphopantetheine, a stable molecule in organisms. Acyl carrier proteins (ACP) (such as ACP synthase and ACP degradation) are also used to produce 4′-phosphopantetheine. This pathway allows for 4′-phosphopantetheine to be replenished in the cell and allows for the conversion to coenzyme A through enzymes, PPAT and PPCK. A 2024 article detailed a plausible chemical synthesis mechanism for the pantetheine component (the main functional part) of coenzyme A in a primordial prebiotic world. Commercial production [edit] Coenzyme A is produced commercially via extraction from yeast, however this is an inefficient process (yields approximately 25 mg/kg) resulting in an expensive product. Various ways of producing CoA synthetically, or semi-synthetically have been investigated, although none are currently operating at an industrial scale. Function [edit] Fatty acid synthesis [edit] Since coenzyme A is, in chemical terms, a thiol, it can react with carboxylic acids to form thioesters, thus functioning as an acyl group carrier. It assists in transferring fatty acids from the cytoplasm to mitochondria. A molecule of coenzyme A carrying an acyl group is also referred to as acyl-CoA. When it is not attached to an acyl group, it is usually referred to as 'CoASH' or 'HSCoA'. This process facilitates the production of fatty acids in cells, which are essential in cell membrane structure. Coenzyme A is also the source of the phosphopantetheine group that is added as a prosthetic group to proteins such as acyl carrier protein and formyltetrahydrofolate dehydrogenase. Energy production [edit] Coenzyme A is one of five crucial coenzymes that are necessary in the reaction mechanism of the citric acid cycle. Its acetyl-coenzyme A form is the primary input in the citric acid cycle and is obtained from glycolysis, amino acid metabolism, and fatty acid beta oxidation. This process is the body's primary catabolic pathway and is essential in breaking down the building blocks of the cell such as carbohydrates, amino acids, and lipids. Regulation [edit] When there is excess glucose, coenzyme A is used in the cytosol for synthesis of fatty acids. This process is implemented by regulation of acetyl-CoA carboxylase, which catalyzes the committed step in fatty acid synthesis. Insulin stimulates acetyl-CoA carboxylase, while epinephrine and glucagon inhibit its activity. During cell starvation, coenzyme A is synthesized and transports fatty acids in the cytosol to the mitochondria. Here, acetyl-CoA is generated for oxidation and energy production. In the citric acid cycle, coenzyme A works as an allosteric regulator in the stimulation of the enzyme pyruvate dehydrogenase. Antioxidant function and regulation [edit] Discovery of the novel antioxidant function of coenzyme A highlights its protective role during cellular stress. Mammalian and bacterial cells subjected to oxidative and metabolic stress show significant increase in the covalent modification of protein cysteine residues by coenzyme A. This reversible modification is termed protein CoAlation (Protein-S-SCoA), which plays a similar role to protein S-glutathionylation by preventing the irreversible oxidation of the thiol group of cysteine residues. Using anti-coenzyme A antibody and liquid chromatography tandem mass spectrometry (LC-MS/MS) methodologies, more than 2,000 CoAlated proteins were identified from stressed mammalian and bacterial cells. The majority of these proteins are involved in cellular metabolism and stress response. Different research studies have focused on deciphering the coenzyme A-mediated regulation of proteins. Upon protein CoAlation, inhibition of the catalytic activity of different proteins (e.g., metastasis suppressor NME1, peroxiredoxin 5, GAPDH, among others) is reported. To restore the protein's activity, antioxidant enzymes that reduce the disulfide bond between coenzyme A and the protein cysteine residue play an important role. This process is termed protein deCoAlation. Thioredoxin A and Thioredoxin-like protein (YtpP), two bacterial proteins, are shown to deCoAlate proteins. Use in biological research [edit] Coenzyme A is available from various chemical suppliers as the free acid and lithium or sodium salts. The free acid of coenzyme A is detectably unstable, with around 5% degradation observed after 6 months when stored at −20 °C, and near complete degradation after 1 month at 37 °C. The lithium and sodium salts of CoA are more stable, with negligible degradation noted over several months at various temperatures. Aqueous solutions of coenzyme A are unstable above pH 8, with 31% of activity lost after 24 hours at 25 °C and pH 8. CoA stock solutions are relatively stable when frozen at pH 2–6. The major route of CoA activity loss is likely the air oxidation of CoA to CoA disulfides. CoA mixed disulfides, such as CoA-S–S-glutathione, are commonly noted contaminants in commercial preparations of CoA. Free CoA can be regenerated from CoA disulfide and mixed CoA disulfides with reducing agents such as dithiothreitol or 2-mercaptoethanol. Non-exhaustive list of coenzyme A-activated acyl groups [edit] See also: Category:Thioesters of coenzyme A Acetyl-CoA fatty acyl-CoA (activated form of all fatty acids; only the CoA esters are substrates for important reactions such as mono-, di-, and triacylglycerol synthesis, carnitine palmitoyl transferase, and cholesterol esterification) Propionyl-CoA Butyryl-CoA Myristoyl-CoA Crotonyl-CoA Acetoacetyl-CoA Coumaroyl-CoA (used in flavonoid and stilbenoid biosynthesis) Benzoyl-CoA Phenylacetyl-CoA Acyl derived from dicarboxylic acids Malonyl-CoA (important in chain elongation in fatty acid biosynthesis and polyketide biosynthesis) Succinyl-CoA (used in heme biosynthesis) Hydroxymethylglutaryl-CoA (used in isoprenoid biosynthesis) Pimelyl-CoA (used in biotin biosynthesis) References [edit] ^ a b c d Dawson RM, Elliott DC, Elliott WH, Jones KM (2002). Data for Biochemical Research (3rd ed.). Clarendon Press. pp. 118–119. 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Oriental Yeast Co., LTD. ^ "Datasheet for lithium salt coenzyme A" (PDF). Oriental Yeast Co., LTD. Bibliography [edit] Nelson DL, Cox MM (2005). Lehninger: Principles of Biochemistry (4th ed.). New York: W .H. Freeman. ISBN 978-0-7167-4339-2. Wikimedia Commons has media related to Coenzyme A. \| v t e Enzyme cofactors \| \| Active forms \| \| \| \| --- \| \| vitamins \| TPP / ThDP (B1) FMN, FAD (B2) NAD+, NADH, NADP+, NADPH (B3) Coenzyme A (B5) PLP / P5P (B6) Biotin (B7) THFA / H4FA, DHFA / H2FA, MTHF (B9) AdoCbl, MeCbl (B12) Ascorbic acid (C) Phylloquinone (K1), Menaquinone (K2) Coenzyme F420 \| \| non-vitamins \| ATP CTP SAMe PAPS GSH Coenzyme B Cofactor F430 Coenzyme M Coenzyme Q Heme / Haem (A, B, C, O) Lipoic Acid Methanofuran Molybdopterin Mycofactocin PQQ THB / BH4 THMPT / H4MPT \| \| metal ions \| Ca2+ Cu2+ Fe2+, Fe3+ Mg2+ Mn2+ Mo Ni2+ Zn2+ \| \| \| Base forms \| vitamins: see vitamins \| Retrieved from " Categories: Coenzymes Metabolism Thiols Hidden categories: Articles with short description Short description is different from Wikidata Chemical articles with multiple compound IDs Multiple chemicals in an infobox that need indexing Chemical articles with multiple CAS registry numbers Articles with changed CASNo identifier Articles with changed EBI identifier ECHA InfoCard ID from Wikidata Articles containing unverified chemical infoboxes Chembox image size set Commons category link is on Wikidata Coenzyme A Add topic
4688	https://www.math-only-math.com/multiplication-table-of-23.html	Multiplication Table of 23 \| Read and Write the Table of 23 \| 23 Times Table Math Only Math Learn math step-by-step. Learning Apps Math Software Subscribe to our ▶️YouTube channel🔴 for the latest videos, updates, and tips. Multiplication Table of 23 Repeated addition by 23’s means the multiplication table of 23. (i) When 2 rows of twenty-three chairs each. By repeated addition we can show 23 + 23 = 46 Then, twenty-three 2 times or 2 twenty-three’s 2 × 23 = 46 Therefore, there are 46 chairs. (ii) When 7 groups having 23 fishes each. By repeated addition we can show 23 + 23 + 23 + 23 + 23 + 23 + 23 = 161 Then, twenty-three 7 times or 7 twenty-three’s 7 × 23 = 161 Therefore, there are 161 fishes. We will learn how to use the number line for counting the multiplication table of 23. Learning Apps Counting Books 10 Save (i) Start at 0. Hop 23, five times. Stop at 115. 5 twenty-three’s are 115 5 × 23 = 115 (ii) Start at 0. Hop 23, eight times. Stop at ____. Thus, it will be 184 8 twenty-three’s are 184 8 × 23 = 184 (iii) Start at 0. Hop 23, eleven times. Stop at ____. Thus, it will be 253 11 twenty-three’s are 253 11 × 23 = 253 How to read and write the table of 23? 10 Save The above chart will help us to read and write the 23 times table. Read 1 twenty-three is 23 2 twenty- three’s are 46 3 twenty- three’s are 69 4 twenty- three’s are 92 5 twenty- three’s are 115 6 twenty- three’s are 138 7 twenty- three’s are 161 8 twenty- three’s are 184 9 twenty- three’s are 207 10 twenty- three’s are 230 11 twenty- three’s are 253 12 twenty- three’s are 276Write 1 × 23 = 23 2 × 23 = 46 3 × 23 = 69 4 × 23 = 92 5 × 23 = 115 6 × 23 = 138 7 × 23 = 161 8 × 23 = 184 9 × 23 = 207 10 × 23 = 230 11 × 23 = 253 12 × 23 = 276 Now we will learn how to do forward counting and backward counting by 23’s. Learning Apps Forward counting by 23’s: 0, 23, 46, 69, 92, 115, 138, 161, 184, 207, 230, 253, 276, 299, 322, 245, 368, 391, 414, 437, 460, 483, 506, 529, 552, 575, …… Backward counting by 23’s: ……, 575, 552, 529, 506, 483, 460, 437, 414, 391, 368, 245, 322, 299, 276, 253, 230, 207, 184, 161, 138, 115, 92, 69, 46, 23, 0 Multiplication Table of 0 Multiplication Table of 2 Multiplication Table of 4 Multiplication Table of 6 Multiplication Table of 8 Multiplication Table of 10 Multiplication Table of 12 Multiplication Table of 14 Multiplication Table of 16 Multiplication Table of 18 Multiplication Table of 20 Multiplication Table of 22 Multiplication Table of 24Multiplication Table of 1 Multiplication Table of 3 Multiplication Table of 5 Multiplication Table of 7 Multiplication Table of 9 Multiplication Table of 11 Multiplication Table of 13 Multiplication Table of 15 Multiplication Table of 17 Multiplication Table of 19 Multiplication Table of 21 Multiplication Table of 23 Multiplication Table of 25 Multiplication Table From Multiplication Table of 23 to HOME PAGE Didn't find what you were looking for? Or want to know more information aboutMath Only Math. Use this Google Search to find what you need. New! Comments Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question. Share this page: What’s this? FacebookXPinterestWhatsAppReddit Share FacebookXPinterestWhatsAppReddit Home Math Blog Preschool Activities Kindergarten Math 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math 9th Grade Math 10th Grade Math 11 & 12 Grade Math Algebra 1 Concepts of Sets Matrix Probability Statistics Logarithms Boolean Algebra Math Coloring Pages Multiplication Table Cool Maths Games Math Flash Cards Online Math Quiz Math Puzzles Binary System Math Dictionary Conversion Chart Homework Sheets Math Problem Ans Free Math Answers Printable Math Sheet Funny Math Answers Employment Test Math Patterns Link Partners About Us Contact Us Site Map Privacy Policy [?]Subscribe To This Site E-mail Address First Name Then Don't worry — your e-mail address is totally secure. I promise to use it only to send you Math Only Math. © and ™ math-only-math.com. All Rights Reserved. 2010 - 2025. This site uses cookies, some of which are required for its operation. Privacy policy. Agree and Continue
4689	https://biblehub.com/topical/q/quarrelsome.htm	\| \| \| \| \| \| \| --- \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- \| \| \| \| \| --- \| \| Bible > Topical > Quarrelsome \| \| \| \| \| \| \| \| ◄ Quarrelsome ► \| \| \| \| Jump to: Webster's • Concordance • Thesaurus • Greek • Library • Subtopics • Terms Topical Encyclopedia The term "quarrelsome" refers to a disposition inclined toward argument, strife, or contention. In the Bible, being quarrelsome is often portrayed negatively, as it disrupts peace and harmony within communities and relationships. The Scriptures provide guidance on avoiding quarrelsome behavior and emphasize the virtues of peace, patience, and understanding. Old Testament References The Old Testament frequently addresses the issue of quarrelsomeness, particularly in the wisdom literature. Proverbs, a book rich in practical advice, warns against the dangers of a contentious spirit. Proverbs 17:14 states, "Starting a quarrel is like breaching a dam; so drop the matter before a dispute breaks out." This proverb highlights the potential for small disagreements to escalate into larger conflicts, urging individuals to seek resolution before tensions rise. Proverbs 20:3 further advises, "It is honorable for a man to resolve a dispute, but any fool will quarrel." Here, the text contrasts the honor found in peacemaking with the folly of engaging in unnecessary disputes. The wisdom literature consistently encourages a demeanor that seeks peace and understanding over conflict. New Testament References In the New Testament, the teachings of Jesus and the apostles continue to discourage quarrelsome behavior. The Apostle Paul, in his letters, often addresses the need for unity and peace within the Christian community. In 2 Timothy 2:24 , Paul instructs, "And a servant of the Lord must not be quarrelsome, but must be kind to everyone, able to teach, and patient." This passage underscores the importance of a gentle and patient spirit, particularly for those in positions of leadership and teaching. James 4:1-2 provides insight into the root causes of quarrels: "What causes conflicts and quarrels among you? Don’t they come from the passions at war within you? You crave what you do not have; you kill and covet, but are unable to obtain it. You quarrel and fight." James identifies internal desires and covetousness as sources of conflict, urging believers to examine their hearts and seek God's wisdom. Character Studies Several biblical characters exemplify the consequences of a quarrelsome nature. In the Old Testament, the account of Korah's rebellion (Numbers 16) illustrates how contentiousness can lead to division and divine judgment. Korah and his followers challenged Moses' leadership, resulting in severe consequences for their insubordination. Conversely, Abigail, the wife of Nabal, is portrayed as a peacemaker in 1 Samuel 25. When her husband’s quarrelsome behavior nearly provokes David to violence, Abigail intervenes with wisdom and humility, averting disaster. Her actions demonstrate the power of a gentle and conciliatory approach in resolving potential conflicts. Practical Application The Bible encourages believers to cultivate a spirit of peace and avoid quarrelsome behavior. Ephesians 4:2-3 advises, "Be completely humble and gentle; be patient, bearing with one another in love. Make every effort to keep the unity of the Spirit through the bond of peace." This passage calls Christians to embody humility, patience, and love, fostering unity within the body of Christ. In personal relationships, believers are urged to practice self-control and seek reconciliation. Proverbs 15:1 offers practical wisdom: "A gentle answer turns away wrath, but a harsh word stirs up anger." By responding with gentleness and understanding, individuals can de-escalate potential conflicts and promote harmony. Overall, the biblical perspective on quarrelsomeness emphasizes the value of peace, the importance of self-examination, and the pursuit of reconciliation in all relationships. Webster's Revised Unabridged Dictionary (a.) Apt to argue; given to contention; easily irritated or provoked to contest; irascible; choleric. Greek 3164. machomai -- to fight ... Word Origin a prim. verb Definition to fight NASB Word Usage argue (1), fight (1), fighting together (1), quarrelsome (1). fight, strive. ... //strongsnumbers.com/greek2/3164.htm - 6k 269. amachos -- abstaining from fighting ... from fighting. Part of Speech: Adjective Transliteration: amachos Phonetic Spelling: (am'-akh-os) Short Definition: not quarrelsome, peaceable Definition ... //strongsnumbers.com/greek2/269.htm - 6k 3943. paroinos -- given to wine, drunken ... Adjective Transliteration: paroinos Phonetic Spelling: (par'-oy-nos) Short Definition: given to wine, drunken Definition: given to wine, drunken, quarrelsome. ... //strongsnumbers.com/greek2/3943.htm - 6k 4131. plektes -- a striker ... bully, striker. From plesso; a smiter, ie Pugnacious (quarrelsome) -- striker. see GREEK plesso. (plekten) -- 2 Occurrences. 4130, 4131. plektes. 4132 . ... //strongsnumbers.com/greek2/4131.htm - 6k Library Psalm 120. Complaint of Quarrelsome Neighbours; Or, a Devout Wish ... ... THE Psalms of David, In Metre. Psalm 120. Complaint of quarrelsome neighbours; or, A devout wish for peace. 1 Thou God of love, thou ... /.../watts/the psalms of david/psalm 120 complaint of quarrelsome.htm Psalm 120 Complaint of Quarrelsome Neighbors; Or, a Devout Wish ... ... THE PSALMS OF DAVID PSALM 120 Complaint of quarrelsome neighbors; or, A devout wish for peace. ... Complaint of quarrelsome neighbors; or, A devout wish for peace. ... /.../the psalms and hymns of isaac watts/psalm 120 complaint of quarrelsome.htm Whether Strife is Always a Sin? ... x) that the word "rixosus [quarrelsome] is derived from the snarling [rictu] of a dog, because the quarrelsome man is ever ready to contradict; he delights in ... /...//christianbookshelf.org/aquinas/summa theologica/whether strife is always a.htm Whether Quarreling is a More Grievous Sin than Flattery? ... 2: Further, there appears to be a certain amount of deceit in flattery, since the flatterer says one thing, and thinks another: whereas the quarrelsome man is ... /.../aquinas/summa theologica/whether quarreling is a more.htm Being Generous and Loving ... It is an honor for a man to avoid strife; Only a fool is quarrelsome. Charcoal for embers, and wood for fire, And a quarrelsome man to kindle strife! ... /.../sherman/the childrens bible/being generous and loving.htm One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Concordance Quarrelsome (9 Occurrences) Romans 1:29 Their hearts overflowed with all sorts of dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY) 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (WEB WEY RSV NIV) 2 Timothy 2:24 and a bondservant of the Lord must not quarrel, but must be inoffensive towards all men, a skilful teacher, and patient under wrongs. (See NAS RSV) Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (YLT) Proverbs 19:13 A foolish son is the destruction of his father; and the bitter arguments of a wife are like drops of rain falling without end. (See NIV) Proverbs 21:9 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 21:19 It is better to be living in a waste land, than with a bitter-tongued and angry woman. (See NIV) Proverbs 25:24 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) \| Quarrels Quarrelsome Man Top of Page Top of Page \| \| \| \| \| \| \| \| --- \| \| Bible > Topical > Quarrelsome \| \| \| \| \| \| \| \| \| \| \| \| ◄ Quarrelsome ► \| \| \| \| Jump to: Webster's • Concordance • Thesaurus • Greek • Library • Subtopics • Terms Topical Encyclopedia The term "quarrelsome" refers to a disposition inclined toward argument, strife, or contention. In the Bible, being quarrelsome is often portrayed negatively, as it disrupts peace and harmony within communities and relationships. The Scriptures provide guidance on avoiding quarrelsome behavior and emphasize the virtues of peace, patience, and understanding. Old Testament References The Old Testament frequently addresses the issue of quarrelsomeness, particularly in the wisdom literature. Proverbs, a book rich in practical advice, warns against the dangers of a contentious spirit. Proverbs 17:14 states, "Starting a quarrel is like breaching a dam; so drop the matter before a dispute breaks out." This proverb highlights the potential for small disagreements to escalate into larger conflicts, urging individuals to seek resolution before tensions rise. Proverbs 20:3 further advises, "It is honorable for a man to resolve a dispute, but any fool will quarrel." Here, the text contrasts the honor found in peacemaking with the folly of engaging in unnecessary disputes. The wisdom literature consistently encourages a demeanor that seeks peace and understanding over conflict. New Testament References In the New Testament, the teachings of Jesus and the apostles continue to discourage quarrelsome behavior. The Apostle Paul, in his letters, often addresses the need for unity and peace within the Christian community. In 2 Timothy 2:24 , Paul instructs, "And a servant of the Lord must not be quarrelsome, but must be kind to everyone, able to teach, and patient." This passage underscores the importance of a gentle and patient spirit, particularly for those in positions of leadership and teaching. James 4:1-2 provides insight into the root causes of quarrels: "What causes conflicts and quarrels among you? Don’t they come from the passions at war within you? You crave what you do not have; you kill and covet, but are unable to obtain it. You quarrel and fight." James identifies internal desires and covetousness as sources of conflict, urging believers to examine their hearts and seek God's wisdom. Character Studies Several biblical characters exemplify the consequences of a quarrelsome nature. In the Old Testament, the account of Korah's rebellion (Numbers 16) illustrates how contentiousness can lead to division and divine judgment. Korah and his followers challenged Moses' leadership, resulting in severe consequences for their insubordination. Conversely, Abigail, the wife of Nabal, is portrayed as a peacemaker in 1 Samuel 25. When her husband’s quarrelsome behavior nearly provokes David to violence, Abigail intervenes with wisdom and humility, averting disaster. Her actions demonstrate the power of a gentle and conciliatory approach in resolving potential conflicts. Practical Application The Bible encourages believers to cultivate a spirit of peace and avoid quarrelsome behavior. Ephesians 4:2-3 advises, "Be completely humble and gentle; be patient, bearing with one another in love. Make every effort to keep the unity of the Spirit through the bond of peace." This passage calls Christians to embody humility, patience, and love, fostering unity within the body of Christ. In personal relationships, believers are urged to practice self-control and seek reconciliation. Proverbs 15:1 offers practical wisdom: "A gentle answer turns away wrath, but a harsh word stirs up anger." By responding with gentleness and understanding, individuals can de-escalate potential conflicts and promote harmony. Overall, the biblical perspective on quarrelsomeness emphasizes the value of peace, the importance of self-examination, and the pursuit of reconciliation in all relationships. Webster's Revised Unabridged Dictionary (a.) Apt to argue; given to contention; easily irritated or provoked to contest; irascible; choleric. Greek 3164. machomai -- to fight ... Word Origin a prim. verb Definition to fight NASB Word Usage argue (1), fight (1), fighting together (1), quarrelsome (1). fight, strive. ... //strongsnumbers.com/greek2/3164.htm - 6k 269. amachos -- abstaining from fighting ... from fighting. Part of Speech: Adjective Transliteration: amachos Phonetic Spelling: (am'-akh-os) Short Definition: not quarrelsome, peaceable Definition ... //strongsnumbers.com/greek2/269.htm - 6k 3943. paroinos -- given to wine, drunken ... Adjective Transliteration: paroinos Phonetic Spelling: (par'-oy-nos) Short Definition: given to wine, drunken Definition: given to wine, drunken, quarrelsome. ... //strongsnumbers.com/greek2/3943.htm - 6k 4131. plektes -- a striker ... bully, striker. From plesso; a smiter, ie Pugnacious (quarrelsome) -- striker. see GREEK plesso. (plekten) -- 2 Occurrences. 4130, 4131. plektes. 4132 . ... //strongsnumbers.com/greek2/4131.htm - 6k Library Psalm 120. Complaint of Quarrelsome Neighbours; Or, a Devout Wish ... ... THE Psalms of David, In Metre. Psalm 120. Complaint of quarrelsome neighbours; or, A devout wish for peace. 1 Thou God of love, thou ... /.../watts/the psalms of david/psalm 120 complaint of quarrelsome.htm Psalm 120 Complaint of Quarrelsome Neighbors; Or, a Devout Wish ... ... THE PSALMS OF DAVID PSALM 120 Complaint of quarrelsome neighbors; or, A devout wish for peace. ... Complaint of quarrelsome neighbors; or, A devout wish for peace. ... /.../the psalms and hymns of isaac watts/psalm 120 complaint of quarrelsome.htm Whether Strife is Always a Sin? ... x) that the word "rixosus [quarrelsome] is derived from the snarling [rictu] of a dog, because the quarrelsome man is ever ready to contradict; he delights in ... /...//christianbookshelf.org/aquinas/summa theologica/whether strife is always a.htm Whether Quarreling is a More Grievous Sin than Flattery? ... 2: Further, there appears to be a certain amount of deceit in flattery, since the flatterer says one thing, and thinks another: whereas the quarrelsome man is ... /.../aquinas/summa theologica/whether quarreling is a more.htm Being Generous and Loving ... It is an honor for a man to avoid strife; Only a fool is quarrelsome. Charcoal for embers, and wood for fire, And a quarrelsome man to kindle strife! ... /.../sherman/the childrens bible/being generous and loving.htm One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Concordance Quarrelsome (9 Occurrences) Romans 1:29 Their hearts overflowed with all sorts of dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY) 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (WEB WEY RSV NIV) 2 Timothy 2:24 and a bondservant of the Lord must not quarrel, but must be inoffensive towards all men, a skilful teacher, and patient under wrongs. (See NAS RSV) Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (YLT) Proverbs 19:13 A foolish son is the destruction of his father; and the bitter arguments of a wife are like drops of rain falling without end. (See NIV) Proverbs 21:9 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 21:19 It is better to be living in a waste land, than with a bitter-tongued and angry woman. (See NIV) Proverbs 25:24 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) \| 269. amachos -- abstaining from fighting ... from fighting. Part of Speech: Adjective Transliteration: amachos Phonetic Spelling: (am'-akh-os) Short Definition: not quarrelsome, peaceable Definition ... //strongsnumbers.com/greek2/269.htm - 6k 3943. paroinos -- given to wine, drunken ... Adjective Transliteration: paroinos Phonetic Spelling: (par'-oy-nos) Short Definition: given to wine, drunken Definition: given to wine, drunken, quarrelsome. ... //strongsnumbers.com/greek2/3943.htm - 6k 4131. plektes -- a striker ... bully, striker. From plesso; a smiter, ie Pugnacious (quarrelsome) -- striker. see GREEK plesso. (plekten) -- 2 Occurrences. 4130, 4131. plektes. 4132 . ... //strongsnumbers.com/greek2/4131.htm - 6k Library Psalm 120. Complaint of Quarrelsome Neighbours; Or, a Devout Wish ... ... THE Psalms of David, In Metre. Psalm 120. Complaint of quarrelsome neighbours; or, A devout wish for peace. 1 Thou God of love, thou ... /.../watts/the psalms of david/psalm 120 complaint of quarrelsome.htm Psalm 120 Complaint of Quarrelsome Neighbors; Or, a Devout Wish ... ... THE PSALMS OF DAVID PSALM 120 Complaint of quarrelsome neighbors; or, A devout wish for peace. ... Complaint of quarrelsome neighbors; or, A devout wish for peace. ... /.../the psalms and hymns of isaac watts/psalm 120 complaint of quarrelsome.htm Whether Strife is Always a Sin? ... x) that the word "rixosus [quarrelsome] is derived from the snarling [rictu] of a dog, because the quarrelsome man is ever ready to contradict; he delights in ... /...//christianbookshelf.org/aquinas/summa theologica/whether strife is always a.htm Whether Quarreling is a More Grievous Sin than Flattery? ... 2: Further, there appears to be a certain amount of deceit in flattery, since the flatterer says one thing, and thinks another: whereas the quarrelsome man is ... /.../aquinas/summa theologica/whether quarreling is a more.htm Being Generous and Loving ... It is an honor for a man to avoid strife; Only a fool is quarrelsome. Charcoal for embers, and wood for fire, And a quarrelsome man to kindle strife! ... /.../sherman/the childrens bible/being generous and loving.htm One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus 3943. paroinos -- given to wine, drunken ... Adjective Transliteration: paroinos Phonetic Spelling: (par'-oy-nos) Short Definition: given to wine, drunken Definition: given to wine, drunken, quarrelsome. ... //strongsnumbers.com/greek2/3943.htm - 6k 4131. plektes -- a striker ... bully, striker. From plesso; a smiter, ie Pugnacious (quarrelsome) -- striker. see GREEK plesso. (plekten) -- 2 Occurrences. 4130, 4131. plektes. 4132 . ... //strongsnumbers.com/greek2/4131.htm - 6k Library Psalm 120. Complaint of Quarrelsome Neighbours; Or, a Devout Wish ... ... THE Psalms of David, In Metre. Psalm 120. Complaint of quarrelsome neighbours; or, A devout wish for peace. 1 Thou God of love, thou ... /.../watts/the psalms of david/psalm 120 complaint of quarrelsome.htm Psalm 120 Complaint of Quarrelsome Neighbors; Or, a Devout Wish ... ... THE PSALMS OF DAVID PSALM 120 Complaint of quarrelsome neighbors; or, A devout wish for peace. ... Complaint of quarrelsome neighbors; or, A devout wish for peace. ... /.../the psalms and hymns of isaac watts/psalm 120 complaint of quarrelsome.htm Whether Strife is Always a Sin? ... x) that the word "rixosus [quarrelsome] is derived from the snarling [rictu] of a dog, because the quarrelsome man is ever ready to contradict; he delights in ... /...//christianbookshelf.org/aquinas/summa theologica/whether strife is always a.htm Whether Quarreling is a More Grievous Sin than Flattery? ... 2: Further, there appears to be a certain amount of deceit in flattery, since the flatterer says one thing, and thinks another: whereas the quarrelsome man is ... /.../aquinas/summa theologica/whether quarreling is a more.htm Being Generous and Loving ... It is an honor for a man to avoid strife; Only a fool is quarrelsome. Charcoal for embers, and wood for fire, And a quarrelsome man to kindle strife! ... /.../sherman/the childrens bible/being generous and loving.htm One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus 4131. plektes -- a striker ... bully, striker. From plesso; a smiter, ie Pugnacious (quarrelsome) -- striker. see GREEK plesso. (plekten) -- 2 Occurrences. 4130, 4131. plektes. 4132 . ... //strongsnumbers.com/greek2/4131.htm - 6k Library Psalm 120. Complaint of Quarrelsome Neighbours; Or, a Devout Wish ... ... THE Psalms of David, In Metre. Psalm 120. Complaint of quarrelsome neighbours; or, A devout wish for peace. 1 Thou God of love, thou ... /.../watts/the psalms of david/psalm 120 complaint of quarrelsome.htm Psalm 120 Complaint of Quarrelsome Neighbors; Or, a Devout Wish ... ... THE PSALMS OF DAVID PSALM 120 Complaint of quarrelsome neighbors; or, A devout wish for peace. ... Complaint of quarrelsome neighbors; or, A devout wish for peace. ... /.../the psalms and hymns of isaac watts/psalm 120 complaint of quarrelsome.htm Whether Strife is Always a Sin? ... x) that the word "rixosus [quarrelsome] is derived from the snarling [rictu] of a dog, because the quarrelsome man is ever ready to contradict; he delights in ... /...//christianbookshelf.org/aquinas/summa theologica/whether strife is always a.htm Whether Quarreling is a More Grievous Sin than Flattery? ... 2: Further, there appears to be a certain amount of deceit in flattery, since the flatterer says one thing, and thinks another: whereas the quarrelsome man is ... /.../aquinas/summa theologica/whether quarreling is a more.htm Being Generous and Loving ... It is an honor for a man to avoid strife; Only a fool is quarrelsome. Charcoal for embers, and wood for fire, And a quarrelsome man to kindle strife! ... /.../sherman/the childrens bible/being generous and loving.htm One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Psalm 120. Complaint of Quarrelsome Neighbours; Or, a Devout Wish ... ... THE Psalms of David, In Metre. Psalm 120. Complaint of quarrelsome neighbours; or, A devout wish for peace. 1 Thou God of love, thou ... /.../watts/the psalms of david/psalm 120 complaint of quarrelsome.htm Psalm 120 Complaint of Quarrelsome Neighbors; Or, a Devout Wish ... ... THE PSALMS OF DAVID PSALM 120 Complaint of quarrelsome neighbors; or, A devout wish for peace. ... Complaint of quarrelsome neighbors; or, A devout wish for peace. ... /.../the psalms and hymns of isaac watts/psalm 120 complaint of quarrelsome.htm Whether Strife is Always a Sin? ... x) that the word "rixosus [quarrelsome] is derived from the snarling [rictu] of a dog, because the quarrelsome man is ever ready to contradict; he delights in ... /...//christianbookshelf.org/aquinas/summa theologica/whether strife is always a.htm Whether Quarreling is a More Grievous Sin than Flattery? ... 2: Further, there appears to be a certain amount of deceit in flattery, since the flatterer says one thing, and thinks another: whereas the quarrelsome man is ... /.../aquinas/summa theologica/whether quarreling is a more.htm Being Generous and Loving ... It is an honor for a man to avoid strife; Only a fool is quarrelsome. Charcoal for embers, and wood for fire, And a quarrelsome man to kindle strife! ... /.../sherman/the childrens bible/being generous and loving.htm One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Psalm 120 Complaint of Quarrelsome Neighbors; Or, a Devout Wish ... ... THE PSALMS OF DAVID PSALM 120 Complaint of quarrelsome neighbors; or, A devout wish for peace. ... Complaint of quarrelsome neighbors; or, A devout wish for peace. ... /.../the psalms and hymns of isaac watts/psalm 120 complaint of quarrelsome.htm Whether Strife is Always a Sin? ... x) that the word "rixosus [quarrelsome] is derived from the snarling [rictu] of a dog, because the quarrelsome man is ever ready to contradict; he delights in ... /...//christianbookshelf.org/aquinas/summa theologica/whether strife is always a.htm Whether Quarreling is a More Grievous Sin than Flattery? ... 2: Further, there appears to be a certain amount of deceit in flattery, since the flatterer says one thing, and thinks another: whereas the quarrelsome man is ... /.../aquinas/summa theologica/whether quarreling is a more.htm Being Generous and Loving ... It is an honor for a man to avoid strife; Only a fool is quarrelsome. Charcoal for embers, and wood for fire, And a quarrelsome man to kindle strife! ... /.../sherman/the childrens bible/being generous and loving.htm One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Whether Strife is Always a Sin? ... x) that the word "rixosus [quarrelsome] is derived from the snarling [rictu] of a dog, because the quarrelsome man is ever ready to contradict; he delights in ... /...//christianbookshelf.org/aquinas/summa theologica/whether strife is always a.htm Whether Quarreling is a More Grievous Sin than Flattery? ... 2: Further, there appears to be a certain amount of deceit in flattery, since the flatterer says one thing, and thinks another: whereas the quarrelsome man is ... /.../aquinas/summa theologica/whether quarreling is a more.htm Being Generous and Loving ... It is an honor for a man to avoid strife; Only a fool is quarrelsome. Charcoal for embers, and wood for fire, And a quarrelsome man to kindle strife! ... /.../sherman/the childrens bible/being generous and loving.htm One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Whether Quarreling is a More Grievous Sin than Flattery? ... 2: Further, there appears to be a certain amount of deceit in flattery, since the flatterer says one thing, and thinks another: whereas the quarrelsome man is ... /.../aquinas/summa theologica/whether quarreling is a more.htm Being Generous and Loving ... It is an honor for a man to avoid strife; Only a fool is quarrelsome. Charcoal for embers, and wood for fire, And a quarrelsome man to kindle strife! ... /.../sherman/the childrens bible/being generous and loving.htm One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Being Generous and Loving ... It is an honor for a man to avoid strife; Only a fool is quarrelsome. Charcoal for embers, and wood for fire, And a quarrelsome man to kindle strife! ... /.../sherman/the childrens bible/being generous and loving.htm One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus One Antidote for Many Ills ... I will tell you the most quarrelsome churches in England, if you will tell me the most lazy churches. It has actually become a proverb now-a-days. ... /.../spurgeon/spurgeons sermons volume 5 1859/one antidote for many ills.htm Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Sermon for the Fourth Sunday after Epiphany ... reconciliation with those who have done them wrong; but the false burn with anger, are envious of others' good fortune, slanderous, quarrelsome, and censorious ... /.../vii sermon for the fourth.htm Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Salvation by Knowing the Truth ... here. In many a village there is a corner where the idle and the quarrelsome gather together; and theology has such corners. It ... /.../spurgeon/spurgeons sermons volume 26 1880/salvation by knowing the truth.htm Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Meekness and Rest ... need of nothing"; instead of mercy we find cruelty; instead of purity of heart, corrupt imaginings; instead of peacemakers we find men quarrelsome and resentful ... //christianbookshelf.org/tozer/the pursuit of god/ix meekness and rest.htm Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Other Sins Forbidden. ... from every evil thing, and from every likeness of it.2. Be not prone to anger, for anger leadeth the way to murder; neither jealous, nor quarrelsome, nor of ... /.../various/the teaching of the twelve apostles/chapter iii name sins forbidden.htm Thesaurus Quarrelsome (9 Occurrences) ... Multi-Version Concordance Quarrelsome (9 Occurrences). ... They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). ... /q/quarrelsome.htm - 8k Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Peaceable (9 Occurrences) ... Noah Webster's Dictionary (a.) Being in or at peace; tranquil; quiet; free from, or not disposed to, war, disorder, or excitement; not quarrelsome. ... /p/peaceable.htm - 9k Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Pugnacious (2 Occurrences) ... (a.) Inclined to fighting; quarrelsome. ... 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (See NAS). ... /p/pugnacious.htm - 7k Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Quarrels (10 Occurrences) /q/quarrels.htm - 9k Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Quarried (3 Occurrences) /q/quarried.htm - 7k Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Quarreling (17 Occurrences) ... (See NIV). Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (See RSV). Genesis ... /q/quarreling.htm - 11k Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Quarrel (22 Occurrences) ... 15. (vt) To compel by a quarrel; as, to quarrel a man out of his estate or rights. 16. (n.) One who quarrels or wrangles; one who is quarrelsome. Int. ... /q/quarrel.htm - 15k Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Ugly (7 Occurrences) ... repulsive; deformed. 2. (superl.) Ill-natured; crossgrained; quarrelsome; as, an ugly temper; to feel ugly. 3. (superl.) Unpleasant ... /u/ugly.htm - 8k Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Overflowed (14 Occurrences) ... dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY). Romans 5:20 ... /o/overflowed.htm - 10k Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Resources What does the Bible say about a contentious or quarrelsome woman? \| GotQuestions.org What are the Christian themes in The Voyage of the Dawn Treader? \| GotQuestions.org When is it right to criticize my pastor? \| GotQuestions.org Quarrelsome: Dictionary and Thesaurus \| Clyx.com Bible Concordance • Bible Dictionary • Bible Encyclopedia • Topical Bible • Bible Thesuarus Romans 1:29 Their hearts overflowed with all sorts of dishonesty, mischief, greed, malice. They were full of envy and murder, and were quarrelsome, crafty, and spiteful. (WEY) 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (WEB WEY RSV NIV) 2 Timothy 2:24 and a bondservant of the Lord must not quarrel, but must be inoffensive towards all men, a skilful teacher, and patient under wrongs. (See NAS RSV) Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (YLT) Proverbs 19:13 A foolish son is the destruction of his father; and the bitter arguments of a wife are like drops of rain falling without end. (See NIV) Proverbs 21:9 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 21:19 It is better to be living in a waste land, than with a bitter-tongued and angry woman. (See NIV) Proverbs 25:24 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) 1 Timothy 3:3 not a drinker, not violent, not greedy for money, but gentle, not quarrelsome, not covetous; (WEB WEY RSV NIV) 2 Timothy 2:24 and a bondservant of the Lord must not quarrel, but must be inoffensive towards all men, a skilful teacher, and patient under wrongs. (See NAS RSV) Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (YLT) Proverbs 19:13 A foolish son is the destruction of his father; and the bitter arguments of a wife are like drops of rain falling without end. (See NIV) Proverbs 21:9 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 21:19 It is better to be living in a waste land, than with a bitter-tongued and angry woman. (See NIV) Proverbs 25:24 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) 2 Timothy 2:24 and a bondservant of the Lord must not quarrel, but must be inoffensive towards all men, a skilful teacher, and patient under wrongs. (See NAS RSV) Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (YLT) Proverbs 19:13 A foolish son is the destruction of his father; and the bitter arguments of a wife are like drops of rain falling without end. (See NIV) Proverbs 21:9 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 21:19 It is better to be living in a waste land, than with a bitter-tongued and angry woman. (See NIV) Proverbs 25:24 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) Titus 3:2 of no one to speak evil, not to be quarrelsome -- gentle, showing all meekness to all men, (YLT) Proverbs 19:13 A foolish son is the destruction of his father; and the bitter arguments of a wife are like drops of rain falling without end. (See NIV) Proverbs 21:9 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 21:19 It is better to be living in a waste land, than with a bitter-tongued and angry woman. (See NIV) Proverbs 25:24 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) Proverbs 19:13 A foolish son is the destruction of his father; and the bitter arguments of a wife are like drops of rain falling without end. (See NIV) Proverbs 21:9 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 21:19 It is better to be living in a waste land, than with a bitter-tongued and angry woman. (See NIV) Proverbs 25:24 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) Proverbs 21:9 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 21:19 It is better to be living in a waste land, than with a bitter-tongued and angry woman. (See NIV) Proverbs 25:24 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) Proverbs 21:19 It is better to be living in a waste land, than with a bitter-tongued and angry woman. (See NIV) Proverbs 25:24 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) Proverbs 25:24 It is better to be living in an angle of the house-top, than with a bitter-tongued woman in a wide house. (See NIV) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) Proverbs 26:21 Like breath on coals and wood on fire, so a man given to argument gets a fight started. (See RSV NIV) Subtopics Quarrelsome Related Terms Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) Quarrelsome Quarries (4 Occurrences) Quarrelsome (9 Occurrences) Quarry (5 Occurrences) Quality (14 Occurrences) Geology Tools (3 Occurrences) Ether (2 Occurrences) Pit (110 Occurrences) Chios (1 Occurrence) City Palestine (1 Occurrence) House (20110 Occurrences) \| \| \| \|
4690	https://chem.libretexts.org/Bookshelves/General_Chemistry/Chemistry_1e_(OpenSTAX)/15%3A_Equilibria_of_Other_Reaction_Classes/15.01%3A_Precipitation_and_Dissolution	15.1.1 15.1.1 15.1.2 15.1.2 15.1.3 15.1.3 15.1.4 15.1.4 15.1.5 15.1.5 15.1.6 15.1.6 15.1.7 15.1.7 15.1.8 15.1.8 15.1.9 15.1.9 15.1.10 15.1.10 15.1.11 15.1.11 15.1.12 15.1.12 BaSO4(s)⇌Ba2+(aq)+SO2−4(aq)Ksp=[Ba2+][SO2−4] Ag2SO4(s)⇌2Ag+(aq)+SO2−4(aq)Ksp=[Ag+]2[SO2−4] Al(OH)3(s)⇌Al2+(aq)+3OH−(aq)Ksp=[Al3+][OH−]3 Pb(OH)Cl(s)⇌Pb2+(aq)+OH−(aq)+Cl−(aq)Ksp=[Pb2+][OH−][Cl−] Skip to main content 15.1: Precipitation and Dissolution Last updated : Oct 27, 2022 Save as PDF 15: Equilibria of Other Reaction Classes 15.2: Lewis Acids and Bases Page ID : 38285 OpenStax OpenStax ( \newcommand{\kernel}{\mathrm{null}\,}) Learning Objectives Write chemical equations and equilibrium expressions representing solubility equilibria Carry out equilibrium computations involving solubility, equilibrium expressions, and solute concentrations The preservation of medical laboratory blood samples, mining of sea water for magnesium, formulation of over-the-counter medicines such as Milk of Magnesia and antacids, and treating the presence of hard water in your home’s water supply are just a few of the many tasks that involve controlling the equilibrium between a slightly soluble ionic solid and an aqueous solution of its ions. In some cases, we want to prevent dissolution from occurring. Tooth decay, for example, occurs when the calcium hydroxylapatite, which has the formula Ca5(PO4)3(OH), in our teeth dissolves. The dissolution process is aided when bacteria in our mouths feast on the sugars in our diets to produce lactic acid, which reacts with the hydroxide ions in the calcium hydroxylapatite. Preventing the dissolution prevents the decay. On the other hand, sometimes we want a substance to dissolve. We want the calcium carbonate in a chewable antacid to dissolve because the CO2−3CO2−3 ions produced in this process help soothe an upset stomach. In this section, we will find out how we can control the dissolution of a slightly soluble ionic solid by the application of Le Chatelier’s principle. We will also learn how to use the equilibrium constant of the reaction to determine the concentration of ions present in a solution. The Solubility Product Constant Silver chloride is what’s known as a sparingly soluble ionic solid (Figure 15.1.115.1.1). Recall from the solubility rules in an earlier chapter that halides of Ag+ are not normally soluble. However, when we add an excess of solid AgCl to water, it dissolves to a small extent and produces a mixture consisting of a very dilute solution of Ag+ and Cl– ions in equilibrium with undissolved silver chloride: AgCl(s)dissolution⇌precipitationAg+(aq)+Cl−(aq) AgCl(s)⇌precipitationdissolutionAg+(aq)+Cl−(aq) This equilibrium, like other equilibria, is dynamic; some of the solid AgCl continues to dissolve, but at the same time, Ag+ and Cl– ions in the solution combine to produce an equal amount of the solid. At equilibrium, the opposing processes have equal rates. The equilibrium constant for the equilibrium between a slightly soluble ionic solid and a solution of its ions is called the solubility product (Ksp) of the solid. Recall from the chapter on solutions and colloids that we use an ion’s concentration as an approximation of its activity in a dilute solution. For silver chloride, at equilibrium: AgCl(s)⇌Ag+(aq)+Cl−(aq) AgCl(s)⇌Ag+(aq)+Cl−(aq) with Ksp=[Ag+(aq)][Cl−(aq)] Ksp=[Ag+(aq)][Cl−(aq)] When looking at dissolution reactions such as this, the solid is listed as a reactant, whereas the ions are listed as products. The solubility product constant, as with every equilibrium constant expression, is written as the product of the concentrations of each of the ions, raised to the power of their stoichiometric coefficients. Here, the solubility product constant is equal to Ag+ and Cl– when a solution of silver chloride is in equilibrium with undissolved AgCl. There is no denominator representing the reactants in this equilibrium expression since the reactant is a pure solid; therefore [AgCl] does not appear in the expression for Ksp. Table 15.1.115.1.1: Common Solubility Products by Decreasing Equilibrium Constants \| Substance \| Ksp at 25 °C \| \| CuCl \| 1.2 × 10–6 \| \| CuBr \| 6.27 × 10–9 \| \| AgI \| 1.5 × 10–16 \| \| PbS \| 7 × 10–29 \| \| Al(OH)3 \| 2 × 10–32 \| \| Fe(OH)3 \| 4 × 10–38 \| Some common solubility products are listed in Table 15.1.115.1.1 according to their Ksp values, whereas a more extensive compilation of products appears in Table E3. Each of these equilibrium constants is much smaller than 1 because the compounds listed are only slightly soluble. A small Ksp represents a system in which the equilibrium lies to the left, so that relatively few hydrated ions would be present in a saturated solution. Example 15.1.115.1.1: Writing Equations and Solubility Products Write the ionic equation for the dissolution and the solubility product expression for each of the following slightly soluble ionic compounds: AgI, silver iodide, a solid with antiseptic properties CaCO3, calcium carbonate, the active ingredient in many over-the-counter chewable antacids Mg(OH)2, magnesium hydroxide, the active ingredient in Milk of Magnesia Mg(NH4)PO4, magnesium ammonium phosphate, an essentially insoluble substance used in tests for magnesium Ca5(PO4)3OH, the mineral apatite, a source of phosphate for fertilizers (Hint: When determining how to break (d) and (e) up into ions, refer to the list of polyatomic ions in the section on chemical nomenclature.) Solution AgI(s)⇌Ag+(aq)+I−(aq)Ksp=[Ag+][I−]AgI(s)⇌Ag+(aq)+I−(aq)Ksp=[Ag+][I−] CaCO3(s)⇌Ca2+(aq)+CO2−3(aq)Ksp=[Ca2+][CO2−3]CaCO3(s)⇌Ca2+(aq)+CO2−3(aq)Ksp=[Ca2+][CO2−3] Mg(OH)2(s)⇌Mg2+(aq)+2OH−(aq)Ksp=[Mg2+][OH−]2Mg(OH)2(s)⇌Mg2+(aq)+2OH−(aq)Ksp=[Mg2+][OH−]2 Mg(NH4)PO4(s)⇌Mg2+(aq)+NH+4(aq)+PO3−4(aq)Ksp=[Mg2+][NH+4][PO3−4]Mg(NH4)PO4(s)⇌Mg2+(aq)+NH+4(aq)+PO3−4(aq)Ksp=[Mg2+][NH+4][PO3−4] Ca5(PO4)3OH(s)⇌5Ca2+(aq)+3PO3−4(aq)+OH−(aq)Ksp=[Ca2+]5[PO3−4]3[OH−]Ca5(PO4)3OH(s)⇌5Ca2+(aq)+3PO3−4(aq)+OH−(aq)Ksp=[Ca2+]5[PO3−4]3[OH−] Exercise 15.1.115.1.1 Write the ionic equation for the dissolution and the solubility product for each of the following slightly soluble compounds: BaSO4 Ag2SO4 Al(OH)3 Pb(OH)Cl Answer a : BaSO4(s)⇌Ba2+(aq)+SO2−4(aq)Ksp=[Ba2+][SO2−4] Answer b : Ag2SO4(s)⇌2Ag+(aq)+SO2−4(aq)Ksp=[Ag+]2[SO2−4] Answer c : Al(OH)3(s)⇌Al2+(aq)+3OH−(aq)Ksp=[Al3+][OH−]3 Answer d : Pb(OH)Cl(s)⇌Pb2+(aq)+OH−(aq)+Cl−(aq)Ksp=[Pb2+][OH−][Cl−] Now we will extend the discussion of Ksp and show how the solubility product constant is determined from the solubility of its ions, as well as how Ksp can be used to determine the molar solubility of a substance. Ksp and Solubility Recall that the definition of solubility is the maximum possible concentration of a solute in a solution at a given temperature and pressure. We can determine the solubility product of a slightly soluble solid from that measure of its solubility at a given temperature and pressure, provided that the only significant reaction that occurs when the solid dissolves is its dissociation into solvated ions, that is, the only equilibrium involved is: MpXq(s)⇌pMm+(aq)+qXn−(aq) MpXq(s)⇌pMm+(aq)+qXn−(aq) In this case, we calculate the solubility product by taking the solid’s solubility expressed in units of moles per liter (mol/L), known as its molar solubility. Example 15.1.215.1.2: Calculation of Ksp from Equilibrium Concentrations We began the chapter with an informal discussion of how the mineral fluorite is formed. Fluorite, CaF2, is a slightly soluble solid that dissolves according to the equation: CaF2(s)⇌Ca2+(aq)+2F−(aq) CaF2(s)⇌Ca2+(aq)+2F−(aq) The concentration of Ca2+ in a saturated solution of CaF2 is 2.1 × 10–4 M; therefore, that of F– is 4.2 × 10–4 M, that is, twice the concentration of Ca2+. What is the solubility product of fluorite? Solution First, write out the Ksp expression, then substitute in concentrations and solve for Ksp: CaF2(s)⇌Ca2+(aq)+2F−(aq) CaF2(s)⇌Ca2+(aq)+2F−(aq) A saturated solution is a solution at equilibrium with the solid. Thus: Ksp=[Ca2+][F−]2=(2.1×10−4)(4.2×10−4)2=3.7×10−11 Ksp=[Ca2+][F−]2=(2.1×10−4)(4.2×10−4)2=3.7×10−11 As with other equilibrium constants, we do not include units with Ksp. Exercise 15.1.215.1.2 In a saturated solution that is in contact with solid Mg(OH)2, the concentration of Mg2+ is 3.7 × 10–5 M. What is the solubility product for Mg(OH)2? Mg(OH)2(s)⇌Mg2+(aq)+2OH−(aq) Mg(OH)2(s)⇌Mg2+(aq)+2OH−(aq) Answer : 2.0 × 10–13 Example 15.1.315.1.3: Determination of Molar Solubility from Ksp The Ksp of copper(I) bromide, CuBr, is 6.3 × 10–9. Calculate the molar solubility of copper bromide. Solution The solubility product constant of copper(I) bromide is 6.3×10–96.3×10–9. The reaction is: CuBr(s)⇌Cu+(aq)+Br−(aq) CuBr(s)⇌Cu+(aq)+Br−(aq) First, write out the solubility product equilibrium constant expression: Ksp=[Cu+][Br−] Ksp=[Cu+][Br−] Create an ICE table (as introduced in the chapter on fundamental equilibrium concepts), leaving the CuBr column empty as it is a solid and does not contribute to the Ksp: At equilibrium: Ksp=[Cu+][Br−]6.3×10−9=(x)(x)=x2x=√(6.3×10−9)=7.9×10−5 Therefore, the molar solubility of CuBr is 7.9 × 10–5 M. Exercise 15.1.3 The Ksp of AgI is 1.5 × 10–16. Calculate the molar solubility of silver iodide. Answer : 1.2 × 10–8 M Example 15.1.4: Determination of Molar Solubility from Ksp, Part II Determination of Molar Solubility from Ksp, Part II The Ksp of calcium hydroxide, Ca(OH)2, is 8.0 × 10–6. Calculate the molar solubility of calcium hydroxide. Solution The solubility product constant of calcium hydroxide is 1.3 × 10–6. The reaction is: Ca(OH)2(s)⇌Ca2+(aq)+2OH−(aq) First, write out the solubility product equilibrium constant expression: Ksp=[Ca2+][OH−]2 Create an ICE table, leaving the Ca(OH)2 column empty as it is a solid and does not contribute to the Ksp: At equilibrium: Ksp=[Ca2+][OH−]21.3×10−6=(x)(2x)2=(x)(4x2)=4x3x=3√1.3×10−64=6.9×10−3 Therefore, the molar solubility of Ca(OH)2 is 6.9 × 10–3 M. Exercise 15.1.4 The Ksp of PbI2 is 1.4 × 10–8. Calculate the molar solubility of lead(II) iodide. Answer : 1.5 × 10–3 M Note that solubility is not always given as a molar value. When the solubility of a compound is given in some unit other than moles per liter, we must convert the solubility into moles per liter (i.e., molarity) in order to use it in the solubility product constant expression. Example 15.1.5 shows how to perform those unit conversions before determining the solubility product equilibrium. Example 15.1.5: Determination of Ksp from Gram Solubility Many of the pigments used by artists in oil-based paints (Figure 15.1.2) are sparingly soluble in water. For example, the solubility of the artist’s pigment chrome yellow, PbCrO4, is 4.6 × 10–6 g/L. Determine the solubility product equilibrium constant for PbCrO4. Solution We are given the solubility of PbCrO4 in grams per liter. If we convert this solubility into moles per liter, we can find the equilibrium concentrations of Pb2+ and CrO2−4, then Ksp: Use the molar mass of PbCrO4 (323.2g1mol) to convert the solubility of PbCrO4 in grams per liter into moles per liter: [PbCrO4]=4.6×10−6gPbCrO41L×1molPbCrO4323.2gPbCrO4 =1.4×10−8molPbCrO41L =1.4×10−8M The chemical equation for the dissolution indicates that 1 mol of PbCrO4 gives 1 mol of Pb2+(aq) and 1 mol of CrO2−4(aq): PbCrO4(s)⇌Pb2+(aq)+CrO2−4(aq) Thus, both [Pb2+] and [CrO2−4] are equal to the molar solubility of PbCrO4: [Pb2+]=[CrO2−4]=1.4×10−8M Solve. Ksp = [Pb2+] [CrO2−4] = (1.4 × 10–8)(1.4 × 10–8) = 2.0 × 10–16 Exercise 15.1.5 The solubility of TlCl [thallium(I) chloride], an intermediate formed when thallium is being isolated from ores, is 3.46 grams per liter at 20 °C. What is its solubility product? Answer : 2.08 × 10–4 Example 15.1.6: Calculating the Solubility of Hg2Cl2 Calomel, Hg2Cl2, is a compound composed of the diatomic ion of mercury(I), Hg2+2, and chloride ions, Cl–. Although most mercury compounds are now known to be poisonous, eighteenth-century physicians used calomel as a medication. Their patients rarely suffered any mercury poisoning from the treatments because calomel is quite insoluble: Hg2Cl2(s)⇌Hg2+2(aq)+2Cl−(aq)Ksp=1.1×10−18 Calculate the molar solubility of Hg2Cl2. Solution The molar solubility of Hg2Cl2 is equal to the concentration of Hg2+2 ions because for each 1 mol of Hg2Cl2 that dissolves, 1 mol of Hg2+2 forms: Determine the direction of change. Before any Hg2Cl2 dissolves, Q is zero, and the reaction will shift to the right to reach equilibrium. Determine x and equilibrium concentrations. Concentrations and changes are given in the following ICE table: Note that the change in the concentration of Cl– (2x) is twice as large as the change in the concentration of Hg2+2 (x) because 2 mol of Cl– forms for each 1 mol of Hg2+2 that forms. Hg2Cl2 is a pure solid, so it does not appear in the calculation. Solve for x and the equilibrium concentrations. We substitute the equilibrium concentrations into the expression for Ksp and calculate the value of x: Ksp=[Hg2+2][Cl−]21.1×10−18=(x)(2x)24x3=1.1×10−18x=3√(1.1×10−184)=6.5×10−7M So the concentrations are [Hg2+2]=6.5×10−7M=6.5×10−7M[Cl−]=2x=2(6.5×10−7)=1.3×10−6M The molar solubility of Hg2Cl2 is equal to [Hg2+2], or 6.5 × 10–7 M. Check the work. At equilibrium, Q = Ksp: Q=[Hg2+2][Cl−]2=(6.5×10−7)(1.3×10−6)2=1.1×10−18 The calculations check. Exercise 15.1.6 Determine the molar solubility of MgF2 from its solubility product: Ksp = 6.4 × 10–9. Answer : 1.2 × 10–3 M Tabulated Ksp values can also be compared to reaction quotients calculated from experimental data to tell whether a solid will precipitate in a reaction under specific conditions: Q equals Ksp at equilibrium; if Q is less than Ksp, the solid will dissolve until Q equals Ksp; if Q is greater than Ksp, precipitation will occur at a given temperature until Q equals Ksp. Using Barium Sulfate for Medical Imaging Various types of medical imaging techniques are used to aid diagnoses of illnesses in a noninvasive manner. One such technique utilizes the ingestion of a barium compound before taking an X-ray image. A suspension of barium sulfate, a chalky powder, is ingested by the patient. Since the Ksp of barium sulfate is 1.1 × 10–10, very little of it dissolves as it coats the lining of the patient’s intestinal tract. Barium-coated areas of the digestive tract then appear on an X-ray as white, allowing for greater visual detail than a traditional X-ray (Figure 15.1.3). Further diagnostic testing can be done using barium sulfate and fluoroscopy. In fluoroscopy, a continuous X-ray is passed through the body so the doctor can monitor, on a TV or computer screen, the barium sulfate’s movement as it passes through the digestive tract. Medical imaging using barium sulfate can be used to diagnose acid reflux disease, Crohn’s disease, and ulcers in addition to other conditions. Predicting Precipitation The equation that describes the equilibrium between solid calcium carbonate and its solvated ions is: CaCO3(s)⇌Ca2+(aq)+CO2−3(aq) We can establish this equilibrium either by adding solid calcium carbonate to water or by mixing a solution that contains calcium ions with a solution that contains carbonate ions. If we add calcium carbonate to water, the solid will dissolve until the concentrations are such that the value of the reaction quotient (Q=[Ca2+][CO2−3]) is equal to the solubility product (Ksp = 4.8 × 10–9). If we mix a solution of calcium nitrate, which contains Ca2+ ions, with a solution of sodium carbonate, which contains CO2−3 ions, the slightly soluble ionic solid CaCO3 will precipitate, provided that the concentrations of Ca2+ and CO2−3 ions are such that Q is greater than Ksp for the mixture. The reaction shifts to the left and the concentrations of the ions are reduced by formation of the solid until the value of Q equals Ksp. A saturated solution in equilibrium with the undissolved solid will result. If the concentrations are such that Q is less than Ksp, then the solution is not saturated and no precipitate will form. We can compare numerical values of Q with Ksp to predict whether precipitation will occur, as Example 15.1.7 shows. (Note: Since all forms of equilibrium constants are temperature dependent, we will assume a room temperature environment going forward in this chapter unless a different temperature value is explicitly specified.) Example 15.1.7: Precipitation of Mg(OH)2 The first step in the preparation of magnesium metal is the precipitation of Mg(OH)2 from sea water by the addition of lime, Ca(OH)2, a readily available inexpensive source of OH– ion: Mg(OH)2(s)⇌Mg2+(aq)+2OH−(aq)Ksp=8.9×10−12 The concentration of Mg2+(aq) in sea water is 0.0537 M. Will Mg(OH)2 precipitate when enough Ca(OH)2 is added to give a [OH–] of 0.0010 M? Solution This problem asks whether the reaction: Mg(OH)2(s)⇌Mg2+(aq)+2OH−(aq) shifts to the left and forms solid Mg(OH)2 when [Mg2+] = 0.0537 M and [OH–] = 0.0010 M. The reaction shifts to the left if Q is greater than Ksp. Calculation of the reaction quotient under these conditions is shown here: Q=[Mg2+][OH−]2=(0.0537)(0.0010)2=5.4×10−8 Because Q is greater than Ksp (Q = 5.4 × 10–8 is larger than Ksp = 8.9 × 10–12), we can expect the reaction to shift to the left and form solid magnesium hydroxide. Mg(OH)2(s) forms until the concentrations of magnesium ion and hydroxide ion are reduced sufficiently so that the value of Q is equal to Ksp. Exercise 15.1.7 Use the solubility products in Table E3 to determine whether CaHPO4 will precipitate from a solution with [Ca2+] = 0.0001 M and [HPO2−4] = 0.001 M. Answer : No precipitation of CaHPO4; Q = 1 × 10–7, which is less than Ksp Example 15.1.8: Precipitation of AgCl upon Mixing Solutions Does silver chloride precipitate when equal volumes of a 2.0 × 10–4-M solution of AgNO3 and a 2.0 × 10–4-M solution of NaCl are mixed? (Note: The solution also contains Na+ and NO−3 ions, but when referring to solubility rules, one can see that sodium nitrate is very soluble and cannot form a precipitate.) Solution The equation for the equilibrium between solid silver chloride, silver ion, and chloride ion is: AgCl(s)⇌Ag+(aq)+Cl−(aq) The solubility product is 1.8 × 10–10 (Table E3). AgCl will precipitate if the reaction quotient calculated from the concentrations in the mixture of AgNO3 and NaCl is greater than Ksp. The volume doubles when we mix equal volumes of AgNO3 and NaCl solutions, so each concentration is reduced to half its initial value. Consequently, immediately upon mixing, [Ag+] and [Cl–] are both equal to: 12(2.0×10−4)M=1.0×10−4M The reaction quotient, Q, is momentarily greater than Ksp for AgCl, so a supersaturated solution is formed: Q=[Ag+][Cl−]=(1.0×10−4)(1.0×10−4)=1.0×10−8>Ksp Since supersaturated solutions are unstable, AgCl will precipitate from the mixture until the solution returns to equilibrium, with Q equal to Ksp. Exercise 15.1.8 Will KClO4 precipitate when 20 mL of a 0.050-M solution of K+ is added to 80 mL of a 0.50-M solution of ClO−4? (Remember to calculate the new concentration of each ion after mixing the solutions before plugging into the reaction quotient expression.) Answer : No, Q = 4.0 × 10–3, which is less than Ksp = 1.07 × 10–2 In the previous two examples, we have seen that Mg(OH)2 or AgCl precipitate when Q is greater than Ksp. In general, when a solution of a soluble salt of the Mm+ ion is mixed with a solution of a soluble salt of the Xn– ion, the solid, MpXq precipitates if the value of Q for the mixture of Mm+ and Xn– is greater than Ksp for MpXq. Thus, if we know the concentration of one of the ions of a slightly soluble ionic solid and the value for the solubility product of the solid, then we can calculate the concentration that the other ion must exceed for precipitation to begin. To simplify the calculation, we will assume that precipitation begins when the reaction quotient becomes equal to the solubility product constant. Example 15.1.9: Precipitation of Calcium Oxalate Blood will not clot if calcium ions are removed from its plasma. Some blood collection tubes contain salts of the oxalate ion, C2O2−4, for this purpose (Figure 15.1.4). At sufficiently high concentrations, the calcium and oxalate ions form solid, CaC2O4•H2O (which also contains water bound in the solid). The concentration of Ca2+ in a sample of blood serum is 2.2 × 10–3 M. What concentration of C2O2−4 ion must be established before CaC2O4•H2O begins to precipitate? Solution The equilibrium expression is: CaC2O4(s)⇌Ca2+(aq)+C2O2−4(aq) For this reaction (Table E3): Ksp=[Ca2+][C2O2−4]=1.96×10−8 CaC2O4 does not appear in this expression because it is a solid. Water does not appear because it is the solvent. Solid CaC2O4 does not begin to form until Q equals Ksp. Because we know Ksp and [Ca2+], we can solve for the concentration of C2O2−4 that is necessary to produce the first trace of solid: Q=Ksp=[Ca2+][C2O2−4]=1.96×10−8 (2.2×10−3)[C2O2−4]=1.96×10−8 [C2O2−4]=1.96×10−82.2×10−3=8.9×10−6 A concentration of [C2O2−4] = 8.9 × 10–6 M is necessary to initiate the precipitation of CaC2O4 under these conditions. Exercise 15.1.9 If a solution contains 0.0020 mol of CrO2−4 per liter, what concentration of Ag+ ion must be reached by adding solid AgNO3 before Ag2CrO4 begins to precipitate? Neglect any increase in volume upon adding the solid silver nitrate. Answer : 4.5 × 10–9 M It is sometimes useful to know the concentration of an ion that remains in solution after precipitation. We can use the solubility product for this calculation too: If we know the value of Ksp and the concentration of one ion in solution, we can calculate the concentration of the second ion remaining in solution. The calculation is of the same type as that in Example 15.1.8—calculation of the concentration of a species in an equilibrium mixture from the concentrations of the other species and the equilibrium constant. However, the concentrations are different; we are calculating concentrations after precipitation is complete, rather than at the start of precipitation. Example 15.1.10: Concentrations Following Precipitation Clothing washed in water that has a manganese [Mn2+(aq)] concentration exceeding 0.1 mg/L (1.8 × 10–6 M) may be stained by the manganese upon oxidation, but the amount of Mn2+ in the water can be reduced by adding a base. If a person doing laundry wishes to add a buffer to keep the pH high enough to precipitate the manganese as the hydroxide, Mn(OH)2, what pH is required to keep [Mn2+] equal to 1.8 × 10–6 M? Solution The dissolution of Mn(OH)2 is described by the equation: Mn(OH)2(s)⇌Mn2+(aq)+2OH−(aq)Ksp=2×10−13 We need to calculate the concentration of OH– when the concentration of Mn2+ is 1.8 × 10–6 M. From that, we calculate the pH. At equilibrium: Ksp=[Mn2+][OH−]2 or (1.8×10−6)[OH−]2=2×10−13 so [OH−]=3.3×10−4M Now we calculate the pH from the pOH: pOH=−log[OH−]=−log(3.3×10−4)=3.48 pH=14.00−pOH=14.00−3.48=10.52 If the person doing laundry adds a base, such as the sodium silicate (Na4SiO4) in some detergents, to the wash water until the pH is raised to 10.52, the manganese ion will be reduced to a concentration of 1.8 × 10–6 M; at that concentration or less, the ion will not stain clothing. Exercise 15.1.10 The first step in the preparation of magnesium metal is the precipitation of Mg(OH)2 from sea water by the addition of Ca(OH)2. The concentration of Mg2+(aq) in sea water is 5.37 × 10–2 M. Calculate the pH at which [Mg2+] is diminished to 1.0 × 10–5 M by the addition of Ca(OH)2. Answer : 10.97 Due to their light sensitivity, mixtures of silver halides are used in fiber optics for medical lasers, in photochromic eyeglass lenses (glass lenses that automatically darken when exposed to sunlight), and—before the advent of digital photography—in photographic film. Even though AgCl (Ksp = 1.6 × 10–10), AgBr (Ksp = 5.0 × 10–13), and AgI (Ksp = 1.5 × 10–16) are each quite insoluble, we cannot prepare a homogeneous solid mixture of them by adding Ag+ to a solution of Cl–, Br–, and I–; essentially all of the AgI will precipitate before any of the other solid halides form because of its smaller value for Ksp. However, we can prepare a homogeneous mixture of the solids by slowly adding a solution of Cl–, Br–, and I– to a solution of Ag+. When two anions form slightly soluble compounds with the same cation, or when two cations form slightly soluble compounds with the same anion, the less soluble compound (usually, the compound with the smaller Ksp) generally precipitates first when we add a precipitating agent to a solution containing both anions (or both cations). When the Ksp values of the two compounds differ by two orders of magnitude or more (e.g., 10–2 vs. 10–4), almost all of the less soluble compound precipitates before any of the more soluble one does. This is an example of selective precipitation, where a reagent is added to a solution of dissolved ions causing one of the ions to precipitate out before the rest. The Role of Precipitation in Wastewater Treatment Solubility equilibria are useful tools in the treatment of wastewater carried out in facilities that may treat the municipal water in your city or town (Figure 15.1.5). Specifically, selective precipitation is used to remove contaminants from wastewater before it is released back into natural bodies of water. For example, phosphate ions (PO2−4) are often present in the water discharged from manufacturing facilities. An abundance of phosphate causes excess algae to grow, which impacts the amount of oxygen available for marine life as well as making water unsuitable for human consumption. One common way to remove phosphates from water is by the addition of calcium hydroxide, known as lime, Ca(OH)2. The lime is converted into calcium carbonate, a strong base, in the water. As the water is made more basic, the calcium ions react with phosphate ions to produce hydroxylapatite, Ca5(PO4)3(OH), which then precipitates out of the solution: 5Ca2++3PO3−4+OH−⇌Ca10(PO4)6⋅(OH)2(s) The precipitate is then removed by filtration and the water is brought back to a neutral pH by the addition of CO2 in a recarbonation process. Other chemicals can also be used for the removal of phosphates by precipitation, including iron(III) chloride and aluminum sulfate. Selective precipitation can also be used in qualitative analysis. In this method, reagents are added to an unknown chemical mixture in order to induce precipitation. Certain reagents cause specific ions to precipitate out; therefore, the addition of the reagent can be used to determine whether the ion is present in the solution. Example 15.1.11: Precipitation of Silver Halides A solution contains 0.0010 mol of KI and 0.10 mol of KCl per liter. AgNO3 is gradually added to this solution. Which forms first, solid AgI or solid AgCl? Solution The two equilibria involved are: AgCl(s)⇌Ag+(aq)+Cl−(aq)Ksp=1.6×10−10 AgI(s)⇌Ag+(aq)+I−(aq)Ksp=1.5×10−16 If the solution contained about equal concentrations of Cl– and I–, then the silver salt with the smallest Ksp (AgI) would precipitate first. The concentrations are not equal, however, so we should find the [Ag+] at which AgCl begins to precipitate and the [Ag+] at which AgI begins to precipitate. The salt that forms at the lower [Ag+] precipitates first. For AgI: AgI precipitates when Q equals Ksp for AgI (1.5 × 10–16). When [I–] = 0.0010 M: Q=[Ag+][I−]=Ag+=1.5×10−16 [Ag+]=1.5×10−160.0010=1.5×10−13 AgI begins to precipitate when [Ag+] is 1.5 × 10–13 M. For AgCl: AgCl precipitates when Q equals Ksp for AgCl (1.6 × 10–10). When [Cl–] = 0.10 M: Qsp=[Ag+][Cl−]=Ag+=1.6×10−10 [Ag+]=1.6×10−100.10=1.6×10−9M AgCl begins to precipitate when [Ag+] is 1.6 × 10–9 M. AgI begins to precipitate at a lower [Ag+] than AgCl, so AgI begins to precipitate first. Exercise 15.1.11 If silver nitrate solution is added to a solution which is 0.050 M in both Cl– and Br– ions, at what [Ag+] would precipitation begin, and what would be the formula of the precipitate? Answer : [Ag+] = 1.0 × 10–11 M; AgBr precipitates first Common Ion Effect As we saw when we discussed buffer solutions, the hydronium ion concentration of an aqueous solution of acetic acid decreases when the strong electrolyte sodium acetate, NaCH3CO2, is added. We can explain this effect using Le Chatelier’s principle. The addition of acetate ions causes the equilibrium to shift to the left, decreasing the concentration of H3O+ to compensate for the increased acetate ion concentration. This increases the concentration of CH3CO2H: [\ce{CH3CO2H + H2O \rightleftharpoons H3O+ + CH3CO2-})] Because sodium acetate and acetic acid have the acetate ion in common, the influence on the equilibrium is called the common ion effect. The common ion effect can also have a direct effect on solubility equilibria. Suppose we are looking at the reaction where silver iodide is dissolved: AgI(s)⇌Ag+(aq)+I−(aq) If we were to add potassium iodide (KI) to this solution, we would be adding a substance that shares a common ion with silver iodide. Le Chatelier’s principle tells us that when a change is made to a system at equilibrium, the reaction will shift to counteract that change. In this example, there would be an excess of iodide ions, so the reaction would shift toward the left, causing more silver iodide to precipitate out of solution. Example 15.1.12: Common Ion Effect Calculate the molar solubility of cadmium sulfide (CdS) in a 0.010-M solution of cadmium bromide (CdBr2). The Ksp of CdS is 1.0 × 10–28. Solution The first thing you should notice is that the cadmium sulfide is dissolved in a solution that contains cadmium ions. We need to use an ICE table to set up this problem and include the CdBr2 concentration as a contributor of cadmium ions: CdS(s)⇌Cd2+(aq)+S2−(aq) Ksp=[Cd2+][S2−]=1.0×10−28 (0.010+x)(x)=1.0×10−28 x2+0.010x−1.0×10−28=0 We can solve this equation using the quadratic formula, but we can also make an assumption to make this calculation much simpler. Since the Ksp value is so small compared with the cadmium concentration, we can assume that the change between the initial concentration and the equilibrium concentration is negligible, so that 0.010 + x ~ 0.010. Going back to our Ksp expression, we would now get: Ksp=[Cd2+][S2−]=1.0×10−28 (0.010)(x)=1.0×10−28 x=1.0×10−26 Therefore, the molar solubility of CdS in this solution is 1.0 × 10–26 M. Exercise 15.1.12 Calculate the molar solubility of aluminum hydroxide, Al(OH)3, in a 0.015-M solution of aluminum nitrate, Al(NO3)3. The Ksp of Al(OH)3 is 2 × 10–32. Answer : 4 × 10–11 Summary The equilibrium constant for an equilibrium involving the precipitation or dissolution of a slightly soluble ionic solid is called the solubility product, Ksp, of the solid. When we have a heterogeneous equilibrium involving the slightly soluble solid MpXq and its ions Mm+ and Xn–: MpXq(s)⇌pMm+(aq)+qXn−(aq) We write the solubility product expression as: Ksp=[Mm+]p[Xn−]q The solubility product of a slightly soluble electrolyte can be calculated from its solubility; conversely, its solubility can be calculated from its Ksp, provided the only significant reaction that occurs when the solid dissolves is the formation of its ions. A slightly soluble electrolyte begins to precipitate when the magnitude of the reaction quotient for the dissolution reaction exceeds the magnitude of the solubility product. Precipitation continues until the reaction quotient equals the solubility product. A reagent can be added to a solution of ions to allow one ion to selectively precipitate out of solution. The common ion effect can also play a role in precipitation reactions. In the presence of an ion in common with one of the ions in the solution, Le Chatelier’s principle applies and more precipitate comes out of solution so that the molar solubility is reduced. Glossary common ion effect : effect on equilibrium when a substance with an ion in common with the dissolved species is added to the solution; causes a decrease in the solubility of an ionic species, or a decrease in the ionization of a weak acid or base molar solubility : solubility of a compound expressed in units of moles per liter (mol/L) selective precipitation : process in which ions are separated using differences in their solubility with a given precipitating reagent solubility product (Ksp) : equilibrium constant for the dissolution of a slightly soluble electrolyte 15: Equilibria of Other Reaction Classes 15.2: Lewis Acids and Bases
4691	https://math.libretexts.org/Workbench/Algebra_and_Trigonometry_2e_(OpenStax)/06%3A_Exponential_and_Logarithmic_Functions/6.07%3A_Exponential_and_Logarithmic_Equations	Published Time: 2022-10-13T01:17:24Z 6.7: Exponential and Logarithmic Equations - Mathematics LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error 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Home 2. Workbench 3. Algebra and Trigonometry 2e (OpenStax) 4. 6: Exponential and Logarithmic Functions 5. 6.7: Exponential and Logarithmic Equations Expand/collapse global location Algebra and Trigonometry 2e (OpenStax) Front Matter 1: Prerequisites 2: Equations and Inequalities 3: Functions 4: Linear Functions 5: Polynomial and Rational Functions 6: Exponential and Logarithmic Functions 7: The Unit Circle - Sine and Cosine Functions 8: Periodic Functions 9: Trigonometric Identities and Equations 10: Further Applications of Trigonometry 11: Systems of Equations and Inequalities 12: Analytic Geometry 13: Sequences, Probability, and Counting Theory 14: Appendix Back Matter 6.7: Exponential and Logarithmic Equations Last updated May 28, 2023 Save as PDF 6.6: Logarithmic Properties 6.8: Exponential and Logarithmic Models Page ID 115079 OpenStax OpenStax ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Learning Objectives 2. Using Like Bases to Solve Exponential Equations 1. USING THE ONE-TO-ONE PROPERTY OF EXPONENTIAL FUNCTIONS TO SOLVE EXPONENTIAL EQUATIONS 2. HOW TO EXAMPLE 1 Solving an Exponential Equation with a Common Base TRY IT#1 Rewriting Equations So All Powers Have the Same Base HOW TO EXAMPLE 2 Solving Equations by Rewriting Them to Have a Common Base TRY IT#2 EXAMPLE 3 Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base TRY IT#3 Q&A EXAMPLE 4 Solving an Equation with Positive and Negative Powers Analysis TRY IT#4 Solving Exponential Equations Using Logarithms HOW TO EXAMPLE 5 Solving an Equation Containing Powers of Different Bases TRY IT#5 Q&A Equations Containing e HOW TO EXAMPLE 6 Solve an Equation of the Form y=Aekt Analysis TRY IT#6 Q&A EXAMPLE 7 Solving an Equation That Can Be Simplified to the Form y=Aekt TRY IT#7 Extraneous Solutions EXAMPLE 8 Solving Exponential Functions in Quadratic Form Analysis TRY IT#8 Q&A Using the Definition of a Logarithm to Solve Logarithmic Equations USING THE DEFINITION OF A LOGARITHM TO SOLVE LOGARITHMIC EQUATIONS EXAMPLE 9 Using Algebra to Solve a Logarithmic Equation TRY IT#9 EXAMPLE 10 Using Algebra Before and After Using the Definition of the Natural Logarithm TRY IT#10 EXAMPLE 11 Using a Graph to Understand the Solution to a Logarithmic Equation TRY IT#11 Using the One-to-One Property of Logarithms to Solve Logarithmic Equations USING THE ONE-TO-ONE PROPERTY OF LOGARITHMS TO SOLVE LOGARITHMIC EQUATIONS HOW TO EXAMPLE 12 Solving an Equation Using the One-to-One Property of Logarithms Analysis TRY IT#12 Solving Applied Problems Using Exponential and Logarithmic Equations EXAMPLE 13 Using the Formula for Radioactive Decay to Find the Quantity of a Substance Analysis TRY IT#13 MEDIA 6.6 Section Exercises Verbal Algebraic Graphical Technology Extensions Learning Objectives In this section, you will: Use like bases to solve exponential equations. Use logarithms to solve exponential equations. Use the definition of a logarithm to solve logarithmic equations. Use the one-to-one property of logarithms to solve logarithmic equations. Solve applied problems involving exponential and logarithmic equations. Figure1Wild rabbits in Australia. The rabbit population grew so quickly in Australia that the event became known as the “rabbit plague.” (credit: Richard Taylor, Flickr) In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions. Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions. Using Like Bases to Solve Exponential Equations The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers b,b,S,S,and T,T,where b>0,b≠1,b>0,b≠1,bS=bTbS=bT if and only if S=T.S=T. In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown. For example, consider the equation 34x−7=32x3.34x−7=32x3.To solve for x,x,we use the division property of exponents to rewrite the right side so that both sides have the common base,3.3.Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for xx: 34x−734x−734x−74x−72xx=32x3=32x31=32x−1=2x−1=6=3Rewrite 3 as 31.Use the division property of exponents.Apply the one-to-one property of exponents.Subtract 2xand add 7 to both sides.Divide by 3.34x−7=32x334x−7=32x31Rewrite 3 as 31.34x−7=32x−1Use the division property of exponents.4x−7=2x−1Apply the one-to-one property of exponents.2x=6Subtract 2xand add 7 to both sides.x=3Divide by 3. USING THE ONE-TO-ONE PROPERTY OF EXPONENTIAL FUNCTIONS TO SOLVE EXPONENTIAL EQUATIONS For any algebraic expressions Sand T,Sand T,and any positive real number b≠1,b≠1, bS=bTif and only ifS=TbS=bTif and only ifS=T HOW TO Given an exponential equation with the form bS=bT,bS=bT,where SS and TT are algebraic expressions with an unknown, solve for the unknown. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form bS=bT.bS=bT. Use the one-to-one property to set the exponents equal. Solve the resulting equation,S=T,S=T,for the unknown. EXAMPLE 1 Solving an Exponential Equation with a Common Base Solve 2x−1=22x−4.2x−1=22x−4. Answer TRY IT#1 Solve 52x=53x+2.52x=53x+2. Rewriting Equations So All Powers Have the Same Base Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property. For example, consider the equation 256=4x−5.256=4x−5.We can rewrite both sides of this equation as a power of 2.2.Then we apply the rules of exponents, along with the one-to-one property, to solve for x:x: 256=4x−528=(22)x−528=22x−108=2x−1018=2xx=9Rewrite each side as a power with base 2.Use the one-to-one property of exponents.Apply the one-to-one property of exponents.Add 10 to both sides.Divide by 2.256=4x−528=(22)x−5Rewrite each side as a power with base 2.28=22x−10Use the one-to-one property of exponents.8=2x−10Apply the one-to-one property of exponents.18=2xAdd 10 to both sides.x=9Divide by 2. HOW TO Given an exponential equation with unlike bases, use the one-to-one property to solve it. Rewrite each side in the equation as a power with a common base. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form bS=bT.bS=bT. Use the one-to-one property to set the exponents equal. Solve the resulting equation,S=T,S=T,for the unknown. EXAMPLE 2 Solving Equations by Rewriting Them to Have a Common Base Solve 8x+2=16x+1.8x+2=16x+1. Answer TRY IT#2 Solve 52x=253x+2.52x=253x+2. EXAMPLE 3 Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base Solve 25x=2–√.25x=2. Answer TRY IT#3 Solve 5x=5–√.5x=5. Q&A Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process? No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined. EXAMPLE 4 Solving an Equation with Positive and Negative Powers Solve 3x+1=−2.3x+1=−2. Answer Analysis Figure 2shows that the two graphs do not cross so the left side is never equal to the right side. Thus the equation has no solution. Figure2 TRY IT#4 Solve 2x=−100.2x=−100. Solving Exponential Equations Using Logarithms Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since log(a)=log(b)log(a)=log(b)is equivalent to a=b,a=b,we may apply logarithms with the same base on both sides of an exponential equation. HOW TO Given an exponential equation in which a common base cannot be found, solve for the unknown. Apply the logarithm of both sides of the equation. If one of the terms in the equation has base 10, use the common logarithm. If none of the terms in the equation has base 10, use the natural logarithm. Use the rules of logarithms to solve for the unknown. EXAMPLE 5 Solving an Equation Containing Powers of Different Bases Solve 5x+2=4x.5x+2=4x. Answer TRY IT#5 Solve 2x=3x+1.2x=3x+1. Q&A Is there any way to solve 2x=3x?2 x=3 x? Yes. The solution is 0.0. Equations Containing e One common type of exponential equations are those with base e.e.This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base ee on either side, we can use the natural logarithm to solve it. HOW TO Given an equation of the form y=Aekt,y=Aekt,solve for t.t. Divide both sides of the equation by A.A. Apply the natural logarithm of both sides of the equation. Divide both sides of the equation by k.k. EXAMPLE 6 Solve an Equation of the Form y=Ae kt Solve 100=20e2t.100=20e2t. Answer Analysis Using laws of logs, we can also write this answer in the form t=ln5–√.t=ln5.If we want a decimal approximation of the answer, we use a calculator. TRY IT#6 Solve 3e0.5t=11.3e0.5t=11. Q&A Does every equation of the formy=Aekty=A e kthave a solution? No. There is a solution when k≠0,k≠0,and when yy and AA are either both 0 or neither 0, and they have the same sign. An example of an equation with this form that has no solution is 2=−3et.2=−3et. EXAMPLE 7 Solving an Equation That Can Be Simplified to the Form y=Ae kt Solve 4e2x+5=12.4e2x+5=12. Answer TRY IT#7 Solve 3+e2t=7e2t.3+e2t=7e2t. Extraneous Solutions Sometimes the methods used to solve an equation introduce anextraneous solution, which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output. EXAMPLE 8 Solving Exponential Functions in Quadratic Form Solve e2x−ex=56.e2x−ex=56. Answer Analysis When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the unique property that when a product is zero, one or both of the factors must be zero. We reject the equation ex=−7ex=−7 because a positive number never equals a negative number. The solution ln(−7)ln(−7)is not a real number, and in the real number system this solution is rejected as an extraneous solution. TRY IT#8 Solve e2x=ex+2.e2x=ex+2. Q&A Does every logarithmic equation have a solution? No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions. Using the Definition of a Logarithm to Solve Logarithmic Equations We have already seen that every logarithmic equation logb(x)=ylogb(x)=y is equivalent to the exponential equation by=x.by=x.We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression. For example, consider the equation log2(2)+log2(3x−5)=3.log2(2)+log2(3x−5)=3.To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for x:x: log2(2)+log2(3x−5)=3log2(2(3x−5))=3log2(6x−10)=323=6x−108=6x−1018=6xx=3Apply the product rule of logarithms.Distribute.Apply the definition of a logarithm.Calculate23.Add 10 to both sides.Divide by 6.log2(2)+log2(3x−5)=3log2(2(3x−5))=3Apply the product rule of logarithms.log2(6x−10)=3Distribute.23=6x−10Apply the definition of a logarithm.8=6x−10Calculate 23.18=6xAdd 10 to both sides.x=3Divide by 6. USING THE DEFINITION OF A LOGARITHM TO SOLVE LOGARITHMIC EQUATIONS For any algebraic expression SS and real numbers bb and c,c,where b>0,b≠1,b>0,b≠1, logb(S)=cif and only ifbc=Slogb(S)=cif and only ifbc=S EXAMPLE 9 Using Algebra to Solve a Logarithmic Equation Solve 2lnx+3=7.2lnx+3=7. Answer TRY IT#9 Solve 6+lnx=10.6+lnx=10. EXAMPLE 10 Using Algebra Before and After Using the Definition of the Natural Logarithm Solve 2ln(6x)=7.2ln(6x)=7. Answer TRY IT#10 Solve 2ln(x+1)=10.2ln(x+1)=10. EXAMPLE 11 Using a Graph to Understand the Solution to a Logarithmic Equation Solve lnx=3.lnx=3. Answer TRY IT#11 Use a graphing calculator to estimate the approximate solution to the logarithmic equation 2x=10002x=1000 to 2 decimal places. Using the One-to-One Property of Logarithms to Solve Logarithmic Equations As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers x>0,x>0,S>0,S>0,T>0T>0 and any positive real number b,b,where b≠1,b≠1, logbS=logbTif and only if S=T.logbS=logbTif and only if S=T. For example, If log2(x−1)=log2(8),then x−1=8.If log2(x−1)=log2(8),then x−1=8. So, if x−1=8,x−1=8,then we can solve for x,x,and we get x=9.x=9.To check, we can substitute x=9x=9 into the original equation:log2(9−1)=log2(8)=3.log2(9−1)=log2(8)=3.In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown. For example, consider the equation log(3x−2)−log(2)=log(x+4).log(3x−2)−log(2)=log(x+4).To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for x:x: log(3x−2)−log(2)=log(x+4)log(3x−22)=log(x+4)3x−22=x+43x−2=2x+8x=10Apply the quotient rule of logarithms.Apply the one to one property of a logarithm.Multiply both sides of the equation by 2.Subtract 2xand add 2.log(3x−2)−log(2)=log(x+4)log(3x−22)=log(x+4)Apply the quotient rule of logarithms.3x−22=x+4Apply the one to one property of a logarithm.3x−2=2x+8Multiply both sides of the equation by 2.x=10Subtract 2xand add 2. To check the result, substitute x=10x=10 into log(3x−2)−log(2)=log(x+4).log(3x−2)−log(2)=log(x+4). log(3(10)−2)−log(2)=log((10)+4)log(28)−log(2)=log(14)log(282)=log(14)The solution checks.log(3(10)−2)−log(2)=log((10)+4)log(28)−log(2)=log(14)log(282)=log(14)The solution checks. USING THE ONE-TO-ONE PROPERTY OF LOGARITHMS TO SOLVE LOGARITHMIC EQUATIONS For any algebraic expressions SS and TT and any positive real number b,b,where b≠1,b≠1, logbS=logbTif and only ifS=TlogbS=logbTif and only ifS=T Note, when solving an equation involving logarithms, always check to see if the answer is correct or if it is an extraneous solution. HOW TO Given an equation containing logarithms, solve it using the one-to-one property. Use the rules of logarithms to combine like terms, if necessary, so that the resulting equation has the form logbS=logbT.logbS=logbT. Use the one-to-one property to set the arguments equal. Solve the resulting equation,S=T,S=T,for the unknown. EXAMPLE 12 Solving an Equation Using the One-to-One Property of Logarithms Solve ln(x2)=ln(2x+3).ln(x2)=ln(2x+3). Answer Analysis There are two solutions:33 or−1.−1.The solution−1−1 is negative, but it checks when substituted into the original equation because the argument of the logarithm functions is still positive. TRY IT#12 Solve ln(x2)=ln1.ln(x2)=ln1. Solving Applied Problems Using Exponential and Logarithmic Equations In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm. One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life.Table 1lists the half-life for several of the more common radioactive substances. \| Substance \| Use \| Half-life \| --- \| gallium-67 \| nuclear medicine \| 80 hours \| \| cobalt-60 \| manufacturing \| 5.3 years \| \| technetium-99m \| nuclear medicine \| 6 hours \| \| americium-241 \| construction \| 432 years \| \| carbon-14 \| archeological dating \| 5,715 years \| \| uranium-235 \| atomic power \| 703,800,000 years \| Table1 We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay: A(t)=A0eln(0.5)TtA(t)=A0eln(0.5)tTA(t)=A0(eln(0.5))tTA(t)=A0(12)tTA(t)=A0eln(0.5)TtA(t)=A0eln(0.5)tTA(t)=A0(eln(0.5))tTA(t)=A0(12)tT where A0A0 is the amount initially present TT is the half-life of the substance tt is the time period over which the substance is studied A(t)A(t)is the amount of the substance present after time tt EXAMPLE 13 Using the Formula for Radioactive Decay to Find the Quantity of a Substance How long will it take for ten percent of a 1000-gram sample of uranium-235 to decay? Answer Analysis Ten percent of 1000 grams is 100 grams. If 100 grams decay, the amount of uranium-235 remaining is 900 grams. TRY IT#13 How long will it take before twenty percent of our 1000-gram sample of uranium-235 has decayed? MEDIA Access these online resources for additional instruction and practice with exponential and logarithmic equations. Solving Logarithmic Equations Solving Exponential Equations with Logarithms 6.6 Section Exercises Verbal 1. How can an exponential equation be solved? 2. When does an extraneous solution occur? How can an extraneous solution be recognized? 3. When can the one-to-one property of logarithms be used to solve an equation? When can it not be used? Algebraic For the following exercises, use like bases to solve the exponential equation. 4. 4−3v−2=4−v4−3v−2=4−v 5. 64⋅43x=1664⋅43x=16 6. 32x+1⋅3x=24332x+1⋅3x=243 7. 2−3n⋅14=2n+22−3n⋅14=2n+2 8. 625⋅53x+3=125625⋅53x+3=125 9. 363b362b=2162−b363b362b=2162−b 10. (164)3n⋅8=26(164)3n⋅8=26 For the following exercises, use logarithms to solve. 11. 9x−10=19x−10=1 12. 2e6x=132e6x=13 13. er+10−10=−42er+10−10=−42 14. 2⋅109a=292⋅109a=29 15. −8⋅10p+7−7=−24−8⋅10p+7−7=−24 16. 7e3n−5+5=−897e3n−5+5=−89 17. e−3k+6=44e−3k+6=44 18. −5e9x−8−8=−62−5e9x−8−8=−62 19. −6e9x+8+2=−74−6e9x+8+2=−74 20. 2x+1=52x−12x+1=52x−1 21. e2x−ex−132=0e2x−ex−132=0 22. 7e8x+8−5=−957e8x+8−5=−95 23. 10e8x+3+2=810e8x+3+2=8 24. 4e3x+3−7=534e3x+3−7=53 25. 8e−5x−2−4=−908e−5x−2−4=−90 26. 32x+1=7x−232x+1=7x−2 27. e2x−ex−6=0e2x−ex−6=0 28. 3e3−3x+6=−313e3−3x+6=−31 For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. 29. log(1100)=−2log(1100)=−2 30. log324(18)=12log324(18)=12 For the following exercises, use the definition of a logarithm to solve the equation. 31. 5log7n=105log7n=10 32. −8log9x=16−8log9x=16 33. 4+log2(9k)=24+log2(9k)=2 34. 2log(8n+4)+6=102log(8n+4)+6=10 35. 10−4ln(9−8x)=610−4ln(9−8x)=6 For the following exercises, use the one-to-one property of logarithms to solve. 36. ln(10−3x)=ln(−4x)ln(10−3x)=ln(−4x) 37. log13(5n−2)=log13(8−5n)log13(5n−2)=log13(8−5n) 38. log(x+3)−log(x)=log(74)log(x+3)−log(x)=log(74) 39. ln(−3x)=ln(x2−6x)ln(−3x)=ln(x2−6x) 40. log4(6−m)=log43mlog4(6−m)=log43m 41. ln(x−2)−ln(x)=ln(54)ln(x−2)−ln(x)=ln(54) 42. log9(2n2−14n)=log9(−45+n2)log9(2n2−14n)=log9(−45+n2) 43. ln(x2−10)+ln(9)=ln(10)ln(x2−10)+ln(9)=ln(10) For the following exercises, solve each equation for x.x. 44. log(x+12)=log(x)+log(12)log(x+12)=log(x)+log(12) 45. ln(x)+ln(x−3)=ln(7x)ln(x)+ln(x−3)=ln(7x) 46. log2(7x+6)=3log2(7x+6)=3 47. ln(7)+ln(2−4x2)=ln(14)ln(7)+ln(2−4x2)=ln(14) 48. log8(x+6)−log8(x)=log8(58)log8(x+6)−log8(x)=log8(58) 49. ln(3)−ln(3−3x)=ln(4)ln(3)−ln(3−3x)=ln(4) 50. log3(3x)−log3(6)=log3(77)log3(3x)−log3(6)=log3(77) Graphical For the following exercises, solve the equation for x,x,if there is a solution . Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. 51. log9(x)−5=−4log9(x)−5=−4 52. log3(x)+3=2log3(x)+3=2 53. ln(3x)=2ln(3x)=2 54. ln(x−5)=1ln(x−5)=1 55. log(4)+log(−5x)=2log(4)+log(−5x)=2 56. −7+log3(4−x)=−6−7+log3(4−x)=−6 57. ln(4x−10)−6=−5ln(4x−10)−6=−5 58. log(4−2x)=log(−4x)log(4−2x)=log(−4x) 59. log11(−2x2−7x)=log11(x−2)log11(−2x2−7x)=log11(x−2) 60. ln(2x+9)=ln(−5x)ln(2x+9)=ln(−5x) 61. log9(3−x)=log9(4x−8)log9(3−x)=log9(4x−8) 62. log(x2+13)=log(7x+3)log(x2+13)=log(7x+3) 63. 3log2(10)−log(x−9)=log(44)3log2(10)−log(x−9)=log(44) 64. ln(x)−ln(x+3)=ln(6)ln(x)−ln(x+3)=ln(6) For the following exercises, solve for the indicated value, and graph the situation showing the solution point. 65. An account with an initial deposit of$6,500$6,500 earns 7.25%7.25%annual interest, compounded continuously. How much will the account be worth after 20 years? 66. The formula for measuring sound intensity in decibels DD is defined by the equation D=10log(II0),D=10log(II0),where II is the intensity of the sound in watts per square meter and I0=10−12I0=10−12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of 8.3⋅1028.3⋅102 watts per square meter? 67. The population of a small town is modeled by the equation P=1650e0.5tP=1650e0.5t where tt is measured in years. In approximately how many years will the town’s population reach 20,000?20,000? Technology For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. 68. 1000(1.03)t=50001000(1.03)t=5000 using the common log. 69. e5x=17e5x=17 using the natural log 70. 3(1.04)3t=83(1.04)3t=8 using the common log 71. 34x−5=3834x−5=38 using the common log 72. 50e−0.12t=1050e−0.12t=10 using the natural log For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. 73. 7e3x−5+7.9=477e3x−5+7.9=47 74. ln(3)+ln(4.4x+6.8)=2ln(3)+ln(4.4x+6.8)=2 75. log(−0.7x−9)=1+5log(5)log(−0.7x−9)=1+5log(5) 76. Atmospheric pressure PP in pounds per square inch is represented by the formula P=14.7e−0.21x,P=14.7e−0.21x,where xx is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.3698.369 pounds per square inch? (Hint: there are 5280 feet in a mile) 77. The magnitude M of an earthquake is represented by the equation M=23log(EE0)M=23log(EE0)where EE is the amount of energy released by the earthquake in joules and E0=104.4E0=104.4 is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing 1.4⋅10131.4⋅1013 joules of energy? Extensions 78. Use the definition of a logarithm along with the one-to-one property of logarithms to prove that blogbx=x.blogbx=x. 79. Recall the formula for continually compounding interest,y=Aekt.y=Aekt.Use the definition of a logarithm along with properties of logarithms to solve the formula for time tt such that tt is equal to a single logarithm. 80. Recall the compound interest formula A=a(1+rk)kt.A=a(1+rk)kt.Use the definition of a logarithm along with properties of logarithms to solve the formula for time t.t. 81. Newton’s Law of Cooling states that the temperature TT of an object at any time t can be described by the equation T=Ts+(T0−Ts)e−kt,T=Ts+(T0−Ts)e−kt,where TsTs is the temperature of the surrounding environment,T0T0 is the initial temperature of the object, and kk is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time tt such that tt is equal to a single logarithm. This page titled 6.7: Exponential and Logarithmic Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform. Back to top 6.6: Logarithmic Properties 6.8: Exponential and Logarithmic Models Was this article helpful? Yes No Recommended articles 6.6: Exponential and Logarithmic EquationsUncontrolled population growth can be modeled with exponential functions. 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Article typeSection or PageAuthorOpenStaxLicenseCC BYLicense Version4.0OER program or PublisherOpenStaxShow Page TOCnoTranscludedyes Tags exponential equations population growth source@ source-math-1510 © Copyright 2025 Mathematics LibreTexts Powered by CXone Expert ® ? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org. Support Center How can we help? Contact Support Search the Insight Knowledge Base Check System Status× contents readability resources tools ☰ 6.6: Logarithmic Properties 6.8: Exponential and Logarithmic Models
4692	https://pressbooks.bccampus.ca/algebraintermediate/chapter/solve-systems-of-linear-equations-with-two-variables/	Skip to content Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices. Systems of Linear Equations Solve Systems of Linear Equations with Two Variables Learning Objectives By the end of this section, you will be able to: Determine whether an ordered pair is a solution of a system of equations Solve a system of linear equations by graphing Solve a system of equations by substitution Solve a system of equations by elimination Choose the most convenient method to solve a system of linear equations Before you get started, take this readiness quiz. For the equation ⓐ Is a solution? ⓑ Is a solution? If you missed this problem, review (Figure). 2. Find the slope and y-intercept of the line If you missed this problem, review (Figure). 3. Find the x- and y-intercepts of the line If you missed this problem, review (Figure). Determine Whether an Ordered Pair is a Solution of a System of Equations In Solving Linear Equations, we learned how to solve linear equations with one variable. Now we will work with two or more linear equations grouped together, which is known as a system of linear equations. System of Linear Equations When two or more linear equations are grouped together, they form a system of linear equations. In this section, we will focus our work on systems of two linear equations in two unknowns. We will solve larger systems of equations later in this chapter. An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations. A linear equation in two variables, such as has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line. To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs that make both equations true. These are called the solutions of a system of equations. Solutions of a System of Equations The solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system. Determine whether the ordered pair is a solution to the system ⓐⓑ ⓐ ⓑ Determine whether the ordered pair is a solution to the system ⓐⓑ ⓐ yes ⓑ no Determine whether the ordered pair is a solution to the system ⓐⓑ ⓐ no ⓑ yes Solve a System of Linear Equations by Graphing In this section, we will use three methods to solve a system of linear equations. The first method we’ll use is graphing. The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system. Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions. Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown. Each time we demonstrate a new method, we will use it on the same system of linear equations. At the end of the section you’ll decide which method was the most convenient way to solve this system. How to Solve a System of Equations by Graphing Solve the system by graphing Solve the system by graphing: Solve the system by graphing: The steps to use to solve a system of linear equations by graphing are shown here. Solve a system of linear equations by graphing. Graph the first equation. Graph the second equation on the same rectangular coordinate system. Determine whether the lines intersect, are parallel, or are the same line. Identify the solution to the system. If the lines intersect, identify the point of intersection. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions. Check the solution in both equations. In the next example, we’ll first re-write the equations into slope–intercept form as this will make it easy for us to quickly graph the lines. Solve the system by graphing: We’ll solve both of these equations for so that we can easily graph them using their slopes and y-intercepts. \| \| \| \| Solve the first equation for y. \| \| Find the slope and y-intercept. \| \| Solve the second equation for y. \| \| Find the slope and y-intercept. \| \| Graph the lines. \| \| Determine the point of intersection. \| The lines intersect at \| \| Check the solution in both equations. \| \| The solution is \| Solve the system by graphing: Solve the system by graphing: In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we’ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions. Solve the system by graphing: \| \| \| \| To graph the first equation, we will use its slope and y-intercept. \| \| To graph the second equation, we will use the intercepts. \| \| \| Graph the lines. \| \| Determine the points of intersection. \| The lines are parallel. Since no point is on both lines, there is no ordered pair that makes both equations true. There is no solution to this system. \| Solve the system by graphing: no solution Solve the system by graphing: no solution Sometimes the equations in a system represent the same line. Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true. There are infinitely many solutions to the system. Solve the system by graphing: \| \| \| \| Find the slope and y-intercept of the first equation. \| \| Find the intercepts of the second equation. \| \| \| Graph the lines. \| \| The lines are the same! Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true. There are infinitely many solutions to this system. \| If you write the second equation in slope-intercept form, you may recognize that the equations have the same slope and same y-intercept. Solve the system by graphing: infinitely many solutions Solve the system by graphing: infinitely many solutions When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same y-intercept. Coincident Lines Coincident lines have the same slope and same y-intercept. The systems of equations in (Figure) and (Figure) each had two intersecting lines. Each system had one solution. In (Figure), the equations gave coincident lines, and so the system had infinitely many solutions. The systems in those three examples had at least one solution. A system of equations that has at least one solution is called a consistent system. A system with parallel lines, like (Figure), has no solution. We call a system of equations like this inconsistent. It has no solution. Consistent and Inconsistent Systems A consistent system of equations is a system of equations with at least one solution. An inconsistent system of equations is a system of equations with no solution. We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent, they each have their own set of solutions. Intersecting lines and parallel lines are independent. If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines. Let’s sum this up by looking at the graphs of the three types of systems. See below and (Figure). \| Lines \| Intersecting \| Parallel \| Coincident \| --- --- \| \| Number of solutions \| 1 point \| No solution \| Infinitely many \| \| Consistent/inconsistent \| Consistent \| Inconsistent \| Consistent \| \| Dependent/ independent \| Independent \| Independent \| Dependent \| Without graphing, determine the number of solutions and then classify the system of equations. ⓐⓑ ⓐ We will compare the slopes and intercepts of the two lines. A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent. ⓑ We will compare the slope and intercepts of the two lines. A system of equations whose graphs are intersect has 1 solution and is consistent and independent. Without graphing, determine the number of solutions and then classify the system of equations. ⓐⓑ ⓐ no solution, inconsistent, independent ⓑ one solution, consistent, independent Without graphing, determine the number of solutions and then classify the system of equations. ⓐⓑ ⓐ no solution, inconsistent, independent ⓑ one solution, consistent, independent Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph. Solve a System of Equations by Substitution We will now solve systems of linear equations by the substitution method. We will use the same system we used first for graphing. We will first solve one of the equations for either x or y. We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy. Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those! After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true. How to Solve a System of Equations by Substitution Solve the system by substitution: Solve the system by substitution: Solve the system by substitution: Solve a system of equations by substitution. Solve one of the equations for either variable. Substitute the expression from Step 1 into the other equation. Solve the resulting equation. Substitute the solution in Step 3 into either of the original equations to find the other variable. Write the solution as an ordered pair. Check that the ordered pair is a solution to both original equations. Be very careful with the signs in the next example. Solve the system by substitution: We need to solve one equation for one variable. We will solve the first equation for y. \| \| \| \| Solve the first equation for y. Substitute for y in the second equation. \| \| Replace the y with \| \| Solve the equation for x. \| \| Substitute into to find y. \| \| The ordered pair is \| \| Check the ordered pair in both equations. \| \| The solution is \| Solve the system by substitution: Solve the system by substitution: Solve a System of Equations by Elimination We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression. The third method of solving systems of linear equations is called the Elimination Method. When we solved a system by substitution, we started with two equations and two variables and reduced it to one equation with one variable. This is what we’ll do with the elimination method, too, but we’ll have a different way to get there. The Elimination Method is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal. For any expressions a, b, c, and d. To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable. Notice how that works when we add these two equations together: The y’s add to zero and we have one equation with one variable. Let’s try another one: This time we don’t see a variable that can be immediately eliminated if we add the equations. But if we multiply the first equation by we will make the coefficients of x opposites. We must multiply every term on both sides of the equation by Then rewrite the system of equations. Now we see that the coefficients of the x terms are opposites, so x will be eliminated when we add these two equations. Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations. Now we’ll see how to use elimination to solve the same system of equations we solved by graphing and by substitution. How to Solve a System of Equations by Elimination Solve the system by elimination: Solve the system by elimination: Solve the system by elimination: The steps are listed here for easy reference. Solve a system of equations by elimination. Write both equations in standard form. If any coefficients are fractions, clear them. Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites. Add the equations resulting from Step 2 to eliminate one variable. Solve for the remaining variable. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. Write the solution as an ordered pair. Check that the ordered pair is a solution to both original equations. Now we’ll do an example where we need to multiply both equations by constants in order to make the coefficients of one variable opposites. Solve the system by elimination: In this example, we cannot multiply just one equation by any constant to get opposite coefficients. So we will strategically multiply both equations by different constants to get the opposites. \| \| \| \| Both equations are in standard form. To get opposite coefficients of y, we will multiply the first equation by 2 and the second equation by 3. \| \| Simplify. \| \| Add the two equations to eliminate y. \| \| Solve for x. \| \| Substitute into one of the original equations. \| \| Solve for y. \| \| Write the solution as an ordered pair. \| The ordered pair is \| \| Check that the ordered pair is a solution to both original equations. \| \| The solution is \| Solve the system by elimination: Solve each system by elimination: When the system of equations contains fractions, we will first clear the fractions by multiplying each equation by the LCD of all the fractions in the equation. Solve the system by elimination: In this example, both equations have fractions. Our first step will be to multiply each equation by the LCD of all the fractions in the equation to clear the fractions. \| \| \| \| To clear the fractions, multiply each equation by its LCD. \| \| Simplify. \| \| Now we are ready to eliminate one of the variables. Notice that both equations are in standard form. \| \| We can eliminate by multiplying the top equation by \| \| Simplify and add. Substitute into one of the original equations. \| \| Solve for . \| \| \| Write the solution as an ordered pair. \| The ordered pair is \| \| Check that the ordered pair is a solution to both original equations. \| \| The solution is \| Solve each system by elimination: Solve each system by elimination: When we solved the system by graphing, we saw that not all systems of linear equations have a single ordered pair as a solution. When the two equations were really the same line, there were infinitely many solutions. We called that a consistent system. When the two equations described parallel lines, there was no solution. We called that an inconsistent system. The same is true using substitution or elimination. If the equation at the end of substitution or elimination is a true statement, we have a consistent but dependent system and the system of equations has infinitely many solutions. If the equation at the end of substitution or elimination is a false statement, we have an inconsistent system and the system of equations has no solution. Solve the system by elimination: This is a true statement. The equations are consistent but dependent. Their graphs would be the same line. The system has infinitely many solutions. After we cleared the fractions in the second equation, did you notice that the two equations were the same? That means we have coincident lines. Solve the system by elimination: infinitely many solutions Solve the system by elimination: infinitely many solutions Choose the Most Convenient Method to Solve a System of Linear Equations When you solve a system of linear equations in in an application, you will not be told which method to use. You will need to make that decision yourself. So you’ll want to choose the method that is easiest to do and minimizes your chance of making mistakes. For each system of linear equations, decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer. ⓐⓑ ⓐ Since both equations are in standard form, using elimination will be most convenient. ⓑ Since one equation is already solved for y, using substitution will be most convenient. For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer. ⓐⓑ ⓐ Since both equations are in standard form, using elimination will be most convenient. ⓑ Since one equation is already solved for x, using substitution will be most convenient. For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer. ⓐⓑ ⓐ Since one equation is already solved for y, using substitution will be most convenient. ⓑ Since both equations are in standard form, using elimination will be most convenient. Key Concepts How to solve a system of linear equations by graphing. Graph the first equation. Graph the second equation on the same rectangular coordinate system. Determine whether the lines intersect, are parallel, or are the same line. Identify the solution to the system. If the lines intersect, identify the point of intersection. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions. 5. Check the solution in both equations. How to solve a system of equations by substitution. 1. Solve one of the equations for either variable. 2. Substitute the expression from Step 1 into the other equation. 3. Solve the resulting equation. 4. Substitute the solution in Step 3 into either of the original equations to find the other variable. 5. Write the solution as an ordered pair. 6. Check that the ordered pair is a solution to both original equations. How to solve a system of equations by elimination. 1. Write both equations in standard form. If any coefficients are fractions, clear them. 2. Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites. 3. Add the equations resulting from Step 2 to eliminate one variable. 4. Solve for the remaining variable. 5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. 6. Write the solution as an ordered pair. 7. Check that the ordered pair is a solution to both original equations. Practice Makes Perfect Determine Whether an Ordered Pair is a Solution of a System of Equations In the following exercises, determine if the following points are solutions to the given system of equations. ⓐ ⓑ ⓐ yes ⓑ no ⓐ ⓑ ⓐ ⓑ ⓐ yes ⓑ no ⓐ ⓑ Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. no solution no solution infinite solutions infinite solutions Without graphing, determine the number of solutions and then classify the system of equations. 1 point, consistent and independent 1 point, consistent and independent infinite solutions, consistent, dependent Solve a System of Equations by Substitution In the following exercises, solve the systems of equations by substitution. Solve a System of Equations by Elimination In the following exercises, solve the systems of equations by elimination. infinitely many infinitely many Choose the Most Convenient Method to Solve a System of Linear Equations In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination. ⓐ ⓑ ⓐ ⓑ ⓐ substitution ⓑ elimination ⓐ ⓑ ⓐ ⓑ ⓐ elimination ⓑ substituion Writing Exercises In a system of linear equations, the two equations have the same intercepts. Describe the possible solutions to the system. Solve the system of equations by substitution and explain all your steps in words: Answers will vary. Solve the system of equations by elimination and explain all your steps in words: Solve the system of equations ⓐ by graphing ⓑ by substitution ⓒ Which method do you prefer? Why? Answers will vary. Self Check After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. If most of your checks were: …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific. …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? …no – I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need. Glossary coincident lines : Coincident lines have the same slope and same y-intercept. consistent and inconsistent systems : Consistent system of equations is a system of equations with at least one solution; inconsistent system of equations is a system of equations with no solution. solutions of a system of equations : Solutions of a system of equations are the values of the variables that make all the equations true; solution is represented by an ordered pair system of linear equations : When two or more linear equations are grouped together, they form a system of linear equations. License Intermediate Algebra Copyright © 2017 by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. Share This Book
4693	https://www.engineeringtoolbox.com/mass-weight-d_589.html	Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications! Mass vs. Weight Mass vs. weight - the Gravity Force. Mass and Weight are two often misused and misunderstood terms in mechanics and fluid mechanics. The fundamental relation between mass and weight is defined by Newton's Second Law . Newton's Second Law can be expressed as F = m a (1) F = force (N, lbf) m = mass (kg, slugs) a = acceleration (m/s2, ft/s2) Mass Mass is a measure of the amount of material in an object, being directly related to the number and type of atoms present in the object. Mass does not change with a body's position, movement or alteration of its shape, unless material is added or removed. an object with mass 1 kg on earth would have the same mass of 1 kg on the moon m = V ρ (1b) where V = volume (m3) ρ = density (kg/m3) Mass is a fundamental property of an object, a numerical measure of its inertia and a fundamental measure of the amount of matter in the object. mass electron 9.1095×10-31 kg mass proton 1.67265×10-27 kg mass neutron 1.67495×10-27 kg Weight Weight is the gravitational force acting on a body mass. The generic expression of Newton's Second Law (1) can be transformed to express weight as a force by replacing the acceleration - a - with the acceleration of gravity - g - as Fg = m ag (2) where Fg = gravitational force - or weight (N, lbf) m = mass (kg, slugs (lbm)) ag = acceleration of gravity on earth (9.81 m/s2, 32.17405 ft/s2) Specific Weight Example - The Weight of a Body on Earth vs. Moon The acceleration of gravity on the moon is approximately 1/6 of the acceleration of gravity on the earth. The weight of a body with mass 1 kg on the earth can be calculated as Fg_earth = (1 kg) (9.81 m/s2) = 9.81 N The weight of the same body on the moon can be calculated as Fg_moon = (1 kg) ((9.81 m/s2) / 6) = 1.64 N The handling of mass and weight depends on the systems of units used. The most common unit systems are the International System - SI the British Gravitational System - BG the English Engineering System - EE One newton is ≈ the weight of one hundred grams - 101.972 gf (gF ) or 0.101972 kgf (kgF or kilopond - kp (pondus is latin for weight)) ≈ halfway between one-fifth and one-fourth of a pound - 0.224809 lb or 3.59694 oz The International System - SI In the SI system the mass unit is the kg and since the weight is a force - the weight unit is the Newton ( N ). Equation (2) for a body with 1 kg mass can be expressed as: Fg = (1 kg) (9.807 m/s2) = 9.807 (N) where 9.807 m/s2= standard gravity close to earth in the SI system As a result: a 9.807 N force acting on a body with 1 kg mass will give the body an acceleration of 9.807 m/s2 a body with mass of 1 kg weights 9.807 N More about the SI System - A tutorial introduction to the SI-system. The Imperial British Gravitational System - BG The British Gravitational System (Imperial System) of units is used by engineers in the English-speaking world with the same relation to the foot - pound - second system as the meter - kilogram - force second system (SI) has to the meter - kilogram - second system. For engineers who deals with forces, instead of masses, it's convenient to use a system that has as its base units length, time, and force , instead of length, time and mass . The three base units in the Imperial system are foot, second and pound-force . In the BG system the mass unit is the slug and is defined from the Newton's Second Law (1) . The unit of mass, the slug , is derived from the pound-force by defining it as the mass that will accelerate with 1 foot per second per second when a 1 pound-force acts upon it: 1 lbf = (1 slug) (1 ft/s2) In other words, 1 lbf (pound-force) acting on 1 slug of mass will give the mass an acceleration of 1 ft/s2. The weight (force) of the mass can be calculated from equation (2) in BG units as Fg (lbf) = (m slugs) (ag ft/s2) With standard gravity ag = 32.17405 ft/s2 the weight (force) of 1 slug mass can be calculated as Fg = (1 slug) (32.17405 ft/s2) 32.17405 lbf The English Engineering System - EE In the English Engineering system of units the primary dimensions are are force, mass, length, time and temperature. The units for force and mass are defined independently the basic unit of mass is pound-mass (lbm) the unit of force is the pound (lb) alternatively pound-force (lbf). In the EE system 1 lbf of force will give a mass of 1 lbm a standard acceleration of 32.17405 ft/s2. Since the EE system operates with these units of force and mass, the Newton's Second Law can be modified to F = m a / gc (3) where gc = a proportionality constant or transformed to weight (force) Fg = m ag / gc (4) The proportionality constant gc makes it possible to define suitable units for force and mass. We can transform (4) to 1 lbf = (1 lbm) (32.174 ft/s2) / gc or gc = (1 lbm) (32.174 ft/s2) / (1 lbf) Since 1 lbf gives a mass of 1 lbm an acceleration of 32.17405 ft/s2 and a mass of 1 slug an acceleration of 1 ft/s2, then 1 slug = 32.17405 lbm Example - Weight versus Mass The mass of a car is 1644 kg. The weight can be calculated: Fg = (1644 kg) (9.807 m/s2) = 16122.7 N = 16.1 kN there is a force (weight) of 16.1 kN between the car and the earth. 1 kg gravitation force = 9.81 N = 2.20462 lbf Weight Converter Make a Shortcut to this Calculator on Your Home Screen? Kg to lb Converter Download and print kg to lb Converter! Unit Converter . Cookie Settings \| \| \| --- \| \| \| \| \| \| \| --- \| \| \| \|
4694	https://kconrad.math.uconn.edu/blurbs/grouptheory/cyclicgp.pdf	SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g ∈G by ⟨g⟩= {gk : k ∈Z}. If G = ⟨g⟩, then G itself is cyclic, with g as a generator. Examples of inﬁnite cyclic groups include Z, with (additive) generator 1, and the group 2Z of integral powers of the real number 2, with generator 2. The most basic examples of ﬁnite cyclic groups are Z/(m) with (additive) generator 1 and µm = {z ∈C× : zm = 1} with generator e2πi/m = cos(2π/m)+i sin(2π/m). Both have size m. The elements of µm lie at the vertices of a regular m-gon on the unit circle and include 1. This is the fundamental geometric way to picture µm. Don’t confuse the groups Z/(m) and (Z/(m))×. The second, as a set, is a subset of the ﬁrst, but it is not a subgroup and it is usually not cyclic (see Theorem A.2). For instance, (Z/(15))× is not cyclic. But many of the groups (Z/(m))× are cyclic, and we will take our examples from such groups which are cyclic. Example 1.1. The group (Z/(23))× has size 22, and it is cyclic with 5 as a generator. (The elements 2 and 3 each have order 11, so they are not generators.) In this handout, we describe the subgroups of a general cyclic group G = ⟨g⟩. Here are the main results, in brief: • Every subgroup of G is cyclic. • If G is inﬁnite then each subgroup has the form ⟨gn⟩for a unique integer n ≥0. • If G is ﬁnite, of size m, then each subgroup has the form ⟨gd⟩, where d is a unique positive divisor of m. For k ∈Z, ⟨gk⟩= ⟨g(k,m)⟩. • The containment relations among subgroups of a cyclic group resemble divisibility relations among integers. 2. Subgroups are always cyclic Let G be a cyclic group. We will show every subgroup of G is also cyclic, taking separately the cases of inﬁnite and ﬁnite G. Theorem 2.1. Every subgroup of a cyclic group is cyclic. Proof. Let G be a cyclic group, with generator g. For a subgroup H ⊂G, we will show H = ⟨gn⟩for some n ≥0, so H is cyclic. The trivial subgroup is obviously of this form (e = g0). So we may suppose H is non-trivial. How are we going to ﬁnd the n ≥1 making H = ⟨gn⟩? The idea is that every element in a subgroup of the form ⟨gn⟩looks like gnk, so n is the smallest positive exponent among the 1 2 KEITH CONRAD elements of this subgroup. With this in mind, given a non-trivial subgroup H ⊂G, deﬁne n to be the smallest positive integer such that gn ∈H. (There are such powers; H contains some non-zero power of g, which becomes a positive power after inversion if necessary.) Now we show any h ∈H is a power of gn. (Any element of ⟨gn⟩lies in H, since gn already does.) The whole group G is generated by g, so h = ga for some a ∈Z. We want n \| a. Well, by the division theorem, a = nq + r for some integers q and r such that 0 ≤r < n. Therefore h = ga = (gn)qgr = ⇒gr = (gn)−qh. Since gn ∈H, this shows gr ∈H. From 0 ≤r < n, the minimality condition deﬁning n implies r can’t be positive, so r = 0. Thus n \| a, so h = ga ∈⟨gn⟩. This proves H = ⟨gn⟩. □ When g has inﬁnite order, any gn with n ̸= 0 also has inﬁnite order, so all non-trivial subgroups of an inﬁnite cyclic group are again inﬁnite cyclic groups. In particular, a subgroup of an inﬁnite cyclic group is again an inﬁnite cyclic group. Theorem 2.1 tells us how to ﬁnd all the subgroups of a ﬁnite cyclic group: compute the subgroup generated by each element and then just check for redundancies. Example 2.2. Let G = (Z/(7))×. We list in the following table the successive powers of an element until we hit 1. a ⟨a mod 7⟩ 1 {1} 2 {2, 4, 1} 3 {3, 2, 6, 4, 5, 1} 4 {4, 2, 1} 5 {5, 4, 6, 2, 3, 1} 6 {6, 1} Each subgroup of (Z/(7))× is cyclic, so it must be in the above table. Taking into account that some subgroups are appearing more than once, (Z/(7))× has four subgroups: {1}, {1, 6}, {1, 2, 4}, and {1, 2, 3, 4, 5, 6}. (We have rewritten the elements in each subgroup in increasing order as integers, for lack of a better idea at this point.) We will see in the next section a better way to systematically enumerate the subgroups of a ﬁnite cyclic group, which will avoid the kinds of redundancies we met in Example 2.2. 3. Enumerating subgroups of a cyclic group By Theorem 2.1, every subgroup of Z (additive!) has the form nZ for some integer n, and we can take n ≥0. Obviously diﬀerent choices of n ≥0 give us diﬀerent subgroups. The story turns out to be the same for any inﬁnite cyclic group. Lemma 3.1. If g is an element of inﬁnite order in a group, then gk = gℓif and only if k = ℓ. If g is an element of ﬁnite order m in a group, then gk = gℓif and only if k ≡ℓmod m. Proof. If gk = gℓand k ̸= ℓ, we may suppose k < ℓ. Then gℓ−k = e, with ℓ−k ̸= 0, so g has ﬁnite order. Thus, contrapositively, if g has inﬁnite order and gk = gℓthen k = ℓ. SUBGROUPS OF CYCLIC GROUPS 3 If g has ﬁnite order m and gk = gℓ, then gk−ℓ= e, so m \| (k −ℓ) and thus k ≡ℓmod m. Conversely, if k ≡ℓmod m, then k = ℓ+ mj, so gk = gℓ+mj = gℓ(gm)j = gℓej = gℓ. □ Theorem 3.2. Each non-trivial subgroup of an inﬁnite cyclic group has two generators, which are inverses of each other. Fixing one generator g of the whole group, we can write each subgroup in the form ⟨gn⟩for a unique n ≥0. Proof. Theorem 2.1 tells us any subgroup has the form ⟨gn⟩for an integer n. Since ⟨gn⟩= ⟨g−n⟩, we can take n ≥0. (Actually, the proof of Theorem 2.1 produced the n as a non-negative integer already, so we didn’t really need this last argument to bring us to the case n ≥0.) As long as we can prove gn is the only generator of ⟨gn⟩with a positive exponent, then gn and g−n (its inverse) are the only generators of ⟨gn⟩at all. If our subgroup is trivial, we must use n = 0, in which case the desired conclusions are all easy to check. Suppose we are working with non-trivial subgroups. Then we want to show, if n and n′ are positive, that ⟨gn⟩= ⟨gn′⟩only when n = n′. When those subgroups are equal, gn and gn′ are both powers of each other, so gn = gn′s and gn′ = gnt for some s and t in Z. By Lemma 3.1, n = n′s and n′ = nt, so n and n′ divide each other. Therefore they are equal (since both are positive, so no sign ambiguity occurs). □ Corollary 3.3. In an inﬁnite cyclic group, with generator g, ⟨gn⟩⊂⟨gn′⟩if and only if n′ \| n. Notice the divisibility is in reverse order to the inclusion. This makes sense, e.g., ⟨g6⟩⊂ ⟨g2⟩since g6 is a power of g2, while 2 \| 6 (not 6 \| 2). Proof. The condition ⟨gn⟩⊂⟨gn′⟩is the same as gn being a power of gn′, say gn = gn′s for some s. By Lemma 3.1, n = n′s, so n′ \| n. □ Example 3.4. If ⟨g⟩is an inﬁnite cyclic group, the subgroups inside of ⟨g6⟩are those of the form ⟨g6s⟩. The subgroups which contain ⟨g6⟩are ⟨g⟩, ⟨g2⟩, ⟨g3⟩, and ⟨g6⟩. Now we turn to the ﬁnite case. Theorem 3.5. In a ﬁnite cyclic group, each subgroup has size dividing the size of the group. Conversely, given a positive divisor of the size of the group, there is a subgroup of that size. Proof. Let G = ⟨g⟩, with size m. Any subgroup has the form ⟨gk⟩for some k. The size of this subgroup is the order of gk, which is m/(k, m) by the handout on orders of elements in a group. Therefore the size of each subgroup divides m. Given a positive divisor d of m, we can write down an element of order d in terms of the chosen generator of G: gm/d has order m/(m/d) = d. Therefore ⟨gm/d⟩has size d. □ There is an essential strengthening of Theorem 3.5: for each divisor of the size of a (ﬁnite) cyclic group, there is exactly one subgroup of that size. To see why, we show how to change the generator of a subgroup into one with a “standard” form (in terms of the choice of a generator for the whole group), and then compare subgroups using such standardized generators (standardized in terms of the choice of a generator for the whole group). Theorem 3.6. Let G = ⟨g⟩be a ﬁnite cyclic group, with size m. For k ∈Z, ⟨gk⟩= ⟨g(k,m)⟩. In particular, any subgroup of G has the form ⟨gd⟩where d is a positive divisor of m. Diﬀerent values of d give subgroups with diﬀerent sizes, so there is just one subgroup of G having a given size. 4 KEITH CONRAD Proof. We know by Theorem 2.1 that any subgroup of G is cyclic. Write it as ⟨gk⟩. We will show gk and g(k,m) are both powers of each other, so they generate the same subgroup. Since (k, m) \| k, gk is certainly a power of g(k,m). To show g(k,m) is a power of gk, we use Bezout’s identity: (k, m) = kx + my for some x, y ∈Z. This tells us g(k,m) = gkxgmy = gkx, where we used gm = e to make the second simpliﬁcation. (Here m = \|G\|, and we saw in class that g\|G\| = e in any ﬁnite abelian group, such as a ﬁnite cyclic group.) Now we know any subgroup is generated by gd, where d is some positive divisor of m. How large is such a subgroup ⟨gd⟩? Since d \| m, gd has order m/d, so \|⟨gd⟩\| = m/d. If d and d′ are diﬀerent positive divisors of m, then ⟨gd⟩̸= ⟨gd′⟩since the two subgroups have diﬀerent sizes. □ Remark 3.7. In the enumeration of subgroups of G according to the divisors of m = \|G\|, the trivial subgroup corresponds to the divisor m. Corollary 3.8. In a ﬁnite cyclic group, two elements generate the same subgroup if and only if the elements have the same order. Proof. The order of an element is the size of the subgroup it generates, so elements generat-ing the same subgroup have the same order. In the other direction, elements with the same order generate subgroups with the same size, and these subgroups are equal since there is just one subgroup with any possible size (Theorem 3.6). □ Remark 3.9. The conclusion of Corollary 3.8 is completely false for ﬁnite non-cyclic groups: equal order hardly has to imply elements generate the same subgroup. Consider (Z/(15))× = {1, 2, 4, 7, 8, 11, 13, 14 mod 15}. The largest order of an element is 4, so this group is not cyclic. There are several subgroups of size 2, such as {1, 14}, {1, 4}, {1, 11}. Corollary 3.10. Let G be a ﬁnite cyclic group. For subgroups H and H′, H ⊂H′ if and only if \|H\| \| \|H′\|. Writing G = ⟨g⟩, m = \|G\|, H = ⟨gn⟩, and H′ = ⟨gn′⟩, ⟨gn⟩⊂⟨gn′⟩if and only if (n′, m) \| (n, m). Proof. Both H and H′ are cyclic, by Theorem 2.1. Thus, if H is a subgroup of H′, then \|H\| \| \|H′\| by Theorem 3.5. Conversely, suppose \|H\| \| \|H′\|. Applying Theorem 3.5 to H′, there is a subgroup K ⊂H′ whose size is \|H\|. Then K will also be a subgroup of G with size \|H\|. Since G can only have one subgroup of any possible size, by Theorem 3.6, K = H. We now translate this result into the language of explicit generators and exponents. Set G = ⟨g⟩, m = \|G\|, H = ⟨gn⟩, and H′ = ⟨gn′⟩. Then ⟨gn⟩⊂⟨gn′⟩if and only if the order of gn divides the order of gn′. This is equivalent to m/(n, m) dividing m/(n′, m), which is the same as (n′, m) \| (n, m). □ Remark 3.11. When we describe containment of subgroups in terms of their sizes, smaller subgroups correspond to divisors. When we describe containment of subgroups in terms of exponents in the generators, smaller subgroups then correspond to multiples, which resembles Corollary 3.3. SUBGROUPS OF CYCLIC GROUPS 5 Example 3.12. Let G = (Z/(11))×. This group has size 10, and is cyclic. One of its generators is 2. k 1 2 3 4 5 6 7 8 9 10 2k mod 11 2 4 8 5 10 9 7 3 6 1 The subgroups of (Z/(11))× are therefore ⟨2⟩of order 10, ⟨22⟩of order 5, ⟨25⟩of order 2, ⟨210⟩of order 1. In the following diagram we draw the subgroups as a lattice, with the whole group at the top and the trivial subgroup at the bottom. The index of a smaller subgroup in a larger subgroup is indicated on the line connecting the subgroups. ⟨2⟩ 2 5 ⟨22⟩ 5 ⟨25⟩ 2 ⟨210⟩ The top subgroup is (Z/(11))×, the bottom one is {1}, and the others have order 2 and 5. Which of these subgroups is ⟨3⟩? We will answer this in three ways. Method 1: Compute the order of 3 in (Z/(11))×. It is 5: 32 ≡9 mod 11, 33 ≡5 mod 11, 34 ≡4 mod 11, 35 ≡1 mod 11. (We could have found this from the previously computed table of powers of 2: since 3 ≡ 28 mod 11, 3 has order 10/(8, 10) = 5.) There is just one subgroup of size 5, namely ⟨22⟩, so this is ⟨3⟩. Method 2: The table of powers of 2 shows 3 ≡28 mod 11, so 3 generates the same subgroup of (Z/(11))× as 2(8,10) = 22. Method 3: Write out the intermediate subgroups explicitly: ⟨22⟩= {22, 24, 26, 28, 210} = {4, 5, 9, 3, 1}, ⟨25⟩= {25, 210} = {10, 1}. In this list, we see 3 shows up in just once, so that has to be the subgroup generated by 3. This is the worst of the three methods, in terms of eﬃciency. If we changed the choice of generator for the group, then our labels for the subgroups would change. For instance, take 8 = 23 as the generator for (Z/(11))×. Then the subgroups have generators 8d for d ∈{1, 2, 5, 10}. These are the elements 81 = 8, 82 = 9, 85 = 10, 810 = 1. ⟨2⟩ 2 5 ⟨8⟩ 2 5 ⟨22⟩ 5 ⟨25⟩ 2 ⟨82⟩ 5 ⟨85⟩ 2 ⟨210⟩ ⟨810⟩ 6 KEITH CONRAD We can avoid generators and label the subgroups simply by their size: there is one of size 1, one of size 2, one of size 5, and one of size 10. A converse to Theorem 3.6 is also true: a ﬁnite group which has at most one subgroup of any size is cyclic. The proof is omitted since it needs more information about the subgroups of ﬁnite groups (Lagrange’s theorem). While some computational aspects of cyclic groups depend on the particular generator being used, the intrinsic group-theoretical properties of the cyclic group certainly don’t depend on the choice of a generator. For instance, the statements of Theorems 2.1 and 3.5, the very end of Theorem 3.6, and Corollary 3.8 don’t depend at all on the choice of a generator for their meaning. Example 3.13. Let G = (Z/(49))×. This is a group of size 42. (Its elements are represented by integers not divisible by 7, so it has 49 −7 = 42 members.) The reader can check 3 is a generator of (Z/(49))×. The subgroups of (Z/(49))× are therefore ⟨3d⟩where d runs through the divisors of 42: ⟨3⟩, ⟨32⟩, ⟨33⟩, ⟨36⟩, ⟨37⟩, ⟨314⟩, ⟨321⟩, ⟨342⟩. ⟨3⟩ 2 3 7 ⟨32⟩ 3 ⟨33⟩ ⟨37⟩ 3 ⟨36⟩ 7 ⟨314⟩ 3 ⟨321⟩ 2 ⟨342⟩ Which of these subgroups is ⟨312⟩? It is ⟨3(12,42)⟩= ⟨36⟩. That is, 312 and 36 generate the same subgroup. Which of these subgroups is ⟨6⟩? To answer this we compute the order of 6 mod 49. A tedious calculation shows the order is 14. The subgroup ⟨3d⟩with size 14 is ⟨342/14⟩= ⟨33⟩, so ⟨6⟩= ⟨33⟩. Example 3.14. Let G = µm. When d \| m, the complex d-th roots of unity are m-th roots of unity, so µd ⊂µm. Conversely, if µd ⊂µm, then \|µd\| \| \|µm\| by Theorem 3.5, so d \| m. Theorem 3.6 tells us a ﬁnite cyclic group has just one subgroup of each possible size, so the subgroups of µm are exactly the groups µd as d runs over the positive divisors of m. Example 3.15. The group (Z/(13))× is cyclic (a generator is 2), with size 12. The sub-groups have size 1,2,3,4,6, and 12. The subgroup of size 4 contains just the subgroups of sizes 1, 2, and 4, and the subgroup of size 3 is contained in the subgroups of sizes 3, 6, and 12. Using the explicit generator of the whole group, we get generators for the subgroups, which are listed below. SUBGROUPS OF CYCLIC GROUPS 7 d Subgp. of size d 1 ⟨1⟩ 2 ⟨26⟩ 3 ⟨24⟩ 4 ⟨23⟩ 6 ⟨22⟩ 12 ⟨2⟩ Here is a diagram of the subgroups of (Z/(13))×. ⟨2⟩ 2 3 ⟨22⟩ 3 2 ⟨23⟩ 2 ⟨24⟩ 3 ⟨26⟩ 2 ⟨212⟩ 4. Abstract versus concrete cyclic groups The properties we have discussed for cyclic groups and their subgroups have relied on nothing about such groups other than their sizes. This is in the nature of things: cyclic groups of the same size are structurally more or less the same thing. Without being com-pletely precise (yet) about what this means, let’s see why any inﬁnite cyclic group essentially resembles Z and any ﬁnite cyclic group of size m essentially resembles Z/(m). Suppose G is a cyclic group. Choose a generator, say g. We can write each element of G in the form gk for an integer k. Lemma 3.1 tells us that if G is inﬁnite then gk determines k as an integer, while if G is ﬁnite of size m then gk determines k as an integer modulo m: gk = gℓif and only k ≡ℓmod m. In either case, the way gk and gℓmultiply coincides with the way the exponents add: gkgℓ= gk+ℓ. This suggests we can set up a correspondence between G and Z or some Z/(m), as follows: • If G is inﬁnite, let gk ∈G correspond to k ∈Z. • If \|G\| = m, let gk ∈G correspond to k mod m ∈Z/(m). These correspondences are meaningful only because of Lemma 3.1, which lets us extract knowledge of the exponent from knowledge of the power as either an integer (when G is inﬁnite) or as an integer modulo m (when \|G\| = m). The multiplication in G corresponds to the addition in Z or Z/(m), depending on the size of G. Therefore we are able to translate all the behavior of an abstract cyclic group into behavior in the concrete additive groups of integers or integers modulo m. By following the exponents, any abstract cyclic group becomes one of the basic cyclic groups Z or Z/(m) for some m > 0. (Note m = 1 corresponds to the trivial cyclic group of size 1.) Therefore any two abstract cyclic groups of the same size are very similar kinds of groups, since they have the same “model” group built out of integers. 8 KEITH CONRAD (Since Z = Z/(0), we can actually use the notation Z/(m) in all cases, taking m = 0 in the case of inﬁnite cyclic groups. Note congruence modulo 0 is just ordinary equality.) While we have argued that any two cyclic groups of the same size are essentially in-terchangeable insofar as their abstract mathematical features are concerned, there is an important practical consideration: it is easy to write down a generator of the additive cyclic groups Z and Z/(m) (use 1 in either case), but other cyclic groups need not have a generator that can be written down easily. An important example to keep in mind in this respect is (Z/(p))×, where p is prime. This group has size p −1 and it can be proved to be cyclic (but we don’t include a proof here). To this day, no method of ﬁnding a generator of (Z/(p))× is known to be more eﬃcient than essentially trying 2, then 3, and so on. Who cares? Well, the diﬃculty of breaking a certain public key crytosystem (due to El Gamal) depends on the diﬃculty of working with generators of (Z/(p))×. Appendix A. More applications Our ﬁrst application of the structure of subgroups of cyclic groups is a non-constructive proof of Bezout’s identity. The proof here is completely independent of Euclid’s algorithm, which is how the identity is proved in a constructive way. Theorem A.1. For any a, b ∈Z, we have aZ + bZ = (a, b)Z, aZ ∩bZ = [a, b]Z. In particular, there are x and y such that ax + by = (a, b). Proof. Given two subgroups H and K of Z, we can construct two other subgroups out of them: H + K (the set of sums, one term from H and the other term from K) and H ∩K. Any subgroup of Z is cyclic, so we must have aZ + bZ = cZ, aZ ∩bZ = dZ for some integers c and d. Without loss of generality, c and d are non-negative. We will show c = (a, b) and d = [a, b]. The elements of cZ are the multiples of c. Since a = a·1+b·0, we obtain c \| a. Similarly, c \| b, so c is a (non-negative) common divisor of a and b. Since c ∈aZ + bZ, we get c = ax + by for some x and y in Z. Then any common divisor of a and b is a common divisor of ax + by = c, which means c = (a, b). That d = [a, b] is left to the reader. □ Here is a neat application of Theorem 3.6: a strong constraint on those m for which (Z/(m))× could be a cyclic group. Theorem A.2. If (Z/(m))× is cyclic, then m must be 2, 4, an odd prime power, or twice an odd prime power. Proof. The idea is this: for any m other than 2, 4, an odd prime power, or twice an odd prime power, we will construct two diﬀerent subgroups of (Z/(m))× with size 2. There can’t be two subgroups of the same size in a cyclic group, by Theorem 3.6, so (Z/(m))× is not cyclic! One subgroup of (Z/(m))× with size 2 is generated by −1. We will ﬁnd an element of order 2 other than −1. Such a “fake” square root of 1 modulo m generates a second subgroup of size 2. SUBGROUPS OF CYCLIC GROUPS 9 First suppose m is a power of 2 bigger than 4: m = 2s with s ≥3. Then (1 + 2s−1)2 ≡ 1 mod 2s and 1 + 2s−1 ̸≡±1 mod 2s since s > 2, so 1 + 2s−1 mod 2s has order 2 and is not −1 mod 2s. The moduli m left to consider are those that are not a prime power or twice an odd prime power. Let m be such a number (m could be 15, 90, and so on). Then there is a way of writing m = ab where (a, b) = 1 and a, b > 2. (For instance, let a be the largest power of some odd prime dividing m and take for b the complementary divisor, e.g., 90 is 9 · 10 or 5 · 18.) By Bezout, we can write 1 = ax + by for some x, y ∈Z. Let t = ax −by = 1 −2by = −1 + 2ax. We will show t2 ≡1 mod m and t ̸≡±1 mod m. (Our choice of t may seem strange at ﬁrst, although those who have seen some number theory will recognize in the next paragraph that our choice was dictated with the Chinese remainder theorem in mind, for moduli a and b.) The diﬀerent formulas for t show t ≡−1 mod a and t ≡1 mod b, so t2 ≡1 mod a and t2 ≡1 mod b. Then a and b both divide t2 −1, so (since a and b are relatively prime) m \| (t2 −1). We have t2 ≡1 mod m. To show t ̸≡±1 mod m, we argue by contradiction. If t ≡1 mod m, then t ≡1 mod a. Already we have t ≡−1 mod a, so 1 ≡−1 mod a. This is impossible, since a > 2. Similarly, since t ≡1 mod b and b > 2, we can’t have t ≡−1 mod m. □ Corollary A.2 actually describes exactly those m ≥2 for which (Z/(m))× is not cyclic: when m is 2, 4, an odd prime power, or twice an odd prime power, then (Z/(m))× is cyclic. The proof of that is omitted.
4695	https://amesweb.info/Materials/Linear_Thermal_Expansion_Coefficient_of_Steel.aspx	Linear Thermal Expansion Coefficient of Steel AmesWeb LINEAR THERMAL EXPANSION COEFFICIENT OF STEEL Linear thermal expansion coefficients of various steels are given in the following chart. Room Temperature Linear Thermal Expansion Coefficient Values for Steels Material Temp.Coef. of Thermal Expansion (CTE) 10-6 (°C)-1 10-6 (°F)-1 Plain Carbon and Low Alloy Steels AISI 1010, Annealed 0-100°C / 32-212°F 12.2 6.8 AISI 1020, Annealed 0-100°C / 32-212°F 11.7 6.5 AISI 1025, Annealed 0-100°C / 32-212°F 12.0 6.7 AISI 1040, Annealed 0-100°C / 32-212°F 11.2 6.2 AISI 1045, Annealed 0-100°C / 32-212°F 11.6 6.4 Alloy Steels AISI 4140, Oil Hardened, tempered 0-100°C / 32-212°F 12.3 6.8 AISI 4340, Oil Hardened, tempered 600 °C 20-100°C / 68-212°F 12.3 6.8 Stainless Steels Grade 304, Annealed 0-100°C / 32-212°F 17.2 9.6 Grade 316, Annealed 0-100°C / 32-212°F 15.9 8.8 Grade 405 0-100°C / 32-212°F 10.8 6.00 Grade 440C 0-100°C / 32-212°F 10.08 5.60 PH 15-7 Mo, Annealed 21-93ºC / 70-200ºF 14.4 8.0 17-4 PH, Annealed 20°C / 68°F 10.8 6.00 17-7 PH, Annealed 21-93ºC / 70-200ºF 15.3 8.5 Tool Steels D2 tool steel, Annealed 20-100°C / 68-212°F 10.5 5.81 T1 tool steel 20-200°C / 68-390°F 9.7 5.4 M2 tool steel 20-100°C / 68-212°F 10.1 5.6 W1 tool steel 20-100°C / 68-212°F 10.4 5.76 O6 tool steel, Annealed 25-450°C / 77-842°F 13.65 7.6 Reference: Cverna, F., & ASM International. (2002). ASM Ready Reference: Thermal Properties of Metals (Materials Data Series). Materials Park, Ohio: ASM International. About us\| Contact us\| Disclaimer \| Privacy Policy Copyright © 2013-2023
4696	https://en.wikipedia.org/wiki/Radius_of_curvature	Radius of curvature - Wikipedia Jump to content [x] Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search [x] Appearance Donate Create account Log in [x] Personal tools Donate Create account Log in Pages for logged out editors learn more Contributions Talk Contents move to sidebar hide (Top) 1 Definition 2 FormulaToggle Formula subsection 2.1 In two dimensions 2.2 In n dimensions 2.3 Derivation 3 ExamplesToggle Examples subsection 3.1 Semicircles and circles 3.2 Ellipses 4 ApplicationsToggle Applications subsection 4.1 Stress in semiconductor structures 5 See also 6 References 7 Further reading 8 External links [x] Toggle the table of contents Radius of curvature [x] 18 languages العربية বাংলা Deutsch Eesti Español فارسی Français 한국어 हिन्दी עברית മലയാളം 日本語 Português Suomi தமிழ் اردو Tiếng Việt 中文 Edit links Article Talk [x] English Read Edit View history [x] Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikidata item Appearance move to sidebar hide From Wikipedia, the free encyclopedia Radius of the circle which best approximates a curve at a given point This article is about the general mathematical concept. For its optical applications, see Radius of curvature (optics). Radius of curvature and center of curvature In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof. Definition [edit] In the case of a space curve, the radius of curvature is the length of the curvature vector. In the case of a plane curve, then R is the absolute value of R≡\|d s d φ\|=1 κ,{\displaystyle R\equiv \left\|{\frac {ds}{d\varphi }}\right\|={\frac {1}{\kappa }},} where s is the arc length from a fixed point on the curve, φ is the tangential angle and κ is the curvature. Formula [edit] In two dimensions [edit] Further information: Curvature §Plane curves If the curve is given in Cartesian coordinates as y(x), i.e., as the graph of a function, then the radius of curvature is (assuming the curve is differentiable up to order 2) R=\|(1+y′2)3 2 y″\|,{\displaystyle R=\left\|{\frac {\left(1+y'^{\,2}\right)^{\frac {3}{2}}}{y''}}\right\|\,,} where y′=d y d x,{\textstyle y'={\frac {dy}{dx}}\,,}y″=d 2 y d x 2,{\textstyle y''={\frac {d^{2}y}{dx^{2}}},} and \|z\| denotes the absolute value of z. If the curve is given parametrically by functions x(t) and y(t), then the radius of curvature is R=\|d s d φ\|=\|(x˙2+y˙2)3 2 x˙y¨−y˙x¨\|{\displaystyle R=\left\|{\frac {ds}{d\varphi }}\right\|=\left\|{\frac {\left({{\dot {x}}^{2}+{\dot {y}}^{2}}\right)^{\frac {3}{2}}}{{\dot {x}}{\ddot {y}}-{\dot {y}}{\ddot {x}}}}\right\|} where x˙=d x d t,{\textstyle {\dot {x}}={\frac {dx}{dt}},}x¨=d 2 x d t 2,{\textstyle {\ddot {x}}={\frac {d^{2}x}{dt^{2}}},}y˙=d y d t,{\textstyle {\dot {y}}={\frac {dy}{dt}},} and y¨=d 2 y d t 2.{\textstyle {\ddot {y}}={\frac {d^{2}y}{dt^{2}}}.} Heuristically, this result can be interpreted as R=\|v\|3\|v×v˙\|,{\displaystyle R={\frac {\left\|\mathbf {v} \right\|^{3}}{\left\|\mathbf {v} \times \mathbf {\dot {v}} \right\|}}\,,} where \|v\|=\|(x˙,y˙)\|=R d φ d t.{\displaystyle \left\|\mathbf {v} \right\|={\big \|}({\dot {x}},{\dot {y}}){\big \|}=R{\frac {d\varphi }{dt}}\,.} In n dimensions [edit] If γ: ℝ → ℝ n is a parametrized curve in ℝ n then the radius of curvature at each point of the curve, ρ: ℝ → ℝ, is given by ρ=\|γ′\|3\|γ′\|2\|γ″\|2−(γ′⋅γ″)2.{\displaystyle \rho ={\frac {\left\|{\boldsymbol {\gamma }}'\right\|^{3}}{\sqrt {\left\|{\boldsymbol {\gamma }}'\right\|^{2}\,\left\|{\boldsymbol {\gamma }}''\right\|^{2}-\left({\boldsymbol {\gamma }}'\cdot {\boldsymbol {\gamma }}''\right)^{2}}}}\,.} As a special case, if f(t) is a function from ℝ to ℝ, then the radius of curvature of its graph, γ(t) = (t, f (t)), is ρ(t)=\|1+f′2(t)\|3 2\|f″(t)\|.{\displaystyle \rho (t)={\frac {\left\|1+f'^{\,2}(t)\right\|^{\frac {3}{2}}}{\left\|f''(t)\right\|}}.} Derivation [edit] Let γ be as above, and fix t. We want to find the radius ρ of a parametrized circle which matches γ in its zeroth, first, and second derivatives at t. Clearly the radius will not depend on the position γ(t), only on the velocity γ′(t) and acceleration γ″(t). There are only three independent scalars that can be obtained from two vectors v and w, namely v · v, v · w, and w · w. Thus the radius of curvature must be a function of the three scalars \|γ′(t)\|2, \|γ″(t)\|2 and γ′(t) · γ″(t). The general equation for a parametrized circle in ℝ n is g(u)=a cos⁡(h(u))+b sin⁡(h(u))+c{\displaystyle \mathbf {g} (u)=\mathbf {a} \cos(h(u))+\mathbf {b} \sin(h(u))+\mathbf {c} } where c ∈ ℝ n is the center of the circle (irrelevant since it disappears in the derivatives), a,b ∈ ℝ n are perpendicular vectors of length ρ (that is, a · a = b · b = ρ 2 and a · b = 0), and h: ℝ → ℝ is an arbitrary function which is twice differentiable at t. The relevant derivatives of g work out to be \|g′\|2=ρ 2(h′)2 g′⋅g″=ρ 2 h′h″\|g″\|2=ρ 2((h′)4+(h″)2){\displaystyle {\begin{aligned}\|\mathbf {g} '\|^{2}&=\rho ^{2}(h')^{2}\\mathbf {g} '\cdot \mathbf {g} ''&=\rho ^{2}h'h''\\|\mathbf {g} ''\|^{2}&=\rho ^{2}\left((h')^{4}+(h'')^{2}\right)\end{aligned}}} If we now equate these derivatives of g to the corresponding derivatives of γ at t we obtain \|γ′(t)\|2=ρ 2 h′2(t)γ′(t)⋅γ″(t)=ρ 2 h′(t)h″(t)\|γ″(t)\|2=ρ 2(h′4(t)+h″2(t)){\displaystyle {\begin{aligned}\|{\boldsymbol {\gamma }}'(t)\|^{2}&=\rho ^{2}h'^{\,2}(t)\{\boldsymbol {\gamma }}'(t)\cdot {\boldsymbol {\gamma }}''(t)&=\rho ^{2}h'(t)h''(t)\\|{\boldsymbol {\gamma }}''(t)\|^{2}&=\rho ^{2}\left(h'^{\,4}(t)+h''^{\,2}(t)\right)\end{aligned}}} These three equations in three unknowns (ρ, h′(t) and h″(t)) can be solved for ρ, giving the formula for the radius of curvature: ρ(t)=\|γ′(t)\|3\|γ′(t)\|2\|γ″(t)\|2−(γ′(t)⋅γ″(t))2,{\displaystyle \rho (t)={\frac {\left\|{\boldsymbol {\gamma }}'(t)\right\|^{3}}{\sqrt {\left\|{\boldsymbol {\gamma }}'(t)\right\|^{2}\,\left\|{\boldsymbol {\gamma }}''(t)\right\|^{2}-{\big (}{\boldsymbol {\gamma }}'(t)\cdot {\boldsymbol {\gamma }}''(t){\big )}^{2}}}}\,,} or, omitting the parameter t for readability, ρ=\|γ′\|3\|γ′\|2\|γ″\|2−(γ′⋅γ″)2.{\displaystyle \rho ={\frac {\left\|{\boldsymbol {\gamma }}'\right\|^{3}}{\sqrt {\left\|{\boldsymbol {\gamma }}'\right\|^{2}\;\left\|{\boldsymbol {\gamma }}''\right\|^{2}-\left({\boldsymbol {\gamma }}'\cdot {\boldsymbol {\gamma }}''\right)^{2}}}}\,.} Examples [edit] Semicircles and circles [edit] For a semi-circle of radius a in the upper half-plane with R=\|−a\|=a,{\textstyle R=\|-a\|=a\,,} y=a 2−x 2 y′=−x a 2−x 2 y″=−a 2(a 2−x 2)3 2.{\displaystyle {\begin{aligned}y&={\sqrt {a^{2}-x^{2}}}\y'&={\frac {-x}{\sqrt {a^{2}-x^{2}}}}\y''&={\frac {-a^{2}}{\left(a^{2}-x^{2}\right)^{\frac {3}{2}}}}\,.\end{aligned}}} For a semi-circle of radius a in the lower half-plane y=−a 2−x 2.{\displaystyle y=-{\sqrt {a^{2}-x^{2}}}\,.} The circle of radius a has a radius of curvature equal to a. Ellipses [edit] An ellipse (red) and its evolute (blue). The dots are the vertices of the ellipse, at the points of greatest and least curvature. In an ellipse with major axis 2 a and minor axis 2 b, the vertices on the major axis have the smallest radius of curvature of any points, R=b 2 a{\textstyle R={b^{2} \over a}}; and the vertices on the minor axis have the largest radius of curvature of any points, R = ⁠a 2/b⁠. The radius of curvature of an ellipse as a function of the geocentric coordinate t{\displaystyle t} with tan⁡t=y x{\displaystyle \tan t={\frac {y}{x}}} is R(t)=(b 2 cos 2⁡t+a 2 sin 2⁡t)3/2 a b.{\displaystyle R(t)={\frac {(b^{2}\cos ^{2}t+a^{2}\sin ^{2}t)^{3/2}}{ab}}.}It has its minima at t=0{\displaystyle t=0} and t=180∘{\displaystyle t=180^{\circ }} and its maxima at t=±90∘{\displaystyle t=\pm 90^{\circ }}. Applications [edit] For the use in differential geometry, see Cesàro equation. For the radius of curvature of the Earth (approximated by an oblate ellipsoid); see also: arc measurement Radius of curvature is also used in a three part equation for bending of beams. Radius of curvature (optics) Thin films technologies Printed electronics Minimum railway curve radius AFM probe Stress in semiconductor structures [edit] Stress in the semiconductor structure involving evaporated thin films usually results from the thermal expansion (thermal stress) during the manufacturing process. Thermal stress occurs because film depositions are usually made above room temperature. Upon cooling from the deposition temperature to room temperature, the difference in the thermal expansion coefficients of the substrate and the film cause thermal stress. Intrinsic stress results from the microstructure created in the film as atoms are deposited on the substrate. Tensile stress results from microvoids (small holes, considered to be defects) in the thin film, because of the attractive interaction of atoms across the voids. The stress in thin film semiconductor structures results in the buckling of the wafers. The radius of the curvature of the stressed structure is related to stress tensor in the structure, and can be described by modified Stoney formula. The topography of the stressed structure including radii of curvature can be measured using optical scanner methods. The modern scanner tools have capability to measure full topography of the substrate and to measure both principal radii of curvature, while providing the accuracy of the order of 0.1% for radii of curvature of 90 meters and more. See also [edit] Base curve radius Bend radius Degree of curvature (civil engineering) Osculating circle Track transition curve References [edit] ^Weisstien, Eric. "Radius of Curvature". Wolfram Mathworld. Retrieved 15 August 2016. ^ abKishan, Hari (2007). Differential Calculus. Atlantic Publishers & Dist. ISBN9788126908202. ^ abcdLove, Clyde E.; Rainville, Earl D. (1962). Differential and Integral Calculus (Sixth ed.). New York: MacMillan. ^"Controlling Stress in Thin Films". Flipchips.com. Retrieved 2016-04-22. ^"On the determination of film stress from substrate bending: Stoney's formula and its limits"(PDF). Qucosa.de. Archived from the original(PDF) on 2017-08-08. Retrieved 2016-04-22. ^Peter Walecki. "Model X". Zebraoptical.com. Archived from the original on 2016-03-17. Retrieved 2016-04-22. Further reading [edit] do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall. ISBN0-13-212589-7. External links [edit] The Geometry Center: Principal Curvatures 15.3 Curvature and Radius of CurvatureArchived 2021-04-29 at the Wayback Machine Weisstein, Eric W."Principal Curvatures". MathWorld. Weisstein, Eric W."Principal Radius of Curvature". MathWorld. \| v t e Various notions of curvature defined in differential geometry \| \| Differential geometry of curves \| Curvature Torsion of a curve Frenet–Serret formulas Radius of curvature Affine curvature Total curvature Total absolute curvature \| \| Differential geometry of surfaces \| Principal curvatures Gaussian curvature Mean curvature Darboux frame Gauss–Codazzi equations First fundamental form Second fundamental form Third fundamental form \| \| Riemannian geometry \| Curvature of Riemannian manifolds Riemann curvature tensor Ricci curvature Scalar curvature Sectional curvature \| \| Curvature of connections \| Curvature form Torsion tensor Cocurvature Holonomy \| Retrieved from " Categories: Differential geometry Curvature (mathematics) Curves Integral calculus Multivariable calculus Theoretical physics Radii Hidden categories: Articles with short description Short description is different from Wikidata Webarchive template wayback links This page was last edited on 22 July 2025, at 18:47(UTC). 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4697	https://www.lanecc.edu/sites/default/files/migrated-files/math/95_reviewsheets.pdf	REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts which are taught in the specified math course. The sheets present concepts in the order they are taught and give examples of their use. WHY THESE SHEETS ARE USEFUL – • To help refresh your memory on old math skills you may have forgotten. • To prepare for math placement test. • To help you decide which math course is best for you. HOW TO USE THESE SHEETS – • Students who successfully review spend from four to five hours on this material. We recommend that you cover up the solutions to the examples and try working the problems one by one. Then, check your work by looking at the solution steps and the answer. KEEP IN MIND – • These sheets are not intended to be a short course. You should use them to simply help you determine at what skill level in math you should begin study. For many people, the key to success and enjoyment of learning math is in getting started at the right place. You will, most likely, be more satisfied and comfortable if you start onto the path of math and science by selecting the appropriate beginning stepping stone. 1) Solve linear equations. Solve: 3z 5 −z −3 2 = z + 2 3 2) Solve an equation or formula for a specified variable. a) Solve for y: x = 2 3 (y+ 4) b) Solve for x: rx = 11x + 7 3) Use formulas and mathematical models. 2 The mathematical model V = C(1−rt) is a relationship among the quantities C (original cost of a piece of equipment), r (percent steady depreciation rate expressed as a decimal), t (age, in years, of the equipment), and V (current value). What is the value after 5 years of a weaving loom that was originally purchased for $2100 and has steadily decreased in value at a rate of 6% per year? 4) Solve an application problem by setting up a linear equation. a) (Percent) The price of an item is increased by 40%. When the item does not sell, it is reduced by 40% of the increased price. If the final price is $220.00, what was the original price? What single percent reduction of the original price is equivalent to increasing by 40% then reducing by 40%? b) (Interest) A real estate agent receives a commission of 8% of the selling price of a house. What should be the selling price so that the seller can get $87,400? 5) Find the area, perimeter, volume, and surface area of common shapes (rectangle, square, triangle, circle, rectangular solid). a) For the rectangular solid shown above left, find each of the following and write proper units. i) Perimeter of the top surface (shaded) ii) Area of the top surface (shaded) iii) Volume of the solid (box) iv) Total surface area including all six surfaces. b) For the triangle above, find its area and perimeter. c) For the semicircle above, find its area and perimeter. 2 ft 15” 13 “ 12” 4 ft 10 cm 5 ft 14” 3 6) Find the x- and y-intercepts of a line from its equation and from a graph. a) Find the coordinates of the x- and y-intercepts of the line A in the graph below. b) Find the x- and y-intercepts of the line whose equation is 3x −2y = 6 . c) Sketch the line whose equation is 3x −2y = 6 by using its x- and y-intercepts. 7) Determine the slope of a line by counting and the ratio rise/run, and by the slope formula. a) Determine the slope of a line that passes through the points (0, 2) and (5, 4). b) Determine the slope of line A and line B shown on the graph at right. c) Determine the slope of a horizontal line that passes through the point (5, –20). 8) Sketch a line having a specified slope. Sketch an example of a line for each given slope: –2/3, 2, 0, undefined. 9) Given a linear equation in two variables, convert it to slope-intercept form and determine the slope and y-intercept of the graph of the line. Write 2x + 3y = 18 in slope-intercept form, then find the slope and y-intercept of its line. 10) Determine if two lines are parallel or perpendicular or neither. a) Plot the four points (1, 4), (3,2), (4, 6), and (2, 8) and join them consecutively with straight lines segments. Use slopes to show that a pair of opposite sides of the figure are parallel and to show that a pair of adjacent sides are not perpendicular. b) Determine whether these two lines are parallel, perpendicular, or neither: 3x – 2y = 6 and 2x + 3y = –6. 11) Given the graph of a vertical or horizontal line write its equation; given its equation sketch its graph. a) What is the equation of the line B shown in the graph above? b) Sketch the graph of the line whose equation is y = –2. 12) Write the equation of a specified line using the slope-intercept form and point-slope form. a) If a line passes through the points (2, 5) and (–3, 6) , find an equation of the line in point-slope form and in slope-intercept form. b) Find the equation of a line which passes through (1, –3) and is perpendicular to the line 2x – y = 1. A B B A X Y 10 10 20 20 -10 -10 -20 -20 4 13) Interpret information depicted in graphs. Suppose in a city riders pay a certain amount for each time they ride a bus. For frequent users a monthly pass is available for purchase. With a pass each ride costs 50 cents less than the regular price. The total monthly cost with and without a pass are represented by the two graphs at right. The horizontal axis represents the number of times a bus is used each month. a) If a person anticipates using the bus 15 times in a month, should a pass be purchased? b) For 15 rides in a month, what is the difference in total cost with and without a pass? c) What is the break-even point, that is, for how many rides would the total cost be the same? d) What is the cost of a monthly pass? 14) Interpret the practical meaning of a slope and intercepts in a linear application problem. a) Consider the line representing a rider with a monthly pass in the graph above. i) What is the value and units of the slope of the line? ii) What is the practical meaning of this slope? iii) What is the practical meaning of the vertical-axis intercept (y-intercept)? b) The model N = 34t +1549 describes the number of nurses N, in thousands, in the United States t years after 1985. Find the vertical-axis intercept (N-intercept) and the slope of this linear model. Explain in a clear sentence or two what each of these numbers means in practical terms. 15) Decide whether an ordered pair is a solution of a linear system. Given the system consisting of x – 3y = 2 and y = 6 – x, is (8, 2) a solution? Is (2, 0)? Is (5, 1)? 16) Solve a linear system graphically. a) Solve the system consisting of the linear equations y = (1/2)x + 3 and y = 7 by graphing each equation. b) Solve the following system graphically: x – 3y = 2 and y = 6 – x. 17) Solve a linear system algebraically by the substitution method and the addition method. a) Solve the system by the substitution method. 4x + y = 5 and 2x – 3y = 13. b) Solve the system by the addition method. x – 2y = 5 and 5x – y = –2. 18) Solve an application problem using a linear system of equations. a) (Interest) Suppose $20,000 was invested, part in a stock expected to pay 7.5% interest over 2 years and the rest in bonds paying 5.4% interest for 2 years. If the total interest earned in 2 years was $2496, how much was invested in the stock and in the bonds? Select and define variables, set up a system of equations, and solve. 5 10 15 20 25 10 20 30 40 5 10 20 30 40 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 10 20 30 40 5 10 20 30 40 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 10 5 10 With pass Without pass Number of bus rides per month Cost in one month (dollars) 5 b) (Mixture) One container has an acid solution containing 30% acid and a second contains 60% acid. If parts of the two are to be mixed together in order to create 60 gallons of a solution which is 50% acid, how many gallons from each container should be used? 19) Use integer exponents including those which are positive, negative, or zero. Simplify. a) (−3x 2 y 5) 2 b) (5x 3 y)(4x −2 y 3 ) −2(2xy) 0 c) 25a −8b 2 75a 3b −4 20) Use scientific notation. Be able to do by hand and with a calculator. Simplify by hand and express the result using scientific notation. (3600)(0.03) (120,000)(0.00090) 21) Recognize the words monomial, binomial, and trinomial, degree, and leading coefficient when describing a polynomial. a) Write the polynomial x 2 −4x 3 +9x −12x 4 −6 in descending powers of x. Find the degree of the polynomial and the value of the leading coefficient. b) Write an example of a trinomial whose degree is 6 with a leading coefficient of 5. There are many possibilities. 22) Add and subtract polynomials. a) Add 5x 3 −9x 2 −8x +11 and 17x 3 −5x 2 + 4x −3. b) Subtract 5x 3 −9x 2 −8x +11 from 17x 3 −5x 2 + 4x −3. 23) Multiply polynomials. Find each product. a) € −2x 2(5x 3 −8x 2 + 7x −3) b) € (x + 5)(x 2 −5x + 25) c) € (5x + 3)(7x −1) d) € (2x −3)(2x + 3) 24) Divide a polynomial by a monomial. Simplify. a) x + 4 x b) 6x 7 −3x 4 + x 2 3x 3 25) Factor out the greatest common factor (GCF) of a polynomial. Factor out the GCF in each expression. a) 2a 2b −5ab 2 +7a 2b 2 b) 7(x −3) + y(x −3) c) −3x 2 + 27x 26) Factor by the grouping method. a) Factor the polynomial x 3 −3x 2 + 4x−12 by the grouping method. b) Factor the trinomial 12x 2 −5x −2 by the grouping method (also known as the a-c method). 27) Factor quadratic trinomials completely. Factor: a) 6x 2 −x −35 b) 15w 3 −25w 2 +10w 6 28) Factor a polynomial that is a difference of squares. Factor 36x 2 −49y 2 29) Factor a polynomial that is a difference of cubes or a sum of cubes. a) Factor x 3 −8y 3 b) Factor 125b 3 + 64 30) Solve a quadratic equation by factoring and applying the zero-factor property. Solve each of the following quadratic equations. a) 6x 2 −x −35 = 0 b) 25 = 30x −9x 2 c) 3x 2 = 12x 31) Recognize and use vocabulary associated with quadratic equations. Given the quadratic equation 3x 2 = x −5. Fill in the blanks with the appropriate responses. a) In standard form the equation is written as ____. b) The degree of the quadratic expression is . The leading coefficient is _. c) The constant term of the quadratic expression is . d) The coefficients a, b, and c are, respectively, , _ , and _ . 32) Determine the values of the variable for which a rational expression is undefined. The expression 3 x2 −1 is not defined for what value(s) of x? 33) Simplify a rational expression by reducing to lowest terms. Simplify. a) 3a + 9 a2 + 6a + 9 b) 4 −x 2x −8 34) Multiply and divide rational expressions. a) Perform the operation indicated and simplify, if possible. 5x 2 −5 3x + 12 ⋅x + 4 x −1 b) Perform the operation indicated. x 2 −y 2 (x + y)2 ÷ x −y 4x + 4y 35) Add and subtract rational expressions. Perform the operation indicated. a) x + 5y x + y + x −3y x + y b) 5 2t −8 − 3 t −4 c) 1 x + 2 x −5 7 36) Simplify a complex fraction. Simplify: 1+ 4 x x 2 37) Solve an equation containing algebraic fractions. Solve each equation. Remember to check for extraneous solutions. a) € 1 x −2 3 = 5 6x b) 3 x2 −9 + 4 x + 3 = 1 38) Solve an application problem involving a rational expression. When a manufacturer produces x number of its product, the average cost in dollars per unit is modeled by C = 5x + 5000 x . Determine the number of units it must produce in order for the average cost of the item to be $9.75. According to this model, can the average cost be made to drop to $4.00 per item? 39) Use function notation to evaluate a function. a) If h(x) =2x 2 +3x −1, find h(0), h(–4), h(c), and h(5r). b) Given the temperature function g(F) = (5/9)(F – 32), complete the input-output table below. 40) Write an interval of numbers using inequality notation, interval notation, and graph on a number line. a) If −1 ≤x < 6 , write this interval using interval notation, and graph it on a number line. b) Let the value of the variable x be at most 10. Represent the values of x using inequality notation and interval notation, then sketch on a number line. 41) Solve a linear inequality and graph the solution on a number line. a) Solve −4x +6 ≥10 and sketch the solution on a number line. b) Solve 7(2x −1) < 9x+11 and express the solution using interval notation. 42) Simplify a radical expression containing perfect powers of numbers without using a calculator. Simplify the following. a) € 16 4 b) € −1 64 3 43) Use a scientific calculator for evaluating radicals and rational exponents. Use your calculator to compute each of the following. a) € 500 4 b) 19 6/ 5 44) Express a radical using rational exponents, and vice versa. a) Rewrite € L2m4 5 using rational exponents. b) Express (8x) 2 / 3 using a radical. F –13 4 32 212 451 g(F) 8 45) Simplify an expression containing rational exponents. a) Evaluate the following without using a calculator. i) 36 1/ 2 ii) 5 6 / 3 iii) 9 3 / 2 b) Simplify and express answer using a rational exponent. (7y 1 / 3)(2y 1 / 4 ) 46) Use the product rule for radicals; simplify square root radicals. Simplify a) € 2xy 3 22x 3y 2 b) € 32x13 47) Use the quotient rule for radicals; use the quotient rule to simplify an expression. Simplify a) € 54x 3 6x b) € 2x 5 9 48) Rationalize a denominator of one term involving a square root. Rationalize the denominator in each expression. a) € 1 2 b) € 5 6 49) Add and subtract like radicals. Simplify € 4 12 −2 75 50) Apply the distributive property with an expression containing radicals. Simplify. a) € 2 3(5+ 2) b) € ( 2 + 7)( 3 + 5) 51) Solve an equation containing radicals. Solve the equation. (Check for extraneous solutions.) a) € 2 + 3x −2 = 7 b) € x = 6x + 7 −2 c) € x 2 + 6x 3 + 2 = 0 52) Solve an application problem involving radicals. The formula € s = 30 fd models the relationship between the distance d, in feet, that a car skids after the brakes are applied, the condition of the pavement f, a number, and the speed of the car s, in miles per hour, just before the brakes were applied. Suppose the length of a skid is 47.5 feet and the road was slightly damp so that f is taken to be 0.63. What was the speed of the vehicle according to the model? 53) Write the square root of a negative number in terms of i, the imaginary unit. Express € −45 in terms of i. 54) Solve a quadratic equation by the square root method (extracting square roots). Solve a) 3x 2 −24 = 0 b) (3y −1) 2 = 16 55) Write imaginary solutions to quadratic equations using complex numbers. Solve a) x 2 + 16 = 0 b) (2x −3) 2 = −4 9 56) Solve a quadratic equation by completing the square. Solve by completing the square. Leave your answers in exact form. x 2 + 6x −5 = 0 57) Find the distance between two points of a rectangular coordinate system by applying the Pythagorean Theorem or the Distance Formula. Determine the distance between the points (4, 3) and (–6, 2). 58) Find the discriminant of a quadratic equation and predict the number and type of solutions. Compute the discriminant, and then predict the number and type of solutions the equation will have. a) −2x 2 + 6x = −1 b) 4y 2 + 2y + 5 = 0 59) Solve a quadratic equation using the quadratic formula. Solve each equation using the quadratic formula. Leave your answers in exact form simplified. a) −2x 2 + 6x = −1 b) t 2 = −6t −13 60) Solve a variety of equations that lead to a quadratic equation. Solve each equation by the algebraic method of your choice. a) (x −10)(x −2) = −7 b) 5 x +1 + x −1 4 = 2 61) Solve an application problem requiring setting up a quadratic equation. a) A rectangular garden has a path 50 feet long along the diagonal. The length of the garden is twice as long as its width. Find the dimensions of the garden exactly and also approximately rounded to the nearest tenth. b) The height of a sign in the shape of a triangle is 4 inches less than its base. Find the base and height of the sign if its area is 100 square inches. Round off your answer to the nearest tenth of an inch. 62) Graph a quadratic function by constructing a table of values and plotting points. Given f(x) = x 2 −6x , make a table of values selecting x values on both sides of x = 3, and construct graph. 63) Solve a quadratic equation graphically. The graph of the quadratic function y = x 2 −2x −5 is shown below. Use the graph to estimate the solutions of the equation x 2 −2x −5 = 0 to one decimal point. 64) Use vocabulary associated with the graph of a quadratic function. Refer to the graph at right. Fill in the blanks. a) The shape of the graph is called a _ . b) The line whose equation is x =1 is called the axis (or line) of ___. c) The coordinates of the vertex of this graph are __ . d) The vertex point (1, –6) is also described as the __ (minimum or maximum?) point of the parabola. 2 4 -2 -4 2 4 -2 -4 -6 10 65) Identify the leading coefficient of a quadratic function and determine whether its parabolic graph opens upward or downward. Determine the orientation of each parabola. a) y = 3x 2 −18x −5 b) d = 80 + 64t −16t 2 66) Find the intercepts of the graph of a parabola algebraically. Find the x- and y-intercepts of the parabola given by the equation y = 2x 2 + x −5. Write each intercept in coordinate form. Use calculator to round off answers to the nearest two decimals. 67) Find the equation of the axis of symmetry and the coordinates of the vertex of a parabola. Find the equation of the axis of symmetry of the parabola and the coordinates of the vertex of the parabola given by the equation f(x) = x 2 −2x −8. 68) Use the intercepts, the vertex, and symmetrical points as an aid in sketching the graph of a parabola. a) Sketch the graph of the parabola whose equation is y = x 2 −2x −8 using its intercepts, its vertex, its axis of symmetry, and any additional points, if helpful. b) Sketch the graph of y = 3x 2 −8x + 4. 69) Find the maximum or minimum point on a parabola algebraically. Determine whether the parabola given by y = 3x 2 −18x −5 has a maximum or minimum point. Then find the coordinates of that point. 70) Find the vertex and intercepts of a parabola and interpret their meanings in an application. A toy rocket is launched from a platform and rises straight up. The height h, in feet, of the rocket at any time t, in seconds, after launch is modeled by the formula h = −16t 2 + 80t + 5. a) How high did the rocket travel? b) How many seconds did it take to reach the highest point? c) How many seconds after launch does the rocket hit the ground? d) Without actually graphing, what is the vertical axis-intercept? What is the practical meaning of this intercept? 71) Evaluate an exponential expression or function. If f (x ) = 5(2 x ) , find f(–1) and f(3). 72) Solve several types of application problems involving exponential functions. a) (Periodic compound interest) Suppose you have $6000 to invest for 5 years. Compare investing the amount in an account giving 5.6% compounded semiannually (two times per year) with an account giving 5.3% monthly. Which is better and how much more of a return do you get? b) (Continuous compounding) If $6000 is invested for 5 years in a program offering 5.3% compounded continuously, how much interest is earned? c) (Changing population) The increasing population of a town can be modeled by the function P(t) = 1200 (1.016) t where t is number of years since 1970. What was the population in 1970? According to this model, what is the predicted population for 2010? 11 73) Change an equation in exponential form to logarithmic form, and vice versa. a) Write the statement log2 64 = 6 in exponential form. b) Change log A = B to exponential form. c) Express the equation ln x = 3.5 in exponential form. d) Write each of the following in logarithmic form: 3 4 = 81 and 100 = e x . e) Fill in the blank: If 5 3 = 125, it means that log5 125 = _ . f) If 1.7 = log2T , 2 must be raised to what exponent to get T? 74) Evaluate a logarithm using a scientific calculator. Find each of the following to 4 decimal places. If an answer does not exist, write “Does not exist” log 2.25 log 225 ln 7.06 ln 706 log (–40)
4698	https://math.stackexchange.com/questions/359937/counterexample-in-propositional-logic	Counterexample in propositional logic - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Counterexample in propositional logic Ask Question Asked 12 years, 5 months ago Modified12 years, 5 months ago Viewed 1k times This question shows research effort; it is useful and clear 4 Save this question. Show activity on this post. There is this lemma: Let Σ⊂Prop(A)Σ⊂Prop(A) and p,q∈Prop(A)p,q∈Prop(A). Then Σ⊨p⟹Σ⊨p∨q Σ⊨p⟹Σ⊨p∨q. I can't figure out a counterexample for the opposite implication (Prop(A)Prop(A) denotes the set of propositions and A A is a set of propositional atoms. Thanks for help. -pizet logic examples-counterexamples propositional-calculus Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Apr 13, 2013 at 13:33 M.Sina 1,730 1 1 gold badge 17 17 silver badges 33 33 bronze badges asked Apr 12, 2013 at 22:50 pizetpizet 1,175 1 1 gold badge 9 9 silver badges 21 21 bronze badges Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 8 Save this answer. Show activity on this post. Let q=¬p q=¬p. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Apr 12, 2013 at 22:56 André NicolasAndré Nicolas 515k 47 47 gold badges 584 584 silver badges 1k 1k bronze badges 4 I think this is the smallest complete answer, I have seen in the site, Andre. However, that "Let" could be also ommitted. ;-) +1 Mikasa –Mikasa 2013-04-13 05:21:12 +00:00 Commented Apr 13, 2013 at 5:21 There have been shorter answers, a famous one by did, for example. About the suggestion to let go of let, I think one is not supposed to start a sentence with math symbols.André Nicolas –André Nicolas 2013-04-13 05:25:38 +00:00 Commented Apr 13, 2013 at 5:25 Thanks. Yes. you are right. We cannot start such that in Maths at least.Mikasa –Mikasa 2013-04-13 05:27:44 +00:00 Commented Apr 13, 2013 at 5:27 @BabakS. There is math.stackexchange.com/a/74383/462 and math.stackexchange.com/a/180649/462 (and look at the comments on that one.)Andrés E. Caicedo –Andrés E. Caicedo 2013-04-13 15:24:49 +00:00 Commented Apr 13, 2013 at 15:24 Add a comment\| This answer is useful 4 Save this answer. Show activity on this post. Σ⊨p∨q⟹?Σ⊨p(c o n v e r s e)(c o n v e r s e)Σ⊨p∨q⟹?Σ⊨p What if p p is false and q q is true?: Suppose, e.g., Suppose q=¬p Suppose q=¬p Then your stated lemma: Σ⊨p⟹Σ⊨p∨q Σ⊨p⟹Σ⊨p∨q certainly holds. But its converse (highlighted) certainly fails to hold. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Apr 13, 2013 at 13:26 answered Apr 12, 2013 at 22:56 amWhyamWhy 211k 198 198 gold badges 283 283 silver badges 505 505 bronze badges 2 Thank you very much, it's quite clear.pizet –pizet 2013-04-12 23:07:41 +00:00 Commented Apr 12, 2013 at 23:07 @amWhy: Yes, very clear amd it is nice to get that feedback from OP! +1 Amzoti –Amzoti 2013-04-13 02:29:08 +00:00 Commented Apr 13, 2013 at 2:29 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. 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4699	https://brainly.com/question/57927255	[FREE] Solve the equation by using the quadratic formula. List the solutions, separated by - brainly.com 1 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +87,3k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +17,2k Ace exams faster, with practice that adapts to you Practice Worksheets +7,4k Guided help for every grade, topic or textbook Complete See more / Mathematics Expert-Verified Expert-Verified Solve the equation by using the quadratic formula. List the solutions, separated by commas. 4 r 2+3 r−9=0 r=□ 1 See answer Explain with Learning Companion NEW Asked by 2k23seasonone • 02/13/2025 0:00 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 1183259437 people 1183M 0.0 0 Upload your school material for a more relevant answer To solve the quadratic equation 4 r 2+3 r−9=0 using the quadratic formula, we will follow these steps: The quadratic formula is given by: r=2 a−b±b 2−4 a c where a, b, and c are the coefficients of the quadratic equation a x 2+b x+c=0. In our equation, a=4, b=3, and c=−9. Calculate the Discriminant: First, find the discriminant, which is given by b 2−4 a c. Discriminant=3 2−4⋅4⋅(−9) =9+144=153 Calculate the Roots: Now, plug the values into the quadratic formula to calculate the roots. For the first root (+ case in the formula): r 1=8−3+153 For the second root (− case in the formula): r 2=8−3−153 Solutions: By evaluating these expressions, we find the roots are approximately: r 1≈1.1712 r 2≈−1.9212 Therefore, the solutions for the equation 4 r 2+3 r−9=0 are approximately 1.1712 and −1.9212, listed as 1.1712,−1.9212. Answered by GinnyAnswer •8M answers•1.2B people helped Thanks 0 0.0 (0 votes) Expert-Verified⬈(opens in a new tab) 0.0 0 Upload your school material for a more relevant answer To solve the quadratic equation 4 r 2+3 r−9=0, we use the quadratic formula to find the roots. The discriminant is 153, leading to two solutions: approximately 1.1712 and −1.9212. Thus, the final answers are 1.1712,−1.9212. Explanation To solve the quadratic equation 4 r 2+3 r−9=0 using the quadratic formula, we will follow these steps: The quadratic formula is given by: r=2 a−b±b 2−4 a c where a, b, and c are the coefficients of the quadratic equation a x 2+b x+c=0. For our equation, we identify the coefficients as follows: a=4 b=3 c=−9 Calculate the Discriminant: To start, we need to find the discriminant, which is calculated using the formula b 2−4 a c. Discriminant=3 2−4⋅4⋅(−9) =9+144=153 Calculate the Roots: Now, we'll substitute the values into the quadratic formula to find the roots: For the first root (the positive case in the formula): r 1=8−3+153 For the second root (the negative case in the formula): r 2=8−3−153 Solutions: By evaluating these expressions: The first root r 1 is approximately: r 1≈1.1712 The second root r 2 is approximately: r 2≈−1.9212 Therefore, the solutions for the equation 4 r 2+3 r−9=0 are approximately 1.1712 and −1.9212, listed as 1.1712,−1.9212. Examples & Evidence For example, if you have the quadratic equation x 2−5 x+6=0, applying the quadratic formula leads to solutions of x=2 and x=3. Similarly, practice with different values for a, b, and c will solidify your understanding of how the quadratic formula works. The quadratic formula is a standard method in algebra for solving quadratic equations and is universally acknowledged in mathematics education. The process of calculating the discriminant and using it to find the roots is well-documented in algebra textbooks. Thanks 0 0.0 (0 votes) Advertisement 2k23seasonone has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Mathematics solutions and answers Community Answer Solve the equation using the quadratic formula. List the solutions, separated by commas. [tex][ 4r^2 - r + 12 = 8 ][/tex] [tex][ r = ][/tex] Community Answer Solve the equation using the quadratic formula. List the solutions, separated by commas. [tex][ \begin{array}{l} 4r^2 - 3r + 7 = 4 \ r = \square \end{array} ][/tex] Community Answer Solve the equation by the quadratic formula. List the solutions. [tex][ 4r^2 + 3r + 12 = 2 ][/tex] [tex][ r = \square ][/tex] Community Answer Solve the equation using the quadratic formula. List the solutions, separated by commas. [tex][ \begin{array}{l} 2p^2 - 3p + 17 = 9 \ p = \end{array} ][/tex] Community Answer Solve the equation using the quadratic formula. List the solutions, separated by commas. Enter exact solutions. [tex][ 2r^2 - 3r - 10 = 0 ][/tex] [tex][ r = ][/tex] tex[/tex] Community Answer Question 12 Solve equation by using the quadratic formula. List the solutions, separated by commas. 3r^(2)+r-1=0 T Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD New questions in Mathematics Determine the vertex of the function f(x)=3 x 2−6 x+13. What is an equivalent equation solved for y 2?A. y 2=m x 2−x 1+y 1B. y 2=m x 2−x 1−y 1C. y 2=m(x 2−x 1)+y 1D. y 2=m(x 2−x 1)−y 1 The inverse of the logarithmic function f(x)=lo g 0.5 xis f−1(x)=0.5 x. What values of a,b, and c complete the table for the inverse function? \| x \| -2 \| -1 \| 0 \| 1 \| 2 \| :---: :---: :---: \| \| y \| 4 \| a \| b \| 0.5 \| c \| Solve the equation for a. K=4 a+9 ab Let P=(x,y) be a point on the graph of y=x 2−9. Express the distance, d, from P to the origin as a function of the point's x-coordinate. Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss

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